Die Dozentinnen und Dozenten der Mathematik (Hrsg.)
Seminarberichte Nr. 87
Mathematik und Informatik
Seminarberichte aus dem Fachbereich Mathematik der FernUniversität 87 – 2015
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NOLI TURBARE CIRCULOS MEOS – A MATHEMATICAL TRIBUTE TO REINHARD BORGER¨
WALTER THOLEN
1. A Brief Curriculum Vitae On June 6, 2014, Reinhard B¨orgerpassed away, after persistent heart complications. He had taught at Fernuniversit¨at in Hagen, Germany, for over three decades where he had received his Dr. rer. nat. (Ph.D.) in 1981, with a thesis [15] on notions of connectedness, written under the direction of Dieter (“Nico”) Pumpl¨un. He had continued to work on mathematical problems until just hours before his death. Reinhard was an extraordinarily talented mathematician, with a broad and deep un- derstanding of many areas of mathematics, combined with an equally deep mathematical intuition. His quick grasp of any kind of subject, as evidenced especially by his comments in seminar settings, invariably impressed his colleagues, friends and acquaintances. His colleagues at Fernuniversit¨at will confirm that, over the last couple of decades, there was virtually no Ph.D student at their department who, no matter which problem she or he was working on, did not profit tremendously from Reinhard’s generous and insightful advice. In this article I hope to give an impression of Reinhard’s specific mathematical interests and the breadth of his work, paying special attention to his early papers and unpublished works that may not be easily accessible. Born on August 19, 1954, Reinhard went to school in Gevelsberg (near Hagen) before beginning his mathematics studies at the Westf¨alischeWilhelms-Universit¨at in M¨unster in 1972. A year later he won a runner-up prize at the highly competitive federal Jugend forscht competition. No surprise then that, alongside his fellow student Gerd Faltings, soon to become famous as a Fields Medal recipient, he was quickly recognized as an exceptional student for all his talent, broad mathematical interests, and his unapologetic defense of his Christian-conservative values in a university environment that was still very much reverberating the 1968 leftist movements. Quite visibly, mathematics seemed to always be on his mind, and he often seemed to appear out of nowhere at lectures, seminars or informal gatherings. These sudden appearances quickly earned him his nickname Geist (ghost), a name that he willingly adopted for himself as well. His trademark ability to then launch pointed and often unexpected, but always polite, questions, be it on mathematics or any other issue, quickly won him the respect of all. Reinhard’s interest in category theory started early during his studies in M¨unsterwhen, supported by a scholarship of the Studienstiftung des Deutschen Volkes, he took Pumpl¨un’s course on the subject that eventually led him to write his 1977 Diplomarbeit (M.Sc. thesis) 1 - 2 -
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about congruence relations on categories [3]. For his doctoral studies he accepted a scholar- ship from the Cusanuswerk and followed Pumpl¨unfrom M¨unsterto Hagen where Pumpl¨un had accepted an inaugural chair at the newly founded Fernuniversit¨atin 1975. After the completion of his doctoral degree in 1981, with a thesis that received an award from the Gesellschaft der Freunde der Fernuniversit¨at, he assumed a number of assistantships, at the University of Karlsruhe (under the direction of Diethard Pallaschke), at the University of Toledo (Ohio, USA), and back at Fernuniversit¨at(under the direction of Holger Petersson and Dieter Pumpl¨un).For his Habilitationsschrift [31], which earned him the venia legendi in 1989, he developed a categorical approach to integration theory. Beginning in 1990 he worked as a Hochschuldozent at Fernuniversit¨at,interrupted by a visiting appointment as associate professor at York University in Toronto (Canada) in 1993, and in 1995 he was appointed Außerplanm¨aßigerProfessor at Fernuniversit¨at,a position that he kept until his premature death in 2014. Of course, this linearization of his career path cannot do justice to Reinhard’s mathe- matical work that never followed a straight career-oriented line but rather resembled the zig-zags of his multiple interests. But, as I will try to show in the remainder of this article, there are trajectories in his papers and notes that follow recurring themes of particular interest to him, some of which he unfortunately was not able to lead to a conclusion. For some more personal remarks I refer to the “Farewell” section at the end of this article. The description of Reinhard’s mathematical work that follows is organized as follows. After a brief account in Section 2 of his work up to the completion of his M.Sc. thesis, I recall some of his early contributions to the development of categorical topology (Section 3), before describing in Section 4 some aspects of his Ph.D. thesis and the work that emanated from it. Section 5 sketches the work on integration theory in his Habilitationsschrift, and Section 6 highlights some of his more isolated mathematical contributions. For Reinhard’s substantial contributions in the area of convexity theory we refer to the article [87] by his coauthor Ralf Kemper that immediately follows this article. The References at the end of this article first list, in approximate chronological order, Reinhard B¨orger’swritten mathematical contributions, including unpublished or incom- plete works, to the extent I was able to trace them, followed by an alphabetical list of references to other works cited in this and Ralf Kemper’s article [87]. Acknowledgements. I am indebted to my former colleagues Nico Pumpl¨unand Holger Petersson at Fernuniversit¨atfor their strong encouragement, helpful critical reading and very good practical advice during the long preparation of this article. Sincere thanks are also due to Ottmar Loos and Diethard Pallaschke for their help in recovering information and materials that may easily have been lost without their invaluable efforts, and to Ralf Kemper for his kind cooperation in our joint effort in presenting Reinhard B¨orger’s work. Andrea B¨orger greatly helped by recalling some of Reinhard’s contributions outside math- ematics, for which I am very grateful. Last, but not least, I thank Andrei Duma as the Managing Editor of the Seminarberichte for his patience and his work on this volume. - 3 -
NOLI TURBARE CIRCULOS MEOS – A MATHEMATICAL TRIBUTE TO REINHARD BORGER¨ 3
2. First Steps The earliest written mathematical work of Reinhard that I am aware of and that may still be of interest today, is the three-page mimeographed note [1] giving a sufficient condition for the non-existence of a cogenerating (also called coseparating) set of objects in a category K. While the existence of such a set in the category of R-modules and, in particular, of abelian groups, is standard, none of the following categories can possess one: fields; skew fields; (commutative; unital) rings; groups; semigroups; monoids; small categories. Reinhard’s theorem, found when he was still an undergraduate student, gives a unified reason for this, as follows. Theorem 2.1. Let K have (strong epi,mono)-factorizations and admit a functor U to Set that preserves monomorphisms. If, for every cardinal number κ, there is a simple object A in K with the cardinality of UA at least κ, then there is no cogenerating set in K. (He defined an object A to be simple if the identity morphism on A is not constant while every strong epimorphism with domain A must be constant or an isomorphism; a morphism f is constant if for all parallel morphisms x, y composable with f one has fx = fy.) When asked by Nico Pumpl¨unat the time how he found this theorem, Reinhard replied that he just kept negating the existence assertion, which made a bystander recite Mephistoteles from Goethe’s Faust: “Ich bin der Geist der stets verneint” (“I am the spirit of perpetual negation”; Geist in German has the double meaning of ghost and spirit). It turned out that the principle behind the theorem (existence of arbitrarily large simple objects, without the notion of simplicity having been coined yet in categorical terms) was already known and used by John Isbell, which is why no attempt was made to publish Reinhard’s note, although it would clearly have been a useful addition to any standard text on category theory. Reinhard returned to the theme of the existence of cogenerators repeatedly throughout his career, see [16, 34, 44, 45, 49, 50]. In 1975 Reinhard and I discussed various generalizations of the notion of right adjoint functor that had appeared in the literature at the time, in particular Kaput’s [85] locally adjunctable functors that I had also treated in my thesis [103]. We tightened that notion to strongly locally right adjoint and proved, among other things, preservation of connected limits by such functors. Our paper [2] was presented at the “Categories” conference in Oberwolfach in 1976, and we discussed it with Yves Diers who was working on a slightly stricter notion for his thesis [72] that today is known under the name multi-right adjoint functor. Diers’ only further requirement to our strong local right adjointness was that the local adjunction units of an object, known as its spectrum, must form a set. Without this size restriction, Reinhard and I had already given in [2] a complete characterization of the spectrum of an object, as follows. Theorem 2.2. For a strongly locally right adjoint functor U : A / X and an object X ∈ X , its spectrum is the only full subcategory of the comma category (X ↓ U) that is a groupoid, coreflective, and closed under monomorphisms. These works actually precede Reinhard’s M.Sc. thesis [3] whose starting point was a notion presented in Pumpl¨un’scategories course, called (uniquely) normal equivalence - 4 -
4 WALTER THOLEN
relation ∼ on the class of morphisms of a category K, requiring the existence of a (uniquely determined) composition law for the equivalence classes that makes K/∼ a category and the projection P : K / K/∼ a functor. Reinhard showed that the behaviour of a compatible equivalence relation ∼ on the morphism class of a category K (so that u ∼ u0 and v ∼ v0 implies uv ∼ u0v0 whenever the composites are defined) requires great caution, giving the following fine analysis: Theorem 2.3. Each of the following statements on an equivalence relation ∼ on the class of morphisms of a category K implies the next, but none of these implications is reversible:
• ∼ is compatible, and 1A ∼ 1B only if A = B, for all objects A, B ∈ K; • ∼ is compatible, and for all u : A / B, v : C / D with 1B ∼ 1C , there are u0 : A0 / B0, v0 : C0 / D0 with u ∼ u0, v ∼ v0 and B0 = C0; • ∼ is uniquely normal; • ∼ is normal; • there is a functor F with domain K inducing ∼ (so that u ∼ u0 ⇐⇒ F u = F u0); • ∼ is compatible. He made the (perfectly valid) case that, of these properties, being induced by a functor is the most natural one from various perspectives. The paper [5] gives a summary of his M.Sc. thesis in English which, among other things, provides first evidence of one of Reinhard’s particular mathematical strengths, namely his ability to construct intricate (counter)examples.
3. Semi-topological functors and total cocompleteness Br¨ummer’s[69], Shukla’s [99], Hoffmann’s [80] and Wischnewsky’s [108] theses and Wyler’s [110, 109], Manes’ [90] and Herrlich’s [76, 77] seminal papers triggered the devel- opment of what became known as Categorical Topology, with various groups in Germany, South Africa, the United States and other countries working intensively throughout the 1970s on axiomatizations of “topologically behaved” functors and their generalizations and properties; see [70] for a survey. Reinhard and I, long before he started working on his doctoral dissertation, were very much part of this effort. Here are some examples of results that he has influenced the most. Topologicity of a functor P : A / X may be defined by the sole requirement that initial liftings of (arbitrarily large) so-called P -structured sources exist, without the a- priori assumption of faithfulness of P . (This is Br¨ummer’s[69] definition, although he did not use the name topological for such functors in his thesis.) Herrlich realized that faithfulness is a consequence of the definition, with a proof that made essential use of the smallness of hom-sets for the categories in question. Reinhard’s spontaneous idea then was to use a Cantor-type diagonal argument instead that works also for not necessarily locally small categories. In [8] we came up with a general theorem that not only proves the faithfulness of topological and, more generally, semi-topological functors [104, 81, 105] , but that also entails Freyd’s theorem that a small category with (co)products must be, up to categorical equivalence, a complete lattice, and that in fact reproduces Cantor’s original - 5 -
NOLI TURBARE CIRCULOS MEOS – A MATHEMATICAL TRIBUTE TO REINHARD BORGER¨ 5
theorem about the cardinality of a a set being always exceeded by that of its power set, as follows:
Theorem 3.1. Consider a (possibly large) family (ti : Ai / C)i∈I of morphisms and an object B in a category K, such that any family (hi : Ai / B)i∈I factors as hi = hti (i ∈ I) for some h : C / B. If there is a surjection I / K(C,B), then for any morphisms f, g : C / B one has fti = gti for some i ∈ I. Earlier Hong [83] had introduced the notion of a topologically algebraic functor P : A / X , by requiring that all P -structured sources (X / PBi)i∈I in X factor through a P -initial source (A /Bi)i∈I in A via a P -epimorphic morphism e : X /PA. (Topologicity is characterized by the fact that e may always be chosen to be an identity morphism.) It was clear a priori that such functors are semi-topological functors [105], which are characterized as the restrictions of topological functors to full reflective subcategories of their domains, but the converse question was very much under scrutiny at the time. Reinhard [6] and a team led by Horst Herrlich [78] had independently constructed somewhat artificial examples (involving categories without good completeness properties), showing the non-equivalence of the two concepts of interest, before in [9] we published a Set-based example:
Theorem 3.2. There is a category A and a semi-topological functor P : A / Set which fails to be topologically algebraic. Moreover, P has a fibre-small MacNeille completion but fails to have a universal completion in the sense of Herrlich [77]. The paper [9] contains another little-known theorem that clearly shows Reinhard’s math- ematical trades. It gives an easy sufficient condition for initial sources to be monic (the converse implication had been addressed in [84]), a property that can distinguish “rich algebraic” categories (like that of groups or rings) from “poor” ones (monoids, semigroups or pointed sets), but that is also applicable outside the realm of algebra (for instance to the category of real or complex Banach spaces and its linear operators of norm at most 1):
Theorem 3.3. For a category A, let P : A / Set be a functor represented by an object G such that there is an epimorphic endomorphism of G different from 1G. (More generally, it suffices to assume that the family of non-identity endomorphisms of G be epimorphic.) Then every P -initial source in A is monomorphic. Semi-topological functors in their various incarnations remained a topic of Reinhard’s and my joint investigation for considerable time, in particular in conjunction with strong (co)completeness properties of the participating categories, as witnessed by our papers [16, 34, 35, 38]. In [106] I had shown that the fundamental property of totality (or total cocompletness) introduced by Street and Walters [101] lifts from X to A along a semi- topological P : A / X , and in [67] total categories with a (strong) generating set of objects were characterized as the categories admitting a semi-topological (and conservative) functor into some small discrete power of Set. For our paper [34] Reinhard constructed an incredible example: Theorem 3.4. There is a total category A with a (single-object) strong generator but no regularly generating set of objects. A is cowell-powered with respect to regular epimorphisms - 6 -
6 WALTER THOLEN
but not with respect to strong epimorphisms; A does not admit co-intersections of arbitrarily large families of strong epimorphisms.The colimit closure B of the strong generator in A fails to be complete since it doesn’t even possess a terminal object. Since totality entails a very strong completeness property, called hypercompleteness by Reinhard (see [16]), the colimit closure B in the example above fails badly to inherit totality from its ambient category A. A comparison with the following affirmative result on totality of colimit closures obtained in [38] demonstrates how “tight” this example is: Theorem 3.5. Let the cocomplete category B be the colimit closure of a small full subcat- egory G, and assume that every extremal epimorphisms in B is the colimit of a chain of regular epimorphisms of length at most α, for some fixed ordinal α. Then B is total and admits large co-intersections of strong epimorphisms, and G is strongly generating in A.
4. Connectedness, coproducts, and ultrafilters Reinhard’s doctoral dissertation [15] relates various categorical notions of connectedness studied throughout the 1970s with each other, adds new concepts and gives some surprising applications. Starting points for him were the notions of component subcategory (initiated by Herrlich [75] and developed further by Preuß [91], Strecker [100] and Tiller [107]), of left- constant subcategory (also initiated by Herrlich [75] in generalization of the correspondence between torsion and torsion-free classes and fully characterized within the category of topological spaces by Arhangel’skii and Wiegandt [68]), and the notion of strongly locally coreflective [2] or multi-coreflective [72] subcategory (already mentioned in Section 2 in the dual situation and applied in topology by Salicrup [97]). Let us concentrate here on a more category-intrinsic approach to connectedness to which Reinhard greatly contributed and which led him to make significant contributions to preser- vation properties of coproducts in abstract and concrete categories. The starting point is the easy observation that a topological space X is (not empty and) connected if, and only ` if, every continuous map X / i∈I Yi into a topological sum factors uniquely through exactly one coproduct injection; in other words, if the covariant hom-functor Top / Set represented by X preserves coproducts. Trading Top for any category K with coproducts Hoffmann [80] called such objects X Z-objects, Reinhard preferred the name coprime, while most people will nowadays use the term connected in K. More specifically, for a cardinal number α, let us call X α-connected in K if the hom-functor of X preserves coproducts indexed by a set of cardinality ≤ α. In his thesis Reinhard was the first to explore this concept deeply in the dual category of the category Rng of unital (but not necessarily commutative) rings. α-connectedness Q of a ring R now means that every unital homomorphism f : β<α Sβ / R depends only on exactly one coordinate (so that it factors uniquely through precisely one projection of the direct product). While it is easy to see that, without loss of generality, one may assume here that every ring Sβ is the ring Z of integers, and that the finitely-connected (i.e., α-connected, for every finite α) rings are precisely those that traditionally are called connected (i.e., those rings that have no idempotent elements other than 0 and 1), Reinhard - 7 -
NOLI TURBARE CIRCULOS MEOS – A MATHEMATICAL TRIBUTE TO REINHARD BORGER¨ 7
unravelled several surprises in the infinite case. Calling a ring ultraconnected when it is ℵ0-connected, he proved in [15] (see also [21]) that the countable case governs the arbitrary infinite case precisely when there are no uncountable measurable cardinals: Theorem 4.1. If there are no uncountable measurable cardinals, then the connected ob- jects in Rngop are precisely the ultraconnected rings. If there are uncountable measurable cardinals, then there are no ultraconnected objects in Rngop. The field R of real numbers is ultraconnected, and so is every subring of an ultraconnected ring. But none of the following connected rings is ultraconnected: the cyclic rings of m cardinality p (p prime, m ≥ 1), the ring Zp of p-adic integers and its field of fractions Qp, and the field C of complex numbers. The Theorem remains valid if Rng is traded for the category of commutative unital rings. Its proof makes essential use of a general categorical result that Reinhard had first presented at a meeting on “Categorical Algebra and Its Applications” held in Arnsberg (Germany) in 1979 (see [13]): Theorem 4.2. For a category K with an initial object and α-indexed coproducts (α an infinite cardinal), a functor F : K / Set preserves such coproducts if, and only if, F preserves β-indexed coproducts for every measurable β ≤ α. He only subsequently learned that Trnkov´a[102] had proved this theorem earlier in the special case that also the domain of F is Set. In [25], keeping the general domain K, he went on to expand it further to functors with target categories other than Set. The significance of the existence of measurable cardinals (i.e., of cardinals α on which there is a non-principal ultrafilter that is closed under forming intersections of less than α of its elements) certainly contributed to Reinhard’s fascination with ultrafilters which recurred in many of his papers. He discovered several peculiarities related to them, such as the fact that a fixed point-free endomap of a discrete topological space extends to a fixed point-free endomap of its Stone-Cechˇ compactification; see [18]. More importantly, let us mention here in particular his characterization of the ultrafilter functor of Set that assigns to every set X the set of ultrafilters on X, first given in [12] and later published in [25], as being terminal amongst all endofunctors of Set that preserve finite coproducts. Consequently, its monad structure (which has the compact Hausdorff spaces as its Eilenberg-Moore algebras), is uniquely determined. In [32] he proved that, for a category K with finite coproducts, the finite-coproduct- preserving functors K / Set form a full coreflective subcategory of the (meta-)category [K, Set], giving an explicit construction of the coreflector even in the case when Set is traded for a category in which finite coproducts commute with connected limits. I should point out that the themes touched upon in, or emerging from, Reinhard’s thesis very much reverberate in today’s research. Let me conclude this section with a prime example in this regard. One of the standard notions of category theory today is that of an extensive category, a term introduced by Carboni, Lack and Walters in [71]: a category K with (finite) coproducts and pullbacks is (finitely) extensive if (finite) coproducts are universal (i.e., stable under pullback) and disjoint (i.e., the pullback of - 8 -
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any two coproduct injections with distinct labels is the initial object). This is a typically geometric property shared by Set and Top, while a pointed extensive category must be trivial. Every elementary topos is finitely extensive, and Grothendieck topoi (i.e., the localizations of presheaf categories) may be characterized as those Barr-exact categories with a generating set of objects that are extensive. In a (finitely) extensive category the (finitely) connected objects are characterized as a topologist would expect: they are precisely the coproduct-indecomposable objects, i.e., those non-initial objects X with the property that whenever X is presented as a coproduct of Y and Z, one of Y,Z must be initial. Reinhard started his studies of the universality and disjointness properties of coproducts years before the appearance of [71]. His initial account [26] went through a multi-year period of refinement, extension and correction before it finally got published in [46]. But his first account already contains all the ingredients to the proof of a refined analysis of the notion of (finite) extensitivity that is missing from [71]; it shows that universality almost implies disjointness, as follows: Theorem 4.3. A category with (finite) coproducts and pullbacks is (finitely) extensive if, and only if, non-empty (binary) coproducts are universal and pre-initial objects are initial. (A pre-initial object admits at most one morphism into any other object, while an initial object admits exactly one. A streamlined proof of the Theorem is contained in [82].) The dual of the category of commutative unital rings is finitely extensive, and Reinhard gave an example showing that commutativity is essential here, although Rngop still has the disjointness property.
5. Measure and Integration Given the wide range of his mathematical interests, it is hardly surprising that a large part of Reinhard’s work addresses analytic themes, which are also at the core of his Habilita- tionsschrift [31], titled “A categorical approach to integration theory” (written in German, with the preprint [28] giving a compressed English version of it). The seeds for his interest in developing such a theory may have been sawn early on during his student times when Diethard Pallaschke introduced him to Semadeni’s book [98] which uses categorical lan- guage and tools in functional analysis. Before Reinhard started his work in this area, there had been only few attempts to present measure and integration theory in a categorically satisfactory fashion, with limited follow-up work; among others, see [88, 89, 74]. Of these, Reinhard’s approach may be seen as a further development of Linton’s early work. The starting point in his approach is the elementary, but crucial, observation that inte- gration of simple functions is given by a universal property. Specifically, for a Boolean alge- bra B (with top and bottom elements 1 and 0) and a real vector space A, the space M(B,A) of charges µ : B / A (i.e., of maps µ with µ(u ∨ v) = µ(u) + µ(v) for all u, v ∈ B with u ∧ v = 0) is representable when considered as a functor in A, so that for the fixed Boolean ∼ algebra B there is a real vector space EB with M(B, −) = HomR(EB, −): VecR / Set. Hence, there is a charge χB : B / EB such that any charge µ : B / A factors as µ = l · χB, for a uniquely determined R-linear map l : EB / A. For a set algebra B - 9 -
NOLI TURBARE CIRCULOS MEOS – A MATHEMATICAL TRIBUTE TO REINHARD BORGER¨ 9
of a set Ω, EB is the space of simple functions, and χB assigns to a subset of Ω in B its characteristic function. In particular then, for A = R and a charge µ, the corresponding map l assigns to a simple function its integral with respect to µ. Since every bounded measurable function is the uniform limit of simple functions, it is clear that one must provide for a “good” convergence setting to arrive at a satisfactory integration theory, and Reinhard formulates the following necessary steps to this end: 1. express the integration of simple functions categorically in sufficient generality; 2. provide for a “convenient convergence environment”, by replacing the category of sets by a suitable category of topological spaces; 3. test the categorical theory obtained against classical approaches to, and results in, integration theory. Unfortunately, as Reinhard explains in the 18-page introduction to his Habilitationsschrift, this obvious roadmap is loaded with specific obstacles. The “simple integration theory” sketched above relies crucially on the fact that the symmetric monoidal-closed category VecR lives over the Cartesian-closed category Set, with the left adjoint L to the forgetful functor V : VecR / Set preserving the monoidal structure: L(X × Y ) = L(X) ⊗ L(Y ) for all sets X,Y . Since the category Top fails to be Cartesian closed and can therefore not replace Set, the first question then is which subtype of topological or analytic structure one should add on both sides of the adjunction without losing its “monoidal well-behavedness”. A good replacement candidate for Set is the Cartesian-closed category SeqHaus of sequential Hausdorff spaces (in which every sequentially closed subset is actually closed). However, since even its finite (categorical) products generally carry a finer topology than the product topology, vector space objects in SeqHaus may fail to be topological vector spaces. To overcome this and other “technical” obstacles, Reinhard restricts himself to considering only vector spaces in which convergence to 0 may be tested with convex neighbourhoods of 0, thus replacing the functor V above by the forgetful functor SCS / SeqHaus of sequentially convex spaces. Reassuringly, SCS is still big enough to contain all Banach spaces (real or complex), even all locally convex Fr´echet spaces. His general categorical setting and theory, which substantially uses and contributes to Eilenberg’s and Kelly’s enriched category theory [73, 86], is centred around a right-adjoint functor V : A / X with a (semi-)additive category A where, for simplicity, I assume here that both A and X be finitely complete and cocomplete. For every Boolean algebra object B in X and every A in A he gives a categorical construction of the set M(B,A) of A-valued measures on B. As described in the elementary case of set-based charges, a representation of M(B, −): A / Set defines a universal measure χB : B / EB, where EB plays the role of L(∞)(B) in concrete situations, and the factorization of an arbitrary measure µ through χB defines the integral with respect to µ. Multiplicativity of measures, a property that Reinhard defines in this abstract setting, requires a symmetric monoidal structure on A and the well-behavedness of the left adjoint L of V with respect to that structure on A and the Cartesian structure of X . Under mild hypotheses he then shows that the universal measure is automatically multiplicative and that E, considered as a functor B / R to the category R of commutative monoid objects in the additive category A, is left adjoint. As a particular consequence then, E preserves binary coproducts, a fact - 10 -
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that may be interpreted as Fubini’s Theorem, as one may explain for the specific categories considered earlier. Indeed, for A = SCS, X = SeqHaus, a Boolean algebra object B in X is now called a sequential Hausdorff Boolean algebra, and a commutative monoid object R in A gives a commutative sequentially convex algebra. The fact that the functor E : SHBool / SCA preserves binary coproducts implies that, for B0,B1 in SHBool, an element in E(B0 ⊗B1), i.e., an integrable functionoid on the coproduct B0 ⊗ B1 in SHBool, may be considered a “functionoid in two variables”, and its “iterated integral” with respect to measures µ0, µ1 on B0,B1 respectively, coincides with its integral with respect to the (real-valued) “product measure” on the coproduct B0 ⊗ B1 in SHBool determined by µ0, µ1. This is only a coarse and partial sketch of the work presented in his Habilitationsschrift. Reinhard kept working on refining and extending his integration theory till the end of his life. Beyond his published article [61] there are preliminary versions of a planned monograph on categorical integration theory of 2006 (see [57]) and 2010 (see [62]) which await some editorial work before they will hopefully be made available to a wider audience.
6. Across Mathematics In the previous sections I have tried to give an impression of Reinhard’s contributions to category theory and its applications to algebra, topology and analysis. But I haven’t touched upon many of his other contributions (as listed in the References) that have no apparent connection to the type of work mentioned so far, for example in number theory (algebraic or analytic) and topology (general or algebraic), of which I can mention here only very few examples. They should underline his fascination with“concrete” objects and problems, his mastery of which was as strong as that of “abstract” mathematical theories. Take, for example, the intricate proof of his solution [39] to the problem of “How to make a path injective” that cleverly utilizes the order of the real unit interval I = [0, 1]:
Theorem 6.1. Let ϕ : I / X be a continuous path from a to b in a Hausdorff space X, a 6= b. Then there exist an injective continuous path ψ : I / X from a to b, a closed subset A ⊆ I and a continuous order-preserving map p : I / I with p(A) = I and ψ · p|A = ϕ. In [53] he constructs “A non-Jordan measurable regularly open subset of the unit in- terval”, and in [33] he exploits the role of rational numbers in R to give a surprisingly easy example of a “reasonable” connected Hausdorff space in which every point has a hereditarily disconnected neighbourhood. In fact, he proves the following theorem. Theorem 6.2. There is a topology on the set of real numbers finer than the Euclidean topology, making it a connected Hausdorff space that is the union of two hereditarily dis- connected open subspaces. His proof takes less than a page and “adds” just a little elementary number theory to everybody’s knowledge of the topology of the real line. Quite a different side of number theory is displayed in Reinhard’s informal discussion note [30] that was sparked by the observation 6! · 7! = 10! and the quest for other integer solutions x, y, z of x! · y! = z! - 11 -
NOLI TURBARE CIRCULOS MEOS – A MATHEMATICAL TRIBUTE TO REINHARD BORGER¨ 11
with 1 ≤ x ≤ y. Hence, after discarding the “trivial” solutions 1, y, y with y ≥ 1 and x, x! − 1, x! with x ≥ 3 he asked whether the set S of non-trivial solutions is finite or, in fact, contains any triple other than 6, 7, 10. His note, which asks for input from specialist number theorists, does not settle this question, but it does provide the following constraint on members of S that he obtained with analytic methods:
Theorem 6.3. Any non-trivial integer solution to x! · y! = z! with 1 ≤ x ≤ y must satisfy √ x 2 2 − x < y. As a consequence, there is no non-trivial integer solution to that equation with x = y.
Further examples of Reinhard’s number-theoretic contributions include his display of “A geometric theory of Henselian local rings” [42] and his treatment of “Infinitary linear combinations over valued fields” [47].
7. Farewell As a former colleague and frequent coauthor I belong to the many privileged people with whom Reinhard generously shared the depth and breadth of his mathematical knowl- edge and ideas. They include his teachers as much as his students and the accidental acquaintance at a conference, all of whom may have experienced his initial shyness that, however, could quickly give way to a spark in his eyes when confronted with an interesting mathematical question, usually followed by a rapid flow of pointed remarks that were often difficult to comprehend at first. Reinhard’s premature death is surely a great loss to all of us. Despite his superior talents Reinhard was a fundamentally modest person, with firm be- liefs in Christian values. He saw no conflict between science and his religion, the principles of which he consistently upheld as a letter writer to papers and author of non-mathematical articles. His life-long dedicated engagement in local parish work as well as his contributions to national organizations addressing social and environmental issues, especially regarding the impact of individual car traffic, may not have been as visible to the people around him as they deserved to be. For example, in spite of having known him since his early university student times, it took me years to understand that his passion for railways and especially the use of local trains and public transport were rooted in much more than just a hobby. Reinhard hardly ever talked much about himself, neither about his accomplishments nor his problems. His mathematical coworkers would rarely hear from him about his en- gagements outside mathematics, even when these were professionally related to his math- ematical activities, such as his ambition to learn the Czech language. Only when asked directly would one hear the proud father speak about his three sons Lukas, Simon and Jonas. He fought hard to overcome the consequences of a devastating stroke some seven years before his death, especially as he was looking forward to celebrate later in 2014 his sixtieth birthday and the thirtieth anniversary of his wedding to Andrea B¨orger.Sadly, he lost that battle. - 12 -
12 WALTER THOLEN
References [1] R. B¨orger. Nichtexistenz von Cogeneratormengen. Typescript, 3 pp, Westf¨alische Wilhelms- Universit¨at,M¨unster1974 (estimated). [2] R. B¨orgerand W. Tholen. Abschw¨achungen des Adjunktionsbegriffs. Manuscripta Math., 19(1):19–45, 1976. [3] R. B¨orger. Kongruenzrelationen auf Kategorien. Diplomarbeit (Master’s thesis), Westf¨alische Wilhelms-Universit¨at,M¨unster,1977. [4] R. B¨orger.Fundamentalgruppoide. Seminarberichte, 1:77–84, Fernuniversit¨at,Hagen 1976. [5] R. B¨orger.Factor categories and totalizers. Seminarberichte, 3:131–157, Fernuniversit¨at,Hagen 1977. [6] R. B¨orger.Semitopologisch 6= topologisch-algebraisch. Preprint, Fernuniversit¨at,Hagen 1977. [7] R. B¨orger.Universal topological completions of semi-topological functors over Ens need not exist. Preprint, Fernuniversit¨atHagen, 1978. [8] R. B¨orgerand W. Tholen. Cantors Diagonalprinzip f¨urKategorien. Math. Zeitschrift, 160(2):135–138, 1978. [9] R. B¨orgerand W. Tholen. Remarks on topologically algebraic functors. Cahiers Topologie G´eom. Diff´erentielle, 20(2):155–177, 1978. [10] R. B¨orger.A Galois adjunction describing component categories. Tagungsberichte, Nordwestdeutsches Kategorienseminar, Universit¨atBielefeld, 1978. [11] R. B¨orger.Legitimacy of certain topological completions. Categorical Topology (Proc. Internat. Conf., Free Univ. Berlin, 1978). Lecture Notes in Math. 719, pp. 18-23, Springer, Berlin 1979. [12] R. B¨orger.A characterization of the ultrafilter monad. Seminarberichte, 6:173–176, Fernuniversit¨at, Hagen 1980. [13] R. B¨orger. Preservation of coproducts by set-valued functors. Seminarberichte, 7:91–106, Fernuniver- sit¨at,Hagen 1980. [14] R. B¨orger.On the left adjoint from complete upper semilattices to frames. Preprint, Feruniversit¨at, Hagen 1981(estimated). [15] R. B¨orger.Kategorielle Beschreibungen von Zusammenhangsbegriffen. Doctoral Dissertation, Fernuni- versit¨at,Hagen 1981. [16] R. B¨orger,W. Tholen, M.-B. Wischnewsky, and H. Wolff. Compact and hypercomplete categories. J. Pure Appl. Algebra., 21(2):71–89, 1981. [17] R. B¨orger.Compact rings are profinite. Seminarberichte, 13:91–100, Fernuniversit¨at,Hagen 1982. [18] R. B¨orger.A funny category of ultrafilters. Seminarberichte, 17:2019–212, Fernuniversit¨at,Hagen 1983. [19] R. B¨orger.Connectivity spaces and component categories. In: Categorical Topology (Toledo, Ohio, 1983). Sigma Ser. Pure Math., 5:71–89, Heldermann, Berlin 1984. [20] R. B¨orger and W. Tholen. Concordant-dissonant and monotone-light. In: Categorical Topology (Toledo, Ohio, 1983). Sigma Ser. Pure Math., 5:90–107, Heldermann, Berlin 1984. [21] R. B¨orgerand M. Rajagopalan. When do all ring homomorphisms depend only on one coordinate? Archiv Math. (Basel), 45(3):223–228, 1985. [22] R. B¨orger.What are monad actions? Seminarberichte, 23:5–8 Fernuniversit¨at,Hagen 1985. [23] R. B¨orger.Multiorthogonality in categories. Seminarberichte, 23:9–40, Fernuniversit¨at,Hagen 1985. [24] R. B¨orger. p-adic valued measures are atomic. Seminarberichte, 25:1–8, Fernuniversit¨at,Hagen 1986. [25] R. B¨orger.Coproducts and ultrafilters. J. Pure Appl. Algebra, 46(1):35–47, 1987. [26] R. B¨orger.Disjoint and universal coproducts I, II. Seminarberichte, 27:13–34, 35–46, Fernuniversit¨at, Hagen 1987. [27] R. B¨orger.Multicoreflective subcategories and coprime objects. Topology Appl., 33:127–142, 35–46, 1989. [28] R. B¨orger.Integration over sequential Boolean algebras. Seminarberichte, 33:27–66, Fernuniversit¨at, Hagen 1989. - 13 -
NOLI TURBARE CIRCULOS MEOS – A MATHEMATICAL TRIBUTE TO REINHARD BORGER¨ 13
[29] R. B¨orgerand W. Tholen. Factorizations and colimit closures. Seminarberichte, 34:13–58, Fernuni- versit¨at,Hagen 1989. [30] R. B¨orger.On the equation x!y! = z!. Preprint, Fernuniversit¨atHagen, 1989 (estimated). [31] R. B¨orger.Ein kategorieller Zugang zur Integrationstheorie. Habilitationsschrift, Fernuniversit¨at,Ha- gen,1989. [32] R. B¨orger.On categories of coproduct preserving functors. Preprint, Fernuniversit¨at,Hagen 1990 (estimated). [33] R. B¨orger. A connected Hausdorff union of two open heriditarily disconnected sets. Seminarberichte, 37:33–34, Fernuniversit¨at,Hagen 1990. [34] R. B¨orgerand W. Tholen. Total categories and solid functors. Canad. J. Math., 42(2):213–229, 1990. [35] R. B¨orger and W. Tholen. Strong, regular and dense generators. Cahiers Topologie G´eom. Diff´erentielle, 32(3):257–276, 1991 [36] R. B¨orger.Fubini’s theorem from a categorical viewpoint. Category Theory at Work (Bremen 1990), Res. Exp. Math 18:367–375, Heldermann, Berlin 1991. [37] R. B¨orger.Making factorizations compositive. Comment. Math. Univ. Carolinae 32(4):749-759, 1991. [38] R. B¨orgerand W. Tholen. Totality of colimit closures. Comment. Math. Univ. Carolinae 32(4):761- 768, 1991. [39] R. B¨orger.How to make a path injective? Recent Developments of General Topology and its Applica- tions (Berlin 1992), Math. Res. 67:57-59, Akademie-Verlag, Berlin 1992. [40] R. B¨orgerand R. Kemper. Normed totally convex spaces. Comm. Algebra, 21(9):57-59, 1993. [41] R. B¨orger,W. Tholen and A. Tozzi. Lexicographic sums and fibre-faithful maps. Appl. Categ. Struc- tures, 1(1):59–83, 1993. [42] R. B¨orger.A geometric theory of Henselian local rings. Seminarberichte, 43:1–4, Fernuniversit¨at, Hagen 1993. [43] R. B¨orger.Implicit field operations. Seminarberichte, 46:25–30, Fernuniversit¨at,Hagen 1993. [44] R. B¨orgerand R. Kemper. Cogenerators for convex spaces. Appl. Categ. Structures, 2(1):1–11, 1994. [45] R. B¨orgerand R. Kemper. There is no cogenerator for totally convex spaces. Cahiers Topologie G´eom. Diff´erentielle, 35(4):335-338, 1994. [46] R. B¨orger.Disjointness and related properties of coproducts. Acta Univ. Carolin. Math. Phys., 35(1):5–18, 1994. [47] R. B¨orger.Infinitary linear combinations over valued fields. Seminarberichte, 53:1–14, Fernuniversit¨at Hagen, 1995. Revised version in Seminarberichte, 58:1–19, Fernuniversit¨at,Hagen 1997. [48] R. B¨orger.Connectivity properties of sequential Boolean algebras. 23rd Winter School on Abstract Analysis, 1995, Acta Univ. Carolin. Math. Phys. 36(2):43-63, 1995. [49] R. B¨orgerand R. Kemper. A cogenerator for preseparated superconvex spaces. Appl. Categ. Structures, 4(4):361–370, 1996. [50] R. B¨orger.Non-existence of a cogenerator for orderd vector spaces. Quaestiones Math., 20(4):587–590, 1997. [51] R. B¨orger.On the characterization of commutative W ∗-algebras. Seminarberichte, 61:1–14, Fernuni- versit¨at,Hagen 1997. [52] R. B¨orger.On suprema of continuous functions. Seminarberichte, 63:63–68, Fernuniversit¨at,Hagen 1998. [53] R. B¨orger.A non-Jordan measurable regularly open subset of the unit interval. Arch. Math. (Basel) 73(4):262–264, 1999. [54] R. B¨orger.When can points in convex sets be separated by affine maps? J. Convex Anal. 8(2):409–264, 2001. [55] R. B¨orger.On the powers of a Lindel¨ofspace. Seminarberichte, 73:1–2, Fernuniversit¨at,Hagen 1998. [56] R. B¨orger.The tensor product of orthomodular posets. Categorical Structures and Their Applications, pp 29–40, World Sci. Publ., River Edge (NJ) 2004. - 14 -
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[57] R. B¨orger.Vector integration by a universal property. Monograph (incomplete, unpublished), 196 pp., 2006. [58] R. B¨orger.Joins and meets of symmetric idempotents. Appl. Categ. Structures, 15(5–6):493–497, 2007. [59] J. Ad´amek,R. B¨orger,S. Milius, J. Velebil. Iterative algebras: how iterative are they? Theory Appl. Categ., 19(5):61–92, 2007. [60] R. B¨orgerand R. Kemper. Infinitary linear combinations in reduced cotorsion modules. Cahiers Topol. G´eom.Diff´er. Cat´eg., 50(3):189–210, 2009. [61] R. B¨orger.A categorical approach to integration. Theory. Appl. Categ., 23(12):243–250, 2010. [62] R. B¨orger.What is an integral? Monograph (incomplete, unpublished; long version of August 2010, 128 pp; short version of October 2010, 59 pp), 2010. [63] R. B¨orgerand A. Pauly. How does universality of coproducts depend on the cardinality? Topology Proc., 37:177–180, 2011. [64] R. B¨orger.Continuous selections, free vector lattices and formal Minkowski differences. J. Convex Anal., 18(3):855–864, 2011. [65] R. B¨orger.Measures and idempotents in the non-commutative situation. Tatra Mt. Math. Publ., 49:49–58, 2011.
[66] J. Ad´amek,H. Herrlich, and G. E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Wiley, New York 1990. [67] J. Ad´amekand W. Tholen. Total categories with generators. J. of Algebra, 133(1):63–78, 1990. [68] A.V. Arhangel’skii and R. Wiegandt. Connectedness and disconnectedness in topology. General Topol- ogy Appl. 5:9–33, 1975. [69] G.C.L. Br¨ummer. A categorical study of initiality in uniform topology. Thesis, University of Cape Town, 1971. [70] G.C.L. Br¨ummer.Topological categories. General Topology and Appl., 4:125-142, 1974. [71] A. Carboni, S. Lack and R.F.C Walters Introduction to extensive and distributive categories. J. Pure Appl. Algebra, 84:145–158, 1993. [72] Y. Diers. Cat´egorieslocalisables. Thesis. Universit´ede Paris VI, 1977. [73] S. Eilenberg and G.M. Kelly. Closed categories. In: Proceedings of the Coference on Categorical Algebra La Jolla 1965, pp 421–562, Springer, Berlin 1966. [74] M. Giry. A categorical approach to probability theory. Lecture Notes in Math. 915:68–85, Springer, Berlin 1982. [75] H. Herrlich. Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78, Springer, Berlin 1968. [76] H. Herrlich. Topological functors. General Topology and Appl., 4:125-142, 1974. [77] H. Herrlich. Initial completions. Math. Z., 150:101–110, 1976. [78] H. Herrlich, R. Nakagawa, G.E. Strecker, T. Titcomb. Equivalence of topologically-algebraic and semitopological functors. Canad. J. Math., 32:34–39, 1980. [79] H. Herrlich, G.E. Strecker. Category Theory. Allyn and Bacon, Boston 1973. [80] R.-E. Hoffmann. Die kategorielle Auffassung der Initial- und Finaltopologie. Thesis. Ruhr-Universit¨at, Bochum 1972. [81] R.-E. Hoffmann. Note on semi-topological functors. Math. Z., 160:9–74, 1977. [82] D. Hofmann, G. J. Seal, and W. Tholen, editors. Monoidal Topology: A Categorical Approach to Order, Metric, and Topology, volume 153 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge 2014. [83] Y.H. Hong. Studies on categories of universal topological algebras. Thesis, McMaster University, Hamilton 1974. [84] S.S. Hong. Categories in which every mono-source is initial. Kyungpook Math. J. 15:133-139, 1975. [85] J.J. Kaput. Locally adjunctable functors. Ill. J. Math., 16:86-94, 1972. - 15 -
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[86] G.M. Kelly. Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series 64, Cambridge University Press, Cambridge 1982. Republished in: Reprints in Theory and Applications of Categories, 10:1–136, 2005. http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf. [87] R. Kemper. In grateful memory of Reinhard B¨orger.In: this volume. [88] F.W. Lawvere. The category of probabilistic mappings. Typescript, 12 pp, 1962. Available at: ://ncat- lab.org/nlab/files/lawvereprobability1962.pdf. [89] F.E.J. Linton. Functorial measure theory. In: Proceedings of the Conference on Functional Analysis Irvine 1966, pp 36–49. Thompson, Washington D.C. 1968. [90] E.G. Manes. A pullback theorem for triples in a lattice fibering with applications to algebra and analysis. Algebra Universalis 2:7–17, 1971. [91] G. Preuß. Uber¨ den E-Zusammenhang und seine Lokalisation. Thesis, Freie Universit¨at,Berlin 1967. [92] D. Pumpl¨un.Regulary ordered Banach spaces and positively convex spaces. Results in Mathematics 7:85–112, 1984. [93] D. Pumpl¨un.Banach spaces and superconvex modules. In: Behara, Fritsch, Lintz (eds): Symp. Gaus- siana, pp. 323–338, de Gruyter, Berlin 1995. [94] D. Pumpl¨unand H. R¨ohrl. Banach spaces and totally convex spaces I. Comm. Algebra., 12(8):953– 1019, 1984. [95] D. Pumpl¨unand H. R¨ohrl.Banach spaces and totally convex spaces II. Comm. Algebra., 13(5):1047– 1113, 1985. [96] D. Pumpl¨unand H. R¨ohrl. Convexity theories IV. Klein-Hilbert Parts in convex modules. Appl. Categ. Structures., 3, 173–200, 1995. [97] G. Salicrup. Local monocoreflectivity in topological categories. Lecture Notes in Math. 915:293–309, Springer, Berlin 1982. [98] Z. Semadeni. Banach spaces of continuous functions, Vol. 1. PWN Polish Scientific Publishers, Warsaw 1971 [99] W. Shukla. On top categories. Thesis, Indian Institute of Technology, Kanpur 1971. [100] G.E. Strecker. Component properties and factorizations. Math. Centre Tracts 52:123–140, 1974. [101] R. Street and R. F. C. Walters. Yoneda structures on 2-categories. J. of Algebra, 50(2):350–379, 1978. [102] V. Trnkov´a.On descriptive classification of Set-functors. Comm. Math. Univ. Carolinae 12:143–174 and 345–357. [103] W. Tholen. Relative Bildzerlegungen und algebraische Kategorien. Thesis, Westf¨alische Wilhelms- Universit¨at,M¨unster,1974. [104] W. Tholen. M-functors. Mathematik Arbeitspapiere, 7:178–185, Universit¨atBremen, 1976. [105] W. Tholen. Semi-topological functors I. J. Pure Appl. Algebra, 15(1):53–73, 1979. [106] W. Tholen. Note on total categories. Bull. Australian Math. Soc., 21:169–173, 1980. [107] J.A. Tiller. Component subcategories. Quaestiones Math. 4:19–40, 1980. [108] M.B. Wischnewsky. Partielle Algebren in Initialkategorien. Thesis, Ludwig-Maximilians-Universit¨at, M¨unchen 1972. [109] O. Wyler. Top categories and categorical topology. General Topology and Appl., 1:17–28, 1971. [110] O. Wyler. On the categories of topological algebra. Archiv Math. (Basel), 22:7–17, 1971.
Walter Tholen, Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3 E-mail address: [email protected] - 16 - - 17 -
In grateful memory of Reinhard Börger
Ralf Kemper
Pumplün and Röhrl ([5], [6]) introduced the category TC of totally convex spaces, which are the Eilenberg-Moore-algebras for the monad induced by the (closed) unit ball functor from the category of Banach spaces over the field K R or C = with non-expansive linear maps to the category of sets as follows. N P Put Ω : {α (αi )i N K i αi 1}. An Ω-algebra is a set X together with a = = ∈ ∈ | | | É mapping Ω Set (X N, X ), α α (α Ω). A morphism from the Ω-algebra X → 7→ X ∈ to the Ω-algebra Y is a set mapping f : X Y satisfying for any α Ω f α → ∈ ◦ X = α f N, where f N : X N Y N is defined componentwise. Y ◦ → N P For an Ω-algebra X , α Ω and x : (xi )i N X define i αi xi : αX (x). Put j j N ∈ j = ∈ ∈ = δ : (δ )i N K , where δ is the Kronecker-symbol (i, j N). = i ∈ ∈ i ∈ DEFINITION ([5], 2.2). (i) An Ω-algebra (X ,(α ,α Ω)) is called a totally convex X ∈ space if and only if X and the following two axioms are satisfied: 6= ; N P j (TC 1) For all (xi )i N X and j N i δ xi x j holds. ∈ ∈ ∈ i = i N (TC 2) For all α,β Ω (i N) and (x j )j N X ∈ ∈ ∈ ∈
X ¡X i ¢ X¡X i ¢ αi βj x j αi βj x j i j = j i holds.
(ii) The full subcategory of the category of Ω-algebras which is determined by all totally convex spaces is denoted by TC.
Later Pumplün introduced the categories PC of positively convex spaces ([8]) N P (where Ω is replaced by Ω+ : {α (αi )i N [0,1] i αi 1}) and SC of super- = = ∈ ∈ | É N P convex spaces ([9]) (where Ω is replaced by ∂Ω+ : {α (αi )i N [0,1] i αi = = ∈ ∈ | = 1}). The “finitary versions” of TC, PC and SC (i.e. the categories obtained by restriction to finitary operations) are AC (absolutely convex spaces), PC f in (fi- nitely positively convex spaces) and Conv (convex spaces).
For A TC (AC,PC,PC resp.) and x A the norm of x is defined by ∈ f in ∈ x : in f {λ [0,1] y A : x λy}. k k = ∈ | ∃ ∈ =
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2
For A as above the interior Å A is Å : {x A x 1} {αy α [0,1), y A}. ⊆ = ∈ | k k < = | ∈ ∈ An x A is called normed if αx α x for all α [0,1], and A is called normed ∈ k k = k k ∈ if all x A are normed. For some spaces (for example finitely positively convex ∈ spaces) one should probably use the word “semi-norm” instead of “norm”, be- cause there x 0 in general does not imply x 0 ([8], 3.3(iii)). Nevertheless, k k = = we stick to the established name. In TC this property is fulfilled ([5], 6.9). In [1] Reinhard and I characterized normed spaces without mentioning the norm explicitly. Theorem ([1], 1.5): A TC (AC,PC,PC resp.) is normed if and only if it satis- ∈ f in fies the following condition:
x A y Å α (0,1) : αx αy x y. ∀ ∈ ∀ ∈ ∀ ∈ = =⇒ =
Corollary ([1], 1.6): All products and all subspaces of normed spaces in TC (AC, PC,PC f in resp.) are normed. If B is a subspace of a normed space A it is surprising that B is normed, because the norms on A and B may behave quite differently. From general categorical reasons (c.f. e.g. [7] , 37.1) it follows that in TC (AC,PC,PC f in resp.) the normed objects form a regular-epireflective subcategory. But this does not give much in- sight into the concrete structure of the reflection, i.e. of the congruence relation induced by the reflection map. A concrete construction of the reflection is given in the following way. For A PC and a,b A let a b if and only if 1 a 1 b. ∈ f in ∈ ≈ 2 = 2 An equivalence relation on A is defined by ∼ a b if and only if a b or there exists a c Å with a b c. ∼ = ∈ ≈ ≈ The relation is a congruence relation on A as a finitely positively convex space. ∼ If even A PC (AC,TC resp.), then is also compatible with this structure ([1], ∈ ∼ 2.2), and A/ is normed ([1], 2.3). ∼ Theorem ([1], 2.4): Let A PC (PC, AC,TC resp.). The projection p : A A/ ∈ f in → ∼ is the reflection map for the reflection from PC f in (PC, AC,TC resp.) into the subcategory of normed objects. In [2] Reinhard and I constructed cogenerators for the categories Conv, AC and for the category of spherical and regular finitely positively convex spaces in the following way. For C Conv and 0 C define C ] : C {0}. C ] has a structure as finitely po- ∈ ∉ = ∪ ] N P sitively convex space. For α Ω+f in,xi C (i ) define i αi xi as follows: P ∈ ∈ ∈ P If i αi 1 or if 0 αi0 for some i0 N with xi0 0, put i αi xi 0. Other- < < P ∈ P = = wise α ∂Ω+ and we define α x : α y , where y C (i N) is such ∈ f in i i i = i i i i ∈ ∈ that x y for all i supp(α); since α 0 for all i N with x 0, the result i = i ∈ i = ∈ i 6= does not depend on the choice of the yi by a well-known computation rule ([5], 2.4).
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3
Theorem ([2], 2.3) R] is a cogenerator of Conv, i.e. for every D Conv and a,b ∈ ∈ D with a b, there exists a Conv-morphism f : D R] with f (a) f (b). 6= → 6= For A PC f in an element a A is called spherical if αa 0 for all α [0,1), ∈ ∈ sph = ∈ and A is called spherical if all x A are spherical. PC is the full subcategory ∈ f in of spherical finitely positively convex spaces. An A PC f in is called regular if reg∈ for all x A, the relation x 0 implies x 0. PC is the full subcategory of ∈ k k = = f in regular finitely positively convex spaces. sph reg Corollary ([2], 2.4) R] is a cogenerator of PC PC . f in ∩ f in sph reg For every D PC PC ∈ f in ∩ f in R(D): {(γ,d) K D ( γ 1,d D \ {0}) or (γ 0,d 0)} = ∈ × | | | = ∈ = = has a structure as absolutely convex space. For (γ ,d ) R(D)(i N) and α i i ∈ ∈ ∈ Ωf in define X X X αi (yi ,di ): (0,0) if αi γi 1 or αi di 0 i = | i | < i | | = and X ¡X X ¢ αi (yi ,di ): αi γi , αi di i = i i | | otherwise. Theorem ([2], 2.5) The set {R(R]),K } is a cogenerator of AC. In [4] Reinhard and I constructed a cogenerator for the category of preseparated superconvex spaces, and described separated convex spaces, i.e. convex spaces for which the morphisms into the unit interval separates points. Preseparated convex spaces, defined in [10], 4.9, are such convex spaces D ful- filling a well known cancellation law, i.e. for all elements x, y,z D,α (0,1) ∈ ∈ αx (1 α)z αy (1 α)z implies x y. + − = + − = For a preseparated convex space D let ϕ : D V be an affine injective mapping → into a real vector space V . Theorem ([49], 2.3) Let D be a convex space. With the above notations the follo- wing statements are equivalent:
1. D is separated.
2. D is preseparated and ϕ(D) ϕ(D) V is linearly bounded. − ⊆ 3. Let u,v,x , y D (n N). n n ∈ ∈ ¡ 1 ¢ 1 ¡ 1 ¢ 1 If for all n N 1 u xn 1 v yn, then u v. ∈ − n + n = − n + n =
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REFERENCES 4
Corollary ([4], 1.) Every separated convex space can be embedded into the unit ball of a real normed vector space.
The category of separated convex spaces has a cogenerator, but there is no co- generator in some categories of linearly bounded convex spaces.
Theorem ([4], 2.4) Neither the category of linearly bounded convex spaces nor the category of preseparated linearly bounded convex spaces has a cogenerator.
Theorem ([4], 3.5) Let D be a preseparated superconvex space. Then there exists a real Banach space B and an injective SC-morphism ϕ : D Oˆ(B). → This yields a cogenerator for the category of preseparated superconvex spaces.
Corollary ([4], 2.) A superconvex space is preseparated if and only if it is sepa- rated, especially, the unit intervall [0, 1], [0, 1] with its canonical superconvex structure, is a cogenerator for the category of preseparated superconvex spaces.
sph reg In [3] Reinhard and I proved in contrast to AC, PC PC and Conv the f in ∩ f in following
Theorem ([3], Theorem): None of the categories TC, PC and SC has a cogenera- tor.
References
[1] R. Börger and R. Kemper. Normed totally convex spaces. Comm. Algebra, 21(9):57-59, 1993.
[2] R. Börger and R. Kemper. Cogenerators for convex spaces. Appl. Categ. Structures, 2(1):1–11, 1994.
[3] R. Börger and R. Kemper. There is no cogenerator for totally convex spaces. Cahiers Topologie Géom. Différentielle, 35(4):335-338, 1994.
[4] R. Börger and R. Kemper. A cogenerator for preseparated superconvex spaces. Appl. Categ. Structures, 4(4):361–370, 1996.
[5] D. Pumplün and H. Röhrl. Banach spaces and totally convex spaces I Comm. Alg., 12(8):953-1019, 1984.
[6] D. Pumplün and H. Röhrl. Banach spaces and totally convex spaces II Comm. Alg., 13(5):1047-1113, 1985.
[7] H. Herrlich, G.E. Strecker. Category theory Allyn and Bacon, Boston, 1973.
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REFERENCES 5
[8] D. Pumplün. Regulary ordered Banach spaces and positively convex spaces Results in Mathematics, Vol. 7, 85-112, 1984.
[9] D. Pumplün. Banach spaces and superconvex modules Symp. Gausiana, Behara, Fritsch, Lintz (eds.), de Gruyter, 323-338, 1995.
[10] D. Pumplün and H. Röhrl. Convexity theories IV.Klein-Hilbert Parts in con- vex modules Appl. Cat. Struct., 3, 173-200, 1995.
Ralf Kemper, Fakultät für Mathematik /Department of Mathematics, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld E-mail address: [email protected]
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Banach Limits revisited
D. Pallaschke and D. Pumpl¨un∗
Dedicated to Reinhard B¨orger, a brilliant and enthusiastic mathematician full of new ideas.
Introduction: Most, if not all, publications where Banach limits are investigated take place in an order unit normed real linear space. Order unit normed linear spaces are a special type of regularly ordered normed linear spaces and therefore the first section is a short collection of the fundamental results on this type of normed linear spaces, for the reader’s convenience. The connection between order unit normed linear spaces and base normed linear spaces within the category of regularly ordered normed linear spaces is described in §2. §3, at last, contains the results on Banach limits in an arbitrary order unit normed linear space. It is shown that the original results on Banach limits are valid in a for greater range.
AMS Subject Classication: 46M99, 26A16, 46B40.
Keywords: Order unit normed spaces, base normed spaces, Banach limits
1 Regularly ordered normed linear spaces
′′ ′′ An ordered normed linear space E with order ≤ , norm E and order cone C(E) is called regularly ordered iff the cone C(E) is E-closed and proper and E is a Riesz norm, i.e. if
∗D. Pallaschke, Institute of Operations Research, University of Karlsruhe –KIT, D-76128 Karlsruhe, Germany, e-mail: [email protected] D. Pumpl¨un, Faculty of Mathematics and Computer Science, FernUniversitaet Hagen, D-58084 Hagen, Germany, e-mail: [email protected]
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(Ri 1) For x, y ∈ E, −y ≤ x ≤ y implies x E ≤ y E, i.e. E is absolutely monotone, and
(Ri 2) For x ∈ E with x E < 1 there exists a y ∈ E with y E < 1 and −y ≤ x ≤ y hold. (see [1], [5]).
Lemma 1.1 Let, for an ordered linear space E with proper and E-closed cone C(E), (Ri 1) hold. Then each of the following two conditions is equiv- alent to (Ri 2) (Ri 3) For x ∈ E and ε > 0 there exists a y ∈ E such that −y ≤ x ≤ y and y E < x E + ε hold. (Ri 4) For any x ∈ E
x E = inf { y E | − y ≤ x ≤ y } ,
holds. Proof. The proof is straightforward. Condition (Ri 2) implies that C(E) generates E. If (Ri 2) holds, then for x ∈ E and ε > 0 there is y ∈ E with x −y ≤ ≤ y, x E + ε hence (Ri 3) is proved for y0 := ( x E + ε) y.
(Ri 3) implies that C(E) generates E and (Ri 1) implies
x E ≤ inf { y E | − y ≤ x ≤ y } =: x 0.
Because of (Ri 3) , for x ∈ E and ε > 0 there is a y ∈ E such that −y ≤ x ≤ y and y E < x E + ε, for any ε > 0 and hence x 0 − ε ≤ x E ≤ x 0 proving (Ri 4). Moreover, (Ri 4) obviously implies (Ri 2) which completes the proof.
In [5] K. Ch. Min introduced regularly ordered normed spaces as a natural and canonical generalization of Riesz spaces . A crucial point in this general- ization was the definition of the corresponding homomorphisms compatible and most closely related to the structure of these spaces, such that, in ad- dition, the set of these special homomorphisms is again a regularly ordered normed linear space in a canonical way. This is done by
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Definition 1.2 If Ei, i = 1, 2 are regularly ordered linear spaces a bounded linear mapping f : E1 −→ E2 is called positive iff f(C(E1)) ⊂ C(E2) holds. A bounded linear mapping is called regular iff it can be expressed as the difference of two positive linear mappings ([5]).
The set
Reg-Ord(E1,E2) := {f | f : E1 −→ E2 regular linear mapping } is a linear space by the obvious operations. One introduces the cone
C( Reg-Ord(E1,E2)) := {f | f ∈ Reg-Ord(E1,E2),
f(C(E1)) ⊂ C(E2) } which is obviously proper and generates Reg-Ord(E1,E2). One often writes x1 ≥ 0 as abbreviation for x1 ∈ C(E1) and consequently calls an f ∈ Reg-Ord(E1,E2) positive and writes f ≥ 0, if f(x1) ≥ 0, for x1 ≥ 0 in E1, i.e. x1 ∈ C(E1). The positive part of the unit ball in a regularly ordered space E with norm E is denoted by
∆(E) = C(E) ∩ (E) , (E) = {x | x ∈ E and E ≤ 1 }.
Lemma 1.3 Let Ei be regularly ordered normed linear spaces with norm i and cone C(Ei), i = 1, 2. If g ∈ C( Reg-Ord(E1,E2)) and g ∞ denotes the usual supremum norm, then
g ∞ := sup { g(x1) 2 | x1 ∈ ∆(E1) } holds.
Proof. For x1 ∈ E1 with x1 1 < 1 there is y1 ∈ E1, y1 < 1, with −y1 ≤ x1 ≤ y1 which implies
−g(y1) ≤ g(x1) ≤ g(y1) and g(x1) 2 ≤ g(y1) 2,
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hence
g ∞ = sup { g(x1) 2 | x1 1 ≤ 1 }
≤ sup { g(y1) 2 | y1 1 ≤ 1 , y1 ∈ C(E1) }
= sup { g(y1) 2 | y1 ∈ ∆(E1) }
≤ sup { g(y1) 2 | y1 1 ≤ 1 } = g ∞.
∗ Now, we proceed to define the norm in the space Reg-Ord (E1,E2) by ∗ f := inf { g ∞ | − g ≤ f ≤ g } . (∗). ∗ Proposition 1.4 For regularly ordered normed spaces E1,E2, is a Riesz norm on Reg-Ord (E1,E2) and makes Reg-Ord (E1,E2) a regularly or- ∗ dered normed linear space. For f ≥ 0 f = f ∞ holds and in general
∗ f ∞ ≤ f .
Proof. The proof that ∗ is a seminorm is straightforward. In order ∗ to show that ∞ ≤ one starts with f, g ∈ Reg-Ord (E1,E2) and −g ≤ f ≤ g. Let x1 1 < 1 and −y1 ≤ x1 ≤ y1 with y1 1 < 1, x1, y1 ∈ E1. Then x1 + y1 ≥ 0 follows and
−g(x1 + y1) ≤ f(x1 + y1) ≤ g(x1 + y1). (i)
Using g − f ≥ 0 and y1 − x1 ≥ 0 one obtains in the same way
−g(y1 − x1) ≤ f(y1 − x1) ≤ g(y1 − x1) and, multiplying by −1
−g(y1 − x1) ≤ f(x1 − y1) ≤ g(y1 − x1) (ii).
Adding (i) and (ii) yields
−g(y1) ≤ f(x1) ≤ g(y1) hence f(x1) 2 ≤ g(y1) 2 and
f ∞ = sup { f(x1) 2 | x1 1 ≤ 1}
≤ sup { g(y1) 2 | y1 1 ≤ 1 and − y1 ≤ x1 ≤ y1 } ≤ g ∞.
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∗ ∗ Now (∗) yields f ∞ ≤ f , i.e. is a norm. If f ≥ 0 then
∗ f := inf { g ∞ | 0 ≤ f ≤ g } ≤ f ∞,
∗ ∗ hence f = f ∞ and is a Riesz norm because of Lemma 1.1, (Ri 4) and the definition of (∗).C ( Reg-Ord (E1,E2)) is obviously ∞-closed ∗ ∗ and therefore also -closed because of f ∞ ≤ f .
In the following ∗ will always denote this norm of regular linear mappings. Note that Reg-Ord is a symmetric, complete and cocomplete monoidal closed category and the inner hom-functor Reg-Ord ( , ) has as an ad- joint, the tensor product ([5]).
2 Order unit and base ordered normed linear spaces
The order unit normed linear spaces are a special type of regularly ordered normed linear spaces , as are the base normed linear spaces ([5], [16]). For investigating a special type of mathematical objects, however, it is always best to use the type of mappings most closely related to the special structure of the objects (the Bourbaki Principle). Hence, for investigating order unit normed spaces we do not look at the full subcategory of Reg-Ord generated by the order unit normed spaces but introduce a more special type of regular linear mappings. The same method, by the way, has been successful for another type of regularly ordered spaces, namely the base normed (Banach) spaces (cp. [10], [11], [5]).
Definition 2.1 For two order unit normed linear spaces Ei with order unit ei, i = 1, 2, define
Bs0(E1,E2) := {f | f ∈ Reg-Ord (E1,E2), f ≥ 0 and f(e1) = e2 } and C0(E1,E2) := R+Bs0(E1,E2).
Proposition 2.2 Let E1,E2 be order unit spaces with order units e1, e2. Then
(i) Bs0(E1,E2) is a ∞-closed convex base of C0(E1,E2) and Bs0(E1,E2) ⊂ ∞(E1,E2), the ∞- closed unit ball of Reg-Ord (E1,E2) in the supremum norm ∞.
(ii) C0(E1,E2) is a ∞- closed proper subcone of C (Reg-Ord(E1,E2)) .
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Proof. (1) Let fn ∈ Bs0(E1,E2) , n ∈ N, and f ∈ Reg-Ord (E1,E2) with limn→∞ fn − f ∞ = 0. Then, for x1 ∈ E1, f(x1) = limn→∞ fn(x1) follows which implies f ≥ 0 and f(e1) = e2, i.e. f ∈ Bs0(E1,E2) showing that Bs0(E1,E2) is ∞- closed. Now, f ∈ Bs0(E1,E2) and −e1 ≤ x1 ≤ e1 imply −e2 ≤ f(x1) ≤ e2 i.e. f ∞ ≤ 1 and even f ∞ = 1 because f(e1) = e2. Hence Bs0(E1,E2) ⊂ ∞(E1,E2) follows, even Bs0(E1,E2) ⊂ ∂ ( ∞(E1,E2)) .
n n
Let αifi, αi ≥ 0, 1 ≤ i ≤ n, αi = 1, be a convex combination of fi ∈ i=1 i=1 Bs0(XE1,E2). Then X n n n
αifi ≥ 0 and αifi (e1) = αie2 = e2, i=1 i=1 ! i=1 X X X follows, i.e. n
αifi ∈ Bs0(E1,E2), i=1 X which proves that Bs0(E1,E2) is convex.
Now αf = βg with α, β > 0 and f, g ∈ Bs0(E1,E2) implies αe2 = αf(e1) = βg(e1) = βe2, and α = β, i.e. Bs0(E1,E2) is a ∞-closed base of C(E1,E2) and 0 ∈/ Bs0(E1,E2).
(ii) This follows from (i) ( see [3], 3.9 p.128).
Corollary 2.3 For order unit normed linear spaces Ei, i = 1, 2,
Ord-Unit (E1,E2) = C(E1,E2) − C(E1,E2) is a base-normed ordered linear space with base Bs0(E1,E2) and base norm denoted by 0.C0(E1,E2) and Bs0(E1,E2) are closed in the base norm 0.
Proof. That Ord-Unit (E1,E2) is a base normed space follows from Propo- sition 2.2 and the definition. That base and cone are base normed closed follows from the fact that they are ∞- closed (see Proposition 2.2 ) and because the ∞- topology is weaker than the 0- topology (see Proposi- tion 2.2 and [3], 3.8.3, p.121).
Remark 2.4 If Reg-Ord (E1,E2) is a Banach space, with the norm ∞, because Ei, i = 1, 2, are Banach spaces, then Bs0(E1,E2) is superconvex ( see [5], [11]) and Reg-Ord (E1,E2) is a base normed Banach space ( see [5], [16], [3]).
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Definition 2.5 The order unit normed linear spaces together with the linear mappings f : E1 −→ E2 with f ∈ Ord-Unit (E1,E2) constitute the category Ord-Unit of order-unit normed linear spaces which is a not full subcategory of Reg-Ord.
There is an equally important subcategory of Reg-Ord, the category of based normed linear spaces. Definition 2.6 A base normed ordered linear space “base normed linear space” for short, is a regular ordered linear space E with proper closed cone C(E) and norm E which is induced by a base Bs(E) of C(E) (see [16], [3]). If Ei, i = 1, 2 are base normed linear spaces, put
Bs(E1,E2) := {f | f : E1 −→ E2 linear and f(Bs(E1)) ⊂ Bs(E2) } .
The elements of Bs(E1,E2) are monotone mappings, Bs(E1,E2) is a base set in Reg-Ord(E1,E2) and it is ∞-closed. Let C(E1,E2) denote the proper closed cone generated by Bs(E1,E2).
BN-Ord(E1,E2) := C(E1,E2) − C(E1,E2) is a base normed space of special mappings from E1 to E2. The base normed linear spaces and these linear mappings form the not full subcategory BN- Ord of Reg-Ord (see [6], [9], [11]), which is therefore a closed category.
What remains in this connection is to investigate special morphisms par- ticularly adapted to these subcategories between spaces belonging to two different of these subcategories Ord-Unit and BN-Ord. We start this with investigating the intersection of these subcategories.
Proposition 2.7 Let (E,C(E), E) be a regular ordered normed linear space. Then E is a base and order unit norm iff (E,C(E), E) is iso- morphic to (R, [0, ∞[, | |) by a regular positive isomorphism. Proof. If e ∈ E is the order unit and if we omit the index E at the norm, then trivially e = 1 and e > 0 hold. Let b ∈ Bs(E) and assume b = e. As Bs(E) ⊂ (E) = [−e, e] (see [3]), 0 < b < e holds and d := e − b > 0 follows or e = b + d, which implies 1 = e = b + d = b + d , because is additive on C(E). This implies d = 0 and hence e = b which gives a contradiction. Therefore Bs(E) = {e} and the assertion follows as C(E) = R+Bs(E) = R+e and E = C(E) − C(E) = Re.
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Hence, the isomorphism is i : E −→ R defined by i(αe) = α, α ∈ R.
It should be noted that this isomorphism is an isomorphism in the category Ord-Unit of order unit normed spaces and also in BN-Ord. So, loosely speaking, Ord-Unit ∩ BN-Ord = {R}. Now the “general connection” between Ord-Unit and BN-Ord is investi- gated via the morphisms:
Proposition 2.8 If E1 is a base normed and E2 an order unit normed linear space, then Reg-Ord(E1,E2) is an order unit normed linear space.
Proof. Define ε : C(E1) −→ C(E2) by ε(Bs(E1)) = {e2} and extend ε positive linearly by ε(αx1) = αe2, for α ≥ 0, x1 ∈ Bs(E1), to ε : C(E1) −→ C(E2) which can be uniquely extended to ε : E1 −→ E2, a monotone, linear mapping in Reg-Ord in the usual way. Obviously, ε ∞ = ε , with the Reg-Ord norm, as ε is a positive mapping. Take a g ∈ Reg-Ord(E1,E2) with g ≤ 1, i.e. g ( (E1)) ⊂ (E2) = [−e2, e2] and hence −e2 ≤ g(b1) ≤ e2 for b1 ∈ Bs(E1) or −ε(b1) ≤ g(b1) ≤ ε(b1) whence −ε(c1) ≤ g(c1) ≤ ε(c1) for c1 ∈ C(E1). For arbitrary x1 ∈ E1, x1 = c1 − d1, c1, d1 ∈ C(E1) −ε(c1) ≤ g(c1) ≤ ε(c1) and −ε(d1) ≤ −g(d1) ≤ ε(d1) follows implying −ε(x1) ≤ g(x1) ≤ ε(x1) for x1 ∈ E1 or −ε ≤ g ≤ ε. This means, for arbitrary g = 0, that − g ε ≤ g ≤ g ε. This shows that ε is an order unit in Reg-Ord (E1,E2). Denoting the order unit norm by 0 g 0 ≤ g follows.
This is a slightly different version of the proof of Theorem 1 in Ellis [3].
Surprisingly a corresponding result also holds if E1 ∈ Ord-Unit and E2 ∈ BN-Ord :
Proposition 2.9 If E1 is an order unit and E2 is a base normed ordered linear space, then Reg-Ord(E1,E2) is a base normed ordered linear space.
Proof. Define
Bs0(E1,E2) := {f | f ∈ C ( Reg-Ord(E1,E2)) and f(e1) ∈ Bs(E2) } where e1 denotes the order unit of E1. One shows first that Bs0(E1,E2) is a base set. For this, let g ∈ C ( Reg-Ord(E1,E2)) g = 0, i.e. g > 0 and g(e1) > 0, implying for
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1 f := g g(e1) 2 that is f > 0, and f(e1) 2 = 1, hence f(e1) ∈ Bs(E2). this implies that Bs0(E1,E2) = ∅. For f, g ∈ Bs0(E1,E2) and 0 ≤ α ≤ 1, obviously αf +(1−α)g ∈ Bs0(E1,E2) and Bs0(E1,E2) is convex. Besides, the above proof shows, that any g ∈ C ( Reg-Ord(E1,E2)) , g > 0, can be written as g = αf with α ≥ 0, f ∈ Bs0(E1,E2). Obviously 0 ∈/ Bs0(E1,E2) and if αf = βg with α, β ≥ 0 and f, g ∈ Bs(E1,E2) implying αf(e1) = βg(e1) from which α = β follows because of f(e1), g(e1) ∈ Bs(E1,E2) and finally f = g.
It is interesting that by defining the subspaces Ord-Unit(E1,E2) and BN-Ord(E1,E2) of Reg-Ord(E1,E2) for order unit or base normed spaces E1,E2, respectively , one gets a number of results which for the bigger space Reg-Ord(E1,E2) have either not yet been proved or were more difficult to prove because the assumptions for Reg-Ord(E1,E2) are weaker (see [4], [2]). The Propositions 2.8 and 2.9 are an exception because here the general space Reg-Ord(E1,E2) has the special structure of an order unit or base normed spaces, respectively.
There are different ways to generalize the structure of R in many fields of mathematics. In Analysis one is primarly interested in aspects of order, norm and convergence. Now, essentially, R with 1, the usual order and the absolute value (considered as a norm) forms the intersection Ord-Unit ∩ BN-Ord = {R}, which both generalize R in different (dual) directions. The above re- sults seem to indicate that the order unit spaces are at least as important as generalizations of R as the base normed spaces while in many publications the latter type seems to play the dominant role. Propositions 2.8 and 2.9 are particularly interesting because the hom-spaces have a special structure if the arguments do not belong to the same of the two subcategories Ord-Unit and BN-Ord.
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3 Banach Limits
For the introduction of Banach Limits we first prove, following a proof method of W. Roth in [13], Theorem 2.1, a special variant of the Hahn- Banach Theorem for order unit normed linear spaces.
Theorem 3.1 (Hahn-Banach Theorem for Order Unit Spaces) Let E be an order unit normed space with order unit e, ordering cone C(E) and norm E and let the following conditions be satisfied: (i) p : E −→ R is a sublinear monotone function with p(e) = 1, (ii) S : E −→ E is a surjective positive linear mapping.
(iii) For any x ∈ E, the set mapping Tx is a right inverse of S : E −→ E with Tx(S(x)) = x. Tx is monotone and −Tx(y) = T−x(−y),Tx(y+z) = Tx(y)+T0(z), x, y, z ∈ E. (iv) G is a muliplicative group G of positive automorphisms of E.
(v) For any x, y ∈ E and for every γ ∈ G, p(S(x)) = p(x), p(Tx(y)) = p(y) and p(γ(x)) = p(x) hold. Then there exists a positive linear functional µ : E −→ R with: a) µ(x) ≤ p(x) and µ(e) = 1, b) µ(S(x)) = µ(x),
c) µ(Tz(x)) = µ(x), d) µ(γ(x)) = µ(x), for x, z ∈ E and γ ∈ G. Proof. Define
Mp := {s | s : E −→ R, sublinear and monotone with − p(−x) ≤ s(x) ≤ p(x),
s(S(x)) = s(Tz(x)) = s(γ(x)) = s(x), for all x, z ∈ E and γ ∈ G} .
Obviously p ∈ Mp holds , hence Mp = ∅. A partial order ”≤” is defined in Mp by putting, for s1, s2 ∈ Mp,
s1 ≤ s2 if and only if s1(x) ≤ s2(x) for all x ∈ E.
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Let O ⊂ Mp be a non-empty, with respect to ”≤” totally ordered subset and define s0(x) = inf{s(x) | s ∈ O }, x ∈ E.
As −p(−x) ≤ s(x) ≤ p(x) for all s ∈ O s0 is well defined and finite and
−p(−x) ≤ s0(x) ≤ p(x) holds.
If x ≤ y then s(x) ≤ s(y) for all s ∈ O and hence for all x ≤ y s0(x) ≤ s0(y) follows, i.e. s0 is monotone.
Let x, y ∈ E then for all s ∈ O s0(x + y) ≤ s(x + y) ≤ s(x) + s(y) and hence s0(x + y) ≤ s0(x) + s0(y).
As obviously s0(αx) = αs0(x) for α ≥ 0 it follows that s0 is sublinear. Also s0(x) trivially satisfies the conditions a) - d) as well as s0(e) = 1. Conse- quently, s0 ∈ Mp and s0 is a lower bound of O in Mp. Zorn’s Lemma now implies the existence of (at least) one minimal element in Mp with respect to ≤ which will be denoted by µ.
Define for x0 ∈ E :
α0(x0) = sup{−p(−x1) − µ(x2) | x1, x2 ∈ E and x1 ≤ x0 + x2}. As, for x ∈ E, −p(−x) ≤ µ(x) ≤ p(x)
−p(−x1) ≤ µ(x1) ≤ µ(x0 + x2) ≤ µ(x0) + µ(x2),
−p(x1) − µ(x2) ≤ µ(x0)
and α0(x0) ≤ µ(x0) ≤ p(x0) follows. (1)
Taking x1 = x0 and x2 = 0 in the defining equation of α0 yields
−p(−x0) ≤ α0(x0) ≤ p(x0) (2) implying α0(e) = 1. (2a)
Now, the remaining equations in the assertion will be proved for α0(x). Take the inequality x1 ≤ S(x0) + x2 from the defining equation of α0(S(x0)), then
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Tx0 (x1) ≤ Tx0 (S(x0) + x2) = x0 + T0(x2) contributing
−p (−Tx0 (x1)) − µ (T0(x2)) = −p (T−x0 (−x1)) − µ(x2) = −p(−x1) − µ(x2) to α0(x0). Conversely, x1 ≤ x0 + x2 leads to
S (x1) ≤ S (x0 + x2) = S (x0) + S (x2) contributing
−p (−S(x1)) − µ (S(x2)) = −p(−x1) − µ(x2) to the definition of α0 (S(x0)) and one gets
α0 (S(x0)) = α0(x0). (3a)
To show the invariance of α0(x0) under Tz, z ∈ E, start with x1 ≤ x0 + x2 from α0(x0). Then
Tz(x1) ≤ Tz (x0 + x2) = Tz(x0) + T0(x2) contributing
−p (−Tz(x1)) − µ (T0(x2)) = −p(−x1) − µ(x2) to α0 (Tz(x0)) .
An inequality x1 ≤ Tz(x0) + x2 of α0 (Tz(x0)) leads to
S(x1) ≤ S (Tz(x0) + x2) = x0 + S(x2) and −p (−S(x1)) − µ (S(x2)) = −p(−x1) − µ(x2) as contribution to α0(x0). Hence
α0 (Tz(x0)) = α0(x0), z ∈ E. (3b)
Verbatim, this proof carries over to the equation
α0 (γ(x0)) = α0(x0), z ∈ E, γ ∈ G. (3c)
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A new function is now introduced by
µ˜(x) := inf{µ(y) + λα0(x0) | λ ≥ 0 and x ≤ y + λx0 , x0, y ∈ E }. (4)
If λ = 0 then x ≤ y + λx0 is x ≤ y and therefore
−p(−x) ≤ µ(x) ≤ µ(y) = µ(y) + 0α0(x0).
x y For λ > 0, x ≤ y + λx0 implies λ ≤ x0 + λ and yields x y −p − λ − µ λ ≤ α0(x0) because of the definition of α0(x0). Hence
−p(−x) ≤ µ(y) + λα0(x0) for λ > 0 and −p(−x) ≤ µ˜(x) (5) follows which implies in, particular,µ ˜(x) ≥ 0 for x ≥ 0, as −p(−x) is positive.
Taking y = x and λ = 0 one has
−p(−x) ≤ µ˜(x) ≤ µ(x) ≤ p(x) (6) in particular −p(−x) ≤ µ˜(x) ≤ p(x). If x1 ≤ x2 and x2 ≤ y2 + λx0 in (4) then µ˜(x1) ≤ µ˜(x2) (7) follows i.e. monotonicity.
Consider now, for xi ∈ E, i = 1, 2,
µ˜(xi) = inf{µ(yi) + λiα0(x0) | yi ∈ E, λi ≥ 0 and xi ≤ yi + λix0 }, then
µ˜(x1 +x2) ≤ µ(y1 +y2)+(λ1 +λ2)α0(x0) ≤ µ(y1)+λ1α0(x0)+µ(y2)+λ2α0(x0) i.e. µ˜(x1 + x2) ≤ µ˜(x1) +µ ˜(x2). Now for σ > 0,
µ˜(σx) := inf{µ(yσ) + λσα0(x0) | yσ ∈ E, λσ ≥ 0 and σx ≤ yσ + λσx0 } and
σµ˜(x) := inf{µ(σy) + σλα0(x0) | y ∈ E, λ ≥ 0 and σx ≤ σy + λσx0 }.
13 - 36 -
The mapping (y, λ) → (σy, σλ)) , σ > 0, is, for fixed σ > 0, bijective, therefore
µ˜(σx) = σµ˜(x) , σ ≥ 0, (8) holds because for σ = 0, one hasµ ˜(0x) =µ ˜(0) = 0. Soµ ˜ is sublinear.
We now show thatµ ˜ also satisfies the equations of the assertions of the theorem. Take x ≤ y + λx0 from the defining set ofµ ˜(x). Then
S(x) ≤ S(y + λx0) = S(y) + λS(x0) follows, contributing µ(S(y)) + λα0(S(x0)) = µ(y) + λα0(x0) toµ ˜(S(x)). Conversely, S(x) ≤ y+λx0 contributing µ(y)+λα0(x0) toµ ˜(S(x)) yields by applying Tx
x ≤ Tx(y + λx0) = Tx(y) + T0(λx0).
Hence
µ (Tx(y)) + 1 α0 (T0(λx0)) = µ(y) + 1 α0(λx0) = µ(y) + λα(x0).
This implies µ˜(S(x)) =µ ˜(x) (9). The proof of the remaining two equations of the assertions follows almost verbatim this pattern of proof and one gets:
µ˜ (Tz(x)) =µ ˜(x), x, z ∈ E
µ˜(γ(x)) =µ ˜(x), x ∈ E, γ ∈ G and (6) impliesµ ˜(e) = 1. Henceµ ˜ ∈ Mp is proved which implies
µ˜ = µ (10) because of (6) and the minimality of µ ∈ Mp.
Now, looking again at the definition (4) ofµ ˜ and putting y = 0, λ = 1 and x = x0 one gets µ˜(x0) ≤ α0(x0) (11)
14 - 37 -
which together with (1) yieldsµ ˜(x0) ≤ α0(x0) ≤ µ(x0) and in combination with (1) and (10) gives
α0(x) = µ(x), x ∈ E. (12)
Now, for x0, y0 ∈ E,
α0(x0) = sup{−p(−x1) − µ(x2) | x1, x2 ∈ E and x1 ≤ x0 + x2}.
and
α0(y0) = sup{−p(−y1) − µ(y2) | y1, y2 ∈ E and y1 ≤ y0 + y2}.
and since −p(−x) and −µ(x) are superlinear, one has
−p(−x1) −µ(x2) −p(−y1) −µ(y2) ≤ −p(−x1 −y1) −µ(x2 + y2) ≤ α0(x0 + y0)
which implies α0(x0)+α0(y0) ≤ α0(x0 +y0) that is α0(x) is superadditive and because of (12) positively homogeneous, i.e. superlinear. This implies that µ(x) is linear because of (12) and satisfies all the equations in the assertion, which completes the proof.
Banach limits are almost always defined as continuous extensions of a con- tinuous linear functional in an order unit normed space. Hence, for the introduction of Banach limits we need Theorem 3.1 in a continuous form. Surprisingly Theorem 3.1 already contains all the necessary continuity con- ditions as the following Corollary shows:
Corollary 3.2 Let the assertions (i) -(v) of Theorem 3.1 be satisfied and put Up := {x | p(x) = −p(−x) } and λ0(x) := p(x) for x ∈ Up. Then
(i) e ∈ Up and Up is an isometrical order unit normed subspace of E which is closed.
(ii) p : E −→ R is continuous and λ0 : Up −→ R is a positive, continuous linear functional with λ0(e) = 1.
(iii) Any µ : E −→ R satisfying Theorem 3.1 is a positive, continuous linear extension of λ0.
15 - 38 -
Proof. (i): Obviously, e ∈ Up and, hence, Up = {0}. For x ∈ Up ⊂ E, − x Ee ≤ x ≤ x Ee holds and this is an inequality in E and Up which proves (i). (ii): p(x) and −p(−x) are both monotone, p(x) sublinear and −p(−x) ≤ p(x) superlinear. As for x ∈ Up, p(x) = −p(−x) holds, p(x) is linear and positive on Up. If −e ≤ x ≤ e then −1 ≤ p(x) ≤ 1, and |p(x)| ≤ 1 s. th. norm of p is p = sup{|p(x)| | − e ≤ x ≤ e } = 1 and p is in 0 continuous hence also for any x ∈ E and |p(x)| ≤ x E, x ∈ E. Hence, we get for any µ in Theorem 3.1 − x E ≤ −p(−x) ≤ µ(x) ≤ p(x) ≤ x E implying the continuity of µ and µ ≤ 1, even µ = 1 because µ(e) = 1. In particular, this holds also for λ0.
It is remarkable that with respect to the continuity properties, the conti- nuity of S, Tx, x ∈ Ei, and γ ∈ G, do not play any role.
Definition 3.3 With the notations of Corollary 3.2 any such µ is called a Banach limit of λ0.
One defines
Ban-Lim(E, e, p, S, T ,G) := {µ | µ Banach limit of λ0 } .
Ban-Lim(E, e, p, S, T ,G), of course, depends on the parameters S, T , G, but in order to make the notation for the following not too cumbersome we will mostly omit them and write simply Ban-Lim(E, e, p).
Proposition 3.4 For Ban-Lim(E, e, p) the following statements hold:
(i) Ban-Lim(E, e, p) is a convex subset of Bs(E′) of the base normed Ba- nach space E′, the dual space of E.
(ii) Ban-Lim(E, e, p) is weakly -*-closed, weakly closed and also ′-closed, ′ ′ ′ where E denotes the usual dual norm of E of E . (iii) Ban-Lim(E, e, p) is a superconvex base set contained in Bs(E′).
n
Proof. (i): Let (α1, ..., αn) ∈ Ω := (α1, ..., αn) | αi ≥ 0, αi = 1 , n ∈ N , ( i=1 ) X be the set of all abstract convex combinations, then , for λi ∈ Ban-Lim(E, e, p),
16 - 39 -
n n
1 ≤ i ≤ n, obviously αiλi ∈ Ban-Lim(E, e, p), as αiλi(x) ≥ 0 for i=1 i=1 n nX n X x ≥ 0, αiλi ≤ p , αiλi(e) = αi = 1 and obviously all other equa- i=1 i=1 i=1 tions inX Theorem 3.1 areX satisfied, too.X
(ii): One first proves that Ban-Lim(E, e, p) is ′-closed, because from this , the other two assertions of (ii) then follow at once. If λn ∈ Ban-Lim(E, e, p), ′ ′ n ∈ N, and there is a λ∗ ∈ E with λn − λ∗ → 0 for n → ∞, then |λn(x) − λ∗(x)| → 0, i.e. limn→∞ λn(x) = λ∗(x) follows for x ∈ E, which implies λ∗(x) ≥ 0, for x ≥ 0, λ∗(e) = 1, λ∗(x) ≤ p(x), x ∈ X, and λ∗(x) = λ0(x) for x ∈ Up. Analogously, one shows the other equations in Theorem 3.1 for λ∗ because they hold for the λn(x), x ∈ E, n ∈ N.
(iii): Obviously, Ban-Lim(E, e, p) ⊂ Bs(E′) holds, hence Ban-Lim(E, e, p) is bounded and because of (ii) ′-closed. Then, it is a general result that Ban-Lim(E, e, p) is superconvex (see [8], Theorem 2.5).
Because Ban-Lim(E, e, p) is as a subset of Bs(E′) trivially a base set
C(Ban-Lim(E, e, p)) := R+Ban-Lim(E, e, p) is a proper cone and
Ban-Lim(E, e, p) := C(Ban-Lim(E, e, p)) − C(Ban-Lim(E, e, p)) is a base normed ordered linear space. To simplify notation, we will write (E, p) instead of (E, e, p) in the following.
Theorem 3.5 If the norm induced by Ban-Lim(E, p) in Ban-Lim(E, p) is ′ denoted by Ban and the topology induced by the weak-*-topology σ(E , E) B L ∗ in an- im(E, p) by τB, then
∗ (Ban-Lim(E, p), Ban, C(Ban-Lim(E, p)), τB) is a compact, base normed Saks space (see [8], Theorem 3.1) and an isomet- rical subspace of (E′, ′,C(E), σ(E′,E)) .
Proof. As Ban-Lim(E, p) is a weakly-*-closed base set and a subset of Bs(E′) which is weakly-*-compact because of Alaoglu-Bourbaki it is also weakly-*-compact and the space Ban-Lim(E, p) generated by the closed
17 - 40 -
cone C(Ban-Lim(E, p)) (see [3], Theorem 3.8.3, [8], Theorem 3.2, [7]) is a compact, base normed Saks space. The last assertion is obvious.
The result of Theorem 3.5 is essentially the definition of a functor from any category with objects satisfying the assertions of Theorem 3.1 to the category of compact, base normed Saks spaces ( [8] , Theorem 3.1). This functor will be investigated by the authors in a forthcoming paper.
References
[1] E. B. Davis (1968) The structure and ideal theory of the predual of a Banach lattices , Trans. AMS 131 , 544 - 555.
[2] E. J. Ellis (1966) Linear operators in partially ordered normed vector spaces, Journ. London Math. Soc. 41 , 323 - 332.
[3] G. Jameson (1970) Ordered Linear Spaces, Lect. Notes Math. 141, Springer, Berlin.
[4] V. L. Klee, jr (1954) Invariant extensions of linear functionals, Pacific Journal of Mathematics 4 (1) , 37 - 46.
[5] K. Ch. Min (1983) An exponential law for regular ordered Banach spaces, Cahiers de topologie et g´eometrie differentielle cat´egoriques Tome 24 (3), 279 - 298.
[6] D. Pumplun,¨ (2011) Banach spaces and superconvex modules, Symp. Gaussiana, M. Behara et al. (eds.), de Gruyter (1995), 323 - 338.
[7] D. Pumplun,¨ (2003) Positively convex modules and ordered normed lin- ear spaces, Journ. Convex Anal. 10 , 109 - 127.
[8] D. Pumplun,¨ (2011) A universal compactification of topological positively convex sets and modules, Journ. Convex Anal. 8 (1) , 255 -267.
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[9] D. Pumplun,¨ H.Rohrl¨ (1999) The Eilenberg-Moore algebras of base normed spaces, Proc. Symp. Cat. Top., Univ. Cape Town, Ba- naschewski, Gilmour, Herrlich (eds.), Cape Town, 187 - 200.
[10] D. Pumplun,¨ (1999): Elemente der Kategorientheorie, Hochschultaschen- buch, Spektrum Akademischer Verlag, Heidelberg, Berlin.
[11] D. Pumplun¨ (2002) The metric completion of convex sets and modules, Results. Math. 41 , 346 - 360.
[12] Z. Semadeni, (1973) Monads and their Eilenberg-Moore algebras in func- tional analysis, Queen’s Papers Pure Appl. Math. 33 , 255 -267.
[13] W. Roth (2000) Hahn-Banach type theorems for locally convex cones. J. Austral. Math. Soc. Ser. A . 68 (1) , 104 - 125.
[14] J.D. Weston (1957) The decomposition of continuous linear functionals into non-negative components, Math. Scand. 5 , 54 -56.
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[16] Yau Chuen Wong and , Kung Fu Ng, (1973): Partially ordered topologi- cal vector spaces. Oxford Mathematical Monographs, Clarendon Press, Oxford.
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OUTER AUTOMORPHISMS OF ALGEBRAIC GROUPS AND A SKOLEM-NOETHER THEOREM FOR ALBERT ALGEBRAS
SKIP GARIBALDI AND HOLGER P. PETERSSON
Dem Andenken Reinhard B¨orgers gewidmet
Abstract. We construct outer automorphisms of simple algebraic groups of 3 type D4 over an arbitrary field that become inner when raised to the third power. The key to the proof is a Skolem-Noether theorem for cubic ´etale subalgebras of Albert algebras which is of independent interest. Necessary and sufficient conditions for a simply connected group of outer type A to admit outer automorphisms of order 2 are also given.
Contents 1. Introduction 1 2. Jordan algebras 4 3. Cubic Jordan algebras 7 4. The weak and strong Skolem-Noether properties 11 5. Cubic Jordan algebras of dimension 9 12 6. Norm classes and strong equivalence 16 7. Albert algebras: proof of Theorem B 18 3 8. Outer automorphisms for type D4: proof of Theorem A 23 9. Outer automorphisms for type A 25 References 27
1. Introduction An algebraic group H defined over an algebraically closed field F is a disjoint union of connected components. The component H◦ containing the identity element is a normal subgroup in H that acts via left multiplication on each of the other components. Picking an F -point x in a connected component X of H gives an isomorphism of varieties with a H◦-action H◦ −→∼ X via h 7→ hx. When F is not assumed to be algebraically closed, the identity component H◦ is still defined as an F -subgroup of H, but the other components need not be. Suppose X is a connected subvariety of H such that, after base change to the algebraic closure Falg of F , X × Falg is a connected component of H × Falg. Then, by the previous paragraph, X is a H◦-torsor, but X may have no F -points. We remark that the question of whether X has an F -point comes up in describing the embedding of the category of compact real Lie groups into the category of linear algebraic groups over R, see [Ser93, §5].
2010 Mathematics Subject Classification. Primary 20G41; Secondary 11E72, 17C40, 20G15. 1 - 44 -
2 SKIP GARIBALDI AND HOLGER P. PETERSSON
1.1. Outer automorphisms of algebraic groups. We will focus on the case where H = Aut(G) and G is semisimple, which amounts to asking about the existence of outer automorphisms of G. This question has previously been studied in [MT95], [PT04b], [Gar12], [CKT12], [CEKT13], and [KT14]. Writing ∆ for the Dynkin diagram of G endowed with the natural action by the Galois group Gal(Fsep/F ) gives an exact sequence of group schemes
1 −−−−→ Aut(G)◦ −−−−→ Aut(G) −−−−→α Aut(∆) as in [DG70, Chap. XXIV, Th. 1.3 and §3.6] or [Spr98, §16.3], hence a natural map α(F ) : Aut(G)(F ) → Aut(∆)(F ). Note that Aut(∆)(Falg) is identified with the connected components of G × Falg in such a way that Aut(∆)(F ) is identified with those components that are defined over F . We ask: is α(F ) onto? That is, which of the components of Aut(G) that are defined over F also have an F -point? Sending an element g of G to conjugation by g defines a surjection G → Aut(G)◦, so the F -points Aut(G)◦(F ) are called inner automorphisms. The F -points of the other components of Aut(G) are called outer. Therefore, our question may be rephrased as: Is every automorphism of the Dynkin diagram induced from an F - automorphism of G? One can quickly observe that α(F ) need not be onto, for example, with the group SL(A) where A is a central simple algebra of odd exponent, where an outer automorphism would amount to an isomorphism of A with its opposite algebra. This is a special case of a general cohomological obstruction. Namely, writing Z for the scheme-theoretic center of the simply connected cover of G, G naturally defines 2 an element tG ∈ H (F,Z) called the Tits class as in [Tit71, 4.2] or [KMRT98, 31.6]. (The cohomology used in this paper is fppf.) For every character χ: Z → Gm, the 2 image χ(tG) ∈ H (F, Gm) is known as a Tits algebra of G; for example, when G = SL(A), Z is identified with the group of (deg A)-th roots of unity, the group of characters is generated by the natural inclusion χ : Z,→ Gm, and χ(tSL(A)) is the class of A. (More such examples are given in [KMRT98, §27.B].) This example illustrates also the general fact: tG = 0 if and only if EndG(V ) is a field for every irreducible representation V of G. The group scheme Aut(∆) acts on H2(F,Z), and it was shown in [Gar12, Th. 11] that this provides an obstruction to the surjectivity of α(F ), namely:
(1.1.1) im [α: Aut(G)(F ) → Aut(∆)(F )] ⊆ {π ∈ Aut(∆)(F ) | π(tG) = tG}.
It is interesting to know when equality holds in (1.1.1), because this information is critical for determining fibers of some maps in Galois cohomology, see [Gar12] for details. Certainly, equality need not hold, for example when G is semisimple (take G to be the product of the compact and split real forms of G2) or when G is neither simply connected nor adjoint (take G to be the split group SO8, for which |im α| = 2 but the right side of (1.1.1) has 6 elements). However, when G is simple and simply connected or adjoint, it is known that equality holds in (1.1.1) when G has inner type or for some fields F . Therefore, one might optimistically hope that the following is true:
Conjecture 1.1.2. If G is an absolutely simple algebraic group that is simply connected or adjoint, then equality holds in (1.1.1). - 45 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 3
2 The remaining open cases are where G has type An for odd n ≥ 3 (the case 2 3 2 where n is even is Cor. 9.1.2), Dn, D4, and E6. Most of this paper is dedicated to settling one of these four cases.
3 Theorem A. If G is a simple algebraic group of type D4 over a field F , then equality holds in (1.1.1). One can ask also for a stronger property to hold: Question 1.1.3. Suppose π is in α(Aut(G)(F )). Does there exist a φ ∈ Aut(G)(F ) so that α(φ) = π and φ and π have the same order? This question, and a refinement of it where one asks for detailed information about the possible φ’s, was considered for example in [MT95], [PT04b], [CKT12], [CEKT13], and [KT14]. It was observed in [Gar12] that the answer to the question is “yes” in all the cases where the conjecture is known to hold. However, [KT14] gives 3 an example of a group G of type D4 that does not have an outer automorphism of order 3, yet the conjecture holds for G by Theorem A. That is, combining the results of this paper and [KT14] gives the first example where the conjecture holds for a group but the answer to Question 1.1.3 is “no”, see Example 8.3.1 In other sections of the paper, we translate the conjecture for groups of type A into one in the language of algebras with involution as in [KMRT98], give a criterion for the existence of outer automorphisms of order 2 (i.e., prove a version for type A of the main result of [KT14]), and exhibit a group of type 2A that does not have an outer automorphism of order 2. 1.2. Skolem-Noether Theorem for Albert algebras. In order to prove The- orem A, we translate it into a statement about Albert F -algebras, 27-dimensional exceptional central simple Jordan algebras. We spend the majority of the paper working with Jordan algebras. Let J be an Albert algebra over a field F and suppose E,E0 ⊆ J are cubic ´etalesubalgebras. It is known since Albert-Jacobson [AJ57] that in general an isomorphism ϕ: E → E0 cannot be extended to an automorphism of J. Thus the Skolem-Noether Theorem fails to hold for cubic ´etalesubalgebras of Albert algebras. In fact, even in the important special case that E = E0 is split and ϕ is an automorphism of E having order 3, obstructions to the validity of this result may be read off from [AJ57, Th. 9]. Our objective in this part of the paper is to provide a way out of this impasse which consists in replacing the automorphism group of J by its structure group and in allowing the isomorphism ϕ to be twisted by the right multiplication of a norm-one element in E. More precisely, referring to our notational conventions in Sections 1.3−3 below, we will establish the following result. Theorem B. Let ϕ: E →∼ E0 be an isomorphism of cubic ´etalesubalgebras of an Albert algebra J over a field F . Then there exists an element w ∈ E satisfying 0 NE(w) = 1 such that ϕ◦Rw : E → E can be extended to an element of the structure group of J. Note that no restrictions on the characteristic of F will be imposed. In order to prove Theorem B, we first derive its analogue (in fact, a substantial general- ization of it, see Th. 5.2.7 below) for absolutely simple Jordan algebras of degree 3 and dimension 9 in place of J. This generalization is based on the notions of - 46 -
4 SKIP GARIBALDI AND HOLGER P. PETERSSON weak and strong equivalence for isotopic embeddings of cubic ´etalealgebras into cubic Jordan algebras (4.1) and is derived here by elementary manipulations of the two Tits constructions. After a short digression into norm classes for pairs of isotopic embeddings in § 6, Theorem B is established by combining Th. 5.2.7 with a density argument and the fact that an isotopy between absolutely simple nine-dimensional subalgebras of an Albert algebra can always be extended to an element of its structure group (Prop. 7.2.4).
1.3. Conventions. Throughout this paper, we fix a base field F of arbitrary char- acteristic. All linear non-associative algebras (in particular, all composition alge- bras) are tacitly assumed to contain an identity element. If C is such an algebra, we × write Rv : C → C for the right multiplication by v ∈ C, and C for the collection of invertible elements in C, whenever this makes sense. For a field extension (or any commutative associative algebra) K over F , we denote by CK := C ⊗ K the scalar extension (or base change) of C from F to K, unadorned tensor products always being taken over F . In other terminological and notational conventions, we mostly follow [KMRT98]. In fact, the sole truly significant deviation from this rule is presented by the theory of Jordan algebras: while [KMRT98, Chap. IX] confines itself to the linear version of this theory, which works well only over fields of charac- 1 teristic not 2 or, more generally, over commutative rings containing 2 , we insist on the quadratic one, surviving as it does in full generality over arbitrary commutative rings. For convenience, we will assemble the necessary background material in the next two sections of this paper.
2. Jordan algebras The purpose of this section is to present a dictionary for the standard vocabulary of arbitrary Jordan algebras. Our main reference is [Jac81].
2.1. The concept of a Jordan algebra. By a (unital quadratic) Jordan algebra over F , we mean an F -vector space J together with a quadratic map x 7→ Ux from J to EndF (J) (the U-operator) and a distinguished element 1J ∈ J (the unit or identity element) such that, writing
{xyz} := Vx,yz := Ux,zy := (Ux+z − Ux − Uz)y for the associated triple product, the equations
U1J = 1J ,
(2.1.1) UUxy = UxUyUx (fundamental formula),
UxVy,x = Vx,yUx hold in all scalar extensions. We always simply write J to indicate a Jordan algebra over F , U-operator and identity element being understood. A subalgebra of J is an F -subspace containing the identity element and stable under the operation Uxy; it is then a Jordan algebra in its own right. A homomorphism of Jordan algebras over F is an F -linear map preserving U-operators and identity elements. In this way we obtain the category of Jordan algebras over F . By definition, the property of being a Jordan algebra is preserved by arbitrary scalar extensions. In keeping with the conventions of Section 1.3, we write JK for the base change of J from F to a field extension K/F . - 47 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 5
2.2. Linear Jordan algebras. Assume char(F ) 6= 2. Then Jordan algebras as de- fined in 2.1 and linear Jordan algebras as defined in [KMRT98, § 37] are virtually the same. Indeed, let J be a unital quadratic Jordan algebra over F . Then J becomes 1 an ordinary non-associative F -algebra under the multiplication x · y := 2 Ux,y1J , and this F -algebra is a linear Jordan algebra in the sense that it is commutative and satisfies the Jordan identity x · ((x · x) · y) = (x · x) · (x · y). Conversely, let J be a linear Jordan algebra over F . Then the U-operator Uxy := 2x · (x · y) − (x · x) · y and the identity element 1J convert J into a unital quadratic Jordan algebra. The two constructions are inverse to one another and determine an isomorphism of cat- egories between unital quadratic Jordan algebras and linear Jordan algebras over F .
2.3. Ideals and simplicity. Let J be a Jordan algebra over F . A subspace I ⊆ J is said to be an ideal if UI J + UJ I + {IIJ} ⊆ J. In this case, the quotient space J/I carries canonically the structure of a Jordan algebra over F such that the projection J → J/I is a homomorphism. A Jordan algebra is said to be simple if it is non-zero and there are no ideals other than the trivial ones. We speak of an absolutely simple Jordan algebra if it stays simple under all base field extensions. (There is also a notion of central simplicity which, however, is weaker than absolute simplicity, although the two agree for char(F ) 6= 2.)
2.4. Standard examples. First, let A be an associative F -algebra. Then the vector space A together with the U-operator Uxy := xyx and the identity element + + 1A is a Jordan algebra over F , denoted by A . If A is simple, then so is A [MZ88, 15.5]. Next, let (B, τ) be an F -algebra with involution, so B is a non-associative algebra over F and τ : B → B is an F -linear anti-automorphism of period 2. Then Symd(B, τ) = {x + τ(x) | x ∈ B} ⊆ H(B, τ) := Sym(B, τ) = {x ∈ B | τ(x) = x} are subspaces of B, and we have Symd(B, τ) = H(B, τ) for char(F ) 6= 2 but not in general. Moreover, if B is associative, then Symd(B, τ) and H(B, τ) are both subalgebras of B+, hence are Jordan algebras which are simple if (B, τ) is simple as an algebra with involution [MZ88, 15.5].
2.5. Powers. Let J be a Jordan algebra over F . The powers of x ∈ J with integer 0 1 n+2 n exponents n ≥ 0 are defined recursively by x = 1J , x = x, x = Uxx . Note for J = A+ as in 2.4, powers in J and in A are the same. For J arbitrary, they satisfy the relations n 2m+n m n p m+n+p m n mn (2.5.1) Uxm x = x , {x x x } = 2x , (x ) = x , hence force X F [x] := F xn n≥0 to be a subalgebra of J. In many cases — e.g., if char(F ) 6= 2 or if J is simple (but not always [Jac81, 1.31, 1.32]) — there exists a commutative associative F -algebra R, necessarily unique, such that F [x] = R+ [McC70, Prop. 1], [Jac81, Prop. 4.6.2]. By abuse of language, we simply write R = F [x] and say R is a subalgebra of J. In a slightly different, but similar, vein we wish to talk about ´etalesubalgebras of a Jordan algebra. This is justified by the fact that ´etale F -algebras are comletely - 48 -
6 SKIP GARIBALDI AND HOLGER P. PETERSSON determined by their Jordan structure. More precisely, we have the following simple result. Lemma 2.5.2. Let E,R be commutative associative F -algebras such that E is finite-dimensional ´etale and E+ = R+ as Jordan algebras. Then E = R as com- mutative associative algebras. Proof. Extending scalars if necessary, we may assume that E as a (unital) F -algebra is generated by a single element x ∈ E. But since the powers of x in E agree with + + those in E = R , hence with those in R, the assertion follows. 2.6. Inverses and Jordan division algebras. Let J be a Jordan algebra over F . An element x ∈ J is said to be invertible if the U-operator Ux : J → J is bijective −1 −1 (equivalently, 1J ∈ Im(Ux)), in which case we call x := Ux x the inverse of x in J. Invertibility and inverses are preserved by homomorphisms. It follows from the fundamental formula (2.1.1) that, if x, y ∈ J are invertible, then so is Uxy −1 −1 n −1 −n and (Uxy) = Ux−1 y . Moreover, setting x := (x ) for n ∈ Z, n < 0, we have (2.5.1) for all m, n, p ∈ Z. In agreement with earlier conventions, the set of invertible elements in J will be denoted by J ×. If J × = J \{0} 6= ∅, then we call J a Jordan division algebra. If A is an associative algebra, then (A+)× = A×, and the inverses are the same. Similarly, if (B, τ) is an associative algebra with involution, then Symd(B, τ)× = Symd(B, τ) ∩ B×, H(B, τ)× = H(B, τ) ∩ B×, and, again, in both cases, the inverses are the same.
2.7. Isotopes. Let J be a Jordan algebra over F and p ∈ J ×. Then the vector (p) space J together with the U-operator Ux := UxUp and the distinguished element (p) −1 1J := p is a Jordan algebra over F , called the p-isotope (or simply an isotope) of J and denoted by J (p). We have J (p)× = J × and (J (p))(q) = J (Upq) for all q ∈ J ×, which implies (J (p))(q) = J for q := p−2. Passing to isotopes is functorial in the following sense: If ϕ: J → J 0 is a homomorphism of Jordan algebras, then so is ϕ: J (p) → J 0(ϕ(p)), for any p ∈ J ×. Let A be an associative algebra over F and p ∈ (A+)× = A×. Then right mul- + (p) ∼ + tiplication by p in A gives an isomorphism Rp :(A ) → A of Jordan algebras. On the other hand, if (B, τ) is an associative algebra with involution, then so is (B, τ (p)), for any p ∈ H(B, τ)×, where τ (p) : B → B via x 7→ p−1τ(x)p stands for the p-twist of τ, and (p) ∼ (p) (2.7.1) Rp : H(B, τ) −→ H(B, τ ) is an isomorphism of Jordan algebras.
2.8. Homotopies and the structure group. If J, J 0 are Jordan algebras over 0 F , a homotopy from J to J 0 is a homomorphism ϕ: J → J 0(p ) of Jordan algebras, 0 0× 0 −1 for some p ∈ J . In this case, p = ϕ(1J ) is uniquely determined by ϕ. Bijec- tive homotopies are called isotopies, while injective homotopies are called isotopic embeddings. The set of isotopies from J to itself is a subgroup of GL(J), called the structure group of J and denoted by Str(J). It consists of all linear bijections ] ] η : J → J such that some linear bijection η : J → J satisfies Uη(x) = ηUxη for all x ∈ J. The structure group contains the automorphism group of J as a sub- group; more precisely, Aut(J) is the stabilizer of 1J in Str(J). Finally, thanks to × the fundamental formula (2.1.1), we have Uy ∈ Str(J) for all y ∈ J . - 49 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 7
3. Cubic Jordan algebras In this section, we recall the main ingredients of the approach to a particularly important class of Jordan algebras through the formalism of cubic norm structures. Our main references are [McC69] and [JK73]. Systematic use will be made of the following notation: given a polynomial map P : V → W between vector spaces V,W over F and y ∈ V , we denote by ∂yP : V → W the polynomial map given by the derivative of P in the direction y. 3.1. Cubic norm structures. By a cubic norm structure over F we mean a quadruple X = (V, c, ], N) consisting of a vector space V over F , a distinguished element c ∈ V (the base point), a quadratic map x 7→ x] from V to V (the adjoint), with bilinearization x × y := (x + y)] − x] − y], and a cubic form N : V → F (the norm), such that, writing
T (y, z) := (∂yN)(c)(∂zN)(c) − (∂y∂zN)(c)(y, z ∈ V ) for the (bilinear) trace of X and T (y) := T (y, c) for the linear one, the equations (3.1.1) c] = c, N(c) = 1 (base point identities), (3.1.2) c × x = T (x)c − x (unit identity), ] (3.1.3) (∂yN)(x) = T (x , y) (gradient identity), (3.1.4) x]] = N(x)x (adjoint identity) hold in all scalar extensions. A subspace of V is called a cubic subnorm structure of X if it contains the base point and is stable under the adjoint map.; it may then canonically be regarded as a cubic norm structure in its own right. A homo- morphism of cubic norm structures is a linear map of the underlying vector spaces preserving base points, adjoints and norms. A cubic norm structure X as above is said to be non-singular if V has finite dimension over F and the bilinear trace T : V × V → F is a non-degenerate symmetric bilinear form. If X and Y are cubic norm structures over F , with Y non-singular, and ϕ: X → Y is a surjective linear map preserving base points and norms, then ϕ is an isomorphism of cubic norm structures [McC69, p. 507]. 3.2. The associated Jordan algebra. Let X = (V, c, ], N) be a cubic norm structure over F and write T for its bilinear trace. Then the U-operator ] (3.2.1) Uxy := T (x, y)x − x × y and the base point c convert the vector space V into a Jordan algebra over F , denoted by J(X) and called the Jordan algebra associated with X. The construction of J(X) is clearly functorial in X. We have 2 (3.2.2) N(Uxy) = N(x) N(y) (x, y ∈ J). Jordan algebras isomorphic to J(X) for some cubic norm structure X over F are said to be cubic. For example, let J be a Jordan algebra over F that is generically algebraic (e.g., finite-dimensional) of degree 3 over F . Then X = (V, c, ], N), where V is the vector space underlying J, c := 1J , ] is the numerator of the inversion map, and N := NJ is the generic norm of J, is a cubic norm structure over F satisfying J = J(X); in particular, J is a cubic Jordan algebra. In view of this correspondence, we rarely distinguish carefully between a cubic norm structure and its associated Jordan algebra. Non-singular cubic Jordan algebras, i.e., Jordan - 50 -
8 SKIP GARIBALDI AND HOLGER P. PETERSSON algebras arising from non-singular cubic norm structures, by [McC69, p. 507] have no absolute zero divisors, so Ux = 0 implies x = 0. 3.3. Cubic ´etale algebras. Let E be a cubic ´etale F -algebra. Then Lemma 2.5.2 allows us to identify E = E+ as a generically algebraic Jordan algebra of degree 2 + 3 (with U-operator Uxy = x y), so we may write E = E = J(V, c, ], N) as in 3.2, where c = 1E is the unit element, ] is the adjoint and N = NE is the norm of + E = E . We also write TE for the (bilinear) trace of E. The discriminant (algebra) of E will be denoted by ∆(E); it is a quadratic ´etale F -algebra [KMRT98, 18.C]. 3.4. Isotopes of cubic norm structures. Let X = (V, c, ], N) be a cubic norm structure over F . An element p ∈ V is invertible in J(X) if and only if N(p) 6= 0, in which case p−1 = N(p)−1p]. Moreover, X(p) := (V, c(p),](p),N (p)),
(p) −1 ](p) −1 ] (p) with c := p , x := N(p)Up x , N := N(p)N, is again a cubic norm structure over F , called the p-isotope of X. This terminology is justified since the associated Jordan algebra J(X(p)) = J(X)(p) is the p-isotope of J(X). We also note that the bilinear trace of X(p) is given by (p) (3.4.1) T (y, z) = T (Upy, z) (y, z ∈ X) in terms of the bilinear trace T of X. Combining the preceding considerations with 3.1, we conclude that the structure group of a non-singular cubic Jordan algebra agrees with its group of norm similarities. 3.5. Cubic Jordan matrix algebras. Let C be a composition algebra over F , with norm nC , trace tC , and conjugation v 7→ v¯ := tC (v)1C − v. For any diagonal matrix Γ = diag(γ1, γ2, γ3) ∈ GL3(F ), the pair −1 t Mat3(C), τΓ , τΓ(x) := Γ x¯ Γ(x ∈ Mat3(C)), is a non-associative F -algebra with involution, allowing us to consider the subspace Her3(C, Γ) := Symd Mat3(C), τΓ = {x + τΓ(x) | x ∈ Mat3(C)} ⊆ Mat3(C), which is easily seen to agree with the space of elements x ∈ Mat3(C) that are Γ- hermitian (x = Γ−1x¯tΓ) and have scalars down the diagonal (the latter condition being automatic for char(F ) 6= 2). In terms of the usual matrix units eij ∈ Mat3(C), 1 ≤ i, j ≤ 3, we therefore have X Her3(C, Γ) = (F eii + C[jl]), the sum on the right being taken over all cyclic permutations (ijl) of (123), where
C[jl] := {v[jl] | v ∈ C}, v[jl] := γlvejl + γjve¯ lj.
Now put V := Her3(C, Γ) as a vector space over F , c := 13 (the 3 × 3 unit matrix) and define adjoint and norm on V by
] X x := αjαl − γjγlnC (vi) eii + − αivi + γivjvl [jl] , X N(x) := α1α2α3 − γjγlαinC (vi) + γ1γ2γ3tC (v1v2v3) - 51 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 9
P for all x = (αieii +vi[jl]) in all scalar extensions of V . Then X := (V, c, ], N) is a cubic norm structure over F . Henceforth, the symbol Her3(C, Γ) will stand for this cubic norm structure but also for its associated cubic Jordan algebra. We always abbreviate Her3(C) := Her3(C, 13). 3.6. Albert algebras. Writing Zor(F ) for the split octonion algebra of Zorn vec- tor matrices over F , the cubic Jordan matrix algebra Her3(Zor(F )) is called the split Albert algebra over F . By an Albert algebra over F , we mean an F -form of Her3(Zor(F )), i.e., a Jordan algebra over F (necessarily absolutely simple and non- singular of degree 3 and dimension 27) that becomes isomorphic to the split Albert algebra when extending scalars to the separable closure. Albert algebras are either reduced, hence have the form Her3(C, Γ) as in 3.5, C an octonion algebra over F (necessarily unique), or are cubic Jordan division algebras. 3.7. Associative algebras of degree 3 with unitary involution. By an asso- ciative algebra of degree 3 with unitary involution over F we mean a triple (K, B, τ) with the following properties: K is a quadratic ´etale F -algebra, with norm nK , trace tK and conjugation ιK , a 7→ a¯, B is an associative algebra of degree 3 over K and τ : B → B is an F -linear involution that induces the conjugation of K via restric- tion. All this makes obvious sense even in the special case that K =∼ F × F is split, as do the generic norm, trace and adjoint of B, which are written as NB,TB,], re- spectively, connect naturally with the involution τ and agree with the corresponding notions for the cubic Jordan algebra B+. In particular, H(B, τ) is a Jordan algebra of degree 3 over F whose associated cubic norm structure derives from that of B+ via restriction. An associative algebra (K, B, τ) of degree 3 with unitary involution over F is said to be non-singular if the corresponding cubic Jordan algebra B+ has this property, equivalently, if B is free of finite rank over K and TB : B × B → K is a non-degenerate symmetric bilinear form in the usual sense. 3.8. The second Tits construction. Let (K, B, τ) be an associative algebra of degree 3 with unitary involution over F and suppose we are given invertible elements u ∈ H(B, τ), µ ∈ K such that NB(u) = nK (µ). We put V := H(B, τ) ⊕ Bj as the external direct sum of H(B, τ) and B as vector spaces over F to define base point, adjoint and norm on V by the formulas
(3.8.1) c := 1B + 0 · j, ] ] ] −1 (3.8.2) x := (v0 − vuv¯) + (¯µv¯ u − v0v)j, (3.8.3) N(x) := NB(v0) + µNB(v) +µ ¯NB(v) − TB v0, vuτ(v) for x = v0 + vj, v0 ∈ H(B, τ), v ∈ B (and in all scalar extensions as well). Then we obtain a cubic norm structure X := (V, c, ], N) over F whose associated cubic Jordan algebra will be denoted by J := J(K, B, τ, u, µ) := J(X) and has the bilinear trace T (x, y) = TB(v0, w0) + TB vuτ(w) + TB wuτ(v) (3.8.4) = TB(v0, w0) + tK TB vuτ(w) for x as above and y = w0 + wj, w0 ∈ H(B, τ), w ∈ B. It follows that, if (K, B, τ) is non-singular, then so is J. Note also that the cubic Jordan algebra H(B, τ) identifies with a subalgebra of J through the initial summand. - 52 -
10 SKIP GARIBALDI AND HOLGER P. PETERSSON
If, in addition to the above, (B, τ) is central simple as an algebra with involution over F , then K is the centre of B, J(B, τ, u, µ) := J(K, B, τ, u, µ) is an Albert algebra over F , and all Albert algebras can be obtained in this way. More precisely, every Albert algebra J over F contains a subalgebra isomorphic to H(B, τ) for some central simple associative algebra (B, τ) of degree 3 with unitary involution over F , and every homomorphism H(B, τ) → J can be extended to an isomorphism from J(B, τ, u, µ) to J, for some invertible elements u ∈ H(B, τ), µ ∈ K satisfying NB(u) = nK (µ). Our next result is a variant of [PR84b, Prop. 3.9] which extends the isomorphism (2.7.1) in a natural way. Lemma 3.8.5. Let (K, B, τ) be a non-singular associative algebra of degree 3 with unitary involution over F and suppose u ∈ H(B, τ), µ ∈ K are invertible elements × (p) satisfying NB(u) = nK (µ). Then, given any p ∈ H(B, τ) , writing τ for the p- (p) ] (p) twist of τ in the sense of 2.7 and setting u := p u, µ := NB(p)µ, the following statements hold. (p) (p) × (p) (p) (p) (a) u ∈ H(B, τ ) , NB(u ) = nK (µ ) and H(B, τ ) = H(B, τ)p. (b) The map
(p) ∼ (p) (p) (p) −1 Rˆp : J(K, B, τ, u, µ) −→ J(K, B, τ , u , µ ), v0 + vj 7−→ v0p + (p vp)j, is an isomorphism of cubic Jordan algebras.
−1 −1 ] ] ] Proof. (a) From p = NB(p) p we conclude pp = NB(p)1B = p p, which ] 2 implies the first assertion, but also the second since NB(p ) = NB(p) . The third one follows from (2.7.1). (b) By (a), (3.4.1) and 3.8, the map Rˆp is a linear bijection between non-singular cubic Jordan algebras preserving base points. By 3.1, it therefore suffices to show that it preserves norms as well. Writing N (resp. N 0) for the norm of J(K, B, τ, u, µ) (p) (p) (p) (resp. J(K, B, τ , u , µ ), we let v0 ∈ H(B, τ), v ∈ B and compute, using (3.8.3),
0 0 −1 (N ◦ Rˆp)(v0 + vj) = N (v0p + (p vp)j)
= NB(p)NB(v0) + NB(p)µNB(v) + NB(p)¯µNB(v) −1 ] (p) −1 − TB v0pp vpp uτ (p vp) = NB(p) NB(v0) + µNB(v) +µ ¯NB(v) − TB v0vuτ(v)
(p) = N (v0 + vj), as desired. Remark 3.8.6. The lemma holds without the non-singularity condition on (K, B, τ) but the proof is more involved. If the quadratic ´etale F -algebra K in 3.8 is split, there is a less cumbersome way of describing the output of the second Tits construction.
3.9. The first Tits construction. Let A be an associative algebra of degree 3 × over F and µ ∈ F . Put V := A ⊕ Aj1 ⊕ Aj2 as the direct sum of three copies of A as an F -vector space and define base point, adjoint and norm on V by the formulas - 53 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 11 c := 1A + 0 · j1 + 0 · j2, ] ] −1 ] ] (3.9.1) x := (v0 − v1v2) + (µ v2 − v0v1)j1 + (µv1 − v2v0)j2, −1 (3.9.2) N(x) := NA(v0) + µNA(v1) + µ NA(v2) − TA(v0v1v2) for x = v0 + v1j1 + v2j2, v0, v1, v2 running over all scalar extensions of A. Then X := (V, c, ], N) is a cubic norm structure over F , with bilinear trace given by
(3.9.3) T (x, y) = TA(v0, w0) + TA(v1, w2) + TA(v2, w1) for x as above and y = w0 + w1j1 + w2j2, w0, w1, w2 ∈ A. The associated cubic Jordan algebra will be denoted by J(A, µ) := J(X). The Jordan algebra A+ identifies with a cubic subalgebra of J(A, µ) through the initial summand, and if A is central simple, then J(A, µ) is an Albert algebra, which is either split or division. Now let (K, B, τ) be an associative algebra of degree 3 with unitary involution × × over F and suppose µ ∈ K , u ∈ H(B, τ) satisfy nK (µ) = NB(u). If K = F × F is split, then (B, τ) identifies with (A × Aop, ε) for some associative algebra A of degree 3 over F , where ε denotes the exchange involution. Moreover, µ = (α, β), −1 where α ∈ F is invertible, β = α NB(u), and there exists a canonical isomorphism J := J(K, B, τ, u, µ) =∼ J(A, α) =: J 0 matching H(A × Aop, ε) canonically with A+ as subalgebras of J, J 0, respectively. On the other hand, if K is a field, the preceding considerations apply to the base change from F to K and then yield an isomorphism ∼ J(K, B, τ, u, µ)K = J(B, µ).
4. The weak and strong Skolem-Noether properties As we have pointed out in 1.2, extending an isomorphism between cubic ´etale subalgebras of an Albert algebra J to an automorphism on all of J will in gen- eral not be possible. Working with elements of the structure group rather than automorphisms, our Theorem B above is supposed to serve as a substitute for this deficiency. Unfortunately, however, this substitute suffers from deficiencies of its own since the natural habitat of the structure group is the category of Jordan algebras not under homomorphisms but, instead, under homotopies. Fixing a cubic Jordan algebra J over our base field F and a cubic ´etale F -algebra E throughout this section, we therefore feel justified in phrasing the following formal definition.
4.1. Weak and strong equivalence of isotopic embeddings. (a) Two isotopic embeddings i, i0 : E → J in the sense of 2.8 are said to be weakly equivalent if there exist an element w ∈ E of norm 1 and an element ϕ ∈ Str(J) such that the diagram
E / E Rw (4.1.1) i0 i J ϕ / J commutes. They are said to be strongly equivalent if ϕ ∈ Str(J) can furthermore be chosen so that the diagram commutes with w = 1 (i.e., Rw = IdE). Weak and strong equivalence clearly define equivalence relations on the set of isotopic embeddings from E to J. (b) The pair (E,J) is said to satisfy the weak (resp. strong) Skolem-Noether property for isotopic embeddings if any two isotopic embeddings from E to J are - 54 -
12 SKIP GARIBALDI AND HOLGER P. PETERSSON weakly (resp. strongly) equivalent. The weak (resp. strong) Skolem-Noether prop- erty for isomorphic embeddings is defined similarly, by restricting the maps i, i0 to be isomorphic embeddings instead of merely isotopic ones. Remark 4.1.2. In 4.1 we have defined four different properties, depending on whether one considers the weak or strong Skolem-Noether property for isotopic or isomor- phic embeddings. Clearly the combination weak/isomorphic is the weakest of these four properties and strong/isotopic is the strongest. In the case where J is an Albert algebra, Theorem B is equivalent to saying that the pair (E,J) satisfies the weakest combination, the weak Skolem-Noether property for isomorphic embeddings. On the other hand, suppose i, i0 : E → J are isomorphic embeddings and ϕ ∈ Str(J) makes (4.1.1) commutative with w = 1. Then ϕ fixes 1J and hence is an automorphism of J. But such an automorphism will in general not exist [AJ57, Th. 9], and if it doesn’t the pair (E,J) will fail to satisfy the strong Skolem-Noether property for isomorphic embeddings. In view of this failure, we are led quite naturally to the following (as yet) open question: Does the pair (E,J), with J absolutely simple (of degree 3), always (4.1.3) satisfy the weak Skolem-Noether property for isotopic embeddings?
(p1) This is equivalent to asking whether, given two cubic ´etalesubalgebras E1 ⊆ J , (p2) × E2 ⊆ J for some p1, p2 ∈ J , every isotopy η : E1 → E2 allows a norm-one element w ∈ E1 such that the isotopy η ◦ Rw : E1 → E2 extends to an element of the structure group of J. Regrettably, the methodological arsenal assembled in the present paper, consisting as it does of rather elementary manipulations involving the two Tits constructions, does not seem strong enough to provide an affirmative answer to this question. But in the case where J is absolutely simple of dimension 9 — i.e., the Jordan algebra of symmetric elements in a central simple associative algebra of degree 3 with unitary involution over F [MZ88, 15.5] — we will show in Th. 5.2.7 below that the weak Skolem-Noether property for isotopic embeddings does hold. This result, in turn, will be instrumental in proving Theorem B in section §7. In phrasing Open Question 4.1.3, we could have gone one step further by bringing the theory of Jordan pairs [Loo75] into play. We will not do so since our methods do not readily adapt to the Jordan pair setting. Instead, we will confine ourselves to making the following remark. Remark 4.1.4. Assume in 4.1.3 that E =∼ F × F × F is split. Giving an isotopic embedding from E to J is then equivalent to giving a frame, necessarily of length 3, in the Jordan pair V := (J, J). But following Loos [Loo91, Cor. 3 of Th. 2], the diagonal Peirce components of two ordered frames in V can be matched by some element in the elementary group of V, i.e., in a certain subgroup of the structure group of J, and this fact is easily seen to translate into the commutative diagram (4.1.1) after an appropriate choice of w ∈ E× (possibly not of norm 1) and ϕ ∈ Str(J).
5. Cubic Jordan algebras of dimension 9 Our goal in this section will be to answer Question 4.1.3 affirmatively in case J is a nine-dimensional absolutely simple cubic Jordan algebra over F . Before we will be able to do so, a few preparations are required. - 55 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 13
5.1. Quadratic and cubic ´etalealgebras. (a) If K and L are quadratic ´etale algebras over F , then so is
K ∗ L := H(K ⊗ L, ιK ⊗ ιL), where ιK and ιL denote the conjugations of K and L, respectively. The composition (K,L) 7→ K ∗ L corresponds to the abelian group structure of H1(F, Z/2Z), which classifies quadratic ´etale F -algebras. (b) Next suppose L and E are a quadratic and cubic ´etale F -algebras, respec- tively. Then E ⊗ L may canonically be viewed as a cubic ´etale L-algebra, whose norm, trace, adjoint will again be denoted by NE,TE,], respectively. On the other hand, E ⊗ L may also be viewed canonically as a quadratic ´etale E-algebra, whose norm, trace and conjugation will again be denoted by nL, tL, and ιL, x 7→ x¯, re- spectively. We may and always will identify E ⊆ E ⊗ L through the first factor and then have E = H(E ⊗ L, ιL). 5.2. The ´etaleTits process. [PT04a, 1.3] Let L, resp. E, be a quadratic, resp cubic, ´etalealgebra over F and as in 3.3 write ∆(E) for the discriminant of E, which is a quadratic ´etale F -algebra. With the conventions of 5.1 (b), the triple (K, B, τ) := (L, E ⊗ L, ιL) is an associative algebra of degree 3 with unitary invo- lution over F in the sense of 3.7 such that H(B, τ) = E. Hence, if u ∈ E and b ∈ L are invertible elements satisfying NE(u) = nL(b), the second Tits construction 3.8 leads to a cubic Jordan algebra
J(E, L, u, b) := J(K, B, τ, u, b) = J(L, E ⊗ L, ιL, u, b) that belongs to the cubic norm structure (V, c, ], N) where V = E ⊕ (E ⊗ L)j as a vector space over F and c, ], N are defined by (3.8.1)–(3.8.3) in all scalar extensions. The cubic Jordan algebra J(E, L, u, b) is said to arise from E, L, u, b by means of the ´etaleTits process. There exists a central simple associative algebra (B, τ) of degree 3 with unitary involution over F uniquely determined by the condition that J(E, L, u, b) =∼ H(B, τ), and by [PR84b, Th. 1], the centre of B is isomorphic to ∆(E) ∗ L (cf. 5.1 (a)) as a quadratic ´etale F -algebra. For convenience, we now recall three results from [PT04a] that will play a cru- cial role in providing an affirmative answer to Question 4.1.3 under the conditions spelled out at the beginning of this section. Theorem 5.2.1. ([PT04a, 1.6]) Let E be a cubic ´etale F -algebra, (B, τ) a central simple associative algebra of degree 3 with unitary involution over F and suppose i is an isomorphic embedding from E to H(B, τ). Writing K for the centre of B and setting L := K ∗ ∆(E), there are invertible elements u ∈ E, b ∈ L satisfying NE(u) = nL(b) such that i extends to an isomorphism from the ´etaleTits process algebra J(E, L, u, b) onto H(B, τ). Theorem 5.2.2. ([PT04a, 3.2]) Let E,E0 and L, L0 be cubic and quadratic ´etale algebras, respectively, over F and suppose we are given invertible elements u ∈ E, 0 0 0 0 0 0 u ∈ E , b ∈ L, b ∈ L satisfying NE(u) = nL(b), NE0 (u ) = nL0 (b ). We write J := J(E, L, u, b) = E ⊕ (E ⊗ L)j, J 0 := J(E0,L0, u0, b0) = E0 ⊕ (E0 ⊗ L0)j0 as in 5.2 for the corresponding ´etaleTits process algebras and let ϕ: E0 →∼ E be an isomorphism. Then, for an arbitrary map Φ: J 0 → J, the following conditions are equivalent. (i) Φ is an isomorphism extending ϕ. - 56 -
14 SKIP GARIBALDI AND HOLGER P. PETERSSON
(ii) There exist an isomorphism ψ : L0 →∼ L and an invertible element y ∈ 0 0 E ⊗ L such that ϕ(u ) = nL(y)u, ψ(b ) = NE(y)b and 0 0 0 0 0 (5.2.3) Φ(v0 + v j ) = ϕ(v0) + y(ϕ ⊗ ψ)(v ) j 0 0 0 0 0 for all v0 ∈ E , v ∈ E ⊗ L . Proposition 5.2.4. ([PT04a, 4.3]) Let E be a cubic ´etale F -algebra and α, α0 ∈ F ×. Then the following conditions are equivalent. (i) The first Tits constructions J(E, α) and J(E, α0) ( cf. 3.9) are isomorphic. (ii) J(E, α) and J(E, α0) are isotopic. 0ε × (iii) α ≡ α mod NE(E ) for some ε = ±1. (iv) The identity of E can be extended to an isomorphism from J(E, α) onto 0 J(E, α ). Our next aim will be to derive a version of Th. 5.2.1 that works with isotopic rather than isomorphic embeddings and brings in a normalization condition already known from [KMRT98, (39.2)]. Proposition 5.2.5. Let (B, τ) be a central simple associative algebra of degree 3 with unitary involution over F and write K for the centre of B. Suppose further that E is a cubic ´etale F -algebra and put L := K ∗ ∆(E). Given any isotopic embedding i: E → J := H(B, τ), there exist elements u ∈ E, b ∈ L such that NE(u) = nL(b) = 1 and i can be extended to an isotopy from J(E, L, u, b) onto J. Proof. By 2.8, some invertible element p ∈ J makes i: E → J (p) an isomorphic embedding. On the other hand, invoking 2.7 and writing τ (p) for the p-twist of τ, it follows that (p) ∼ (p) Rp : J −→ H(B, τ ) (p) is an isomorphism of cubic Jordan algebras, forcing i1 := Rp ◦ i: E → H(B, τ ) to be an isomorphic embedding. Hence Th. 5.2.1 yields invertible elements u1 ∈ E, b1 ∈ L such that NE(u1) = nL(b1) and, adapting the notation of 3.8 to the present set-up in an obvious manner, i1 extends to an isomorphism 0 ∼ (p) η1 : J(E, L, u1, b1) = E ⊕ (E ⊗ L)j1 −→ H(B, τ ). 0 ∼ (p) Thus η1 := Rp−1 ◦η1 : J(E, L.u1, b1) → J is an isomorphism, which may therefore be viewed as an isotopy from J(E, L, u1, b1) onto J extending i. Now put u := −1 3 ¯ −1 −1 × NE(u1) u1, b := b1b1 and y := u1 ⊗ b1 ∈ (E ⊗ L) to conclude NE(u) = nL(b) = 1 as well as nL(y)u1 = u, NE(y)b1 = b. Applying Th. 5.2.2 to ϕ := 1E, ψ := 1L, we therefore obtain an isomorphism ∼ Φ: J(E, L, u, b) −→ J(E, L, u1, b1), v0 + vj1 7−→ v0 + (yv)j of cubic Jordan algebras, and η := η1 ◦ Φ: J(E, L, u, b) → J is an isotopy of the desired kind. Lemma 5.2.6. Let L, resp. E be a quadratic, resp. cubic ´etale algebra over F and suppose we are given elements u ∈ E, b ∈ L satisfying NE(u) = nL(b) = 1. Then −1 w := u ∈ E has norm 1 and Rw : E → E extends to an isomorphism ∼ (u) Rˆw : J(E, L, 1E, b) −→ J(E, L, u, b) , v + xj 7−→ (vw) + xj of cubic Jordan algebras. - 57 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 15
Proof. This follows immediately from Lemma 3.8.5 for (K, B, τ) := (L, E ⊗ L, ιL), µ := b and p := u. We are now ready for the main result of this section. Theorem 5.2.7. Let (B, τ) be a central simple associative algebra of degree 3 with unitary involution over F and E a cubic ´etale F -algebra. Then the pair (E,J) satifies the weak Skolem-Noether property for isotopic embeddings in the sense of 4.1 (b). Proof. Given two isotopic embeddings i, i0 : E → J, we must show that they are weakly equivalent. In order to do so, we write K for the centre of B as a quadratic ´etale F -algebra and put L := K ∗∆(E). Then Prop. 5.2.5 yields elements u, u0 ∈ E, b, b0 ∈ L satisfying 0 0 (5.2.8) NE(u) = NE(u ) = nL(b) = nL(b ) = 1 such that the isotopic embeddings i, i0 can be extended to isotopies (5.2.9) η : J(E, L, u, b) = E ⊕ (E ⊗ L)j −→ J, η0 : J(E, L, u0, b0) = E ⊕ (E ⊗ L)j0 −→ J, respectively. We now distinguish the following two cases. Case 1: L =∼ F × F is split. As we have noted in 3.9, there exist elements α, α0 ∈ F × and isomorphisms Φ: J(E, L, u, b) −→∼ J(E, α), Φ0 : J(E, L, u0, b0) −→∼ J(E, α0) extending the identity of E. Thus (5.2.9) implies that Φ◦η−1 ◦η0 ◦Φ0−1 : J(E, α0) → J(E, α) is an isotopy, and applying Prop. 5.2.4, we find an isomorphism θ : J(E, α0) →∼ J(E, α) extending the identity of E. But then ϕ := η ◦ Φ−1 ◦ θ ◦ Φ0 ◦ η0−1 : J −→ J is an isotopy, hence belongs to the structure group of J, and satisfies 0 −1 0 0−1 0 ϕ ◦ i = η ◦ Φ ◦ θ ◦ Φ ◦ η ◦ η |E = η|E = i. Thus i and i0 are even strongly equivalent. Case 2: L is a field. Since J(E, L, u, b) and J(E, L, u0, b0) are isotopic (via η0−1 ◦η), so are their scalar extensions from F to L. From this and 3.9 we therefore conclude that J(E⊗L, b) and J(E⊗L, b0) are isotopic over L. Hence, by Prop. 5.2.4, 0ε (5.2.10) b = b NE(z) × for some ε = ±1 and some z ∈ (E ⊗ L) . Now put ϕ := 1E, ψ := ιL and 0 × 0−1 0 y := u ⊗ 1L ∈ (E ⊗ L) . Making use of (5.2.8) we deduce nL(y)u = u , 0−1 ¯0 NE(y)b = b . Hence Th. 5.2.2 shows that the identity of E can be extended to an isomorphism θ : J(E, L, u0, b0) −→∼ J(E, L, u0−1, b0−1), 0−1 0−1 0 0 −1 and we still have NE(u ) = nL(b ) = 1. Thus, replacing η by η ◦ θ if necessary, we may assume ε = 1 in (5.2.10), i.e., 0 (5.2.11) b = b NE(z). × 0 0 Next put ϕ := 1E, ψ := 1L and y := z ∈ (E ⊗ L) , u1 := nL(y)u , b1 := NE(y)b = b (by (5.2.11)). Taking L-norms in (5.2.11) and observing (5.2.8), we conclude 0 NE(y)NE(y) = nL NE(z)) = 1, and since u1 = yyu¯ , this implies NE(u1) = 1. Hence Th. 5.2.2 yields an isomorphism ∼ 0 0 θ : J(E, L, u1, b1) −→ J(E, L, u , b ) - 58 -
16 SKIP GARIBALDI AND HOLGER P. PETERSSON extending the identity of E, and replacing η0 by η0 ◦ θ if necessary, we may and from now on will assume (5.2.12) b = b0. −1 Setting w := u and consulting Lemma 5.2.6, we have NE(w) = 1 and obtain a commutative diagram E / E / J _ Rw _ i 5
η J(E, L, 1E, b) / J(E, L, u, b), Rˆw where η◦Rˆw : J(E, L, 1E, b) → J is an isotopy and the isotopic embeddings i, i◦Rw from E to J are easily seen to be weakly equivalent. Hence, replacing i by i ◦ Rw and η by η ◦ Rˆw if necessary, we may assume u = 1E. But then, by symmetry, we 0 may assume u = 1E as well, forcing 0 η, η : J(E, L, 1E, b) −→ J to be isotopies extending i, i0, respectively. Thus ϕ := η ◦ η0−1 ∈ Str(J) satisfies 0 0−1 0 0 ϕ◦i = η◦η ◦η |E = η|E = i, so i and i are strongly, hence weakly, equivalent. 6. Norm classes and strong equivalence 6.1. Let (B, τ) be a central simple associative algebra of degree 3 with unitary involution over F and E a cubic ´etale F -algebra. Then the centre, K, of B and the discriminant, ∆(E), of E are quadratic ´etale F -algebras, as is L := K ∗ ∆(E) (cf. 5.1 (a)). To any pair (i, i0) of isotopic embeddings from E to J := H(B, τ) we × × will attach an invariant, belonging to E /nL((E ⊗ L) ) and called the norm class of (i, i0), and we will show that i and i0 are strongly equivalent if and only if their norm class is trivial. In order to achieve these objectives, a number of preparations will be needed. We begin with an extension of Th. 5.2.2 from isomorphisms to isotopies. Proposition 6.1.1. Let E,E0 and L, L0 be cubic and quadratic ´etalealgebras, respectively, over F and suppose we are given invertible elements u ∈ E, u0 ∈ E0, 0 0 0 0 b ∈ L, b ∈ L satisfying NE(u) = nL(b), NE0 (u ) = nL0 (b ). We write J := J(E, L, u, b) = E ⊕ (E ⊗ L)j, J 0 := J(E0,L0, u0, b0) = E0 ⊕ (E0 ⊗ L0)j0 as in 5.2 for the corresponding ´etaleTits process algebras and let ϕ: E0 →∼ E be an 0 −1 isotopy. Then, letting Φ: J → J be an arbitrary map and setting p := ϕ(1E0 ) ∈ E×, the following conditions are equivalent. (i) Φ is an isotopy extending ϕ. (ii) There exist an isomorphism ψ : L0 →∼ L and an invertible element y ∈ 0 ] −3 0 E ⊗ L such that ϕ(u ) = nL(y)p p u, ψ(b ) = NE(y)b and 0 0 0 0 0 (6.1.2) Φ(v0 + v j ) = ϕ(v0) + y(ϕ ⊗ ψ)(v ) j 0 0 0 0 0 for all v0 ∈ E , v ∈ E ⊗ L . 0 Proof. ϕ1 := Rp ◦ ϕ: E → E is an isotopy preserving units, hence is an isomor- phism. By 5.2 we have
J := J(E, L, u, b) = J(L, E ⊗ L, ιL, u, b), - 59 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 17
(p) ] (p) and in obvious notation, setting u := p u, b := NE(p)b, Lemma 3.8.5 yields an isomorphism (p) ∼ (p) (p) (p) (p) Rˆp : J −→ J1 := J(L, E ⊗ L, ιL, u , b ) = J(E, L, u , b ),
v0 + vj 7−→ (v0p) + vj1 0 Thus Rˆp : J → J1 is an isotopy and Φ1 := Rˆp ◦ Φ is a map from J to J1. Since ϕ1 preserves units, this leads to the following chain of equivalent conditions.
Φ is an isotopy extending ϕ ⇐⇒ Φ1 is an isotopy extending ϕ1
⇐⇒ Φ1 is an isotopy extending ϕ1 and preserving units
⇐⇒ Φ1 is an isomorphism extending ϕ1. × By Th. 5.2.2, therefore, (i) holds if and only if there exist an element y1 ∈ (E ⊗L) 0 0 (p) 0 (p) and an isomorphism ψ : L → L such that ϕ1(u ) = nL(y1)u , ψ(b ) = NE(y1)b and 0 0 0 0 0 Φ1(v0 + v j ) = ϕ1(v0) + y1(ϕ1 ⊗ ψ)(v ) j1 0 0 0 0 0 0 for all v0 ∈ E , v ∈ E ⊗ L . Setting y := y1p, and observing (ϕ1 ⊗ ψ)(v ) = (ϕ ⊗ ψ)(v0)p for all v0 ∈ E0 ⊗ L0, it is now straightforward to check that the preceding equations, in the given order, are equivalent to the ones in condition (ii) of the theorem. Lemma 6.1.3. ([PT04a, Lemma 4.5]) Let L (resp. E) be a quadratic (resp. a cubic) ´etale F -algebra. Given y ∈ E ⊗ L such that c := NE(y) satisfies nL(c) = 1, 0 0 0 there exists an element y ∈ E ⊗ L satisfying NE(y ) = c and nL(y ) = 1. 6.2. Notation. For the remainder of this section we fix a central simple associative algebra (B, τ) of degree 3 with unitary involution over F and a cubic ´etale F -algebra E. We write K for the centre of B, put J := H(B, τ) and L := K ∗ ∆(E) in the sense of 5.1. Theorem 6.2.1. Let i: E → J be an isotopic embedding and suppose w ∈ E has norm 1. Then the isotopic embeddings i and i ◦ Rw from E to J are strongly × equivalent if and only if w ∈ nL((E ⊗ L) ).
Proof. By Prop. 5.2.5, we find invertible elements u ∈ E, b ∈ L such that NE(u) = nL(b) and i extends to an isotopy η : J1 := J(E, L, u, b) → J. On the other hand, i and i ◦ Rw are strongly equivalent by definition (cf. 4.1) if and only if there exists an element Ψ ∈ Str(J) making the central square in the diagram _ J1 o ? E / E / J1 Rw i i (6.2.2) η η ( J / J.v Ψ commutative, equivalently, the isotopy ϕ := Rw : E → E can be extended to an el- −1 ement of the structure group of J1. By Prop. 6.1.1 (with p = w ), this in turn hap- −1 ] 3 pens if and only if some invertible element y ∈ E ⊗ L has uw = nL(y)(w ) w u = 4 2 ¯ −1 nL(y)w u, i.e., w = nL(w y), and eitherNE(y) = 1 or NE(y) = bb . Replacing y 2 by w y, we conclude that i and i ◦ Rw are strongly equivalent if and only ¯ −1 (6.2.3) some y ∈ E ⊗ L satisfies (i) nL(y) = w and (ii) NE(y) ∈ {1, bb }. - 60 -
18 SKIP GARIBALDI AND HOLGER P. PETERSSON
In particular, for i and i ◦ Rw to be strongly equivalent it is necessary that w ∈ × nL((E ⊗ L) ). Conversely, let this be so. Then some y ∈ E ⊗ L satisfies condition (i) of (6.2.3), so we have w = nL(y) and nL(NE(y)) = NE(nL(y)) = NE(w) = 1. 0 0 Hence Lemma 6.1.3 yields an element y ∈ E ⊗ L such that NE(y ) = NE(y) and 0 0−1 nL(y ) = 1. Setting z := yy ∈ E ⊗ L, we conclude nL(z) = nL(y) = w and 0 −1 NE(z) = NE(y)NE(y ) = 1, hence that (6.2.3) holds for z in place of y. Thus i and i ◦ Rw are strongly equivalent. 6.3. Norm classes. Let i, i0 : E → J be isotopic embeddings. By Th. 5.2.7, there exist a norm-one elements w ∈ E as well as an element ϕ ∈ Str(J) such that the left-hand square of the diagram E / E o E Rw Rv i0 i i0 J / JJo ϕ ψ commutes. Given another norm-one element v ∈ E and another element ψ ∈ Str(J) such that the right-hand square of the above diagram commutes as well, then the 0 0 isotopic embeddings i and i ◦ Rv−1w from E to J are strongly equivalent (via −1 × ψ ◦ ϕ), and Th. 6.2.1 implies w ≡ v mod nL((E ⊗ L) ). Thus the class of × w mod nL((E ⊗ L) ) does not depend on the choice of w and ϕ. We write this class as [i, i0] and call it the norm class of (i, i0); it is clearly symmetric in i, i0. We say i, i0 have trivial norm class if 0 × × [i, i ] = 1 in E /nL((E ⊗ L) ). For three isotopic embeddings i, i0, i00 : E → J, it is also trivially checked that [i, i00] = [i, i0][i0, i00]. Corollary 6.3.1. Two isotopic embeddings i, i0 : E → J are strongly equivalent if and only if [i, i0] is trivial. Proof. Let i, i0 : E → J be isotopic embeddings. By Th. 5.2.7, they are weakly 0 equivalent, so some norm-one element w ∈ E makes i and i ◦ Rw strongly equiva- 0 lent. Thus i and i are strongly equivalent if and only if i and i ◦ Rw are strongly × equivalent, which by Th. 6.2.1 amounts to the same as w ∈ nL((E ⊗ L) ), i.e., to 0 i and i having trivial norm class. Remark 6.3.2. When confined to isomorphic rather than isotopic embeddings, Cor. 6.3.1 reduces to [PT04a, Th. 4.2].
7. Albert algebras: proof of Theorem B 7.1. Unfortunately, we have not succeeded in extending Th. 5.2.7, the notion of norm class as defined in 6.3, or Cor. 6.3.1 from absolutely simple Jordan algebras of degree 3 and dimension 9 to Albert algebras. Instead, we will have to be more modest by settling with Theorem B, i.e., with the weak Skolem-Noether property for isomorphic rather than arbitrary isotopic embeddings. Given a cubic ´etalealgebra E and an Albert algebra J over F , the idea of the proof is to factor two isomorphic embeddings from E to J through the same absolutely simple nine-dimensional subalgebra of J, which by structure theory will have the form H(B, τ) for some central simple associative algebra (B, τ) of degree 3 with unitary involution over F , - 61 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 19 allowing us to apply Th. 5.2.7 and reach the desired conclusion. In order to carry out this procedure, a few preparations will be needed. Throughout this section, we fix an arbitrary Albert algebra J and a cubic ´etale algebra E over F .
Lemma 7.1.1. Assume F is algebraically closed and denote by E1 := Diag3(F ) ⊆ + Mat3(F ) the cubic ´etalesubalgebra of diagonal matrices. Then there exists a cubic + + ´etalesubalgebra E2 ⊆ Mat3(F ) such that Mat3(F ) is generated by E1 and E2 as a cubic Jordan algebra over F .
+ Proof. We realize Mat3(F ) as a first Tits construction + J1 := Mat3(F ) = J(E1, 1), with adjoint ], norm N, trace T , and identify the diagonal matrices on the left with E1 viewed canonically as a cubic subalgebra of J(E1, 1). Since F is infinite, we find × an element u0 ∈ E1 satisfying E1 = F [u0]. Letting α ∈ F , we put
y := u0 + αj1 ∈ J1.
Since u0 and j1 generate J1 as a cubic Jordan algebra, so do u0 and y, hence E1 and E2 := F [y]. It remains to show that, for a suitable choice of α, the F -algebra E2 is cubic ´etale.We first deduce from (3.9.1) and (3.9.3) that ] ] 2 y = u0 + (−αu0)j1 + α j2,
T (y) = TE1 (u0), ] ] T (y ) = TE1 (u0), 3 N(y) = NE1 (u0) + α . Thus y has the generic minimum (= characteristic) polynomial 3 2 ] 3 t − TE1 (u0)t + TE1 (u0)t − NE1 (u0) + α ∈ F [t], whose discriminant by [Lan02, IV, Exc. 12(b)] is 2 ] 2 ] 3 3 3 ∆y := TE1 (u0) TE1 (u0) − 4TE1 (u0) − 4TE1 (u0) (NE1 (u0) + α ) 3 2 ] 3 − 27(NE1 (u0) + α ) + 18TE1 (u0)TE1 (u0)(NE1 (u0) + α ) 3 ] 3 6 = ∆u0 − 4TE1 (u0) + 54NE1 (u0) − 18TE1 (u0)TE1 (u0) α − 27α , where ∆u0 6= 0 is the discriminant of the minimum polynomial of u0. Regardless × of the characteristic, we can therefore choose α ∈ F in such a way that ∆y 6= 0, in which case E2 is a cubic ´etale F -algebra. 7.2. Digression: pointed quadratic forms. By a pointed quadratic form over F we mean a triple (V, q, c) consisting of an F -vector space V , a quadratic form q : V → F , with bilinearization q(x, y) = q(x + y) − q(x) − q(y), and an element c ∈ V that is a base point for q in the sense that q(c) = 1. Then V together with the U-operator
(7.2.1) Uxy := q(x, y¯)x − q(x)¯y (x, y ∈ V ), wherey ¯ := q(c, y)c − y, and the unit element 1J := c is a Jordan algebra over F , denoted by J := J(V, q, c) and called the Jordan algebra of the pointed quadratic form (V, q, c). It follows immediately from (7.2.1) that the subalgebra of J generated P by a family of elements xi ∈ J, i ∈ I, is F c + i∈I F xi. - 62 -
20 SKIP GARIBALDI AND HOLGER P. PETERSSON
Lemma 7.2.2. Assume F is infinite and let i, i0 : E → J be isomorphic embed- 0 dings. Then there exist isomorphic embeddings i1, i1 : E → J such that i (resp., 0 0 i ) is strongly equivalent to i1 (resp., i1) and the subalgebra of J generated by 0 i1(E) ∪ i1(E) is absolutely simple of degree 3 and dimension 9. Proof. We proceed in two steps. Assume first that F is algebraically closed. Then E = F × F × F and J = Her3(C) are both split, C being the octonion algebra of + ∼ Zorn vector matrices over F . Note that Mat3(F ) = Her3(F × F ) may be viewed canonically as a subalgebra of J. By splitness of E, there are frames (i.e., complete 0 orthogonal systems of absolutely primitive idempotents) (ep)1≤p≤3,(ep)1≤p≤3 in J P 0 P 0 such that i(E) = F ep, i (E) = F ep. But frames in the split Albert algebra are conjugate under the automorphism group. Hence we find automorphisms ϕ, ψ 0 of J satisfying ϕ(ep) = ψ(ep) = epp for 1 ≤ p ≤ 3. Applying Lemma 7.1.1, we find + a cubic ´etalesubalgebra E2 ⊆ Mat3(F ) ⊆ J that together with E1 := Diag3(F ) = + (ϕ ◦ i)(E) generates Mat3(F ) as a cubic Jordan algebra over F . Again, the cubic P ´etale E2 is split, so we find a frame (cp)1≤p≤3 in J satisfying E2 = F cp. This 0 in turn leads to an automorphism ψ of J sending epp to cp for 1 ≤ p ≤ 3. Then 0 0 0 0 i1 := ϕ ◦ i and i1 := ψ ◦ ψ ◦ i are strongly equivalent to i, i , respectively, and 0 satisfy i1(E) = E1, i1(E) = E2, hence have the desired property. Now let F be an arbitrary infinite field and write F¯ for its algebraic closure. We have E = F [u] for some u ∈ E and put x := i(u), x0 := i0(u) ∈ J. We write k-alg for the category of commutative associative k-algebras with 1, put G := Aut(J) × Aut(J) as a group scheme over F and, given R ∈ k-alg,(ϕ, ϕ0) ∈ G(R), 0 write xm := xm(ϕ, ϕ ), 1 ≤ m ≤ 9, in this order for the elements ] x1 := 1JR , x2 := ϕ(xR), x3 := ϕ(xR), 0 0 ] 0 x4 := ϕ (xR), x5 := ϕ (xR), x6 := ϕ(xR) × ϕ (xR), ] 0 0 ] ] 0 ] x7 := ϕ(xR) × ϕ (xR), x8 := ϕ(xR) × ϕ (xR), x9 := ϕ(xR) × ϕ (x ).
By a result of Br¨uhne(cf. [Pet15, Prop. 6.6]), the subalgebra of JR generated 0 0 by (ϕ ◦ iR)(ER) and (ϕ ◦ iR)(ER) is spanned as an R-module by the elements x1, . . . , x9. Now consider the open subscheme X ⊆ G defined by the condition that X(R), R ∈ k-alg, consist of all elements (ϕ, ϕ0) ∈ G(R) satisfying 0 0 × det TJ xm(ϕ, ϕ ), xn(ϕ, ϕ ) ∈ R . 1≤m,n≤9
By what we have just seen, this is equivalent to saying that the subalgebra of JR 0 0 generated by (ϕ ◦ iR)(ER) and (ϕ ◦ iR)(ER) is a free R-module of rank 9 and has a non-singular trace form. By the preceding paragraph, X(F¯) ⊆ G(F¯) is a non-empty (Zariski-) open, hence dense, subset. On the other hand, by [Spr98, 13.3.9(iii)], G(F ) is dense in G(F¯). Hence so is X(F ) = X(F¯)∩G(F ). In particular, we can find elements ϕ, ϕ0 ∈ Aut(J)(F ) such that the subalgebra J 0 of J generated by (ϕ ◦ i)(E) and (ϕ0 ◦ i0)(E) is non-singular of dimension 9. This property is preserved under base field extensions, as is the property of being generated by two elements. Hence, if J 0 were not absolutely simple, some base field extension of it would split into the direct sum of two ideals one of which would be the Jordan algebra of a pointed quadratic form of dimension 8 [Rac72, Th. 1]. On the other hand, the property of being generated by two elements is inherited by this Jordan 0 0 0 algebra, which by 7.2 is impossible. Thus i1 := ϕ ◦ i and i1 := ϕ ◦ i satisfy all conditions of the lemma. - 63 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 21
Proposition 7.2.3. Let F be a finite field and i: E → J an isomorphic embedding. Writing K := ∆(E) for the discriminant of E, there exists a subalgebra J1 ⊆ J such that ∼ i(E) ⊆ J1 = Her3(K, Γ), Γ := diag(1, −1, 1). Proof. F being finite, the Albert algebra J is necessarily split. Replacing E by i(E) if necessary, we may assume E ⊆ J and that i: E,→ J is the inclusion. We write E⊥ ⊆ J for the orthogonal complement of E in J relative to the bilinear trace and, ⊥ ] for all v ∈ E , denote by qE(v) the E-component of v relative to the decomposition J = E ⊕ E⊥. By [PR84a, Prop. 2.1], E⊥ may be viewed as an E-module in a ⊥ natural way, and qE : E → E is a quadratic form over E. Moreover, combining [PR84a, Cor. 3.8] with a result of Engelberger [Eng02, Prop. 1.2.5], we conclude ⊥ that there exists an element v ∈ E that is invertible in J and satisfies qE(v) = 0. Now [PR84a, Prop. 2.2] yields a non-zero element α ∈ F such that the inclusion E,→ J can be extended to an isomorphic embedding from the ´etalefirst Tits ∼ construction J(E, α) into J. Write J1 ⊆ J for its image. Then E ⊆ J1 = J(E, α), ∼ and from [PR84b, Th. 3] we deduce J(E, α) = Her3(K, Γ) with Γ := diag(1, −1, 1) as above. 0 Proposition 7.2.4. Let J1,J1 be nine-dimensional absolutely simple subalgebras 0 of J. Then every isotopy J1 → J1 can be extended to an element of the structure group of J.
0 × (w) 0 Proof. Let η1 : J1 → J1 be an isotopy. Then some w ∈ J1 makes η1 : J1 → J1 an isomorphism. On the other hand, structure theory yields a central simple associa- tive algebra (B, τ) of degree 3 with unitary involution over F and an isomorphism −1 × ϕ: H(B, τ) → J1 which, setting p := ϕ (w) ∈ H(B, τ) , may be regarded as an isomorphism (p) ∼ (w) ϕ: H(B, τ) −→ J1 . On the other hand, following (2.7.1), (p) ∼ (p) Rp : H(B, τ) −→ H(B, τ ) is an isomorphism as well, and combining, we end up with an isomorphism 0 −1 (p) ∼ 0 ϕ := η1 ◦ ϕ ◦ Rp : H(B, τ ) −→ J1. Writing K for the centre of B and consulting 3.8, we now find invertible elements u ∈ H(B, τ), µ ∈ K satisfying NB(u) = nK (µ) such that ϕ extends to an isomorphism Φ: J(B, τ, u, µ) −→∼ J. 0 (p) 0 0 Similarly, we find invertible elements u ∈ H(B, τ ), µ ∈ K satisfying NB(u ) = 0 0 nK (µ ) such that ϕ extends to an isomorphism Φ0 : J(B, τ (p), u0, µ0) −→∼ J. ]−1 0 −1 0 Next, setting u1 := p u , µ1 := NB(p) µ , we apply Lemma 3.8.5 to obtain an isotopy (p) 0 0 −1 (7.2.5) Rˆp : J(B, τ, u1, µ1) −→ J(B, τ , u , µ ), v0 + vj 7−→ (v0p) + (p vp)j, and combining, we end up with an isotopy ˆ−1 0−1 Rp ◦ Φ ◦ Φ: J(B, τ, u, µ) −→ J(B, τ, u1, µ1). - 64 -
22 SKIP GARIBALDI AND HOLGER P. PETERSSON
Hence [Pet04, Th. 5.2] yields an isomorphism
∼ Ψ: J(B, τ, u, µ) −→ J(B, τ, u1, µ1) inducing the identity on H(B, τ). Thus
0 −1 η := Φ ◦ Rˆp ◦ Ψ ◦ Φ : J −→ J is an isotopy that fits into the diagram
J(B, τ, u1, µ1) O Rˆp Ψ ( J(B, τ, u, µ) J(B, τ (p), u0, µ0) O O
? ? H(B, τ) / H(B, τ (p)) Rp
Φ ϕ ϕ0 Φ0
0 J1 / J1 _ η1 _
Ñ J η / J,
0 whose arrows are either inclusions or isotopies. Now, since η ◦ Φ = Φ ◦ Rˆp ◦ Ψ by definition of η, and Rˆp agrees with Rp on H(B, τ) by (7.2.5), simple diagram chasing shows that η ∈ Str(J) is an extension of η1.
We can now prove Theorem B in a form reminiscent of Th. 5.2.7.
Theorem 7.2.6. Let J be an Albert algebra over F and E a cubic ´etale F -algebra. Then the pair (E,J) satisfies the weak Skolem Noether property for isomorphic embeddings.
Proof. Leit i, i0 : E → J be two isomorphic embeddings. We must show that they are weakly equivalent and first claim that we may assume the following: there exist a central simple associative algebra (B, τ) of degree 3 with unitary involution over ∼ 0 F and a subalgebra J1 ⊆ J such that J1 = H(B, τ) and i, i factor uniquely through 0 J1 to isomorphic embeddings i1 : E → J1, i1 : E → J1. Indeed, replacing the iso- morphic embeddings i, i0 by strongly equivalent ones if necessary, this is clear by Lemma 7.2.2 provided F is infinite. On the other hand, if F is finite, Prop. 7.2.3 0 leads to absolutely simple nine-dimensional subalgebras J1,J1 ⊆ J that are isomor- 0 0 phic and have the property that i, i factor uniquely through J1,J1 to isomorphic 0 0 0 embeddings i1 : E → J1, i1 : E → J1, respectively. But every isomorphism from J1 to J1 extends to an automorphism of J [KMRT98, 40.15], [Pet04, Remark 5.6(b)]. 0 Hence we may assume J1 = J1, as desired. 0 With J1, i1, i1 as above, Th. 5.2.7 yields elements w ∈ E of norm 1 and ϕ1 ∈ 0 Str(J1) such that i1 ◦ Rw = ϕ1 ◦ i1. Using Prop. 7.2.4, we extend ϕ1 to an element ϕ ∈ Str(J) and therefore conclude that the diagram (4.1.1) commutes. - 65 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 23
3 8. Outer automorphisms for type D4: proof of Theorem A In this section, we apply Theorem B to prove Theorem A. 8.1. A subgroup of Str(J). In this subsection, we rely on some facts that are only proved in the literature under the hypothesis char F 6= 2, 3. This hypothesis is not strictly necessary but we adopt it for now in order to ease the writing. The element ψ ∈ Str(J) provided by Theorem B belongs to the subgroup H of Str(J) of elements that normalize E and such that Nψ = N. (Indeed, as ψ ∈ Str(J), there is a µ ∈ F × such that Nψ = µN, but for e ∈ E we have N(ψ(e)) = N(ϕ(ew)) = N(ϕ(e))N(ϕ(w)) = N(e).) We now describe H in the case where J is a matrix Jordan algebra as in §3.5 with Γ = 13 and E is the subalgebra of diagonal matrices. P Fix h ∈ H. The norm N restricts to E as N( αieii) = α1α2α3, so h permutes the three singular points [eii] in the projective variety N|E = 0 in P(E). There is an embedding of the symmetric group on 3 letters, Sym3, in H acting by permuting the eii by their indices, see [Gar06, §3.2] for an explicit formula, and consequently ∼ H = H0 o Sym3, where H0 is the subgroup of H of elements normalizing F eii for × ×3 each i. For w := (w1, w2, w3) ∈ (F ) such that w1w2w3 = 1, it follows that 2 Uw ∈ H (cf. (3.2.2)) sends eii 7→ wi eii. Assuming now that F is algebraically closed, after multiplying h by a suitable Uw, we may assume that h restricts to be the identity on E. The subgroup of such elements of Str(J) is identified with the Spin(C) which acts on the off-diagonal entries in J as a direct sum of the three inequivalent minuscule 8-dimensional representations, see [KMRT98, 36.5, 38.6, 38.7] or [Jac71, p. 18, Prop. 6]. Thus, (1) we may identify H with (RE/F (Gm) · Spin(C)) o Sym3, where Sym3 acts via outer automorphisms on Spin(C) as in [Gar06, §3] or [KMRT98, 35.15]. 8.2. The Tits class. Recall that we view the Dynkin diagram of a group G to be endowed with an action by the absolute Galois group of F , and elements of Aut(∆)(F ) act naturally on H2(F,Z).
Lemma 8.2.1. Let G be a group of type D4 over a field F with Dynkin diagram ∆. If there is a π ∈ Aut(∆)(F ) of order 3 such that π(tG) = tG, then G has type 1 3 D4 or D4 and tG = 0. 2 6 Proof. For the first claim, if G has type D4 or D4, then Aut(∆)(F ) = Z/2 or 1. 1 Now suppose that G has type D4. We may assume that G is simply connected. The center Z of the simply connected cover of G is µ2 × µ2, with automorphism group Sym3 and π acts on Z with order 3. The three nonzero characters χ1, χ2, χ3 : Z → Gm are permuted transitively by π, so by hypothesis the element χi(tG) ∈ 2 H (F, Gm) does not depend on i. As the χi’s satisfy the equations χ1 +χ2 +χ3 = 0 and 2χi = 0 (compare [Tit71, 6.2] or [KMRT98, 9.14]), it follows that χi(tG) = 0 for all i, hence tG = 0 by [Gar12, Prop. 7]. 3 If G has type D4, then there is a unique cyclic cubic field extension E of F 1 2 such that G × E has type D4. By the previous paragraph, restriction H (F,Z) → 2 H (E,Z) kills tG. That map is injective because Z has exponent 2, so tG = 0. In the next result, the harder, “if” direction is the crux case of the proof of Theorem A and is an application of Theorem B. The easier, “only if” direction amounts to [CEKT13, Th. 13.1] or [KT14, Prop. 4.2]; we include it here as a consequence of the (a priori stronger) Lemma 8.2.1. - 66 -
24 SKIP GARIBALDI AND HOLGER P. PETERSSON
Proposition 8.2.2. Let G be a group of type D4 over a field F . The image of α(F ) : Aut(G)(F ) → Aut(∆)(F ) contains an element of order 3 if and only if G 1 3 has type D4 or D4, G is simply connected or adjoint, and tG = 0. 1 Proof. “If” : We may assume that G is simply connected. If G has type D4, then G is Spin(q) for some 3-Pfister quadratic form q, and the famous triality automorphisms of Spin(q) as in [SV68, 3.6.3, 3.6.4] are of order 3 and have image 3 in Aut(∆)(F ) of order 3. So assume G has type D4. Assume for this paragraph that char F 6= 2, 3. There is a uniquely determined 1 cyclic Galois field extension E of F such that G × E has type D4. By hypothesis, there is an Albert F -algebra J with norm form N such that E ⊂ J and we may identify G with the algebraic group with K-points
{g ∈ GL(J ⊗ K) | Ng = N and g|E⊗K = IdE⊗K } for every extension K of F . Take now ϕ to be a non-identity F -automorphism of E and w ∈ E of norm 1 and ψ ∈ Str(J) to be the elements given by Theorem B such that ψ|E = ϕ ◦ Rw. As ψ normalizes E and preserves N, it follows immediately that ψ normalizes G as a subgroup of Str(J). (Alternatively this is obvious from the fact that in subsection 8.1, Spin(C) is the derived subgroup of H◦.) Tracking through the description of H in subsection 8.1, we find that conjugation by ψ is an outer automorphism of G such that ψ3 is inner. In case F has characteristic 2 or 3, one can reduce to the case of characteristic zero as follows. Find R a complete discrete valuation ring with residue field F and fraction field K of characteristic zero. Lifting E to R allows us to construct a quasi-split simply connected group scheme Gq over R whose base change to F is the quasi-split inner form Gq of G. We have maps 1 q ∼ 1 q 1 q H (F,G ) ←− H´et(R, G ) ,→ H (K, G × K) where the first map is an isomorphism by Hensel and the second map is injective by [BT87]. Twisting by a well chosen Gq-torsor, we obtain
1 ∼ 1 1 H (F,G) ←− H´et(R, G) ,→ H (K, G × K) 3 ∼ where G×K has type D4 and zero Tits class and G = G×F . Now in Aut(G)(F ) → Aut(∆)(F ) = Z/3, the inverse image of 1 is a connected component X of Aut(G) defined over F , a G-torsor. Lifting X to H1(K, G × K), we discover that this G- torsor is trivial (by the characteristic zero case of the theorem), hence X is F -trivial, i.e., has an F -point. “Only if” : Let φ ∈ Aut(G)(F ) be such that α(φ) has order 3. In view of the 3 inclusion (1.1.1), Lemma 8.2.1 applies. If G has type D4, then it is necessarily 1 simply connected or adjoint, so assume G has type D4. Then φ lifts to an auto- morphism of the simply connected cover Ge of G, hence acts on the center Z of Ge in such a way that it preserves the kernel of the map Z → G. As Z is isomorphic to µ2 × µ2 and φ acts on it as an automorphism of order 3, the kernel must be 0 or Z, hence G is simply connected or adjoint. 3 8.3. Proof of Theorem A. Let G be a group of type D4, so Aut(∆)(F ) = Z/3; put π for a generator. If π(tG) 6= tG, then the right side of (1.1.1) is a singleton and the containment is trivially an equality, so assume π(tG) = tG. Then tG = 0 by Lemma 8.2.1 and the conclusion follows by Proposition 8.2.2. - 67 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 25
Example 8.3.1. Let F0 be a field with a cubic Galois extension E0. For the split adjoint group PSO8 of type D4 over F , a choice of pinning gives an isomorphism of Aut(PSO8) with PSO8 o Sym3 where Sym3 denotes the symmetric group on 3 letters, such that elements of Sym3 normalize the Borel subgroup appearing in the 1 pinning. Twisting Spin8 by a 1-cocycle with values in H (F0, Sym3) representing q 3 the class of E0 gives a simply connected quasi-split group G of type D4. As in [GMS03, pp. 11, 12], there exists an extension F of F0 and a versal torsor ξ ∈ H1(F,Gq); define G to be Gq × F twisted by ξ. As ξ is versal, the Rost 3 invariant rGq (ξ) ∈ H (F, Z/6Z) has maximal order, namely 6 [GMS03, p. 149]. Moreover, the map α(F ): Aut(G)(F ) → Aut(∆)(F ) = Z/3 is onto by Theorem A. In case char F0 6= 2, 3, then Aut(Γ) for some twisted composition Γ. As rGq (ξ) is not 2-torsion, by [KMRT98, 40.16], Γ is not Hurwitz, and by [KT14], Aut(G)(F ) contains no outer automorphisms of order 3. This is a newly observed phenomenon, in that in all other cases where α(F ) is known to be onto, it is also split.
9. Outer automorphisms for type A
9.1. Groups of type An. We now consider Conjecture 1.1.2 and Question 1.1.3 for groups G of type An. If G has inner type (i.e., is isogenous to SL1(B) for a degree d central simple F -algebra) then equality holds in (1.1.1) and the answer to Question 1.1.3 is “yes” as in [Gar12, p. 232]. So assume that G has outer type and in particular n ≥ 2. The simply connected cover of G is SU(B, τ) for B a central simple K-algebra of degree d := n + 1, where K is a quadratic ´etale F -algebra, and τ is a unitary K/F -involution. (This generalizes the (K, B, τ) defined in §3.7 by replacing 3 by d.) As the center Z of SU(B, τ) is the group scheme (µd)[K] of d-th roots of unity twisted by K in the sense of [KMRT98, p. 418] (i.e., is the Cartier dual of the finite ´etalegroup scheme (Z/d)[K]), every subgroup of Z is characteristic, hence (1.1.1) is an equality for G if and only if it is so for SU(B, τ) and similarly the answers to Question 1.1.3 are the same for G and SU(B, τ). Therefore, we need only treat SU(B, τ) below. The automorphism group Aut(∆)(F ) is Z/2 and its nonzero element π acts on 2 H (F,Z) as −1, hence π(tSU(B,τ)) = −tSU(B,τ) and the right side of (1.1.1) is a singleton (if 2tSU(B,τ) 6= 0) or has two elements (if 2tSU(B,τ) = 0). These cases are distinguished by the following lemma.
Lemma 9.1.1. In case d is even (resp., odd): 2tSU(B,τ) = 0 if and only if B ⊗K B (resp., B) is a matrix algebra over K.
∗ ∗ Proof. The cocenter Z := Hom(Z, Gm) is (Z/d)[K]; put χi ∈ Z for the element corresponding to i ∈ (Z/d)[K]. If d = 2e for some integer e, then the element χe is fixed by Gal(F ) and 2χe = χd = 0, regardless of B or tSU(B,τ). All other χi have 2 stabilizer subgroup Gal(K) and χi(2tSU(B,τ)) ∈ H (K,Z) can be identified with the class of B⊗2i in the Brauer group of K, cf. [KMRT98, p. 378]. The algebra B ⊗K B is a matrix algebra, then, if and only if χi vanishes on 2tSU(B,τ) for all i. This is equivalent to 2tSU(B,τ) = 0 by [Gar12, Prop. 7]. When the degree d of B is odd, B ⊗K B is a matrix algebra if and only if B is such.
Corollary 9.1.2. If G is a group of type An for n even, then equality holds in (1.1.1) and the answer to Question 1.1.3 is “yes”. - 68 -
26 SKIP GARIBALDI AND HOLGER P. PETERSSON
Proof. We may assume that G has outer type and is SU(B, τ). If 2tSU(B,τ) 6= 0, then the right side of (1.1.1) is a singleton and the claim is trivial. Otherwise, by Lemma 9.1.1, B is a matrix algebra, i.e., SU(B, τ) is the special unitary group of a K/F -hermitian form, and the claim follows. 9.2. The algebraic group Aut(SU(B, τ)) has two connected components: the iden- tity component, which is identified with the adjoint group of SU(B, τ), and the other component, whose F -points are the outer automorphisms of SU(B, τ). Theorem 9.2.1. There is an isomorphism between the F -variety of K-linear anti- automorphisms of B commuting with τ and the non-identity componet of SU(B, τ), given by sending an anti-automorphism ψ to the outer automorphism g 7→ ψ(g)−1. Clearly, such an anti-automorphism provides an isomorphism of B with its op- posite algebra, hence can only exist when B has exponent 2. This is a concrete illustration of the inclusion (1.1.1). Proof. First suppose that F is separably closed, in which case we may identify t t K = F × F , B = Md(F ) × Md(F ), and τ(b1, b2) = (b2, b1). A K-linear anti- t −1 t −1 automorphism ψ is, by Skolem-Noether, of the form ψ(b1, b2) = (x1b1x1 , x2b2x2 ) −t for some x1, x2 ∈ PGLd(F ), and the assumption that ψτ = τψ forces that x2 = x1 . As NrdB/K ψ = NrdB/K , it follows that ψ is an automorphism of the variety SU(B, τ), hence φ defined by φ(g) := ψ(g)−1 is an automorphism of the group. As φ acts nontrivially on the center — φ(b) = b−1 for b ∈ K× — φ is an outer automorphism. We have shown that there is a well-defined morphism from the variety of anti- automorphisms commuting with τ to the outer automorphisms of SU(B, τ), and it remains to prove that it is an isomorphism. For this, note that PGLd acts on SU(B, τ) where the group action is just function composition, that this action is the natural action of the identity component of SU(B, τ) on its other connected compo- nent, and that therefore the outer automorphisms are a PGLd-torsor. Furthermore, the first paragraph of the proof showed that the anti-automorphisms commuting −1 with τ also make up a PGLd-torsor, where the actions are related by y.ψ = y .φ for y ∈ PGLd. This completes the proof for F separably closed. For general F , we note that the map ψ 7→ φ is F -defined and gives an isomor- phism over Fsep, hence is an isomorphism over F . 9.3. We do not know how to prove or disprove existence of an anti-automorphism commuting with τ in general, but we can give a criterion for Question 1.1.3 that is 3 analogous to the one given in [KT14] for groups of type D4. Corollary 9.3.1. A group SU(B, τ) of outer type A has an F -defined outer au- tomorphism of order 2 if and only if there exists a central simple algebra (B0, τ0) over F with τ0 an involution of the first kind such that (B, τ) is isomorphic to (B0 ⊗ K, τ0 ⊗ ι), for ι the non-identity F -automorphism of K. Proof. The bijection in Theorem 9.2.1 identifies outer automorphisms of order 2 with anti-automorphisms of order 2. If such a (B0, τ0) exists, then clearly τ0 pro- vides an anti-automorphism of order 2. Conversely, given an anti-automorphism τ0 of order 2, define a semilinear auto- morphism of B via ι := τ0τ. Set B0 := {b ∈ B | ι(b) = b}; it is an F -subalgebra and τ0 restricts to be an involution on B0. - 69 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 27
Example 9.3.2. We now exhibit a (B, τ) with B of exponent 2, but such that SU(B, τ) has no outer automorphism of order 2 over F . The paper [ART79] provides a field F and a division F -algebra C of degree 8 and exponent 2 such that C is not a tensor product of quaternion algebras. Moreover, it provides a quadratic extension K/F contained in C. It follows that C ⊗ K has index 4, and we set B to be the underlying division algebra. As corK/F [B] = 2[C] = 0 in the Brauer group, B has a unitary involution τ. For sake of contradiction, suppose that SU(B, τ) had an outer automorphism of order 2, hence there exists a (B0, τ0) as in Corollary 9.3.1. Then B0 has degree 4, so B0 is a biquaternion algebra. Moreover, C ⊗ B0 is split by K, hence is Brauer- equivalent to a quaternion algebra Q. By comparing degrees, we deduce that C is isomorphic to B0 ⊗ Q, contradicting the choice of C.
2 9.4. Type E6. Results entirely analogous to Theorem 9.2.1, Corollary 9.3.1, and 2 Example 9.3.2 also hold for groups G of type E6, using proofs of a similar flavor. The Dynkin diagram of type E6 has automorphism group Z/2 = {Id, π}, and arguing as in Lemmas 8.2.1 or 9.1.1 shows that π(tG) = tG if and only if tG = 0. So for addressing Conjecture 1.1.2 and Question 1.1.3, it suffices to consider only those groups with zero Tits class, which can be completely described in terms of the hermitian Jordan triples introduced in [GP07, §4] or the Brown algebras studied in [Gar01]. We leave the details to the interested reader. 2 Does Conjecture 1.1.2 hold for every group of type E6? One might hope to imitate the outline of the proof of Theorem A. Does an analogue of Theorem B hold, where one replaces Albert algebras, cubic Galois extensions, and the inclusion of root systems D4 ⊂ E6 by Brown algebras or Freudenthal triple systems, quadratic Galois extensions, and the inclusion E6 ⊂ E7?
Acknowledgements. The first author was partially supported by NSF grant DMS- 1201542 and the Charles T. Winship Fund at Emory University. He thanks TU Dort- mund for its hospitality under the auspices of the Gambrinus Fellowship while some of the research was conducted. The second author thanks the Department of Mathematics and Statistics of the University of Ottawa, where much of this research was initiated, for its support and hospitality.
References [AJ57] A.A. Albert and N. Jacobson, On reduced exceptional simple Jordan algebras, Ann. of Math. (2) 66 (1957), 400–417. MR 0088487 (19,527b) [ART79] S.A. Amitsur, L.H. Rowen, and J.-P. Tignol, Division algebras of degree 4 and 8 with involution, Israel J. Math. 33 (1979), 133–148. [BT87] F. Bruhat and J. Tits, Groupes alg´ebriques sur un corps local. Chapitre III. Compl´ementset applications ´ala cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 671–698. [CEKT13] V. Chernousov, A. Elduque, M-A. Knus, and J-P. Tignol, Algebraic groups of type D4, triality, and composition algebras, Documenta Math. 18 (2013), 413–468. [CKT12] V. Chernousov, M-A. Knus, and J-P. Tignol, Conjugacy class of trialitarian automor- phisms and symmetric compositions, J. Ramanujan Math. Soc. 27 (2012), 479–508. [DG70] M. Demazure and A. Grothendieck, SGA3: Sch´emasen groupes, Lecture Notes in Mathematics, vol. 151–153, Springer, 1970. [Eng02] R. Engelberger, Twisted compositions, Ph.D. thesis, ETH Z¨urich, 2002. [Gar01] S. Garibaldi, Structurable algebras and groups of type E6 and E7, J. Algebra 236 (2001), no. 2, 651–691. - 70 -
28 SKIP GARIBALDI AND HOLGER P. PETERSSON
[Gar06] , Unramified cohomology of classifying varieties for exceptional simply con- nected groups, Trans. Amer. Math. Soc. 358 (2006), no. 1, 359–371. [Gar12] , Outer automorphisms of algebraic groups and determining groups by their maximal tori, Michigan Math. J. 61 (2012), no. 2, 227–237. [GMS03] S. Garibaldi, A. Merkurjev, and J-P. Serre, Cohomological invariants in Galois coho- mology, University Lecture Series, vol. 28, Amer. Math. Soc., 2003. [GP07] S. Garibaldi and H.P. Petersson, Groups of outer type E6 with trivial Tits algebras, Transf. Groups 12 (2007), no. 3, 443–474. [Jac71] N. Jacobson, Exceptional Lie algebras, Lecture notes in pure and applied mathematics, vol. 1, Marcel-Dekker, New York, 1971. [Jac81] N. Jacobson, Structure theory of Jordan algebras, University of Arkansas Lecture Notes in Mathematics, vol. 5, University of Arkansas, Fayetteville, Ark., 1981. MR 634508 (83b:17015) [JK73] N. Jacobson and J. Katz, Generically algebraic quadratic Jordan algebras, Scripta Math. 29 (1973), no. 3-4, 215–227, Collection of articles dedicated to the memory of Abraham Adrian Albert. MR 0404370 (53 #8172) [KMRT98] M.-A. Knus, A.S. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, Colloquium Publications, vol. 44, Amer. Math. Soc., 1998. 3 [KT14] M-A. Knus and J-P. Tignol, Triality and algebraic groups of type D4, arxiv:1409.1718, September 2014. [Lan02] S. Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556 (2003e:00003) [Loo75] O. Loos, Jordan pairs, Springer-Verlag, Berlin, 1975, Lecture Notes in Mathematics, Vol. 460. MR R0444721 (56 #3071) [Loo91] , Finiteness conditions in Jordan pairs, Math. Z. 206 (1991), no. 4, 577–587. MR 1100842 (92d:17030) [McC69] K. McCrimmon, The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495–510. MR 0238916 (39 #276) [McC70] , The Freudenthal-Springer-Tits constructions revisited, Trans. Amer. Math. Soc. 148 (1970), 293–314. MR 0271181 (42 #6064) [MT95] A.S. Merkurjev and J.-P. Tignol, The multipliers of similitudes and the Brauer group of homogeneous varieties, J. reine angew. Math. 461 (1995), 13–47. [MZ88] K. McCrimmon and E. Zelmanov, The structure of strongly prime quadratic Jordan algebras, Adv. in Math. 69 (1988), no. 2, 133–222. MR 946263 (89k:17052) [Pet04] H.P. Petersson, Structure theorems for Jordan algebras of degree three over fields of arbitrary characteristic, Comm. Algebra 32 (2004), no. 3, 1019–1049. MR 2063796 [Pet15] , An embedding theorem for reduced Albert algebras over arbitrary fields, Comm. Algebra 43 (2015), no. 5, 2062–2088. MR 3316839 [PR84a] H.P. Petersson and M.L. Racine, Springer forms and the first Tits construction of exceptional Jordan division algebras, Manuscripta Math. 45 (1984), no. 3, 249–272. MR 85i:17029 [PR84b] , The toral Tits process of Jordan algebras, Abh. Math. Sem. Univ. Hamburg 54 (1984), 251–256. MR 86g:17020 [PT04a] H.P. Petersson and M. Thakur, The ´etaleTits process of Jordan algebras revisited, J. Algebra 273 (2004), no. 1, 88–107. MR 2032452 (2004m:17047) [PT04b] R. Preeti and J.-P. Tignol, Multipliers of improper similitudes, Documenta Math. 9 (2004), 183–204. [Rac72] M.L. Racine, A note on quadratic Jordan algebras of degree 3, Trans. Amer. Math. Soc. 164 (1972), 93–103. MR 0304447 (46 #3582) [Ser93] J-P. Serre, G`ebres, L’Enseignement Math. 39 (1993), 33–85, (reprinted in Oeuvres, vol. IV, pp. 272–324). [Spr98] T.A. Springer, Linear algebraic groups, second ed., Birkh¨auser,1998. [SV68] T.A. Springer and F.D. Veldkamp, On Hjelmslev-Moufang planes, Math. Zeit. 107 (1968), 249–263. [Tit71] J. Tits, Repr´esentationslin´eaires irr´eductiblesd’un groupe r´eductifsur un corps quel- conque, J. Reine Angew. Math. 247 (1971), 196–220. - 71 -
OUTER AUTOMORPHISMS AND A SKOLEM-NOETHER THEOREM 29
Garibaldi: Institute for Pure and Applied Mathematics, UCLA, 460 Portola Plaza, Box 957121, Los Angeles, California 90095-7121, USA E-mail address: [email protected]
Petersson: Fakultat¨ fur¨ Mathematik und Informatik, FernUniversitat¨ in Hagen, D- 58084 Hagen, Germany E-mail address: [email protected] - 72 - - 73 -
Note on a 1-colouring game on paths and cycles
Stephan Dominique Andres Reinhard B¨orger † Fakult¨atf¨urMathematik und Informatik, Fernuniversit¨atin Hagen, Universit¨atsstr.1, 58084 Hagen, Germany [email protected]
Abstract In the 1-colouring achievement game, two players alternately choose pairwise nonadjacent distinct vertices of a given graph. A player looses if he cannot move any more. We characterize the paths and cycles for which the first player has a winning strategy. This answers an open question of Harary and Tuza. MSC 2000: primary 91A46; secondary 05C57 Key words: combinatorial game, 1-colouring achievement game, path, cycle, nim sum
1 Introduction
Consider the following 1-colouring achievement game on a graph G = (V,E). At the beginning of the game, every vertex of the graph is unmarked. Two players move alternately. A move consists in marking an unmarked vertex all of which neighbours are unmarked. The first player who is unable to move looses the game. We call this game G∗. Let n ∈ N and Pn be the path with n vertices, where P0 denotes the empty graph, and, for n ≥ 3, let Cn be the cycle with n vertices. In this note we ∗ ∗ characterize which player has a winning strategy in the games Pn resp. Cn. This answers an open question raised by Harary and Tuza [4] who examined a similar k-colouring game on paths and cycles for k ≥ 2.
2 Combinatorial games and nim sum
A combinatorial game is a 2-player game, where the players move alternately. The game consists of several configurations Si, i = 0, 1, . . . , k, where S0 is the starting configuration, and each configuration is a set of other configurations, the options. The players move alternately, where a move consists in choosing an
†Reinhard B¨orgerpassed away June 6, 2014 - 74 -
option (if any) from the actual configuration. The first player’s actual configu- ration is S0. If a player is unable to move (since the actual configuration is the empty set), the game ends and he looses. We furthermore impose that a combi- natorial game is always finite, i.e. the game ends after a finite number of moves. Note that each configuration of a combinatorial game defines a combinatorial game. In the following, we identify the games with their starting configuration. Let S be the set of configurations of a combinatorial game with starting configuration S0. Sprague [6] (and later Grundy [3]) showed that there is a unique mapping g : S −→ N such that (i) if S0 is an option of S ∈ S, then g(S0) 6= g(S), and (ii) if g(S) > 0 for S ∈ S, then, for any 0 ≤ k < g(S), the configuration S has an option S0 with g(S0) = k.
The number g(S0) is called the Grundy value of the game. So, if S0 is the set {O1,...,Om} of options, then
g(S0) = mex{g(O1), . . . , g(Om)}, (1) where for a finite M ⊆ N the mex is defined as mexM := min(N \ M). The Grundy value describes which player has a winning strategy for the game S: the first player wins if and only if g(S) > 0. The sum S(0) + S(1) of two combinatorial games S(0) and S(1) is the game, where in each move a player chooses some k ∈ {0, 1} and plays in S(k) according to the rules of the respective games. Sprague [6] showed that the Grundy value of a sum S(0) + S(1) is the nim sum g(S(0) + S(1)) = g(S(0)) ⊕ g(S(1)) (2) defined by
n ! n ! n X i X i X i αi2 ⊕ βi2 := (αi + βi mod 2)2 (3) i=0 i=0 i=0 for n ∈ N, αi, βi ∈ {0, 1}. The nim sum was already considered by Bouton [2] for nim games, in a generalized version by Moore [5].
3 The 1-colouring achievement game on paths
In this section we discuss the 1-colouring achievement game.
Lemma 3.1. Let n ∈ N. Then ∗ ∗ g(P0 ) = 0, g(P1 ) = 1, ∗ and, for n ≥ 2, the options of Pn are ∗ ∗ ∗ Pn−2, and Pk + Pn−k−3 for any 0 ≤ k ≤ n − 3. - 75 -
∗ Table 1: The Grundy values of P17n+k n k = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 0 1 1 2 0 3 1 1 0 3 3 2 2 4 0 5 2 1 2 3 3 0 1 1 3 0 2 1 1 0 4 5 2 7 4 2 0 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 3 2 3 3 0 1 1 3 0 2 1 1 0 4 5 3 7 4 4 8 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 5 9 3 3 0 1 1 3 0 2 1 1 0 4 5 3 7 4 6 8 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 7 9 3 3 0 1 1 3 0 2 1 1 0 4 5 3 7 4 8 8 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 9 9 3 3 0 1 1 3 0 2 1 1 0 4 5 3 7 4 10 8 1 1 2 0 3 1 1 0 3 3 2 2 4 4 5 5 11 9 3 3 0 1 1 3 0 2 1 1 0 4 5 3 7 4
Proof. In Pn, the first player may either mark a vertex of degree 1, this leads to a path Pn−2 which can still be marked, or he can mark a vertex of degree 2, this leads to 3 unmarkable vertices in the middle, leaving a path Pk at the left and a path Pn−k−3 at the right. Using Lemma 3.1, (1) and (2), it is possible to calculate the Grundy value ∗ of Pn recursively via
∗ ∗ ∗ ∗ g(Pn ) = mex{g(Pn−2), g(Pk ) ⊕ g(Pn−k−3) | 0 ≤ k ≤ n − 3} (4) for n ≥ 2. The first 204 values are displayed in Table 1. ∗ Table 1 suggests that the sequence of Grundy values g(Pn ) is periodic with period 34, except for 0 ≤ n ≤ 51. In fact, this is true. Lemma 3.2. For all N ≥ 86,
∗ ∗ g(PN ) = g(PN−34).
Proof. For 86 ≤ N ≤ 173 the lemma is true by the above table. If N ≥ 174, ∗ ∗ we have N − 2 ≥ 172 ≥ 86, hence g(PN−2) = g(PN−34−2) by the induction hypothesis. Moreover, for 0 ≤ k ≤ N −3 we have k ≥ 86 or N −k−3 ≥ 86, hence ∗ ∗ ∗ ∗ g(Pk ) = g(Pk−34) or g(PN−k−3) = g(PN−k−3−34) by the induction hyopthesis. ∗ ∗ ∗ ∗ ∗ ∗ Therefore, g(Pk )⊕g(PN−k−3) = g(Pk−34)⊕g(PN−k−3) or g(Pk )⊕g(Pn−k−3) = ∗ ∗ g(Pk ) ⊕ g(PN−k−3−34). Using Lemma 3.1 this means that, for any option of ∗ ∗ PN , there is an option of PN−34 with the same Grundy value. Since, for 0 ≤ k ≤ N − 34 − 3, we have that k ≥ 52 or N − 34 − 3 − k ≥ 52, by a similar ∗ argumentation we conclude that, for any option of PN−34, there is an option of ∗ PN with the same Grundy value. This proves the lemma. Now we can prove our main result: - 76 -
∗ Theorem 3.3. The second player wins the game Pn if and only if (i) n ∈ {0, 14, 34} or (ii) n ≡ c mod 34 with c ∈ {4, 8, 20, 24, 28}.
∗ Proof. The second player wins on Pn if and only if g(Pn ) = 0. Thus the theorem follows from Table 1 and Lemma 3.2.
∗ Theorem 3.4. For n ≥ 3, the first player wins the game Cn if and only if (i) n ∈ {3, 17, 37} or (ii) n ≡ c mod 34 with c ∈ {7, 11, 23, 27, 31}.
∗ ∗ Proof. In the first move of the game Cn the only option is Pn−3. Therefore the first player wins on Cn if and only if the second player wins on Pn−3, thus the theorem follows from Theorem 3.3.
References
[1] E.R. Berlekamp, J.H. Conway, and R.K. Guy, “Winning ways for your mathematical plays”, Academic Press, 1982 [2] C.L. Bouton, Nim, a game with a complete mathematical theory, Ann. of Math. (2) 3 (1901–1902), 35–39 [3] P.M. Grundy, Mathematics and games, Eureka 2 (1939), 6–8 [4] F. Harary and Zs. Tuza, Two graph-colouring games, Bull. Austral. Math. Soc. 48 (1993), 141–149 [5] E.H. Moore, A generalization of the game called Nim, Ann. of Math. (2) 11 (1910), 93–94 [6] R. Sprague, Uber¨ mathematische Kampfspiele, Tohoku Math. J., First Se- ries 41 (1935), 438–444 [7] Zs. Tuza, Graph colorings with local constraints, Discuss. Math. Graph Theory 17 (1997), 161–228
Note added
After having completed this paper we came to know that Theorem 3.3 had been already proved by Berlekamp et al. [1], pages 88–90, i.e. even before the problem was announced as an open problem by Harary and Tuza [4], as was remarked in a survey of Tuza [7], see pages 214–215. - 77 -
LIMITS AND COLIMITS OF QUANTALOID-ENRICHED CATEGORIES AND THEIR DISTRIBUTORS
LILI SHEN AND WALTER THOLEN
Dedicated to the memory of Reinhard B¨orger
Abstract. It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is total and cototal, and so is the category of Q-distributors and Q-Chu transforms.
1. Introduction The importance of (small) categories enriched in a (unital) quantale rather than in an arbitrary monoidal category was discovered by Lawvere [18] who enabled us to look at individual mathe- matical objects, such as metric spaces, as small categories. Through the study of lax algebras [15], quantale-enriched categories have become the backbone of a larger array of objects that may be viewed as individual generalized categories. Prior to this development, Walters [31] had extended Lawvere’s viewpoint in a different manner, replacing the quantale at work by a quantaloid (a term proposed later by Rosenthal [22]), thus by a bicategory with the particular property that its hom-objects are given by complete lattices such that composition from either side preserves suprema; quantales are thus simply one-object quantaloids. Based on the theory of quantaloid-enriched categories developed by Stubbe [25, 26], recent works [16, 21, 27, 28] have considered in particular the case when the quantaloid in question arises from a given quantaloid by a “diagonal construction” whose roots go far beyond its use in this paper; see [12, 13]. Specifically for the one-object quantaloids (i.e., quantales) whose enriched categories give (pre)ordered sets and (generalized) metric spaces, the corresponding small quantaloids of diagonals lead to truly partial structures, in the sense that the full structure is available only on a subset of the ambient underlying set of objects. In the first instance then, this paper aims at exploring the categorical properties of the category Q-Cat of small Q-enriched categories and their Q-functors for a small quantaloid Q. By showing that Q-Cat is topological [2] over the comma category Set/ ob Q (Proposition 2.2) one easily describes small limits and colimits in this category, and beyond. In fact, one concludes that categories of this type are total [24] and cototal, hence possess even those limits and colimits of large diagrams whose existence is not made impossible by the size of the small hom-sets of Q-Cat (see [9]). Our greater interest, however, is in the category Q-Chu whose objects are often called Q-Chu spaces, the prototypes of which go back to [3, 20] and many others (see [4, 10]). Its objects are Q-distributors of Q-categories (also called Q-(bi)modules or Q-profunctors), hence they are compatible Q-relations (or Q-matrices) that have been investigated intensively ever since B´enabou [5] introduced them (see [6, 7]). While when taken as the morphisms of the category whose objects are Q-categories, they make for a in many ways poorly performing category (as already the case Q = 2 shows), when taken as objects of Q-Chu with morphisms given by so-called Q-Chu transforms, i.e., by pairs of Q-functors that behave like adjoint operators, we obtain a category that in terms of the existence of limits and colimits behaves as strongly as Q-Cat itself. In analogy to the property shown in [11] in a different categorical context, we first prove that the domain functor Q-Chu / Q-Cat allows for initial liftings [2] of structured cones over small diagrams
2010 Mathematics Subject Classification. 18D20, 18A30. Partial financial assistance by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. 1 - 78 -
2 LILI SHEN AND WALTER THOLEN
(Theorem 3.4), which then allows for an explicit description of all limits and colimits in Q-Chu over small diagrams. But although the domain functor fails to be topological, just as for Q-Cat we are able to show totality (and, consequently, cototality) of Q-Chu. A key ingredient for this result is the existence proof of a generating set in Q-Chu, and therefore also of a cogenerating set (Theorem 3.8).
2. Limits and colimits in Q-Cat Throughout, let Q be a small quantaloid, i.e., a small category enriched in the category Sup of complete lattices and sup-preserving maps. A small Q-category is given by a set X (its set of objects) and a lax functor a : X / Q, where the set X is regarded as a quantaloid carrying the chaotic structure, so that for all x, y ∈ X there is precisely one arrow x / y, called (x, y). Explicitly then, the Q-category structure on X is given by
• a family of objects |x|X := ax in Q (x ∈ X), • a family of morphisms a(x, y): |x| / |y| in Q (x, y ∈ X), subject to
1|x| ≤ a(x, x) and a(y, z) ◦ a(x, y) ≤ a(x, z)
(x, y, z ∈ X). When one calls |x| = |x|X the extent (or type) of x ∈ X, a Q-functor f : (X, a) / (Y, b) of Q-categories (X, a), (Y, b) is an extent-preserving map f : X / Y such that there is a lax natural transformation a / bf given by identity morphisms in Q; explicitly,
|x|X = |f(x)|Y and a(x, y) ≤ b(f(x), f(y)) for all x, y ∈ X. Denoting the resulting ordinary category by Q-Cat, we have a forgetful functor
ob Q-Cat / Set/ ob Q f f XXY/ Y XXY/ Y ≤ a 7→ b |-| |-| Q ob Q Example 2.1. (1) If Q is a quantale, i.e., a one-object quantaloid, then the extent functions of Q-categories are trivial and Q-Cat assumes its classical meaning (as in [17], where Q is considered as a monoidal (closed) category). Prominent examples are Q = 2 = {0 < 1} and Q = ([0, ∞], ≥, +), where then Q-Cat is the category Ord (sets carrying a reflexive and transitive relation, called order here but commonly known as preorder, with monotone maps) and, respectively, the category Met (sets X carrying a distance function a : X×X /[0, ∞] required to satisfy only a(x, x) = 0 and a(x, z) ≤ a(x, y)+a(y, z) but, in accordance with the terminology introduced by Lawvere [18] and used in [15], nevertheless called metric here, with non-expanding maps f : X / Y , so that b(f(x), f(y)) ≤ a(x, y) for all x, y ∈ X). (2) (Stubbe [27]) Every quantaloid Q gives rise to a new quantaloid DQ whose objects are the morphisms of Q, and for morphisms u, v in Q, a morphism (u, d, v): u / v in DQ, normally written just as d, is a Q-morphism d : dom u / cod v satisfying (d . u) ◦ u = d = v ◦ (v & d), also called a diagonal from u to v:
v&d •••/ •
d u v
••• / • d.u - 79 -
LIMITS AND COLIMITS OF QUANTALOID-ENRICHED CATEGORIES AND THEIR DISTRIBUTORS 3
(Here d . u, v & d denote the internal homs of Q, determined by z ≤ d . u ⇐⇒ z ◦ u ≤ d, t ≤ v & d ⇐⇒ v ◦ t ≤ d for all z : cod u / cod d, t : dom d / dom v.) With the composition of d : u / v with e : v / w in DQ defined by e d = (e . v) ◦ d = e ◦ (v & d), and with identity morphisms u : u / u, DQ becomes a quantaloid whose local order is inherited from Q. In fact, there is a full embedding
Q / DQ, (u : t / s) 7→ (u : 1t / 1s) of quantaloids. We remark that the construction of D works for ordinary categories; indeed it is part of the proper factorization monad on CAT [13]. (2a) For Q = 2, the quantaloid DQ has object set {0, 1}. There are exactly two DQ-arrows 1 / 1, given by 0, 1, and 0 is the only arrow in every other hom-set of DQ; composition is given by infimum. A DQ-category is given by a set X, a distinguished subset A ⊆ X (those elements of X with extent 1) and a (pre)order on A. Hence, a DQ-category structure is a (truly!) partial order, and with those maps that may be monotonely restricted to the distinguished subsets as morphisms. We write ParOrd = DQ-Cat for the resulting category, which contains Ord as a full coreflective subcategory. (2b) For Q = ([0, ∞], ≥, +), the hom-sets of DQ are easily described by DQ(u, v) = {s ∈ [0, ∞] | u, v ≤ s}, with composition given by t s = s − v + t (for t : v / w). A DQ-category structure on a set X consists of functions |-| : X / [0, ∞], a : X × X / [0, ∞] satisfying |x|, |y| ≤ a(x, y), a(x, x) ≤ |x|, a(x, z) ≤ a(x, y) − |y| + a(y, z) (x, y, z ∈ X). Obviously, since necessarily |x| = a(x, x), these conditions simplify to a(x, x) ≤ a(x, y), a(x, z) ≤ a(x, y) − a(y, y) + a(y, z) (x, y, z ∈ X), describing a as a partial metric on X (see [16, 19, 21]1). With non-expanding maps one obtains the category ParMet = DQ-Cat, which contains Met as a full coreflective subcategory: the coreflector restricts the partial metric a on X to those elements x ∈ X with a(x, x) = 0. To see how limits and colimits in the (ordinary) category Q-Cat are to be formed, it is best to first prove its topologicity over Set/ ob Q. Recall that, for any functor U : A /X , a U-structured cone over a diagram D : J / A is given by an object X ∈ X and a natural transformation ξ : ∆X / UD.A lifting of (X, ξ) is given by an object A in A and a cone α : ∆A / D over D with UA = X, Uα = ξ. Such lifting (A, α) is U-initial if, for all cones β : ∆B / D over D and morphisms t : UB / UA in X , there is exactly one morphism h : B / A in A with Uh = t and α · ∆h = β. We call U small-topological [11] if all U-structured cones over small diagrams admit U-initial liftings, and U is topological when this condition holds without the size restriction on diagrams. Recall also the following well-known facts: • Topological functors are necessarily faithful [8], and for faithful functors it suffices to consider discrete cones to guarantee topologicity. • U : A / X is topological if, and only if, U op : Aop / X op is topological.
1Here our terminology naturally extends Lawvere’s notion of metric and is synonymous with “generalized partial metric” as used by Pu-Zhang [21] who dropped finiteness (a(x, y) < ∞), symmetry (a(x, y) = a(y, x)) and separation (a(x, x) = a(x, y) = a(y, y) ⇐⇒ x = y) from the requirement for the notion of “partial metric” as originally introduced by Matthews [19]. - 80 -
4 LILI SHEN AND WALTER THOLEN
• The two properties above generally fail to hold for small-topological functors. However, for any functor U, a U-initial lifting of a U-structured cone that is a limit cone in X gives also a limit cone in A. • Every small-topological functor is a fibration (consider singleton diagrams) and has a fully faithful right adjoint (consider the empty diagram). Proposition 2.2. For every (small) quantaloid Q, the “object functor” Q-Cat / Set/ ob Q is topological.
Proof. Given a (possibly large) family fi :(X, |-|) / (Yi, |-|i)(i ∈ I) of maps over ob Q, where every Yi carries a Q-category structure bi with extent function |-|i, we must find a Q-category structure a on X with extent function |-| such that (1) every fi :(X, a) / (Y, b) is a Q-functor, and (2) for every Q-category (Z, c), any extent preserving map g : Z / X becomes a Q-functor (Z, c) / (X, a) whenever all maps fig are Q-functors (Z, c) / (Yi, bi)(i ∈ I). But this is easy: simply define ^ a(x, y) := bi(fi(x), fi(y)) i∈I for all x, y ∈ X. Hence, a is the ob-initial structure on X with respect to the structured sink (fi : X / (Yi, bi))i∈I . Corollary 2.3. Q-Cat is complete and cocomplete, and the object functor has both a fully faithful left adjoint and a fully faithful right adjoint. Remark 2.4. (1) The set of objects of the product (X, a) of a small family of Q-categories (Xi, ai)(i ∈ I) is given by the fibred product of (Xi, |-|i)(i ∈ I), i.e.,
X = {((xi)i∈I , q) | q ∈ ob Q, ∀i ∈ I(xi ∈ Xi, |xi| = q)},
and (when writing (xi)i∈I instead of ((xi)i∈I , q) and putting |(xi)i∈I | = q) we have ^ a((xi)i∈I , (yi)i∈I ) = ai(xi, yi): |(xi)i∈I | / |(yi)i∈I | i∈I for its hom-arrows. In particular, (ob Q, >) with >(q, r) = > : q / r the top element in Q(q, r) (for all q, r ∈ ob Q), is the terminal object in Q-Cat. (2) The coproduct (X, a) of Q-categories (Xi, ai)(i ∈ I) is simply formed by the coproduct in Set, with all structure to be obtained by restriction: ( a ai(x, y) if x, y ∈ Xi, X = Xi, |x|X = |x|X if x ∈ Xi, a(x, y) = i ⊥ : |x| |y| else. i∈I / In particular, ∅ with its unique Q-category structure is an initial object in Q-Cat. (3) The equalizer of Q-functors f, g :(X, a) / (Y, b) is formed as in Set, by restriction of the structure of (X, a). The object set of their coequalizer (Z, c) in Q-Cat is also formed as in Set, so that Z = Y/ ∼, with ∼ the least equivalence relation on Y with f(x) ∼ g(x), 0 x ∈ X. With π : Y / Z the projection, necessarily |π(y)|Z = |y|Y , and c(π(y), π(y )) is the join of all 0 0 0 0 b(yn, yn) ◦ b(yn−1, yn−1) ◦ · · · ◦ b(y2, y2) ◦ b(y1, y1), 0 0 0 0 0 where |y| = |y1|, |y1| = |y2|,..., |yn−1| = |yn|, |yn| = |y | (yi, yi ∈ Y, n ≥ 1). (4) The fully faithful left adjoint of Q-Cat /Set/ ob Q provides a set (X, |-|) over ob Q with the discrete Q-structure, given by ( 1 if x = y, a(x, y) = |x| ⊥ : |x| / |y| else; while the fully faithful right adjoint always takes > : |x| / |y| as the hom-arrow, i.e., it chooses the indiscrete Q-structure. - 81 -
LIMITS AND COLIMITS OF QUANTALOID-ENRICHED CATEGORIES AND THEIR DISTRIBUTORS 5
Example 2.5. The product of partial metric spaces (Xi, ai)(i ∈ I) provides its carrier set
X = {((xi)i∈I , s) | s ∈ [0, ∞], ∀i ∈ I(xi ∈ Xi, |xi| = s)} with the “sup metric”:
a((xi)i∈I , (yi)i∈I ) = sup ai(xi, yi). i∈I [0, ∞] is terminal in ParMet when provided with the chaotic metric that makes all distances 0, and it is a generator when provided with the discrete metric d: ( 0 if s = t, d(s, t) = ∞ else. Beyond small limits and colimits, Q-Cat actually has all large-indexed limits and colimits that one can reasonably expect to exist. More precisely, recall that an ordinary category C with small hom-sets is (see [9]) • hypercomplete if a diagram D : J / C has a limit in C whenever the limit of C(A, D−) exists in Set for all A ∈ ob C; equivalently: whenever, for every A ∈ ob C, the cones ∆A / D in C may be labeled by a set; • totally cocomplete if a diagram D : J / C has a colimit in C whenever the colimit of C(A, D−) exists in Set for all A ∈ ob C; equivalently: whenever, for every A ∈ ob C, the connected components of (A ↓ D) may be labelled by a set. The dual notions are hypercocomplete and totally complete. It is well known (see [9]) that op •C is totally cocomplete if, and only if, C is total, i.e., if the Yoneda embedding C /SetC has a left adjoint; • total cocompleteness implies hypercompleteness but not vice versa (with Ad´amek’smonadic category over graphs [1] providing a counterexample); • for a solid (=semi-topological [29]) functor A / X , if X is hypercomplete or totally complete, A has the corresponding property [30]; • in particular, every topological functor, every monadic functor over Set, and every full reflective embedding is solid. It is also useful for us to recall [9, Corollary 3.5]: Proposition 2.6. A cocomplete and cowellpowered category with small hom-sets and a generating set of objects is total. Since Q-Cat is topological over Set/ ob Q which, as a complete, cocomplete, wellpowered and cowellpowered category with a generating and a cogenerating set, is totally complete and totally cocomplete, we conclude: Theorem 2.7. Q-Cat is totally complete and totally cocomplete and, in particular, hypercocom- plete and hypercomplete. Remark 2.8. Of course, we may also apply Proposition 2.6 directly to obtain Theorem 2.7 since the left adjoint of Q-Cat / Set/ ob Q sends a generating set of Set/ ob Q to a generating set of Q-Cat, and the right adjoint has the dual property. Explicitly then, denoting for every s ∈ ob Q by {s} the discrete Q-category whose only object has extent s, we obtain the generating set {{s} | s ∈ ob Q} for Q-Cat. Similarly, providing the disjoint unions Ds = {s} + ob Q (s ∈ ob Q) with the identical extent functions and the indiscrete Q-category structures, one obtains a cogenerating set in Q-Cat.
3. Limits and colimits in Q-Chu For Q-categories X = (X, a), Y = (Y, b), a Q-distributor [5] ϕ : X ◦ / Y (also called Q- (bi)module [18], Q-profunctor) is a family of arrows ϕ(x, y): |x| / |y| (x ∈ X, y ∈ Y ) in Q such that b(y, y0) ◦ ϕ(x, y) ◦ a(x0, x) ≤ ϕ(x0, y0) - 82 -
6 LILI SHEN AND WALTER THOLEN for all x, x0 ∈ X, y, y0 ∈ Y . Its composite with ψ : Y ◦ / Z is given by _ (ψ ◦ ϕ)(x, z) = ψ(y, z) ◦ ϕ(x, y). y∈Y Since the structure a of a Q-category (X, a) is neutral with respect to this composition, we obtain the category Q-Dis of Q-categories and their Q-distributors which, with the local pointwise order ϕ ≤ ϕ0 ⇐⇒ ∀x, y : ϕ(x, y) ≤ ϕ0(x, y), is actually a quantaloid. Every Q-functor f : X / Y gives rise to the Q-distributors f\ : X ◦ / Y and f \ : Y ◦ / X with \ f\(x, y) = b(f(x), y) and f (y, x) = b(y, f(x)) \ (x ∈ X, y ∈ Y ). One has f\ a f in the 2-category Q-Dis, and if one lets Q-Cat inherit the order of Q-Dis via \ \ f ≤ g ⇐⇒ f ≤ g ⇐⇒ g\ ≤ f\ ⇐⇒ 1|x| ≤ b(f(x), g(x)) (x ∈ X), then one obtains 2-functors co \ op (−)\ :(Q-Cat) / Q-Dis, (−) :(Q-Cat) / Q-Dis which map objects identically; here “op” refers to the dualization of 1-cells and “co” to the dualization of 2-cells. Example 3.1 (See Example 2.1). (1) A 2-distributor is an order ideal relation; that is, a relation ϕ : X ◦ / Y of ordered sets that behaves like a two-sided ideal w.r.t. the order: x0 ≤ x & xϕy & y ≤ y0 =⇒ x0ϕy0. A [0, ∞]-distributor ϕ : X ◦ / Y introduces a distance function between metric spaces (X, a), (Y, b) that must satisfy ϕ(x0, y0) ≤ a(x0, x) + ϕ(x, y) + a(y, y0) for all x, x0 ∈ X, y, y0 ∈ Y . (2) A D2-distributor ϕ : X ◦ / Y is given by a 2-distributor A ◦ / B where A = {x ∈ X | x ≤ x}, B = {y ∈ Y | y ≤ y} are the coreflections of X, Y , respectively. Likewise, a D[0, ∞]-distributor ϕ : X ◦ / Y is given by a distributor of the metric coreflections of the partial metric spaces X and Y . In our context Q-Dis plays only an auxiliary role for us in setting up the category Q-Chu whose objects are Q-distributors and whose morphisms (f, g): ϕ / ψ are given by Q-functors f :(X, a) / (Y, b), g :(Z, c) / (W, d) such that the diagram
f\ XXY◦ / Y
ϕ◦ ◦ψ (3.i) WZW ◦ / Z g\ commutes in Q-Dis: ψ(f(x), z) = ϕ(x, g(z)) (3.ii) for all x ∈ X, z ∈ Z. In particular, with ϕ = a, ψ = b one obtains that the morphisms (f, g) : 1(X,a) / 1(Y,b) in Q-Chu are precisely the adjunctions f a g :(Y, b) / (X, a) in the - 83 -
LIMITS AND COLIMITS OF QUANTALOID-ENRICHED CATEGORIES AND THEIR DISTRIBUTORS 7
2-category Q-Cat. With the order inherited from Q-Cat, Q-Chu is in fact a 2-category, and one has 2-functors dom : Q-Chu / Q-Cat, (f, g) 7→ f, cod : Q-Chu / (Q-Cat)op, (f, g) 7→ g. In order for us to exhibit properties of Q-Chu, it is convenient to describe Q-Chu transforms, i.e., morphisms in Q-Chu, alternatively, with the help of presheaves, as follows. For every s ∈ ob Q, let {s} denote the discrete Q-category whose only object has extent s. For a Q-category X = (X, a), a Q-presheaf ϕ on X of extent |ϕ| = s is a Q-distributor ϕ : X ◦/ {s}. Hence, ϕ is given by a family of Q-morphisms ϕx : |x| / |ϕ| (x ∈ X) with ϕy ◦ a(x, y) ≤ ϕx (x, y ∈ X). With ^ [ϕ, ψ] = ψx . ϕx, x∈X PX becomes a Q-category, and one has the Yoneda Q-functor
yX = y : X / PX, x 7→ (a(−, x): X ◦/ {|x|}). y is fully faithful, i.e., [y(x), y(y)] = a(x, y)(x, y ∈ X). The point of the formation of PX for us is as follows (see [14, 23]): Proposition 3.2. The 2-functor (−)\ :(Q-Cat)op / Q-Dis has a left adjoint P which maps a Q-distributor ϕ : X ◦ / Y to the Q-functor ϕ∗ : PY / PX, ψ 7→ ψ ◦ ϕ; hence, ∗ _ (ϕ (ψ))x = ψy ◦ ϕ(x, y) y∈Y for all ψ ∈ PY , x ∈ X. In particular, for a Q-functor f : X / Y one has ∗ ∗ ∗ f := (f\) : PY / PX, (f (ψ))x = ψf(x).
Denoting by ϕe : Y / PX the transpose of ϕ : X ◦ / Y under the adjunction, determined \ by ϕe ◦ (yX )\ = ϕ, so that (ϕe(y))x = ϕ(x, y) for all x ∈ X, y ∈ Y , we can now present Q-Chu transforms, as follows: Corollary 3.3. A morphism (f, g): ϕ / ψ in Q-Chu (as in (3.i)) may be equivalently presented as a commutative diagram f ∗ PX o PY O O
ϕe ψe (3.iii)
W o g Z in Q-Cat. Condition (3.ii) then reads as
(ϕe(g(z)))x = (ψe(z))f(x) (3.iv) for all x ∈ X, z ∈ Z. Proof. For all z ∈ Z, ∗ \ f (ψe(z)) = ψe(z) ◦ f\ = yZ (z) ◦ ψ ◦ f\ = yZ (z) ◦ g ◦ ϕ = yY (g(z)) ◦ ϕ = ϕe(g(z)). Theorem 3.4. Let D : J / Q-Chu be a diagram such that the colimit W = colim cod D exists in Q-Cat. Then any cone γ : ∆X / dom D in Q-Cat has a dom-initial lifting Γ : ∆ϕ / D in Q-Chu with ϕ : X ◦ / W , dom Γ = γ. In particular, if γ is a limit cone in Q-Cat, Γ is a limit cone in Q-Chu. - 84 -
8 LILI SHEN AND WALTER THOLEN
Proof. Considering the functors
\ dom (−)\ cod (−) Q-Chu / Q-Cat / Q-Dis, Q-Chu / (Q-Cat)op / Q-Dis, one has the natural transformation
\ κ : (dom(−))\ ◦ / (cod(−)) , κϕ := ϕ (ϕ ∈ ob Q-Chu).
\ By the adjunction of Proposition 3.2, κD : (dom D)\ ◦ / (cod D) corresponds to a natural trans- ∗ formation κDf : cod D /P(dom D)\, and the given cone γ gives a cocone γ : P(dom D)\ /∆PX. Forming the colimit cocone δ : cod D / ∆W one now obtains a unique Q-functor ϕe : W / PX making γ∗ ∆PX o P(dom D)\ O O ∆ϕ e κDg
∆W o cod D δ commute in Q-Cat or, equivalently, making
γ\ ∆X ◦ / (dom D)\
∆ϕ◦ ◦κD ∆W ◦ / (cod D)\ δ\ commute in Q-Dis, with ϕ : X ◦ / W corresponding to ϕe. In other words, we have a cone Γ : ∆ϕ / D with dom ϕ = X, dom Γ = γ, namely Γ = (γ, δ). Given a cone Θ : ∆ψ / D with ψ : Y ◦ / Z in Q-Dis and a Q-functor f : Y / X with γ · ∆f = := dom Θ, the cocone ϑ := cod Θ : cod D / ∆Z corresponds to a unique Q-functor g : W / Z with ∆g · δ = ϑ by the colimit property. As the diagram
γ∗ ∆PX o P(dom D)\ O O ∆f ∗ ∆ϕe ∗ κDg ∆PY t δ O ∆W o cod D
∆ψe ∆g ϑ ∆Z t
∗ shows, the colimit property of W also guarantees f ϕe = ψge (with ψe corresponding to ψ) which, by Corollary 3.3, means that (f, g): ψ / ϕ is the only morphism in Q-Chu with dom(f, g) = f and Γ · ∆(f, g) = Θ.
Corollary 3.5. dom : Q-Chu / Q-Cat is small-topological; in particular, dom is a fibration with a fully faithful right adjoint which embeds Q-Cat into Q-Chu as a full reflective subcategory. cod : Q-Chu / (Q-Cat)op has the dual properties. Proof. With the existence of small colimits guaranteed by Corollary 2.3, dom-initial liftings to small dom-structured cones exist by Theorem 3.4. For the assertion on cod, first observe that every Q-category X = (X, a) gives rise to the Qop-category Xop = (X, a◦), where a◦(x, y) = a(y, x) - 85 -
LIMITS AND COLIMITS OF QUANTALOID-ENRICHED CATEGORIES AND THEIR DISTRIBUTORS 9
(x, y ∈ X). With the commutative diagram
(−)op (Q-Chu)op / Qop-Chu
codop dom Q-Cat / Qop-Cat (−)op one sees that, up to functorial isomorphisms, codop :(Q-Chu)op / Q-Cat coincides with the op op small-topological functor dom : Q -Chu / Q -Cat.
Corollary 3.6. Q-Chu is complete and cocomplete, all small limits and colimits in Q-Chu are preserved by both dom and cod.
Proof. The dom-initial lifting of a dom-structured limit cone in Q-Cat is a limit cone in Q-Chu, which is trivially preserved. Having a right adjoint, dom also preserves all colimits.
Remark 3.7. (1) Let us describe (small) products in Q-Chu explicitly: Given a family of Q-distributors ϕi : Xi ◦ / Yi (i ∈ I), one first forms the product X of the Q-categories Xi = (Xi, ai) as in Remark 2.4(1) with projections pi and the coproduct of the Yi = (Yi, bi) as in Remark 2.4(2) with injections si (i ∈ I). The transposes ϕei then determines a Q- functor ϕe making the left square of
∗ pi (pi)\ PX o PXi XXX◦ / Xi O O
ϕ ϕ ϕ e ϕei ◦ ◦ i Y o Yi YYY ◦ / Yi si \ (si)
commutative, while the right square exhibits ϕ as a product of (ϕi)i∈I in Q-Chu with projections (pi, si), (i ∈ I); explicitly,
ϕ(x, y) = (ϕ(y)) = (ϕ (y) ◦ (p ) ) = (ϕ (y)) = ϕ (x , y) e x ei i \ x ei pi(x) i i
for x = ((xi)i∈I , q) in X and y = si(y) in Yi, i ∈ I. (2) The coproduct of ϕi : Xi ◦ / Yi (i ∈ I) in Q-Chu is formed like the product, expect that the roles of domain and codomain need to be interchanged. Hence, one forms the coproduct X of (Xi)i∈I and the product Y of (Yi)i∈I in Q-Cat and obtains the coproduct ϕ : X ◦ / Y in Q-Chu as in
(si)\ Xi ◦ / X
ϕi ◦ ◦ϕ Y ◦ Y i \ / (ϕi)
so that ϕ(x, y) = ϕi(x, yi) for y = ((yi)i∈I , q) in Y and x = si(x) in Xi, i ∈ I. • The equalizer of (f, g), (f, g): ϕ / ψ in Q-Chu is obtained by forming the equalizer and coequalizer
f g i / / p U / X / Y and W / Z / V f g - 86 -
10 LILI SHEN AND WALTER THOLEN
in Q-Cat, respectively. With χe : V /PU obtained from the coequalizer property making i∗ PU o PX O O
χe ϕe
VVWo p W commutative, Theorem 3.4 guarantees that
(i,p) (f,g) χ / ϕ // ψ (f,g) is an equalizer diagram in Q-Chu, where ∗ χ(x, p(w)) = (χe(p(w))x = (i (ϕe(w)))x = (ϕe(w) ◦ i\)x = ϕ(i(x), w) for all x ∈ U, w ∈ W . • Coequalizers in Q-Chu are formed like equalizers, except that the roles of domain and codomain need to be interchanged. We will now strengthen Corollary 3.6 and show total completeness and total cocompleteness of Q-Chu with the help of Proposition 2.6. To that end, let us observe that, since the limit and colimit preserving functors dom and cod must in particular preserve both monomorphisms and epimorphisms, a monomorphism (f, g): ϕ / ψ in Q-Chu must be given by a monomorphism f and an epimorphism g in Q-Cat, i.e., by an injective Q-functor f and a surjective Q-functor g. Consequently, Q-Chu is wellpowered, and so is its dual (Q-Chu)op =∼ Qop-Chu. The main point is therefore for us to prove: Theorem 3.8. Q-Chu contains a generating set of objects and, consequently, also a cogenerating set. Proof. With the notations explained below, we show that
{ηs : ∅ ◦ / Ds | s ∈ ob Q} ∪ {λt : {t} ◦ / Cb | t ∈ ob Q} is generating in Q-Chu. Here Ds belongs to a generating set of Q-Cat (see Remark 2.8), and a C = P{t} t∈ob Q is a coproduct in Q-Cat (see Remark 2.4(2)) of the presheaf Q-categories of the singleton Q- categories {t} (see Proposition 3.2). From C one obtains Cb by adding an isomorphic copy of each object in C, which may be easily explained for a Q-category (X, a): simply provide the set Xb := X × {1, 2} with the structure |(x, i)| = |x| and a((x, i), (y, j)) = a(x, y) Xb X b for all x, y ∈ X, i, j ∈ {1, 2}. Noting that the objects of P{t} are simply Q-arrows with domain t, we now define λt : {t} ◦ / Cb by ( u if dom u = t, λt(u, i) = ⊥ else for i ∈ {1, 2} and every object t and arrow u in Q. For another element (v, j) in Cb, if dom v = dom u = t one then has
[(u, i), (v, j)] ◦ λt(u, i) = (v . u) ◦ u ≤ v = λt(v, j), and in other cases this inequality holds trivially. Hence, λt is indeed a Q-distributor. - 87 -
LIMITS AND COLIMITS OF QUANTALOID-ENRICHED CATEGORIES AND THEIR DISTRIBUTORS 11
Let us now consider Q-Chu transforms (f, g) 6= (f, g): ϕ / ψ as in
f\ ◦ / (X, a))(◦ / (Y, b) f \ ϕ◦ ◦ψ g\ ◦ / (W, d) ◦ / (Z, c) g\
Case 1: X = ∅ is the initial object of Q-Cat (and Q-Dis). Then g 6= g, and we find s ∈ ob Q and h : W / Ds with hg 6= hg in Q-Cat. Consequently, (1∅, h): ηs / ϕ satisfies (f, g)(1∅, h) 6= (f, g)(1∅, h). Case 2: f 6= f, so that f(x0) 6= f(x0) for some x0 ∈ X. Then, for t := |x0|, e : {t} / X, |x0| 7→ x0, is a Q-functor with fe 6= fe, and it suffices to show that
h : W / C,b w 7→ (ϕ(x0, w), 1) is a Q-functor making (e, h): λt / ϕ a Q-Chu transform. Indeed, 0 0 0 d(w, w ) ≤ ϕ(x0, w ) . ϕ(x0, w) = [h(w), h(w )],
λt(h(w)) = ϕ(x0, w) = ϕ(e(t), w) for all w, w0 ∈ W . Case 3: X 6= ∅ and g 6= g. Then g(z0) 6= g(z0) for some z0 ∈ Z, and with any fixed x0 ∈ X we may alter the previous definition of h : W / Cb by ( (ϕ(x , w), 2) if w = g(z ), h(w) := 0 0 (ϕ(x0, w), 1) else.
The verification for h to be a Q-functor and (e, h): λt / ϕ a Q-Chu transform remain intact, and since hg 6= hg, the proof is complete. Remark 3.9. A generating set in Q-Chu may be alternatively given by
{λ∅ : ∅ ◦ / Cb} ∪ {λt : {t} ◦ / Cb | t ∈ ob Q}, so that in Case 1 one may proceed exactly as in Case 3 only by replacing ϕ(x0, w) with > : q /|w| for any fixed q ∈ ob Q. With Theorem 3.8 we obtain: Corollary 3.10. Q-Chu is totally complete and totally cocomplete and, in particular, hyperco- complete and hypercomplete.
References [1] J. Ad´amek.Colimits of algebras revisited. Bulletin of the Australian Mathematical Society, 17:433–450, 1977. [2] J. Ad´amek,H. Herrlich, and G. E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Wiley, New York, 1990. [3] M. Barr. ∗-Autonomous categories and linear logic. Mathematical Structures in Computer Science, 1:159–178, 1991. [4] J. Barwise and J. Seligman. Information Flow: The Logic of Distributed Systems, volume 44 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1997. [5] J. B´enabou. Les distributeurs. Universit´eCatholique de Louvain, Institute de Mat´ematique Pure et Appliqu´ee, Rapport no. 33, 1973. [6] J. B´enabou. Distributors at work. Lecture notes of a course given at TU Darmstadt, 2000. [7] F. Borceux. Handbook of Categorical Algebra: Volume 1, Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994. [8] R. B¨orgerand W. Tholen. Cantors Diagonalprinzip f¨urKategorien. Mathematische Zeitschrift, 160(2):135–138, 1978. [9] R. B¨orgerand W. Tholen. Total categories and solid functors. Canadian Journal of Mathematics, 42:213–229, 1990. - 88 -
12 LILI SHEN AND WALTER THOLEN
[10] B. Ganter. Relational Galois connections. In S. O. Kuznetsov and S. Schmidt, editors, Formal Concept Analysis, volume 4390 of Lecture Notes in Computer Science, pages 1–17. Springer, Berlin-Heidelberg, 2007. [11] E. Giuli and W. Tholen. A topologist’s view of Chu spaces. Applied Categorical Structures, 15(5-6):573–598, 2007. [12] M. Grandis. Weak subobjects and the epi-monic completion of a category. Journal of Pure and Applied Algebra, 154(1-3):193–212, 2000. [13] M. Grandis. On the monad of proper factorisation systems in categories. Journal of Pure and Applied Algebra, 171(1):17–26, 2002. [14] H. Heymans. Sheaves on Quantales as Generalized Metric Spaces. PhD thesis, Universiteit Antwerpen, Bel- gium, 2010. [15] D. Hofmann, G. J. Seal, and W. Tholen, editors. Monoidal Topology: A Categorical Approach to Order, Metric, and Topology, volume 153 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2014. [16] U. H¨ohleand T. Kubiak. A non-commutative and non-idempotent theory of quantale sets. Fuzzy Sets and Systems, 166:1–43, 2011. [17] G. M. Kelly. Basic Concepts of Enriched Category Theory, volume 64 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1982. [18] F. W. Lawvere. Metric spaces, generalized logic and closed categories. Rendiconti del Seminario Mat´ematico e Fisico di Milano, XLIII:135–166, 1973. [19] S. G. Matthews. Partial metric topology. Annals of the New York Academy of Sciences, 728(1):183–197, 1994. [20] V. Pratt. Chu spaces and their interpretation as concurrent objects. In J. Leeuwen, editor, Computer Science Today, volume 1000 of Lecture Notes in Computer Science, pages 392–405. Springer, Berlin-Heidelberg, 1995. [21] Q. Pu and D. Zhang. Preordered sets valued in a GL-monoid. Fuzzy Sets and Systems, 187(1):1–32, 2012. [22] K. I. Rosenthal. The Theory of Quantaloids, volume 348 of Pitman Research Notes in Mathematics Series. Longman, Harlow, 1996. [23] L. Shen and W. Tholen. Topological categories, quantaloids and Isbell adjunctions. arXiv:1501.00703, 2015. [24] R. Street and R. F. C. Walters. Yoneda structures on 2-categories. Journal of Algebra, 50(2):350–379, 1978. [25] I. Stubbe. Categorical structures enriched in a quantaloid: categories, distributors and functors. Theory and Applications of Categories, 14(1):1–45, 2005. [26] I. Stubbe. Categorical structures enriched in a quantaloid: tensored and cotensored categories. Theory and Applications of Categories, 16(14):283–306, 2006. [27] I. Stubbe. An introduction to quantaloid-enriched categories. Fuzzy Sets and Systems, 256(0):95–116, 2014. Special Issue on Enriched Category Theory and Related Topics (Selected papers from the 33rd Linz Seminar on Fuzzy Set Theory, 2012). [28] Y. Tao, H. Lai, and D. Zhang. Quantale-valued preorders: Globalization and cocompleteness. Fuzzy Sets and Systems, 256(0):236–251, 2014. Special Issue on Enriched Category Theory and Related Topics (Selected papers from the 33rd Linz Seminar on Fuzzy Set Theory, 2012). [29] W. Tholen. Semi-topological functors I. Journal of Pure and Applied Algebra, 15(1):53–73, 1979. [30] W. Tholen. Note on total categories. Bulletin of the Australian Mathematical Society, 21:169–173, 1980. [31] R. F. C. Walters. Sheaves and Cauchy-complete categories. Cahiers de Topologie et G´eom´etrieDiff´erentielle Cat´egoriques, 22(3):283–286, 1981.
Lili Shen, Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3 E-mail address: [email protected]
Walter Tholen, Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3 E-mail address: [email protected] - 89 -
Kan extensions and cartesian monoidal categories
Ross Street∗ Mathematics Department, Macquarie University, NSW 2109 Australia
Dedicated to the memory of Reinhard Börger
2010 Mathematics Subject Classification: 18D10; 18D20; 18A40; 18C10 Key words and phrases: Lawvere theory; monoidal category; enriched category; pointwise Kan extension.
Abstract The existence of adjoints to algebraic functors between categories of models of Lawvere theories follows from finite-product-preservingness surviving left Kan extension. A result along these lines was proved in Appendix 2 of Brian Day’s PhD thesis [1]. His context was categories enriched in a cartesian closed base. A generalization is described here with essentially the same proof. We introduce the notion of carte- sian monoidal category in the enriched context. With an advanced viewpoint, we give a result about left extension along a promonoidal module and further related results.
Contents
1 Introduction 2
2 Weighted colimits 2
3 Cartesian monoidal enriched categories 3
4 Main result 4 ∗The author gratefully acknowledges the support of Australian Research Council Dis- covery Grant DP1094883.
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5 An advanced viewpoint 5
1 Introduction
The pointwise left Kan extension, along any functor between categories with finite products, of a finite-product-preserving functor into a cartesian closed category is finite-product-preserving. This kind of result goes back at least to Bill Lawvere’s thesis [8] and some 1966 ETH notes of Fritz Ulmer. Eduardo Dubuc and the author independently provided Saunders Mac Lane with a proof along the lines of the present note at Bowdoin College in the Northern Hemisphere Summer of 1969. Brian Day’s thesis [1] gave a generalization to categories enriched in a cartesian closed base. Also see Kelly-Lack [7] and Day-Street [3]. Our purpose here is to remove the restriction on the base and, to some extent, the finite products.
2 Weighted colimits
We work with a monoidal category V as used in Max Kelly’s book [9] as a base for enriched category theory. Recall that the colimit of a V -functor F : A −→ X weighted by a V -functor W : A op −→ V is an object
colim(W, F ) = colimA(W A, F A) of X equipped with an isomorphism
X (colim(W, F ),X) =∼ [A op, V ](W, X (F,X))
V -natural in X. Independence of naturality in the two variables of two variable naturality, or Fubini’s theorem [9], has the following expression in terms of weighted colimits.
Nugget 1. For V -functors
op op W1 : A1 −→ V ,W2 : A2 −→ V ,F : A1 ⊗ A2 −→ X , if colim(W2,F (A, −)) exists for each A ∈ A then ∼ colim(W1, colim(W2,F )) = colim(W1 ⊗ W2,F ) .
Here the isomorphism is intended to include the fact that one side exists if and only if the other does. Also (W1 ⊗ W2)(A, B) = W1A ⊗ W2B.
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Proof. Here is the calculation:
∼ op X (colim(W1 ⊗ W2,F ),X) = [(A1 ⊗ A2) , V ](W1 ⊗ W2, X (F,X)) ∼ op op = [A1 , V ](W1, [A2 , V ](W2, X (F,X))) ∼ op = [A1 , V ](W1, X (colim(W2,F ),X))) ∼ = X (colim(W1, colim(W2,F )),X) .
Here is an aspect of the calculus of mates expressed in terms of weighted col- imits. Note that S a T : A −→ C means T op a Sop : A op −→ C op.
Nugget 2. For V -functors W : A op −→ V , G : C −→ X , and a V -adjunction
S a T : A −→ C , there is an isomorphism
colim(WSop,G) =∼ colim(W, GT ) .
Proof. Here is the calculation:
X (colim(W, GT ),X) =∼ [A op, V ](W, X (GT, X)) =∼ [A op, V ](W, X (G, X)T op) =∼ [C op, V ](WSop, X (G, X)) =∼ X (colim(WSop,G),X) .
Recall that a pointwise left Kan extension of a V -functor F : A −→ X along a V -functor J : A −→ B is a V -functor K = LanJ (F ): B −→ X such that there is a V -natural isomorphism ∼ KB = colimA(B(JA, B),FA) .
3 Cartesian monoidal enriched categories
A monoidal V -category A will be called cartesian when the tensor product and unit object have left adjoints. That is, A is a map pseudomonoid in the monoidal 2-category V -Catco in the sense of [5]. Let us denote the tensor product of A by −?− : A ⊗A −→ A with left adjoint ∆ : A −→ A ⊗ A and the unit by N : I −→ A with left adjoint E : A −→ I . (Here I is the unit V -category: it has one object 0 and I (0, 0) = I.) It is clear that these right adjoints make A a comonoidal V -category; that is, a pseudomonoid
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in V -Catop. Since ob : V -Cat −→ Set is monoidal, we see that ∆ : A −→ A ⊗ A is given by the diagonal on objects. We have ∼ A (A, A1 ?A2) = A (A, A1) ⊗ A (A, A2) , where V -functoriality in A on the right-hand side uses ∆. If A is cartesian, the V -functor category [A , V ] becomes monoidal under con- volution using the comonoidal structure on A . This is a pointwise tensor product in the sense that, on objects, it is defined by: (M ∗ N)A = MA ⊗ NA.
On morphisms it requires the use of ∆. Indeed, the Yoneda embedding Y: A op −→ [A , V ] is strong monoidal.
4 Main result
Theorem 3. Suppose J : A −→ B is a V -functor between cartesian monoidal V -categories. Assume also that J is strong comonoidal. Suppose X is a monoidal V -category such that each of the V -functors − ⊗ X and X ⊗ − preserves colimits. Assume the V -functor F : A −→ X is strong monoidal. If the pointwise left Kan extension K : B −→ X of F along J exists then K too is strong monoidal. Proof. Using that tensor in X preserves colimits in each variable, the Fubini The- orem 1, that F is strong monoidal, Theorem 2 with the cartesian property of A , and the cartesian property of B, we have the calculation: ∼ KB1 ⊗ KB2 = colimA1 (B(JA1,B1),FA1) ⊗ colimA2 (B(JA2,B2),FA2) ∼ = colimA1,A2 (B(JA1,B1) ⊗ B(JA2,B2),FA1 ⊗ FA2) ∼ = colimA1,A2 (B(JA1,B1) ⊗ B(JA2,B2),F (A1 ?A2)) ∼ = colimA(B(JA, B1) ⊗ B(JA, B2),FA) ∼ = colimA(B(JA, B1 ?B2),FA) ∼ = K(B1 ?B2) . For the unit part, for similar reasons, we have: N =∼ FN0 ∼ = colim0(I (0, 0),FN0) ∼ = colimA(I (EA, 0),FA) ∼ = colimA(I (EJA, 0),FA) =∼ KN.
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5 An advanced viewpoint
In terminology of [4], suppose H : M −→ N is a monoidal pseudofunctor between monoidal bicategories. The main point to stress here is that the constraints
ΦA,B : HA ⊗ HB −→ H(A ⊗ B) are pseudonatural in A and B. Then we see that H takes pseudomonoids (= monoidales) to pseudomonoids, lax morphisms of pseudomonoids to lax morphisms, oplax morphisms of pseudomonoids to oplax morphisms, and strong morphisms of pseudomonoids to strong morphisms. In particular, this applies to the monoidal pseudofunctor
V -Mod(−, I ): V -Modop −→ V -CAT which takes the V -category A to the V -functor V -category [A , V ]. Now pseu- domonoids in V -Modop are precisely promonoidal (= premonoidal) V -categories in the sense of Day [1, 2]. Therefore, for each promonoidal V -category A , we obtain a monoidal V -category V -Mod(A , I ) = [A , V ] which is none other than what is now called Day convolution since it is defined and analysed in [1, 2]. A lax morphism of pseudomonoids in V -Modop, as written in V -Mod, is a module K : B −→ A equipped with module morphisms
P K B / B ⊗ B B / A φ0 φ +3 K ks K⊗K J J ~ / ⊗ I A P A A satisfying appropriate conditions. In other words, we have
φA1,A2,B : colimB1,B2 (K(A1,B1) ⊗ K(A2,B2),P (B1,B2,B)) =⇒ colimA(K(A, B),P (A1,A2,A)) and
φ0B : JB =⇒ colimA(K(A, B),JA) .
We call such a K a promonoidal module. It is strong when φ and φ0 are invertible. We also have the V -functor
∃K :[A , X ] −→ [B, X ] defined by (∃K )B = colimA(K(A, B),FA) .
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By the general considerations on monoidal pseudofunctors, ∃K is a monoidal V -functor when X = V . However, the same calculations needed to show this explicitly show that it works for any monoidal V -category X for which each of the tensors X ⊗ − and − ⊗ X preserves colimits.
Theorem 4. If K : B −→ A is a promonoidal V -module then ∃K :[A , X ] −→ [B, X ] is a monoidal V -functor. If K is strong promonoidal then ∃K is strong monoidal.
Proof. Although the result should be expected from our earlier remarks, here is a direct calculation. ∼ (∃K F1 ∗ ∃K F2)B = colimB1,B2 (P (B1,B2,B), (∃K F1)B1 ⊗ (∃K F2)B2) ∼ = colimB1,B2 (P (B1,B2,B), colimA1 (K(A1,B1),F1A1) ⊗
colimA2 (K(A2,B2),F2A2)) ∼ = colimB1,B2,A1,A2 (K(A1,B1) ⊗ K(A2,B2) ⊗ P (B1,B2,B), F1A1 ⊗ F2A2)
=⇒ colimA,A1,A2 (K(A, B) ⊗ P (A1,A2,A),F1A1 ⊗ F2A2) ∼ = colimA(K(A, B), colimA1,A2 (P (A1,A2,A),F1A1 ⊗ F2A2)) ∼ = colimA(K(A, B), (F1 ∗ F2)A)) ∼ = ∃K (F1 ∗ F2)B.
The morphism on the fourth line of the calculation is induced by φA1,A2,B and so is invertible if K is strong promonoidal. We also have φ0B : JB =⇒ (∃K J)B.
For the corollaries now coming, assume as above that X is a monoidal V - category such that X ⊗ − and − ⊗ X preserve existing colimits. Also A and B are monoidal V -categories. The monoidal structure on [A op, X ] is convolution op with respect to the promonoidal structure A (A, A1 ?A2) on A ; similarly for [Bop, X ].
Corollary 5. If J : A −→ B is strong monoidal then so is
op op LanJop :[A , X ] −→ [B , X ] .
Proof. Apply Theorem 4 to the module K : Bop −→ A op defined by K(A, B) = B(B,JA). We see that K is strong promonoidal using Yoneda twice and strong monoidalness of J.
Corollary 6. If W : A −→ V is strong monoidal then so is
colim(W, −):[A op, X ] −→ X .
Proof. Take B = I in Theorem4.
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Corollary 7. Suppose A is cartesian monoidal. If F : A −→ X is strong monoidal then so is colim(−,F ):[A op, V ] −→ X .
Proof. Here is the calculation for binary tensoring: ∼ colim(W1 ⊗ W2,F ) = colimA((W1 ⊗ W2)∆A, F A) ∼ = colimA1,A2 (W1A1 ⊗ W2A2,F (A1 ?A2)) ∼ = colimA1,A2 (W1A1 ⊗ W2A2,FA1 ⊗ FA2)) ∼ = colimA1 (W1A1,FA1) ⊗ colimA2 (W2A2,FA2) ∼ = colim(W1,F ) ⊗ colim(W2,F ) .
The unit preservation is easier.
Corollary 8. Suppose A and B are cartesian monoidal and J : A −→ B is strong comonoidal. If F : A −→ X is strong monoidal then so is
LanJ F : B −→ X .
op Proof. Notice that LanJ F is the composite of B(J, 1) : B −→ [A , V ] and colim(−,F ):[A op, V ] −→ X . The first is strong monoidal by hypothesis on J. The second is strong monoidal by Corollary 7.
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References
[1] Brian J. Day, Construction of Biclosed Categories (PhD Thesis, UNSW, 1970)
[2] Brian J. Day, On closed categories of functors, Lecture Notes in Mathematics 137 (Springer-Verlag, 1970) 1–38.
[3] Brian J. Day and Ross Street, Kan extensions along promonoidal functors, Theory and Applications of Categories 1(4) (1995) 72–77.
[4] Brian J. Day and Ross Street, Monoidal bicategories and Hopf algebroids, Advances in Math. 129 (1997) 99–157.
[5] Brian J. Day, Paddy McCrudden and Ross Street, Dualizations and an- tipodes, Applied Categorical Structures 11 (2003) 229–260.
[6] Samuel Eilenberg and G. Max Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra (La Jolla, 1965), (Springer-Verlag,1966) 421–562.
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[7] G. Max Kelly and Stephen Lack, Finite-product-preserving functors, Kan ex- tensions, and strongly-finitary monads, Applied Categorical Structures 1(1) (1993) 84–94.
[8] F. William Lawvere, Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic The- ories, (Ph.D. thesis, Columbia University, 1963); Reports of the Midwest Category Seminar II (1968) 41–61; Reprints in Theory and Applications of Categories 5 (2004) 1–121.
[9] G. Max Kelly, Basic concepts of enriched category theory, London Mathemat- ical Society Lecture Note Series 64 (Cambridge University Press, Cambridge, 1982).
[10] Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (Springer-Verlag, 1971).
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NOWHERE-ZERO FLOWS IN REGULAR MATROIDS AND HADWIGER’S CONJECTURE
LUIS A. GODDYN AND WINFRIED HOCHSTATTLER¨
To the memory of Reinhard B¨orger
Abstract. We present a tool that shows, that the existence of a k-nowhere-zero-flow is compatible with 1-,2- and 3-sums in regular matroids. As application we present a conjecture for regular matroids that is equivalent to Hadwiger’s conjecture for graphs and Tuttes’s 4- and 5-flow conjectures.
Keywords: nowhere zero flow, regular matroid, chromatic number, flow number, total unimodularity
1. Introduction A (real) matrix is totally unimodular (TUM) if each subdeterminant belongs to {0, ±1}. Totally unimodular matrices enjoy several nice properties which give them a fundamental role in combinatorial optimization and matroid theory. In this note we prove that the TUM possesses an attractive property. Let S ⊆ R, and let A be a real matrix. A column vector f is a S-flow of A if Af = 0 and every entry of f is a member of ±S. For any additive abelian group Γ use the notation Γ∗ = Γ \{0}. For a TUM A and a column vector f with entries in Γ, the product Af is a well defined column vector with entries in Γ, by interpreting (−1)γ to be the additive inverse of γ. It is convenient to use the language of matroids. A regular oriented matroid M is an oriented matroid that is representable M = M[A] by a TUM matrix A. Here the elements E(M) of M label the columns of A. Each (signed) cocircuit D = (D+,D−) of M corresponds to a {0, ±1}-valued vector in the row space of A and having minimal support. The +1-entries in this vector constitute the sets D+. It is known [19, Prop. 1.2.5] that two TUMs represent the same oriented matroid if and only if the first TUM can be converted to the second TUM by a succession of the following operations: multiplying a row by −1, adding one row to another, deleting a row of zeros, and permuting columns (with their labels). P For S ⊆ E(M) we use the notation f(S) = e∈S f(e). Let M = M[A] be the regular oriented matroid represented by the TUM A. Let S ⊆ Γ where Γ is an abelian group. An S-flow of M is a function f : E(M) → S for which Af = 0, where f is interpreted to be a vector indexed by the column labels of A. For any S ⊆ Γ we say that a regular matroid M has an S-flow if any of the TUMs that represent M has an S-flow. By the previous paragraph, this property of M is well defined. Since the rows of a TUM A generate the cocycle space of M = M[A], we have that a function f : E(M) → Γ is a flow if and only if for every signed cocircuit D = (D+,D−) we have that f(D) = 0 where f(D) is defined to equal f(D+) − f(D−). Let Γ be a finite abelian group. Let M be a regular oriented matroid, and let F ⊆ E(M) and let f : F → Γ. ∗ Let τΓ(M, f) denote the number of Γ -flows of M which are extensions of f. Theorem 1. Let M be an regular oriented matroid. Let F ⊆ E(M) and let f, f 0 : F → Γ. Suppose that for every minor N of M satisfying E(N) = F , we have that f is a Γ-flow of N if and only f 0 is a Γ-flow of N. 0 Then τΓ(M, f) = τΓ(M, f ). Proof. We proceed by induction on d = |E\F |. If d = 0, then there is nothing to prove. Otherwise let e ∈ E\F . 0 If e is a coloop of M, then τΓ(M, f) = τΓ(M, f ) = 0. If e is a loop of M, then by applying induction to 0 M \ e, we have τΓ(M, f) = τΓ(M, f ) = (|Γ| − 1)τΓ(M\e, f). Otherwise we apply Tutte’s deletion/contraction formula [3] and induction to get 0 0 0 τΓ(M, f ) = τΓ(M/e, f ) − τΓ(M\e, f ) = τΓ(M/e, f) − τΓ(M\e, f) = τΓ(M, f).
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2 LUIS A. GODDYN AND WINFRIED HOCHSTATTLER¨
Corollary 2. Let D be a positively oriented cocircuit of a regular oriented matroid M. Let f, f 0 : D → Γ. 0 0 Suppose that for every S ⊆ D we have that f(S) = 0 if and only if f (S) = 0. Then τΓ(M, f) = τΓ(M, f ). S Proof. Let N be a minor of M satisfying E(N) = D. Then E(N) is a disjoint union i Di of positively oriented cocircuits of N [9, Prop. 9.3.1]. Thus f is a Γ∗-flow of N if and only if f has no zeros, and f(Di) = 0 for each i. The result follows from Theorem 1. Corollary 3. Let M be a regular oriented matroid which has a Γ∗-flow f. (1) Let e ∈ E(M) and γ ∈ Γ∗. Then M has a Γ∗-flow f 0 with f 0(e) = γ. (2) Let D be a signed cocircuit of M of cardinality three. Let f 0 : D → Γ∗ satisfy f 0(D) = 0. Then f 0 extends to a Γ∗-flow of M. 0 ∗ Proof. (1) In any minor N with E(N) = {e}, both f and f {e} are Γ -flows of N if and only if N is a 0 loop. Thus by Theorem 1 τΓ(M, f ) = τΓ(M, f) > 0. (2) Let S ⊂ D. For any e ∈ D we have f 0(D \{e}) = f 0(D) − f 0(e) = −f 0(e) 6= 0. Therefore f 0(S) = 0 if and only if S = D. Since f is a Γ-flow and D is a positively oriented cocircuit of D we have f(D) = 0. Since f(e) 6= 0 for e ∈ D we again have that f(S) = 0 if and only if S = D. It follows from Theorem 0 1 that τΓ(M, f ) = τΓ(M, f) > 0. A k-nowhere zero flow (k-NZF) of a regular oriented matroid M is an S-flow of M for S = {1, 2, . . . , k−1} ⊂ R. We frequently use the following observation of Tutte [15]. Proposition 4. Let Γ be an abelian group of order k, and let S = {1, 2, . . . , k − 1} ⊂ R. Then M has a k-NZF if and only if M has a Γ∗-flow. In particular, the existence of a Γ∗-flow in M depends only on |Γ|. A key step in the proof of Proposition 4 is the conversion of a Γ∗-flow into a k-NZF, where Γ is the group of integers modulo k. By modifying this argument, one can show that the statement of Corollary 3 remains true if each occurrence of the symbol Γ∗ is replaced by the set of integers S = {±1, ±2,..., ±(k − 1)}. We omit the proof of this fact, as it is not needed in this paper.
2. Seymour decomposition We provide here a description of Seymour’s decomposition theorem for regular oriented matroids. We refer the reader to [13] for further details. We first describe three basic types of regular oriented matroids. A oriented matroid is graphic if it can be represented by the {0, ±1}-valued vertex-edge incidence matrix of a directed graph, where loops and multiple edges are allowed. Any {0, ±1}-valued matrix which whose rows span the nullspace of a network matrix is called a dual network matrix. Dual network matrices are also TUM, and an oriented matroid is cographic is it is representable by a dual network matrix. The third class consists of all the all the orientations of one special regular matroid R10. Every orientation of R10 can be represented by the matrix [I|B] where B is obtained by negating a subset of the columns of the following matrix. + 0 0 + − − + 0 0 + (1) + − + 0 0 0 + − + 0 0 0 + − + Here “+” and “−” respectively denote +1 and −1. Let M1, M2 be regular oriented matroids. If E(M1) and E(M2) are disjoint, then the 1-sum M1 ⊕1 M2 is just the direct sum of M1 and M2. The signed cocircuits of M1 ⊕1 M2 are the signed subsets of E(M1)∪E(M2) which are signed cocircuits of either M1 or M2. If M1 ∩ M2 = {e} and e is neither a loop nor a coloop in each Mi, then the 2-sum M1 ⊕2 M2 has element set E(M1)∆E(M2), where “∆” is the symmetric difference operator. A signed cocircuit is a signed subset of E(M1 ⊕2 M2) that is either a signed cocircuit of M1 or M2, or is a signed set of the form + + − − (2) D = (D1 ∆D2 ,D1 ∆D2 ) - 99 -
NOWHERE-ZERO FLOWS IN REGULAR MATROIDS AND HADWIGER’S CONJECTURE 3
+ − + + − − where each (Di ,Di ) is a signed cocircuit of Mi, and e ∈ (D1 ∩ D2 ) ∪ (D1 ∩ D2 ). If M1 ∩ M2 = B and + − B = (B ,B ) is a signed cocircuit of cardinality 3 in each Mi, then the 3-sum M1 ⊕3 M2 has element set E(M1)∆E(M2). A signed cocircuit is a signed subset of E(M1 ⊕3 M2) that is either a signed cocircuit of M1 + − or M2, or a signed subset of the form (2) where each (Di ,Di ) is a signed cocircuit of Mi, with D1 ∩ D2 = ∅ and (B+,B−) equals one of the following ordered pairs: + + + + − − − − ((D1 ∩ B ) ∪ (D2 ∩ B ) , (D1 ∩ B ) ∪ (D2 ∩ B )) − + − + + − + − ((D1 ∩ B ) ∪ (D2 ∩ B ) , (D1 ∩ B ) ∪ (D2 ∩ B )). The oriented version of Seymour’s decomposition theorem [13] and can be derived from [5, Theorem 6.6]. Theorem 5. Every regular oriented matroid M can be constructed by means of repeated application of k-sums, k = 1, 2, 3, starting with oriented matroids, each of which is isomorphic to a minor of M and each of which is either graphic, cographic, or an orientation of R10. We note that Schriver [12] states an equivalent version of Theorem 5 in terms of TUMs, that requires a second representation of R10 in (1) due to his implicit selection of a basis. Here is the main tool of this paper, which we employ in two subsequent applications. Theorem 6. Let k ≥ 2 be an integer and let M be a set of regular oriented matroids that is closed under minors. If every graphic and cographic member of M has a k-NZF, then every matroid in M has a k-NZF.
Proof. Let M ∈ M. We proceed by induction on |E(M)|. If M is an orientation of R10, then M has a 2-NZF since R10 is a disjoint union of circuits, and each circuit is the support of a {0, ±1}-flow in M. If M is graphic or cographic, then we are done by assumption. Otherwise, by Theorem 5, M has two proper minors M1, M2 ∈ M. such that M = M1 ⊕i M2, for some i = 1, 2, 3. By induction, each Mi has a k-NZF. Thus by Proposition 4, both minors have a Γ∗-flow where Γ is any fixed group of order k. By Corollary 3, we may ∗ ∗ assume that these Γ -flows coincide on M1 ∩ M2. Hence the union of these functions is a well defined Γ -flow on M and we are done by another application of Proposition 4.
3. Tutte’s flow Conjectures and Hadwiger’s Conjecture In this section we will present a conjecture that unifies two of Tutte’s Flow Conjectures and Hadwiger’s Conjecture on graph colorings.
Conjecture 7 (H(k)[4]). If a simple graph is not k-colorable, then it must have a Kk+1-minor. While H(1) and H(2) are trivial, Hadwiger proved his conjecture for k = 3 and pointed out that Klaus Wagner proved that H(4) is equivalent to the Four Color Theorem [18, 2, 10]. Robertson, Seymour and Thomas [11] reduced H(5) to the Four Color Theorem. The conjecture remains open for k ≥ 6. Tutte [15] pointed out that the Four Color Theorem is equivalent to the statement that every planar graph admits an 4-NZ-flow. Generalizing this to arbitrary graphs he conjectured that
Conjecture 8 (Tutte’s Flow Conjecture [15]). There is a finite number k ∈ N such that every bridgeless graph admits a k-NZ-flow. and moreover that Conjecture 9 (Tutte’s Five Flow Conjecture [15]). Every bridgeless graph admits a 5-NZ-flow. Note that the latter is best possible as the Petersen graph does not admit a 4-NZ-flow. Conjecture 8 has been proven independently by Kilpatrick [7] and Jaeger [6] with k = 8 and improved to k = 6 by Seymour [14]. Conjecture 9 has a sibling which is a more direct generalization of the Four Color Theorem. Conjecture 10 (Tutte’s Four Flow Conjecture [16, 17]). Every graph without a Petersen-minor admits a 4-NZ-flow. In [16, 17] Tutte cited Hadwiger’s conjecture as a motivating theme and pointed out that while “Hadwiger’s conjecture asserts that the only irreducible chain-group which is graphic is the coboundary group of the complete 5-graph” - 100 -
4 LUIS A. GODDYN AND WINFRIED HOCHSTATTLER¨ Conjecture 10 means that “the only irreducible chain-group which is cographic is the cycle group of the Petersen graph.” The first statement refers to the case where the rows of a totally unimodular matrix A consist of a basis of signed characteristic vectors of cycles of a digraph. Combining these we derive the following formulation in terms of regular matroids. First let us call any integer combination of the rows of A a coflow. Clearly, by duality resp. orthogonality, flows and coflows yield the same concept in regular matroids. Note that the existence of a k-NZ-coflow in a graph is equivalent to k-colorability [16]. Conjecture 11 (Tutte’s Four Flow Conjecture, matroid version). A regular matroid that does not admit a 4-NZ-flow has either a minor isomorphic to the cographic matroid of the K5 or a minor isomorphic to the graphic matroid of the Petersen graph. Equivalently, we have Conjecture 12 (Hadwigers’s Conjecture for regular matroids and k = 4). A regular matroid that is not 4-colorable, i.e. that does not admit a NZ-4-coflow, has a K5 or a Petersen-dual as a minor. Some progress concerning this Conjecture was made by Lai, Li and Poon using the Four Color Theorem
Theorem 13 ([8]). A regular matroid that is not 4-colorable has a K5 or a K5-dual as a minor. Tutte’s Five Flow Conjecture now suggests the following matroid version of Hadwiger’s conjecture: Conjecture 14 (Hadwigers’s Conjecture for regular matroids and k ≥ 5). If a regular matroid is not k- colorable for k ≥ 5, then it must have a Kk+1-minor. Theorem 15. (1) Conjecture 11 is equivalent to Conjecture 10. (2) Conjecture 14 for k = 5 is equivalent to Conjecture 9. (3) Conjecture 14 for k ≥ 6 is equivalent to Conjecture 7.
∗ Proof. (1) By Weiske’s Theorem [4] a graphic matroid has no K5 -minor. Hence Conjecture 11 clearly implies Conjecture 10. The other implication is proven by induction on |E(M)|. Consider a regular matroid M, that is not 4-colorable, i.e. that does not admit a NZ-4-coflow. Clearly, M cannot be isomorphic to R10. If M is graphic, it must have a K5-minor by the Four Color Theorem [2, 10] and an observation of Klaus Wagner [18]. If M is cographic it must have a Petersen-dual-minor by Conjecture 10. Otherwise, by Theorem 5, M has two proper minors M1, M2 ∈ M. such that M = M1 ⊕i M2, for some i = 1, 2, 3 and at least one of them is not 4-colorable by Theorem 6. Using induction we find either a Petersen-dual-minor or a K5-minor in one of the Mi and hence also in M. Thus, Conjecture 10 implies Conjecture 11. (2) We proceed as in the first case using H(5) for graphs [11] instead of the Four Color Theorem. (3) We proceed similar to the first case, with only a slight difference in the base case. If M is graphic, it must have a Kk+1-minor by Conjecture 7. M cannot be cographic by Seymour’s 6-flow-theorem [14]. Remark 16. James Oxley pointed that Theorem 15 could also be proven using splitting formulas for the Tutte polynomial (see e.g. [1]), Seymour’s decomposition and the fact that the flow number as well as the chromatic number are determined by the smallest non-negative integer non-zero of certain evaluations of the Tutte polynomial.
References 1. Artur Andrzejak, Splitting formulas for tutte polynomials, Journal of Combinatorial Theory, Series B 70 (1997), no. 2, 346 – 366. 2. Kenneth I. Appel and Wolfgang Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976), no. 5, 711–712. 3. D.K Arrowsmith and F Jaeger, On the enumeration of chains in regular chain-groups, Journal of Combinatorial Theory, Series B 32 (1982), no. 1, 75–89. 4. Hugo Hadwiger, Uber¨ eine Klassifikation der Streckenkomplexe, Vierteljahresschrift der Naturforschenden Gesellschaft in Z¨urich 88 (1943), 133–142. - 101 -
NOWHERE-ZERO FLOWS IN REGULAR MATROIDS AND HADWIGER’S CONJECTURE 5
5. Winfried Hochst¨attlerand Robert Nickel, The flow lattice of oriented matroids, Contributions to Discrete Mathematics 2 (2007), no. 1, 68–86. 6. F. Jaeger, Flows and generalized coloring theorems in graphs, Journal of Combinatorial Theory, Series B 26 (1979), no. 2, 205 – 216. 7. Peter Allan Kilpatrick, Tutte’s first colour-cycle conjecture., Master’s thesis, University of Cape Town, 1975. 8. Hong-Jian Lai, Xiangwen Li, and Hoifung Poon, Nowhere zero 4-flow in regular matroids, J. Graph Theory 49 (2005), no. 3, 196–204. 9. James G. Oxley, Matroid theory, The Clarendon Press Oxford University Press, New York, 1992. 10. Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas, The Four-Colour theorem, Journal of Combinatorial Theory, Series B 70 (1997), no. 1, 2–44. 11. Neil Robertson, Paul Seymour, and Robin Thomas, Hadwiger’s conjecture for k6-free graphs, Combinatorica 13 (1993), 279–361. 12. Alexander Schrijver, Theory of linear and integer programming, Wiley, June 1998. 13. P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), no. 3, 305–359. MR 579077 (82j:05046) 14. , Nowhere-zero 6-flows, J. Combin. Theory Ser. B 30 (1981), no. 2, 130–135. MR MR615308 (82j:05079) 15. W.T. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math. 6 (1954), 80–91. 16. , On the algebraic theory of graph colorings, Journal of Combinatorial Theory 1 (1966), no. 1, 15 – 50. 17. , A geometrical version of the four color problem, Combinatorial Math. and Its Applications (R. C. Bose and T. A. Dowling, eds.), Chapel Hill, NC: University of North Carolina Press, 1967. 18. K. Wagner, Uber¨ eine Eigenschaft der ebenen Komplexe, Mathematische Annalen 114 (1937), no. 1, 570–590. 19. Neil White, Combinatorial geometries, Cambridge University Press, September 1987. - 102 - - 103 -
Towards a flow theory for the dichromatic number
Winfried Hochst¨attler FernUniversit¨atin Hagen, Fakult¨atf¨urMathematik und Informatik Universit¨atsstr.1, 58084 Hagen, Germany [email protected]
To the memory of Reinhard B¨orger
Abstract We transfer the ideas of analyzing the chromatic number of a graph using nowhere-zero-colorings and -flows to digraphs and the dichromatic number.
1 Introduction
In [4] Victor Neumann-Lara introduced the dichromatic number ~χ(D) of a di- graph D = (V,A) as the smallest integer k such that the vertices of D can be colored with k colors such that each color class induces a directed acyclic graph. We give a characterization of the dichromatic number in terms of coflows of the digraph and develop a flow theory dual to this.
2 Notation
2 Let D = (V,A) be a directed graph. A mapping f = (f1, f2): A → Z is called a Neumann-Lara-flow or NL-flow for short, if both components of f satisfy Kirchhoff’s law of flow conversation X X ∀v ∈ V : fi(a) = fi(a) a∈δ−(v) a∈δ+(v) and furthermore f1(a) = 0 ⇒ f2(a) > 0. An NL-flow is an NL-k flow, if
∀a ∈ A : |f1(a)| < k. ∗ ∗ ∗ 2 A mapping f = (f1 , f2 ): A → Z is called an NL-coflow for short, if for each cycle C of the underlying undirected graph
X ∗ X ∗ fi (a) = fi (a) (1) a∈C+ a∈C−
1 - 104 -
where C+ and C− denote the arcs of C that are traversed in forward resp. backward direction and furthermore
∗ ∗ f1 (a) = 0 ⇒ f2 (a) > 0.
An NL-coflow is an NL-k-coflow, if
∗ ∀a ∈ A : |f1 (a)| < k.
Theorem 1. Let D = (V,A) be a loopless connected directed graph. Then D has an NL-k-coflow if and only if it has dichromatic number at most k.
Proof. Let f ∗ : A → Z2 be an NL-k-coflow. We define a coloring of c as follows. Choose an arbitrary vertex v ∈ V which receives color zero c(v) = 0. Now let w be another vertex and P1 be a (not necessarily directed) v-w-path in D. Then we define the color of w as
X ∗ X ∗ c˜(w) = f1 (a) − f1 (a) + − a∈P1 a∈P1
+ − where P1 and P1 denote the arcs of P1 that are traversed in positive resp. negative direction and claim that this value is independent of the chosen path. Namely, if P2 is another such path, then the concatenation of P1 and P2 tra- versed backwards is a closed tour and hence can be decomposed into circuits of ∗ ∗ D. Since f1 is a coflow in D, f1 sums to zero on all of these circuits. Hence
X ∗ X ∗ X ∗ X ∗ f1 (a) − f1 (a) − f1 (a) + f1 (a) = 0 + − + − a∈P1 a∈P1 a∈P2 a∈P2 andc ˜ is well defined. Now to get c in the proper range we set c(h) =c ˜(h) mod k. We are left to verify that the color classes of this coloring induce acyclic ∗ subdigraphs. Assume we had a directed cycle in one color class. Then f1 ≡ 0 ∗ on this cycle C and thus f2 > 0 on C, hence
X ∗ X ∗ X ∗ f2 (a) > 0 = f2 (a) = f2 (a)
a∈C+ a∈∅ a∈C− contradicting the definition of a coflow. On the other hand if we have a coloring c with colors {0, . . . , k−1} such that each color class induces an acyclic directed graph, we define an NL-k-coflow as ∗ ∗ follows. If a = (v, w) ∈ A is an arc of D we put f1 (a) = c(w) − c(v). Since f1 this way is defined by a potential it vanishes on every cycle and hence satisfies ∗ (1). Let A1 denote the set of arcs which receive an non-zero f1
∗ A1 := {a ∈ A | f1 (a) 6= 0}.
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Since each color class induces an acyclic directed graph, already D \ A1 must be acyclic. Hence using topological sort we find an ordering V = {v1, . . . , vn} of its vertices such that forall a = (vi, vj) ∈ A \ A1 we have i < j. Hence putting
∗ X ~ f2 = 1∂({v1,...,vi}), i=1,...,n−1
where 1∂({v1,...,vi}) denotes the directed characteristic function of the cut defined by {v1, . . . , vi}, we find a function that vanishes on all cycles and is strictly positive on A \ A1.
3 Planar Digraphs
In [3] Neumann-Lara conjectured that the dichromatic number of an orientation of a planar simple graph is bounded by 2. Clearly, an NL-k-flow in a bridgeless planar digraph D is an NL-k-coflow in its dual D∗ and vice versa. Hence, two- colorability of every orientation of a planar digraph is equivalent to the existence of an NZ-2-flow in every planar digraph whose underlying graph is three edge connected. The support of f1 of an NL-2-flow must be an even subgraph E, i.e. an edge disjoint union of not necessarily directed cycles. Contracting E f2 becomes a strictly positive integer vector in the cycle space of D/E, which can be decomposed into a sum of not necessarily disjoint directed cycles. On the other hand, if contracting an even subgraph we have a strictly positive flow, this yields a flow in the original graph, which is strictly positive outside of the even subgraph. On the even subgraph we find a flow using only ±1. Hence Neumann-Lara’s conjecture is equivalent to Conjecture 1. Let D = (G, A) be a three edge connected planar digraph. There exists an even subgraph E ⊆ A such that D/E is strongly connected. Observation 2. Every orientation of the Petersen graph admits an NL-2-flow. Proof. It suffices to show that there always exist two vertex disjoint 5-cycles, the complement of a perfect matching, such that the matching edges are not all oriented the same way with respect to the cycles. Starting with the pentagon and the pentagram we are done, if the complementary matching edges are oriented not all the same way. Therefore, and by symmetry, we may assume that all edges are directed from the pentagram to the pentagon. Now considering the red circuits in Figure 1 and the complementary edge to the uppermost vertex we are done, if not all matching edges are oriented towards the upper cycle, indicated in blue. Using the symmetry of the Petersen graph and rotating the configuration we find two cycles the contraction of which leaves a strongly connected graph. By Tutte’s 4-flow conjecture [5] the Petersen graph is the only cographical obstruction to 4-colorability. Since the Petersen graph is not an obstruction to the existence of an NL-2-flow we are tempted to conjecture
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Figure 1: Any orientation of the Petersen graph has an NL-2-flow
Conjecture 2. Let D = (G, A) be a three edge connected digraph. There exists an even subgraph E ⊆ A such that D/E is strongly connected.
4 Oriented Matroids
There is a natural way to generalize the above to oriented matroids the same way as Tutte’s coloring and flow theory for regular matroids was generalized to oriented matroids by Hochst¨attler,Neˇsetˇril and, later, Hochst¨attlerand Nickel ([1, 2]). Assume we are given an oriented matroid O on a finite set E represented by its covectors. By D we denote its set of cocircuits and for D ∈ D by ~χ(D) its signed characteristic function. Recall that the chromatic number χ(O) of an oriented matroid is defined as the smallest k such that the lattice of coflows ( ) ∗ X F (O) := λD ~χD | λD ∈ Z . (2) D∈D contains a coflow f ∗ ∈ F ∗(O) such that
∀e ∈ E : 0 < |f(e)| < k.
An NL-coflow in an oriented matroid is a tuple (f ∗, f +) ∈ F ∗(O) × O such that ∗ + ∀e ∈ E : f (e) = 0 ⇒ fe = +. The dichromatic number dichr(O) then is defined as the smallest k, such that there exists an NL-coflow (f ∗, f +) such that
∀e ∈ E : |f ∗(e)| < k.
Note that replacing each element in an oriented matroid by a pair of an- tiparallel elements the dichromatic number of the constructed oriented matroid is the chromatic number of the original one. Hence, as in the graphic case the
4 - 107 -
dichromatic number is a proper generalization of the chromatic number. Clearly the chromatic number of the underlying reorientation class of an oriented ma- troid is always an upper bound for the dichromatic number. Furthermore, the dichromatic number is 1 if and only if the oriented matroid is acyclic, meaning that O contains the all +-vector. We observe that the dichromatic number of an oriented matroid which has a cospanning cocircuit is bounded by 2. Proposition 3. Let O be a uniform oriented matroid of rank r ≥ 1 on n elements, which has an independent hyperplane, i.e. a flat F in the underlying matroid such that rg(F ) = r − 1 = |F |. Then dichr(O) ≤ 2. Proof. D := E \ F is a cocircuit of size n − r + 1. In O we choose one of its two orientations. Since F is independent, D is cospanning. Hence, for each i ei ∈ E \ supp(D) for 0 ≤ i ≤ r − 1 we can choose a cocircuit D such that i i supp(D ) \ supp(D) = {ei} consists of a single new element and D (ei) = +. Now we set X = D ◦ D1 ◦ · · · ◦ Dr−1. Then X is a covector without zeroes which is positive outside of supp(D). Hence, setting f ∗ = ~χ(D) yields the NL-2-coflow (f ∗,X).
Corollary 4. If O is a uniform oriented matroid, the orientation of a paving matroid or the cographic matroid of a Hamiltonian graph, then dichr(O) ≤ 2.
References
[1] Winfried Hochst¨attlerand Jaroslav Neˇsetˇril, Antisymmetric flows in ma- troids, Eur. J. Comb. 27 (2006), no. 7, 1129–1134. [2] Winfried Hochst¨attlerand Robert Nickel, On the chromatic number of an oriented matroid, J. Comb. Theory, Ser. B 98 (2008), no. 4, 698–706. [3] V. Neumann-Lara, Vertex colourings in digraphs. Some problems. Tech- nical report, University of Waterloo, 1985. [4] V. Neumann-Lara, The dichromatic number of a digraph, J. Combin. Theory B 33 (1982) 265–270 [5] W.T. Tutte, On the algebraic theory of graph colorings, Journal of Com- binatorial Theory 1 (1966), no. 1, 15 – 50.
5 - 108 - - 109 -
EINE REMINISZENZ AN PROFESSOR REINHARD BORGER¨
Eugen Grycko1, Werner Kirsch2, Tobias M¨uhlenbruch3 1,2,3 Department of Mathematics and Computer Science University of Hagen Universit¨atsstrasse1 D-58084 Hagen, GERMANY
1. Einleitung
Die Klasse der Euklidischen R¨aumebeliebiger (endlicher) Dimension bietet Modelle f¨ur metrische R¨aume. Aus der Differentialgeometrie ist bekannt, dass sich jede differenzierbare Mannigfaltigkeit endlicher Dimension diffeo- morph in einen Euklidischen Raum einbetten l¨asst. In der Topologie lernt man, dass sich jeder endliche Hausdorff-Raum hom¨oomorphin einen Euk- lidischen Raum einbetten l¨asst. In einem der zahlreichen mathematischen Gespr¨ache mit Reinhard B¨orger (*19.08.1954, † 6.06.2014), an das wir uns gerne erinnern, ist die Frage aufge- treten, ob sich jeder endliche metrische Raum isometrisch in einen Euklidis- chen Raum einbetten l¨asst.Zu bemerken ist, dass die Isometrie-Eigenschaft einer Abbildung st¨arker ist als Hom¨oomorphie. Uberrascht¨ hat uns der Beitrag von Reinhard, der darauf hin einen metrischen Raum mit vier Elementen angegeben hat, von dem er zeigen konnte, dass der Raum sich nicht in einen Euklidischen Raum einbetten l¨aßt.Das Beispiel, das wir hier wiedergeben, gibt Anlass zu der schw¨acheren Vermutung, wonach sich jeder endliche metrische Raum isometrisch in einen normierten Raum einbetten l¨aßt.Diese Vermutung ist nach unserer Wahrnehmung offen. Rein- hard hat darauf hingewiesen, dass ein Beweis ihrer G¨ultigkeit die folgende allgemeinere Aussage implizieren w¨urde: Jeder metrische Raum l¨asstsich isometrisch in einen normierten Raum ein- betten. Als Argument f¨urdiese Folgerung wurde das Kompaktheitstheorem der Pr¨adikatenlogik genannt.
1 - 110 -
2. Der B¨orger-Raum
F¨ur j = 1, 2 sei (Xj.dj) ein metrischer Raum. Eine Abbildung T : X1 → X2 heißt Isometrie, falls die Identit¨at
(2.1) d2(T (x),T (y)) = d1(x, y) f¨uralle x, y ∈ X1 erf¨ulltist. Da jede Isometrie insbesondere injektiv und stetig ist, nennen wir eine Ab- bildung T , f¨urdie (2.1) erf¨ulltist, eine isometrische Einbettung von X1 in X2. Um den B¨orger-Raum einzuf¨uhren,betrachten wir die Menge X := {1, 2, 3, 4} mit vier Elementen. Wir versehen X mit einer Metrik d, die sich tabellarisch einf¨uhrenl¨asst:
k \ l 1 2 3 4 1 0 1 2 2 2 1 0 1 1 3 2 1 0 1 4 2 1 1 0
d(k, l) Die Eigenschaften einer Metrik lassen sich f¨ur d direkt verifizieren. 2.1 Satz Der eben eingef¨uhrtemetrische Raum (X, d) l¨asstsich in keinen Euklidischen Raum isometrisch einbetten. Beweis:
Wir nehmen das Gegenteil an. Dann gibt es vier Punkte x1, x2, x3, x4 im Eu- n klidischen Raum R versehen mit der Euklidischen Stadardmetrik dE derart, dass gilt:
(2.2) 2 = dE(x1, x3) = dE(x1, x2) + dE(x2, x3) = 1 + 1 und
(2.3) 2 = dE(x1, x4) = dE(x1, x2) + dE(x2, x4) = 1 + 1.
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(2.2) und (2.3) implizieren: 1 1 x = · (x + x ) = · (x + x ), 2 2 1 3 2 1 4 was wegen x3 6= x4 einen Widerspruch darstellt.
3. Eine isometrische Einbettung des B¨orger-Raums
Wir versehen den Vektorraum R2 mit der Maximum-Norm:
||(x1, x2)||∞ := max{|x1|, |x2|}.
Die Abbildung ι : X → R2 sei tabellarisch definiert:
k 1 2 3 4
ι(k) (0,0) (1,1) (1,2) (2,1)
F¨ur ι l¨asstsich die Identit¨at
||ι(k) − ι(l)||∞ = d(k, l) f¨ur k, l ∈ X direkt verifizieren, d.h. ι ist eine isometrische Einbettung des 2 B¨orger-Raumsin den normierten Raum (R , ||.||∞), was die Vermutung st¨utzt, dass sich n¨amlich jeder endliche metrische Raum in einen normierten Raum einbetten l¨asst.
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Figure 1: A handwritten note from Reinhard B¨orger
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A Note on the Digraph Parameters Lightness and Weight and their Duals Heaviness and Mass
Dedicated to the memory of Reinhard B¨orger
Stephan Dominique Andres Fakult¨atf¨urMathematik und Informatik, Fernuniversit¨atin Hagen Universit¨atsstr.1, 58084 Hagen, Germany [email protected]
Abstract
Let D be a digraph. The weight w(D) of D is the minimum of the sums of the in-degrees of the two end vertices of an arc over all arcs of D. The lightness L(D) of D is the minimum of the maxima of the in-degrees of the two end vertices of an arc over all arcs of D. Dually, we define two other digraph parameters: the mass m(D) of D is the maximum of the sums of the in-degrees of the two end vertices of an arc over all arcs of D, and the heaviness H(D) of D is the maximum of the minima of the in-degrees of the two end vertices of an arc over all arcs of D. In this article some fundamental relations between these four digraph parameters are studied. MSC 2010: 05C20, 05C07 Key words: digraph, heaviness, lightness, mass, weight, minimum in- degree, maximum in-degree
1 Introduction
The tight relation between the two digraph parameters lightness and weight has appeared in 2009 in the context of upper bounds for digraph coloring parameters in a game-theoretic setting [1]. While the weight of a graph is known since some time – and there have been several publications of Kotzig [5], Borodin [2, 3] and others on the weight of planar graphs – the lightness is a relatively new parameter, introduced in the last decade by Zhu [6] and later used by He et al. [4]. The main aim of this paper is to obtain dual parameters which have similar properties. - 114 -
Let D = (V,E) be a digraph. For us, a graph is always a symmetric digraph, i.e. a digraph in which for each arc (v, w) there is also an arc (w, v). In this way our parameters can be defined for digraphs and graphs at the same time. Let d+(v) be the in-degree of vertex v. δ+(D) is the minimum in-degree of D, + and ∆ (D) the maximum in-degree. For an arc e = (v, w), let ve = v be the initial vertex and we = w be the terminal vertex of e. The lightness L(D) of D is defined as + + L(D) = min(max{d (ve), d (we)}). e∈E The weight w(D) of D is defined as
+ + w(D) = min(d (ve) + d (we)). e∈E
Dually, we define the heaviness H(D) of D as
+ + H(D) = max(min{d (ve), d (we)}), e∈E and the mass m(D) of D as
+ + m(D) = max(d (ve) + d (we)). e∈E Whenever it is clear from the context that there is only one digraph under consideration, we will write L, w, H, m, δ+, and ∆+ instead of L(D), w(D), H(D), m(D), δ+(D), and ∆+(D). In the case of graphs we will sometimes write d, δ, and ∆ instead of d+, δ+, and ∆+.
The paper is structured as follows. In Section 2 we discuss the main results, the relation between heaviness and mass as an analog of the relation between lightness and weight. Some observations concerning the relation between all four parameters are given in Section 3.
2 Fundamental relations by residue parameters
Lightness and weight, resp., heaviness and mass, are connected by the following fundamental relations. 1 w ≤ L ≤ w − δ+ (1) 2 2H ≤ m ≤ H + ∆+ (2) Therefore it is convenient to consider the residue parameters
R1 = 2L − w, + R2 = w − δ − L,
S1 = m − 2H, + S2 = H + ∆ − m. - 115 -
Obviously, for regular digraphs (where each vertex has the same in-degree) we have R1 = R2 = S1 = S2 = 0. In [1] the author proved the following critereon to recognize the digraphs with R1 = R2 = 0.
Proposition 1 Let D = (V,E) be a digraph with E 6= ∅. Then the following statements are equivalent:
(i) R1(D) = R2(D) = 0 (ii) L(D) = δ+(D) (iii) w(D) = 2δ+(D) (iv) D contains an arc (v, w) with d+(v) = d+(w) = δ+(D)
Here, we prove the dual assertion.
Proposition 2 Let D = (V,E) be a digraph with E 6= ∅. Then the following statements are equivalent:
(i) S1(D) = S2(D) = 0 (ii) H(D) = ∆+(D) (iii) m(D) = 2∆+(D) (iv) D contains an arc (v, w) with d+(v) = d+(w) = ∆+(D)
Proof: The solution of the system (i) of linear equations is (ii) and (iii). So (i) implies (ii). However, since S1 ≥ 0 and S2 ≥ 0 by (2), one of the statements (ii) and (iii) is redundant. Indeed, if we assume that H = ∆+, then we have
+ 0 ≤ S1 = m − 2H = m − 2∆ , + + 0 ≤ S2 = H + ∆ − m = 2∆ − m, thus m = 2∆+, i.e. (ii) implies (iii). Assume (iii) holds. Then, since E 6= ∅, + + + + there must be an arc e with d (ve) + d (we) = 2∆ , which implies d (ve) = + + d (we) = ∆ , i.e. (iv). If (iv) is true, then by the definition of m and H (ii) and (iii) are true, thus (iv) implies (i). The situation of Proposition 2 is not typical for a digraph. There are even + graphs with arbitrarily large S1, S2, and ∆ . For example, let n, k1, k2 ≥ 1 be integers. Consider the following forest Fn,k1,k2 , consisting of four center vertices v1, v2, v3, v4, with edges between v1 and v2 resp. v3 and v4, and some leaf vertices, so that v1 and v2 have degree n + k2, v3 has degree n, and v4 has + degree n + 2k2 + k1. It is easy to see that ∆ = n + 2k2 + k1, H = n + k2, and - 116 -
Figure 1: The forest F2,1,2
+ m = 2n + 2k2 + k1. Therefore S1 = k1 and S2 = k2. The choice of n makes ∆ arbitrarily large. Figure 1 depicts F2,1,2.
In [1] the author proved that there are also trees with arbitrarily large R1 and R2. With a slight modification of the argument one can show that there are + also graphs with arbitrarily large R1, R2 and δ .
3 Interdual relations
It is obvious that w ≤ m (3)
Thus we obtain
Proposition 3 L ≤ H + ∆+ − δ+.
(1) (3) (2) + + + + Proof: L ≤ w − δ ≤ m − δ ≤ H + ∆ − δ . However, neither L ≤ H nor H ≤ L in general. Consider the complete bipartite graph Ki,i+j on the one hand, which has L = i + j and H = i. On the other hand, the graph which consists of a complete graph Kn and an independent extra edge has L = 1 but H = n − 1.
4 Undirected graphs
In this section we examine lightness, weight, heaviness and mass in the case of certain undirected graphs. It is obvious that
Proposition 4 Let G be an r-regular graph. Then L(G) = H(G) = r, w(G) = m(G) = 2r. - 117 -
Another class of graphs with high regularity is the class of complete multipartite graphs.
Proposition 5 Let k ≥ 4, n1 ≥ n2 ≥ ... ≥ nk and G = Kn1,...,nk . Denote Pk N = i=1 ni. Then
L(G) = N − n2,
w(G) = 2N − n1 − n2,
H(G) = N − nk−1,
m(G) = 2N − nk−1 − nk.
Proof: Let Ai be the ith partite set. A vertex v ∈ Ai has nj neighbours in the partite set Aj, for any j 6= i. Therefore
∀v ∈ Ai : d(v) = N − ni.
This implies that, for all v1 ∈ A1, . . . , vk ∈ Ak,
d(v1) ≤ d(v2) ≤ . . . d(vk).
A light edge has one end vertex in A1 and the other in A2, from which the formula for lightness and weight follows. A heavy edge has one end vertex in Ak and the other in Ak−1, from which the formula for heaviness and mass follows. Note that when choosing graphs from the class of complete multipartite graphs the residue parameters R1 and S1 can be chosen independently, however R2 and S2 are constant. Indeed, for N as in Proposition 5 we have
δ(Kn1,...,nk ) = N − n1, ∆(Kn1,...,nk ) = N − nk. (4) Therefore,
R1(Kn1,...,nk ) = n1 − n2,
R2(Kn1,...,nk ) = 0,
S1(Kn1,...,nk ) = nk−1 − nk,
S2(Kn1,...,nk ) = 0.
This and the results of Section 2 motivate us to ask the following.
4 Open Question 6 For which (r1, r2, s1, s2) ∈ Z does a graph G with
(R1(G),R2(G),S1(G),S2(G)) = (r1, r2, s1, s2) exist?
Even for trees, general results on the difference of the four parameters lightness, weight, heaviness and mass seem to be difficult. The following questions are open. - 118 -
4 Open Question 7 For which (r1, r2, s1, s2) ∈ Z does a tree T with
(R1(T ),R2(T ),S1(T ),S2(T )) = (r1, r2, s1, s2) exist?
Problem 8 Let r1, r2, s1, s2 ≥ 0. Characterize the class of trees T with
(R1(T ),R2(T ),S1(T ),S2(T )) = (r1, r2, s1, s2).
Considering orientations of graphs we might also ask for characterizations. The following problem is motivated by Problem 8.
Problem 9 Let r1, r2, s1, s2 ≥ 0. Characterize the class of orientions T~ of trees with (R1(T~),R2(T~),S1(T~),S2(T~)) = (r1, r2, s1, s2).
Proposition 5 suggests to consider to following problem.
Problem 10 Let r1, r2, s1, s2 ≥ 0. Characterize the class of orientions M~ of complete multipartite graphs with
(R1(M~ ),R2(M~ ),S1(M~ ),S2(M~ )) = (r1, r2, s1, s2).
References
[1] S. D. Andres, Lightness of digraphs in surfaces and directed game chromatic number, Discrete Math. 309 (2009), 3564–3579 [2] O. V. Borodin, Generalization of a theorem of Kotzig and a prescribed coloring of the edges of planar graphs, Math Notes 48 (1990), 1186–1190 [3] O. Borodin, Joint extension of two theorems of Kotzig on 3-polytopes, Combinatorica 13 (1993), 121–125 [4] W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang, and X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002), 307–317 [5] A. Kotzig, From the theory of Euler’s polyhedrons (Russian), Mat.-Fyz. Cas., Slovensk. Akad. Vied 13 (1963), 20–30 [6] X. Zhu, The game coloring number of planar graphs, J. Combin. Theory B 75 (1999), 245–258 - 119 -
Operations with Binders vis-`a-vis Operations with Equations
John Power1,2 and Daniel Schmitter3
Department of Computer Science University of Bath Bath BA2 7AY, United Kingdom
Abstract
We compare the category theoretic semantics for binding signatures by Power and Tanaka with the abstract approach to universal algebra by Hyland. It is striking to see that two different ideas turn out to be so similar. We especially note that both approaches rely heavily on considering a monoid in the monoidal structure induced by a 0-cell in the Kleisli bicategory generated by a pseudo-distributive law of pseudo-monads. We further explain the implications the discovery of those similarities have by considering constructions that were only used in either of the two bodies of work.
Keywords: pseudo-monad, pseudo-distributive law, Kleisli bicategory, substitution monoidal structure, binding signature, initial algebra semantics, algebraic theory.
1 Introduction
Early this century a group in Edinburgh, including John Power and Miki Tanaka, realized the importance of pseudo-distributivities to give a general category theoretic formulation of the substitution structure underlying the category theoretic study of variable binding. Their work [28] eventually not only gave a unified account of Cartesian binders as by Marcelo Fiore, Gordon Plotkin, and Daniele Turi [8] and linear binders as by Miki Tanaka [30], but also extended to other types of binders. Examples include binding structures such as those associated with the Logic of Bunched Implications. Around the same time a group in Cambridge, including Marcelo Fiore, Nicola Gambino, Martin Hyland, and Glynn Winskel, realized that Kleisli bicategories are
1 The author acknowledges EPSRC grant EP/K028243/1 “Coalgebraic Logic Programming for Type In- ference”. 2 Email: [email protected] 3 Email: [email protected] No new data was produced in regard to this paper. - 120 - J. Power, D. Schmitter a rich source of models and contexts in which to understand variants of algebraic theories. Their observations had their origin in Glynn Winskel’s use of presheaf categories and profunctors in the foundation of concurrency [4]. A relation between the methods used was noticed early and led to the joint pa- per [5]. However, work has progressed independently since then. More results have been published on binders [6,7,25,26,27,28,31], fewer on algebraic theories [9,10,11]. Unnoticed until now is the remarkable fact that although the first is motivated by operations with binders, while the second is motivated by operations with equations, formally they have employed almost identical techniques. The central connection between the two subjects is the treatment of substitution. It is induced in both approaches by using the composition structure of a Kleisli bicategory, which is in turn induced by a composite structure in both cases. In order to get such a structure one needs the notion of a pseudo-distributivity of pseudo- monads by Francisco Marmolejo in [21] and previously by Max Kelly in [13]. The nine coherence conditions in [21] were then reworked, reorganized, and extended by Miki Tanaka in [31] (with a condition proven redundant in [22]). The two approaches consider pseudo-distributivities, or liftings respectively, over one specific construction, which is the presheaf construction. Ideally one would like to consider the construction which sends a small category C to the functor category [C op, Set] as a (pseudo-)monad on Cat, the category of small categories. However this is not possible as [C op, Set] is generally not small but only locally small. The two strands of work differ in how this issue is dealt with, which also leads to slightly different definitions of Kleisli bicategories. Table 1 shows the main connections we elaborate on in this paper with Table 2 showing specific examples considered. For the tables we jumped ahead a little bit and used unified notation and not necessarily notation found in the references. This is in accordance with the findings discussed in this paper and is used in order to make the connections more direct. The connections displayed have several important consequences. From a purely technical point of view they allow us to transport techniques and results between the two strands of work. There is a full account of enrichment in the case of binders which is thus far not available in the algebraic case. But due to the discovery of the similarities of the two approaches this body of work becomes readily available for algebraic theories. But at least as important is the conceptual point of view. As the two strands of work have different motivations they have different structures that are natural to consider. For example, in the case of binders the structure associated with the Logic of Bunched Implications made it natural to consider a context that allows for a combination of Cartesian and linear binders, whereas such a structure was not considered for algebraic theories. On the other hand it is natural to consider non-symmetric linear contexts for algebraic theories, which is not one of the first examples to come to mind in the case of binders. In addition there are also things that fall somewhere in between the two points of view mentioned previously. Since one uses 2-monads to generate contexts it makes sense to see what implication pseudo-monad morphisms have on the associated theories, which was however only consider in the work on algebraic theories. Hence it suggests to study the effect of pseudo-monad morphisms in the work on binders as
2 - 121 - J. Power, D. Schmitter
Binding Signatures Abstract Approach to Universal Algebra Motivation Unified account of Abstract account of operations with binders operations with equations (algebraic theories) Presheaf Pseudo-monad on Cat Relative pseudo-monad on construction Cat ֒→ CAT Distributivity Lifting of the pseudo-monad Lifting of the pseudo-monad for presheaves to the for presheaves to the pseudo-algebras of the pseudo-algebras of the context monad context monad (also seen as an extension of the context 2-monad to a Kleisli bicategory) Generating 2-monad on Cat 2-monad on CAT contexts (restricting to a 2-monad on Cat) Substitution Induced by a Kleisli Induced by a Kleisli structure bicategory bicategory
Additional • Initial algebra se- • Kleisli objects (giv- Constructions mantics ing for example • Enrichment (ωCpo Lawvere theories to account for recur- and PROPs) sion) • Extensions of rela- tions between con- texts to theories Table 1 Comparison between the two approaches. well, which is possible due to the technical similarities. Also, noting that monoids and monads inside a bicategory are the same, gives rise to the question about what the importance of the Kleisli object is in the treatment of binding signatures. The organization of the paper mostly follows the horizontal outline given in the first table as follows: Section 2 introduces the motivation behind the two approaches and techniques used therein. In the next two sections we examine the different treatments of the presheaf construction and collect some results about pseudo- distributive laws. In Section 5 we give examples of contexts considered as shown in the second table. These are then used in the next section to give examples of theories and we thereby also consider the substitution structures involved. The final section highlights constructions that were only considered in either strand of work and the implications that they have when interpreted in the other, as well as further points of interest for future research. Some basic definitions used throughout this paper are given in an appendix at the end. 3 - 122 - J. Power, D. Schmitter
Context; Monad Binding Signatures Abstract Approach to for. . . Universal Algebra small categories Cartesian binders (including Algebraic theories (in the with finite λ-calculus) [8] sense of universal products algebra) [18] small symmetric Linear binders [30] (Colored) Operads [23] monoidal categories small monoidal Not considered Non-symmetric operads [16] categories small monoidal Logic of Bunched Not considered (but categories with Implications [24] would be natural) finite products
Table 2 Contexts and examples thereof. 2 Motivation
This section gives background information on both strands of work considered in the rest of this paper.
2.1 Binding Signatures The base of this work is in John Power and Miki Tanaka’s generalization of a categorical treatment of Cartesian and linear binders as found in [8] and [30]. The idea is to use a 2-monad S on Cat to generate contexts and then lift the presheaf construction to the category Ps−S−Alg of pseudo-S-algebras, compatible with the forgetful functor for pseudo-algebras. For intuition it is helpful to consider the motivation for the Cartesian case and untyped λ-calculus, as expressed in the leading example in [8]. Example 2.1 One starts with the category Fop, which is the opposite of the category of finite cardinals (or equivalently a skeleton of the category Setop). The coproduct structure of F gives rise to operations of exchange, weakening, and contraction. One then considers the presheaf category [F, Set], where the value of a presheaf X at n is interpreted as a set of terms modulo α-conversion containing at most n variables. One then constructs a monoidal structure on [F, Set] to model substitution and uses the finite product structure of [F, Set] to model pairing. Afterwards a binding signature is defined to consist of a set O of operations together with an arity function ar: O → N∗. In the case of untyped λ-calculus
t ::= x | λx.t | app(t, t) one has O = {λ, app} with ar(λ) = h1i and ar(app) = h0, 0i corresponding to the operations of λ-abstraction and application (λ-abstraction has one argument and binds one variable and application has two arguments and binds no variables). The 4 - 123 - J. Power, D. Schmitter substitution monoidal structure, the finite product structure, and the definition of a binding signature are then used to define and characterize initial algebra semantics, i.e. the initial presheaf with a monoid structure and an algebra structure for the binding signature subject to a coherence axiom relating the two. The presheaf involved in in this example is Λα : F → Set, defined by
Λα(n) := {[t]α | t ∈ ΛVar ∧ FV(t) ⊆ {x1, . . . , xn}}, i.e. the set of α-equivalence classes of λ-terms over {xi}i∈N+ with free variables in {x1, . . . , xn} for all n ∈ F. The initial algebra semantics is called the initial F-monoid in [8]. The thought behind linear binders is almost the same except for the use of P, the category of finite cardinals with permutations, instead of F. This corresponds to the fact that one is not allowed to copy or discard variables in this setting. These similarities lend themselves to the attempt of giving a more general account of the techniques at play here, as to avoid having to do similar proofs over and over again “just” because of a change of context.
2.2 Algebraic Theories Universal algebra studies theories, i.e. it treats specifications as mathematical ob- jects and not their models. In categorical universal algebra, its modern incarnation, those specifications are often encoded in categories (PROs [19], PROPs [17], Law- vere theories [18]), multi-categories or colored operads [3], or rather differently by monads [20]. All of those approaches have their advantages and disadvantages, e.g. regarding the possibilities of combining them or the space of their models. Further one needs different formulations for different types of arities (e.g. operads capture linear arities while symmetric operads capture linear symmetric arities). As was the case for binders, one often encounters similar constructions in dif- ferent treatments, due to the technique used to express theories or types. The approach proposed by Martin Hyland in [10] sets out to give a unified framework taking care of both of these issues by not only considering specifications as math- ematical objects, but also the types. For example Cartesian contexts are based on the 2-monad for small categories with finite products, whereas symmetric linear contexts are based on the 2-monad for small symmetric monoidal categories. The idea in this approach to algebraic theories is very similar to the one of binding signatures: One uses a 2-monad S on CAT, the category of locally small categories, which restricts to a monad on Cat, to generate contexts and then lifts the presheaf construction to the category Ps−S−Alg, compatible with the forgetful functor for pseudo-algebras. Algebraic theories are then regarded as monads in Kleisli bicategories. More specifically, an algebraic theory becomes a profunctor (see Section 3 for why this happens) M: C −7→ S C for a small category C , often written as a functor M : (S C )op × C → Set. Given c ∈ ob(C ) and c ∈ ob(S C ), interpreting M(c, c) as being the set of formal function symbols with input arity c and output arity c. This reflects the fact that S determines the input arities under consideration. The monoidal structure of the Kleisli bicategory is then used to handle composition, i.e. 5 - 124 - J. Power, D. Schmitter given a function symbol f ∈ M(c, c) and an S-indexed family of function symbols g ∈ S M(C, c) their composite is given in M(µ(C), c), where µ is the multiplication of the monad S. Further explanations can be found in Section 6. For intuition we consider the Cartesian case, i.e. setting S = Tfp (see Example 5.1 for details) and groups. Example 2.2 A specification of groups is given by the abstract clone with the sets Cn being the equivalence classes of terms in n variables, i.e. the free group on n variables. This assignment can be seen as a functor C: F → Set, showing an astonishing similarity to Example 2.1. Noting that Fop is equivalent to the free category with finite coproducts on 1 and writing Tfp for the 2-monad for small op categories with finite products, we can rewrite C as a functor C: (Tfp 1) ×1 → Set, where 1 denotes the category with one object and its identity morphism. This in turn corresponds to a profunctor C: 1 −7→ Tfp 1 and hence a 1-cell in the Kleisli bicategory Kl(Tcoc Tfp). Again as in the previous example, it comes equipped with a monoid structure in the monoidal category Kl(Tcoc Tfp)(1, 1) that corresponds to n the abstract clone composition (Cm) × Cn → Cm. One can also (try to) see this example in the framework for binders, where the binding signature is given by O = {e,−1 , ·} with ar(e) = hi, ar(−1) = h0i, and ar(·) = h0, 0i corresponding to the unit, the inverse, and the multiplication of the group (the unit has no arguments and binds no variables, the inverse has one argument and binds no variables, and multiplication has two arguments and binds no variables). However there is no account for the equations satisfied by groups, i.e. one ends up with algebraic theories consisting of a nullary, an unary, and a binary operation satisfying no equations.
3 Presheaf Construction
As mentioned in the introduction one cannot directly see the presheaf construction as a (pseudo-)monad on Cat due to size reasons. In the following we explain how this issue is dealt with in both cases and how they relate in order to simplify notation for future work. The size issue is, more or less, swept under the carpet in favor of a simpler presentation in [28]. This may be done as there exist techniques to deal with such issues, e.g. as in [14] by assuming the existence of a strongly inaccessible cardinal κ and considering small categories that are cocomplete for diagrams of size less than κ. This enables one to consider the presheaf construction as a pseudo-monad Tcoc on Cat. Additionally, this pseudo-monad has a similar characterization as the examples given in Section 5: For any small category C , a free cocompletion of C is given by the the functor category [C op, Set] as explained in [14]. The treatment of the presheaf construction in Martin Hyland’s paper [10] is more elaborate and we first need to define Kleisli structures in order to explain it and the relation to the previous approach. Kleisli structures are 2-dimensional versions of restricted monads as defined in [1], which generalize the notion of a monad by noting that the definition of a monad in terms of a Kleisli triple can be generalized from being defined on a category C to being defined on a functor J : J → C between 6 - 125 - J. Power, D. Schmitter two categories. In the following let B be any bicategory with a sub-bicategory J and inclusion J: J → B. Definition 3.1 (Kleisli structure) (To avoid confusion in the next definition we use “.” to denote composition.) A Kleisli structure P on J: J → B consists of • a mapping P: ob(J ) → ob(B), • for all A ∈ ob(J ) a morphism ηA ∈ B(A, P A), • for all (A, B) ∈ ob(J )2 a functor (−)# : B(A, P B) → B(P A, P B), • # for all A ∈ ob(J ) an invertible 2-cell κA :(ηA) → 1P A, • for all (A, B) ∈ ob(J )2 and k ∈ ob(B(A, P B)) an invertible 2-cell # ηk : k → k .ηA, and • for all (A, B, C) ∈ ob(J )3, k ∈ ob(B(A, P B)), and l ∈ ob(B(B, P C)) an # # # # invertible 2-cell κl,k :(l .k) → l .k such that • η is natural in k, • κ is natural in k and l, and • the 2-cells satisfy the usual unit and pentagon coherence conditions. With this definition at hand one can define a Kleisli bicategory along the same lines as one can define a Kleisli category from a Kleisli triple. Definition 3.2 (Kleisli bicategory from a Kleisli structure) Given a Kleisli structure P on J: J → B, we define its Kleisli bicategory Kl(P) as follows: • ob(Kl(P)) = ob(J ), • for all (A, B) ∈ ob(Kl(P))2 let Kl(P)(A, B) := B(A, P B), • for all A ∈ ob(Kl(P)) let the identity on Kl(P)(A, A) be ηA ∈ B(A, P A), • for all k ∈ ob(Kl(P)(A, B)) and l ∈ ob(Kl(P)(B,C)) let their composition be l ◦ k := l#.k, • for all k ∈ ob(Kl(P)(A, B)) let the left unit isomorphism λk : ηB ◦ k → k be the # κB .k composite (ηB) .k −→ 1P B.k =∼ k, • for all k ∈ ob(Kl(P)(A, B)) let the right unit isomorphism ρk : k ◦ ηA → k be −1 # ηk k .ηA −→ k, and • for all k ∈ ob(Kl(P)(A, B)), l ∈ ob(Kl(P)(B,C)), and m ∈ ob(Kl(P)(C,D)) let the associativity isomorphism αm,l,k :(m◦l)◦k → m◦(l◦k) be the composite κm,l.k (m#.l)#.k → (m#.l#).k =∼ m#.(l#.k). That the definitions for Kl(P) above satisfy the coherence axioms for a bicategory follows from the coherence conditions of the Kleisli structure P. The structure of interest to us is the structure of presheaves, which we are going to explain in the following. The presheaf Kleisli structure is given on the inclusion J : Cat → CAT by the composite P := U P,ˆ where P:ˆ Cat → COC is the functor that sends a small category C to the presheaf category [C op, Set], U : COC → CAT is the evi-
7 - 126 - J. Power, D. Schmitter dent forgetful functor, and COC denotes the 2-category of all locally small cocom- plete categories, cocontinuous functors between them, and natural transformations. The unit morphisms ηC : C → P C are given by the Yoneda embedding and for K: C → P D the lifting K# :P C → P D is given by a choice of left Kan extension of K along the Yoneda embedding. The 2-dimensional structure is induced by the adjoint equivalence (−)† CAT(C , U D) ⊥ COC(PˆC , D) (−)ηC † for small C and cocomplete D, where (−) is left Kan extension regarded as landing in COC. Details of this construction appear in [10]. The bicategory that one gets by this construction is a familiar one: Its objects are small categories and for small categories C and D we have
Kl(P)(C , D) = [C , P D] = [C , [Dop, Set]] =∼ [Dop × C , Set] = Prof(C , D) and one readily checks that the Kleisli composition corresponds to the composition in Prof, i.e. we get the category of profunctors. These definitions make obvious the connection of the presheaf construction with profunctors. However, since the definition of a Kleisli structure is only given on an inclusion of a sub-bicategory, one does not in general get a multiplication needed to define a monad as in the case of Kleisli triples. This makes this point of view less appealing. Hence it would be good to be able to treat all constructions as in the case of binding signatures. This is indeed possible, since the two points of view are essentially the same. This follows from a similar argument as in [15] by replacing “finitely presentable” by “size less than κ”. We therefore stick to the first interpretation of the presheaf construction for the rest of this paper and rewrite the statements in [10] in these terms. This also simplifies the definition of the Kleisli bicategory as seen in the next definition. Definition 3.3 (Kleisli bicategory from a pseudo-monad) Let B be any bi- category and T any pseudo-monad on B. The Kleisli bicategory of T, denoted by Kl(T), is given by • ob(Kl(T)) := ob(B), • for all (A, B) ∈ ob(B)2, Kl(T)(A, B) := B(A, T B), and • for all (A, B, C) ∈ ob(B)3 the evident composition
B(B, T C) × B(A, T B) → B(T B, T2 C) × B(A, T B) → B(A, T2 C) → B(A, T C)
determined by the action of T on the hom-categories, the composition in B and the multiplication of T with the rest of the bicategory structure determined by the pseudo-monad structure of T.
8 - 127 - J. Power, D. Schmitter
4 Pseudo-Distributivity
To generalize the substitution structure in [8] and [30] one needs a pseudo- distributivity of the context 2-monad S over the presheaf pseudo-monad Tcoc. As in the case of Beck’s Theorem for ordinary monads [2] one would like to get a cor- respondence between distributivities and liftings in order to prove the existence of distributive laws between certain pseudo-monads. This requires one to weaken the axioms in the definition of a distributive law as seen in the following definition. Definition 4.1 (Pseudo-distributivity of pseudo-monads) Let C be any 2- category and S = (S, µS, ηS, τ S, λS, ρS) and T = (T, µT, ηT, τ T, λT, ρT) any pseudo-monads on C .A pseudo-distributive law δ of S over T is a quintuple δ = (δ, µS, µT, ηS, ηT) consisting of • a pseudo-natural transformation δ :ST → TS and • invertible modifications S δ δ S δ T T δ S2 TSTSTS2 ST2 TST T2 S
µS T = µS T µS S µT = µT µT S ⇐ ⇐
ST TS, ST TS, δ δ T S T ηS ηT S ηS T and S ηT ⇒ ⇒ = ηS = ηT STTS, STTS δ δ subject to ten coherence axioms listed in [26]. In the case of algebraic theories it is suggested, for intuition, to also consider extensions of the context 2-monad S to a monad STcoc on Prof. The extension is along the canonical pseudo-functor Cat → Kl(Tcoc) =∼ Prof which takes a functor F : C → D in Cat to F∗, where F∗(d, c) := D(d, F c) for all c ∈ ob(C ) and d ∈ ob(D). As it turns out, there is also a correspondence between pseudo- distributivities and extensions as in the ordinary case. However, the correspondence statements need slight refinements and a lot of care for details. We do not get into these subtleties in this paper, see [5,27] for more details. In the following we state the main definitions and theorems needed in later sections. Definition 4.2 (Lifting) Given two pseudo-monads S and T on a 2-category C , a lifting of the pseudo-monad T to the 2-category Ps−S−Alg of pseudo-S-algebras is a pseudo-monad T on Ps−S−Alg such that US T = T US holds and similarly for the other data, where US denotes the forgetful 2-functor of pseudo-S-algebras. e e Definition 4.3 (Extension) Given two pseudo-monads S and T on a 2-category C , an extension of the pseudo-monad S to the Kleisli bicategory Kl(T) is a pseudo- monad ST on Kl(T) such that ST(−)∗ = (−)∗ S holds and similarly for the other data, where (−)∗ denotes the canonical inclusion C → Kl(T). 9 - 128 - J. Power, D. Schmitter
With these definitions at hand (and taking care of some subtleties concerning transformations between them) one can then prove the following theorems. Theorem 4.4 ([5,27]) Given two pseudo-monads S and T on a 2-category C , the following are equivalent: • a pseudo-distributive law of S over T, • a lifting of T to a pseudo-monad T on Ps−S−Alg, and • an extension of S to a pseudo-monad ST on Kl(T). e Theorem 4.5 ([5,27]) Given a pseudo-distributive law δ :ST → T S of pseudo- monads S and T on Cat, the following hold: • The pseudo-functor TS acquires the structure of a pseudo-monad with multi- plication given by δ µTµS TSTS T−→S TTSS −→ TS, • Ps−TS−Alg is canonically isomorphic to Ps−T−Alg, • Kl(ST) is biequivalent to Kl(T S), and e • the object TS 1 has both canonical pseudo-S-algebra and pseudo-T-algebra structures on it. Note that there is a priori nothing special about the category 1 in the last statement above, as this is true for any small category C . However we will later construct a monoidal structure on [C , TS C ] and letting C = 1 in that setting yields a monoidal structure on T S 1. The combination of the canonical pseudo-S-algebra and pseudo-T-algebra structures combined with this monoidal structure make it special.
5 Contexts
In the following we describe the context monads mentioned in Table 2, give their relations to constructions mentioned previously, and point out some facts about them concerning liftings of the presheaf pseudo-monad.
Example 5.1 Let Tfp be the 2-monad on Cat for small categories with finite prod- ucts, i.e. such that Tfp(C ) is the free category with finite products on C . The 2-category Ps−Tfp−Alg has small categories with finite products as objects, func- tors that preserve finite products up to coherent isomorphism as maps, and natural transformations as 2-cells, i.e. it is the 2-category of small categories with finite products. Taking C = 1, we see that Tfp(C ) is (up to equivalence) the category Setop and hence equivalent to Fop in the notation of [8].
Example 5.2 Let Tsm be the 2-monad on Cat for small symmetric monoidal cat- egories, i.e. such that Tsm(C ) is the free symmetric monoidal category on C . The 2-category Ps−Tsm−Alg has small symmetric monoidal categories as objects, strong symmetric monoidal functors as maps, and symmetric monoidal natural transforma- tions as 2-cells and hence is the 2-category of small symmetric monoidal categories. Taking C = 1, we get that Tfp(C ) is (up to equivalence) the category of finite sets and permutations, denoted by Pop in [30].
10 - 129 - J. Power, D. Schmitter
Example 5.3 Let Tm be the 2-monad on Cat for small monoidal categories, i.e. such that Tm(C ) is the free monoidal category on C . The 2-category Ps−Tm−Alg has small monoidal categories as objects, strong monoidal functors as maps, and monoidal natural transformations as 2-cells and hence is the 2-category of small monoidal categories. Example 5.4 Combining the first two examples by taking the sum of 2-monads we get the 2-monad TBI on Cat for small symmetric monoidal categories with finite products. The 2-category Ps−TBI−Alg has small symmetric monoidal categories with finite products as objects, strong symmetric monoidal functors that preserve finite products up to coherent isomorphism as maps, and symmetric monoidal nat- ural transformations as 2-cells. The objects of TBI(1) are the bunches of Bunched Implications in [29]. We note that [10] uses strict versions of the monads above, i.e. requiring strictly associative products and similarly for the others, however this has little influence as the categories obtained are biequivalent. Hence we decided to unify them to a common description. As shown in [26], Tcoc lifts to Ps−Tsm−Alg, and similar arguments work for the other examples mentioned above. Using the results from the previous section we therefore get, among others, pseudo-monads Tcoc Tfp and Tcoc Tsm. Evaluating them at C = 1, we recover (up to equivalence) the presheaf categories used for Cartesian and linear binders: Tcoc Tfp(1) = [F, Set] and Tcoc Tsm(1) = [P, Set]. It is worth noting that not all 2-monads allow a lift of Tcoc. Obvious examples include monads that give structure that presheaf categories do not possess, such as biproducts. But even for structures that presheaf categories possess it is possible that the structure is not preserved by the Yoneda embedding or by the left Kan extension, with coproducts being such a structure. At the moment there does not seem to exist a characterization of which 2-monads allow a lift of Tcoc and which ones do not.
6 Examples
In this section we give specific examples for some of the context monads considered in the previous section.
6.1 Binding Signatures In order to give an example we first need to consider some additional structures needed. As Kl(T) is a bicategory, its composition determines a monoidal structure on the category Kl(T)(A, A) = B(A, T A) for every object A of the underlying bicategory B. In our case the bicategory B is Cat and, by choosing A = 1, we get a monoidal structure on T 1. However, for the monoidal structure to agree with the structures used in [8] and [30] one has to use the dual of this monoidal structure, i.e. the monoidal structure induced by Kl(T)op. The choice of 1 above corresponds to untyped contexts. Letting A be any set K, we get contexts of type K. For ease of exposition, we stick to the untyped case for the remainder of this section. The corresponding typed statements can be found in [28]. 11 - 130 - J. Power, D. Schmitter
Definition 6.1 (Binding signature) A binding signature for a pseudo-monad S on Cat is a pair Σ = (O, ar) consisting of a set of operations O together with an arity function ar : O → ArS, where an element (k, α, (ni, βi)1≤i≤k) of ArS consists of a natural number k, an object α of the category S k, and a natural number ni and an object βi of the category S(ni + 1) for 1 ≤ i ≤ k.
The number k tells how many terms are involved, the ni’s tell how many variables th get bound in the i term and α and the βi’s determine which sorts of binders are used and how they are combined. Given any pseudo-monad S on Cat, a part (C , C) of any pseudo-S-algebra (C , C, Cµ, Cη), and any object α of the category S k for any small category k (in- k cluding natural numbers), the object α induces a functor αC : C → C by the composition S ×α ev C C k =∼ C k × 1 −→ (S C )k × S k −→α S C −→ C . Proposition 6.2 ([26]) Each binding signature Σ for a pseudo-monad S on Cat induces an endofunctor on [(S 1)op, Set] by
~ ~ ΣX := α[(S 1)op,Set](X(β1S 1(1, −)),.. ., X(βkS 1(1, −))), o∈O ar(o)=(k,α,a(ni,βi)1≤i≤k) where ~1 denotes a list 1,..., 1 of length determined by the context in which it is written. The main result that allowed a characterization of the presheaf of terms gener- ated by a signature as an initial algebra in [8] and [30] involved the description of a canonical strength. It is shown in [26] that for any binding signature Σ the induced functor, which is also called Σ, has a canonical strength
ΣX • Y → Σ(X • Y) with respect to the monoidal structure • for pointed Y. It is further proven in [26] that for any binding signature Σ, the free monad generated by Σ on the category Tcoc S 1, denoted by TΣ, exists and has a canonical strength over pointed objects with respect to •. This implies that the object TΣ 1 op of Tcoc S 1 = [(S 1) , Set] has a canonical monoid structure on it. Definition 6.3 (F-monoid) Let F be a strong (over pointed objects) endofunctor on a monoidal closed category (C , ·,I). An F-monoid is a quadruple (X, µ, ι, h) consisting of a monoid (X, µ, ι) in C and an F-algebra (X, h) such that
tX,X F µ F X · X F(X · X)F X
h · X h
X · X µ X commutes, where t denotes the strength of F. 12 - 131 - J. Power, D. Schmitter
F-monoids with maps given by maps in C that preserve both the F-algebra structure as well as the monoid structure form a category. The characterization of initial algebra semantics follows from: op Theorem 6.4 ([26]) For any binding signature Σ, the object TΣ 1 of [(S 1) , Set] together with its canonical Σ-algebra structure and monoid structure form the initial Σ-monoid. We have all necessary tools at hand now to give an explicit example. We again choose Cartesian binders and untyped λ-calculus as in Section 2 to show how that example works in the more general framework.
Example 6.5 Let S be Tfp and let 2 be defined to have objects x and y. The binding signature Σ for untyped λ-calculus is given as follows: The set of operations is given by O = {λ, app}. The arity for λ-abstraction is given by k = 1, α ∈ Tfp 1 is 1, n1 = 1, and β1 is the object x × y of Tfp 2. The arity for application is given by k = 2, α is the object x × y of Tfp 2, n1 = n2 = 0, and β1 and β2 are 1. Theorem 6.4 then states that the presheaf of terms generated by the binding signature Σ above together with its canonical monoid and algebra structures has an abstract universal characterization as the initial Σ-monoid as in [8].
Replacing Tfp by Tsm in the above example yields a similar result for linear λ-calculus as in [30]. For the 2-monad TBI and the signature for the Logic of Bunched Implications we refer the reader to [26].
6.2 Algebraic Theories As in the previous subsection, one uses the composition of the Kleisli bicategory to get a monoidal structure on the categories Kl(T S)(A, A) =∼ Kl(ST)(A, A) for any object A (assuming the existence of a pseudo-distributivity of S over T). Working in the category Cat, a monad in Kl(Tcoc S)(C , C ) consists of a profunc- tor M : C −7→ S C together with a unit 2-cell ηC ∗ ⇒ M and a composition 2- cell M ⊙ M ⇒ M satisfying the usual equations, where ⊙ denotes composition in Kl(Tcoc S). The idea being that the unit represents variables and that composition gives the interpretation of formal composites as mentioned in Section 2. For S = Tfp one gets algebraic theories in the sense of universal algebra. They op arise as profunctors M : ((Tfp 1) × 1) ≃ F → Set. Note that there was nothing special about the choice of groups in Example 2.2, i.e. given any algebraic theory one sets M(n) to be the set of all n-ary terms generated by the operations subject to the equations of the theory. Similarly, replacing 1 by any set K, one finds many-sorted theories (of type K). For S = Tsm one gets a correspondence between monads M : 1 −7→ Tsm 1 and symmetric operads. Replacing 1 by any set K, one finds colored operads. Similarly Tm gives a correspondence with non-symmetric operads.
7 Additional Constructions and Ideas
An important thing to note is that [28] talks about monoids in a bicategory, whereas [10] talks about monads in a bicategory. Nevertheless, the two notions are exactly 13 - 132 - J. Power, D. Schmitter the same. This has the consequence that it makes sense to talk about the Kleisli object of a monad in the second case, whereas such a thing does not exist for monoids and was therefore not considered for binding signatures. However, there seems to be a major importance to this construction as Kleisli objects turn out to catch notions such as Lawvere theories, PROPs, and PROs in the algebraic case (corresponding to the 2-monads Tfp,Tsm, and Tm respectively). It has to be examined what this construction yields for binding signatures. Further, [10] also considers the implications that pseudo-monad morphisms be- tween the context 2-monads have on the corresponding categories of theories. A pseudo-monad morphism induces a 2-adjunction between the corresponding cate- gories of theories. This should certainly also be considered in the case of binding signatures, as for example the obvious pseudo-monad morphism Tsm → Tfp induces such a 2-adjunction. The interpretation of the pseudo-functors making up this 2-adjunction being that the right adjoint “forgets” from an algebraic theory to a symmetric operad, whereas the left adjoint yields a free algebraic theory generated by a symmetric operad. In the general account for binding signatures there is a rigorous account for enrichment, which directly translates to the algebraic work, where it was not con- sidered. For binding signatures ωCpo-enrichment was considered to give an account for recursion. It is interesting to see what kind of enrichments make sense in the algebraic case and then “translate” them back to binders. The fact that some 2-monads for context generation were only considered in one of the two settings suggests an investigation of them in the other setting. In the case of binders, Tfp gives a context that allows for the operations of contraction, weakening, and exchange whereas Tsm only allows for the operation of exchange. But, inspired by the algebraic approach, one should also considering the further reduction to Tm, which additionally excludes the operation of exchange. In the case of algebraic theories the context 2-monads Tfp and Tsm were also considered but not TBI, which is the sum of the two. It seems natural to ask in what sense a theory for this context can be seen as a combination of an algebraic theory with a symmetric operad. The examples considered thus far are exclusive to either of the two strands of work and it would be good to find common (non-trivial) examples to deepen the understanding of why the two approaches are so similar. As it turns out, λ-calculus, the leading example for binding signatures, is a possible candidate. The reason for this being twofold. On one hand the category theoretic formulation of Engeler- style models in [12] is based on (ordinary) Kleisli categories, which lends itself to a generalization to the setting presented herein. On the other hand, λ-calculus is treated as an algebraic theory with a semi-closed structure in [9,11], which should be captured, in some sense, by the abstract approach to algebras. Of course, given the possibility of extending operations with equations or binders, it is obvious to search for a mechanism that treats operations with equa- tions and binders. Again, studying λ-calculus should yield valuable insights in order to achieve this. Even without having a conceptual reason for why the two approaches studied use so similar techniques, the technical relations should prove valuable. This is of
14 - 133 - J. Power, D. Schmitter course not to say that finding such a conceptual reason has to be out of reach. For the time being we cannot give such a reason but working on the points mentioned above might give the needed insights.
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A Basic Definitions
This appendix contains definitions of terms used throughout the paper. As usual, the structure maps of bicategories have been suppressed in the diagrams. ′ Definition A.1 (Pseudo-functor) Let B and B be any bicategories. A pseudo- ′ functor F from B to B is a triple F = (F, ϕ, ϕ) consisting of ′ • a mapping F: ob(B) → ob(B ), • 2 ′ for all (A, B) ∈ ob(B) a functor FA,B : B(A, B) → B (F A, F B), • for all (A, B, C) ∈ ob(B)3 a natural isomorphism
′ ϕA,B,C : ◦ F A,F B,F C ◦ (F × F) → F ◦◦A,B,C ,
and • ′ for all A ∈ ob(B) an isomorphism ϕA :IF A → FIA as shown in F × F ′ ′ B(B,C) × B(A, B) B (F B, F C) × B (F A, F B)
◦ = ϕ ◦ ⇐
′ B(A, C) B (F A, F C) F 16 - 135 - J. Power, D. Schmitter and 1 ′ IA IF A = ⇐ϕ ′ B(A, A) B (F A, F A) F such that
B(A, B) ′ 1 A,B × IA B( ) F × IF A = ⇐ 1F × ϕ ′ ′ B(A, B) × B(A, A) B (F A, F B) × B (F A, F A) F × F
◦ = ϕ ◦ ⇐
′ B(A, B) B (F A, F B) F is the identity,
B(A, B) ′ IB ×1 B(A,B) IF B × F = ⇐ ϕ × 1F ′ ′ B(B,B) × B(A, B) B (F B, F B) × B (F A, F B) F × F
◦ = ϕ ◦ ⇐
′ B(A, B) B (F A, F B) F is the identity, and
F × F × F ′ ′ ′ B(C,D) × B(B,C) × B(A, B) B (F C, F D) × B (F B, F C) × B (F A, F B)
= 1F × ϕ 1B(C,D) × ◦ ⇐ 1B′(F C,F D) × ◦
′ ′ B(C,D) × B(A, C) B (F C, F D) × B (F A, F C) F × F
◦ = ϕ ◦ ⇐
′ B(A, D) B (F A, F D) F 17 - 136 - J. Power, D. Schmitter equals
F × F × F ′ ′ ′ B(C,D) × B(B,C) × B(A, B) B (F C, F D) × B (F B, F C) × B (F A, F B)
= ϕ × 1F ◦ × 1B(A,B) ⇐ ◦ × 1B′(F A,F B)
′ ′ B(B,D) × B(A, B) B (F B, F D) × B (F A, F B) F × F
◦ = ϕ ◦ ⇐
′ B(A, D) B (F A, F D) F for all A, B, C, D ∈ ob(B). ′ Definition A.2 (Pseudo-natural transformation) Let B and B be any bicat- ′ egories and (F, ϕ, ϕ) and (G, ψ, ψ) any pseudo-functors from B to B .A pseudo- natural transformation σ from F to G consists of • for all A ∈ ob(B) a 1-cell σA :F A → G A and • for all (A, B) ∈ ob(B)2 a natural transformation
′ ′ σA,B : B (σA, 1G B) ◦ G → B (1F A, σB) ◦ F as shown in G ′ B(A, B) B (G A, G B)
′ F = σA,B B (σA, 1G B) ⇐
′ ′ B (F A, F B) ′ B (F A, G B) B (1F A, σB) such that
σA σA F A G A F A G A G f G f F gf = σf F gf = σgf G gf F f ⇐ ⇐ = F C = ϕ F B G B = F C G C ψg,f G B
g,f ⇐
⇐ σB σC
F g = σg G g G g 1F C ⇐ 1G C
F C G C G C σC
18 - 137 - J. Power, D. Schmitter and
σA F A G A
∼ = σA ∼
F 1A = 1F A ⇐ F 1G = 1G A
F A G A σA is the identity for all A, B, C ∈ ob(B), f ∈ ob(C (A, B)), and g ∈ ob(C (B,C)). ′ Definition A.3 (Modification) Let B and B be any bicategories, (F, ϕ, ϕ) and ′ (G, ψ, ψ) any pseudo-functors from B to B , and σ and τ any pseudo-natural trans- formations from F to G.A modification χ from σ to τ is an ob(B)-indexed family of 2-cells χA : σA ⇒ τA such that
σA σA F A G A F A G A
F f = σf G f 1F A = χA 1G A ⇐ ⇐
F B G B = F A G A σB τA
1F B = χB 1G B F f = τf G f ⇐ ⇐
F B G B F B G B τB τB
holds for all A, B ∈ ob(B) and f ∈ ob(B(A, B)). Definition A.4 (Pseudo-monad) Let B be any bicategory. A pseudo-monad T on B is a 6-tuple T = (T, µ, η, τ, λ, ρ) consisting of • a pseudo-functor T: B → B, • a pseudo-natural transformation µ:T → T, • a pseudo-natural transformation η : 1B → T, and • invertible modifications
T µ T η η T T3 T2 TT2 T2 T =⇒ = ⇐ ρ µ T = τ µ λ µ µ ⇐ and 1T 1T
2 T µ T, T T 19 - 138 - J. Power, D. Schmitter such that T2 µ T2 µ T4 T3 T4 T3
= T µ T µ 2 T τ 2 = µ µ ⇐ µ µ µ T T T ⇐ T µ T
T3 = τ T T3 T2 = T3 T2 = τ T2 ⇐ T µ T µ ⇐ µ =
µ T τ µ = µ
⇐ τ
µ T µ T ⇐
2 2 T µ T T µ T and T η T T µ T η T T2 T3 T2 T2 T3
= = ⇐ ⇐ ρ λ T µ T = τ µ = T T µ 1T2 ⇐ 1T2
2 2 µ T µ T T T hold. Definition A.5 (Pseudo-algebra) Let B be any bicategory and T = (T, µ, η, τ, λ, ρ) any pseudo-monad on B.A pseudo-T-algebra is a quadruple (A, a, aµ, aη) consisting of • an object A of B, • a 1-cell a:T A → A, and • invertible modifications
T a ηA T2 A T A A T A
= ⇐a µA = aµ a and η a ⇐ 1A
A T AAa such that
T2 a T2 a T3 A T2 A T3 A T2 A =
= µa
T aµ T a ⇐ T a µT A ⇐ µT A µA T µA = 2 = τ 2 2 aµ T A A T A T A = T A T A ⇐ T A ⇐ µA T a a µ T = aµ a = aµ a
µA ⇐ µA ⇐
T AAa T AAa 20 - 139 - J. Power, D. Schmitter and
T ηA T a T ηA T A T2 A T A T A T2 A
= = ⇐ ⇐ λA µA = aµ a = T a ⇐ T aµ 1TA 1TA a T AAa T AA hold.
Definition A.6 (Pseudo-map) Let (A, a, aµ, aη) and (B, b, bµ, bη) by any pseudo- T-algebras for a pseudo-monad T on a bicategory B.A pseudo-map from (A, a, aµ, aη) to (B, b, bµ, bη) is a pair (f, f a,b) consisting of • a 1-cell f : A → B and • an invertible 2-cell T f T A T B
a = f b
⇐ a,b
AB f such that T2 f T2 f T2 A T2 B T2 A T2 B = T f T b = µf T b µA ⇐ a,b µA µB T a ⇐
T A = aµ T A T B = T A T B = bµ T B ⇐ T f T f ⇐
= b a f b = b ⇐ a,b f
a a ⇐ a,b
A B A B f f and f f AB AB
ηA ηB ηB = ηf ⇐
= T A T B = = T B 1A ⇐ 1B ⇐ aη T f bη
a = f b b
⇐ a,b
A B B f hold. Definition A.7 (Algebra 2-cell) In the setting of the definition above, given two pseudo-maps (f, f a,b) and (g, ga,b) from (A, a, aµ, aη) to (B, b, bµ, bη), an algebra 21 - 140 - J. Power, D. Schmitter
2-cell from (f, f a,b) to (g, ga,b) is a 2-cell χ: f ⇒ g such that
T f T f T A T B T A = T χ T B ⇐ T g = f = ⇐ a,b a b = a ga,b b f ⇐
AB= χ AB ⇐ g g holds. Definition A.8 (Pseudo-monad morphism) Let C be any 2-category and S = (S, µS, ηS, τ S, λS, ρS) and T = (T, µT, ηT, τ T, λT, ρT) any pseudo-monads on C . A pseudo-monad morphism α from S to T is a triple α = (α, αµ, αη) consisting of • a pseudo-natural transformation α:S → T and • invertible modifications
ηS 2 S α α T 2 S STT 1C S
= ⇐α S = T η µ αµ µ and α ⇐ ηT
S α T T such that S2 α S α T α T2 S3 S2 TST2 T3 S µ S S T = T ∼ S µ S αµ S µ =α T µ ⇐
S
= S α α T τ ⇐ S2 S2 ST T2
µS = αµ µT µS ⇐
S α T equals S2 α S α T α T2 S3 S2 T ST2 T3 T S T T µ ∼ S = µ S =µS µ T αµ T µ T ⇐ S α α T S2 ST T2 ⇐= T2 τ T
µS = αµ µT ⇐ µT
S α T, 22 - 141 - J. Power, D. Schmitter
S α α T S α α T S2 STT2 S2 STT2
= S αη
S ⇒ T
S η = S T S ∼ T µ
= ⇒ µ αµ µ = S η T =α T η
S ⇐ S η =
λ λT ⇐ SS T S α TT, 1S α 1T and S α α T S α α T S2 STT2 S2 STT2 αη T
S ⇒ T η S = S T S = T µ
= ∼ ⇒ µ αµ µ = S η =ηS S η T
⇐ S η T =
ρ ρT ⇐ SST S TT 1S α α 1T hold.
23 - 142 - - 143 -
Uber¨ ein Paradoxon bei der Preisgestaltung in Versorgungssystemen
Helmut Meister D-81737 Munchen,¨ Germany E-Mail: [email protected]
Dedicated to Prof. Dr. R. B¨orger (†2014) I remember him as a good companion for a long time as well as an enthusiastic and experienced mathematician who contributed substantially to the development of Category Theory. Nevertheless, he was also interested in many other mathematical topics. For one of the recent interchanges of ideas with him compare [2]. 24. M¨arz 2015
Executive Summary The Shapley Value is one of the most popular concepts from Coope- rative Game Theory (see [3]). It serves as a solution to the problem, how a common numeric value can be distributed to the members of a group in such a way that the contribution of each member is taken into account adequately. The share of each member is defined according to his expected marginal contribution to any possible coalition. The practical disadvanta- ge of this approach results from the definition of the Shapley Value as an expectation value over all permutations of the set of actors. The calculati- on of this expectation value turns out to be an NP-problem in the general case and is often impractical even for groups with few actors. Therefore it makes sense to look for relatively simple calculation schemes and efficient algorithms to overcome these difficulties. One of these approaches can be found in the paper of P. L. Szczepanski, T. Michalak and T. Rahwan ([5]), which is concerned with a closed expression for the Shapley Value in the context of a certain centrality concept within social networks. The Shapley Value has also emerged as one of the considered concepts for risk based capital allocation in the financial sector. Several investigations on this topic based on mathematical methods have been published especially after the financial crisis of 2008 (see for instance [1]). In the following excursus we will focus on the assignment of graduated tariffs within a distribution system of commodities like energy or water supply. The distribution network will not be a matter of the investigation. Rather, we will concentrate on the pricing system deduced by the priciple
1 - 144 -
of responsibility for costs by the members in the distribution system. We will make use of the Shapley Value to allocate the corresponding costs to the members. First, we will discuss a paradox for sales discounts conceded to major customers. Normally, sales discounts result from less expensive production of bigger quantities of certain commodities. By contrast, in the case of energy supply temporary greater demand forces the suppliers to add less efficient power plants to the network. Therefore, the production costs increase disproportionately high. Contrary to expectations, apply- ing the Shapley Value Principle in such a case, leads to lower tariffs for major customers than for customers with low demand. In the same way, disproportionately low increasing production costs cause higher tariffs for this group. One of the handicaps in the analysis turns out to be the numerical difficulty associated with the calculation of the Shapley Value even in our special situation. In the succeeding part, we will therefore also be concer- ned with approximation methods for the expected marginal contribution of the members.
1 Einfuhrung¨
Ein einfaches ¨okonomisches Modell bildet n Verbraucher ab, die (feste und po- sitive) Quantit¨aten q1, . . . , qn eines Gutes fur¨ sich in Anspruch nehmen. Wir k¨onnen diese Quantit¨aten normieren, d.h.
n X qi = 1 (1) i=1 voraussetzen. Das Modell geht von einer fest installierten Versorgungsstruktur und festen Verbrauchsdaten aus. Die Preise des Gutes entwickeln sich entspre- chend einer nichtnegativen reellen Kostenfunktion u : [0, 1] −→ R mit u(0) = 0. Die Funktion u gibt an, welche Kosten bei welcher Absatzmenge x im Intervall [0, 1] entstehen. Auf Energiem¨arkten wird diese Funktion in der Regel monoton wachsend und konvex sein, da die Grundlast durch kostengunstige¨ Energietr¨ager geliefert werden kann, bei h¨oherem Bedarf aber weniger kostengunstige¨ Erzeu- gungsmethoden zum Einsatz kommen. Nach spieltheoretischen Prinzipien kann man in diesem Modell jeder Koalition S von Verbrauchern einen Wert X V (S) = u( qi) ∀S ⊂ {1, . . . , n} (2) i∈S zuweisen, der angibt, welche Kosten die Koalition durch ihre Nachfrage ver- ursacht. Der Marginalbeitrag eines Spielers i zur Kostenentwicklung auf dem Markt, in dem sich bereits eine Koalition S von weiteren Spielern etabliert hat, ist dann gegeben durch
Vmarg(i, S) = V (S + {i}) − V (S). (3)
2 - 145 -
Mit diesem Ansatz kann man den mittleren Marginalbeitrag jedes Verbrau- chers i zur Kostenentwicklung berechnen, wenn man fur¨ alle m¨oglichen Ein- trittsreihenfolgen von Spielern in den Markt den jeweiligen Marginalbeitrag be- stimmt und daruber¨ den Mittelwert bildet. Der auf diese Weise ermittelte Wert entspricht nach spieltheoretischen Gesichtspunkten dem Shapley-Wert Φi jedes Spielers i in dem vorgegebenen kooperativen Spiel. Bezeichnet man die Menge der Permutationen der Spielermenge {1, . . . , n} mit Π, so nimmt der Shapley- Wert Φi jedes Spielers i die Gestalt 1 X Φ = V (i, {j|π(j) < π(i)}) (4) i n! marg π∈Π an. Der Shapley-Wert ist damit der Erwartungswert der Marginalbeitr¨age des Spielers i zu allen Koalitionen aus Spielern, die vor ihm in den Markt ein- getreten sind, wobei die Gleichverteilung auf der Menge aller Permutationen zugrunde gelegt wird. Aus der Spieltheorie ist bekannt, dass der Shapley-Wert effizient ist, d.h., dass n X Φi = V ({1, . . . , n}) (5) i=1 gilt. Die Summe der mittleren Marginalbeitr¨age der Spieler ist folglich gleich den Gesamtkosten der Produktion auf dem Markt. Die Marginalbeitr¨age der Spieler h¨angen ganz entscheidend vom jeweiligen Eintrittszeitpunkt“ in den ” Markt ab, d.h. von den bereits im Markt agierenden Verbrauchern. Im Falle der Konvexit¨at von u ist dieser Marginalbeitrag umso gr¨oßer, je sp¨ater“ der Spieler ” dem Markt beitritt. Dieser Marginalbeitrag h¨angt zus¨atzlich vom jeweiligen Bedarf qi des Verbrauchers ab. Wenn man den mittleren Marginalbeitrag durch den Verbrauch qi des Spielers dividiert, so erh¨alt man einen plausiblen Tarif (Stuckpreis),¨ den man dem Verbraucher zuordnen kann. Der Tarif des Spielers i ist folglich gleich dessen Shapley-Wert Φi dividiert durch den Verbrauch qi. In diesem Zusammenhang stellt sich deshalb die Frage nach der Abh¨angigkeit der bei dieser Methode des Cost Sharings“ ermittelten Tarife der Spieler von ” deren Verbrauch . Dieser Frage wird im Folgenden nachgegangen.
2 Eine Charakterisierung des Ph¨anomens der Großkundenrabatte
Zum Zweck einer mathematischen Analyse betrachten wir die folgende Situati- on: Es befindet sich bereits eine Koalition von Verbrauchern im Energiemarkt, deren Abnahmemenge sich insgesamt zum Wert a aufaddiert. Ein zus¨atzlicher Verbraucher ben¨otigt die Energiemenge q und verursacht damit Zusatzkosten in der H¨ohe von cq(a) = u(a + q) − u(a). (6)
3 - 146 -
Seine Kosten pro Energieeinheit betragen deshalb u(a + q) − u(a) t (a) = . (7) q q Will man den Shapley-Wert des vorliegenden Verbrauchers ermitteln, so ist der Erwartungswert der Zusatzkosten cq(a) uber¨ alle m¨oglichen Werte von a bestim- men. Im Detail muss das nicht geschehen, wenn man lediglich daran interessiert ist, wie der zu erwartende Tarif vom Verbrauch q abh¨angt. Wir betrachten dazu einen weiteren Verbraucher mit einem Energiebedarf p ≤ q und vergleichen die beiden zu erwartenden Tarife. Hierzu w¨ahlen wir eine beliebige Beitrittsreihen- folge aller Verbraucher zum Energiemarkt und untersuchen die Auswirkungen auf die Tarife tp und tq , wenn die Verbraucher p und q ihre Pl¨atze in der Bei- trittsreihenfolge tauschen. Die Verbrauchsmenge im Markt zum Eintrittszeit- punkt des ersten der beiden Verbraucher sei a, die Verbrauchsmenge b sei der Gesamtverbrauch, sobald der letzte der beiden Verbraucher in den Markt ein- getreten ist. Fur¨ x = p, q ergibt sich fur¨ die Summe der beiden Eintrittszeiten die Beziehung
u(a + x) − u(a) u(b) − u(b − x) t (a) + t (b − x) = + (8) x x x x 1 = (u(b) − u(a) − (u(b − x) − u(a + x))) (9) x Notwendig und hinreichend fur¨ die Gultigkeit¨ von
tp(a) + tp(b − p) ≤ tq(a) + tq(b − q) (10) ist damit die Ungleichung u(b) − u(b − p) u(b) − u(b − q) u(a + q) − u(a) u(a + p) − u(a) − ≤ − . (11) p q q p Da die Wahrscheinlichkeiten fur¨ jede Beitrittsreihenfolge der beiden Ver- braucher ubereinstimmen,¨ genugt¨ die Forderung der Gultigkeit¨ von (11) fur¨ alle passenden Werte von a und b, damit die Erwartungswerte der Tarife beider Ver- braucher die Ungleichung E(tp) ≤ E(tq) erfullen.¨ Die beiden Erwartungswerte E(tp) und E(tq) sind identisch mit dem Shapley-Wert pro verbrauchte Einheit (mittlere Kosten pro Einheit) der beiden Konsumenten. Insbesondere gibt es un- ter der genannten Bedingung einen Großkundenrabatt auf dem Markt. Da die angegebene Ungleichung in manchen F¨allen nicht gut zu uberpr¨ ufen¨ ist, bietet sich an, nach eing¨angigeren hinreichenden Bedingungen fur¨ Großkundenrabatte zu suchen. Im n¨achsten Abschnitt wird der Nachweis erbracht, dass es genugt,¨ die Konvexit¨at der ersten Ableitung u0 der Kostenfunktion u zu fordern.
3 Das Paradoxon der Großkundenrabatte
Zum Nachweis des Ph¨anomens Großkundenrabatt“ stellen wir zun¨achst einige ” elementare Sachverhalte fur¨ konvexe Funktionen zusammen.
4 - 147 -
Lemma 3.1. Sei u eine Kostenfunktion mit konvexer Ableitung u0 im Intervall [a, b] mit a ≤ b. Dann gilt die Absch¨atzung
u(b) − u(a) u0(a) + u0(b) ≤ . (12) b − a 2 Beweis. Wir definieren die Funktion h durch x − a h(x) := u0(a) + (u0(b) − u0(a)) (13) b − a Dann ist h eine lineare Funktion mit h(a) = u0(a) und h(b) = u0(b). Wegen der Konvexit¨at von u0 gilt deshalb u0(x) ≤ h(x). Durch Integration der beiden Funktionen im Intervall [a, b] ergibt sich daraus
Z b Z b u(b) − u(a) = u0(x)dx ≤ h(x)dx = (14) a a 1 = (b − a)u0(a) + (b − a)(u0(b) − u0(a)) = (15) 2 1 = (u0(a) + u0(b)) (16) 2
Lemma 3.2. Sei u eine Kostenfunktion mit konvexer Ableitung u0 im Intervall [a, b] mit a ≤ b. Dann gilt die Absch¨atzung
a + b u(b) − u(a) u0( ) ≤ . (17) 2 2
1 Beweis. Wir legen das normierte Lebesgue-Maß b−a λ auf dem Intervall [a, b] zugrunde. Da dieses Maß ein Wahrscheinlichkeitsmaß auf dem Intervall [a, b] ist, folgt aus der Konvexit¨at von u0 die Absch¨atzung
u(b) − u(a) 1 Z b = u0(x)dx ≥ (18) b − a b − a a 1 Z b a + b ≥ u0( xdx) = u0( ). (19) b − a a 2
Mit diesen Vorbereitungen lassen sich die durch
u(a + x) − u(a) g(x) := (20) x und u(b) − u(b − x) h(x) := (21) x
5 - 148 -
auf dem Intervall [a, b] definierten Tarife g und h absch¨atzen. Zun¨achst ergibt sich lim g(x) = u0(a), lim h(x) = u0(b), (22) x→0 x→0 sofern u differenzierbar ist. Die Funktionen g und h lassen sich also stetig auf den Rand des Intervalls fortsetzen. Ziel ist eine genauere Analyse der Tariffunk- tionen g und h. Im folgenden Lemma wird das Ergebnis dieser Kurvendiskussion festgehalten. Lemma 3.3. Sei u eine zweimal differenzierbare Kostenfunktion mit konvexer Ableitung u0 im Intervall [a, b] mit a ≤ b. Weiter seien die Funktionen g und h wie oben definiert. Dann sind die Funktionen g und h konvex. Fur¨ diese Funktionen gilt außerdem u(b) − u(a) lim (g(x) + h(x)) ≥ (23) x→0 b − a 1 b − a b − a u(b) − u(a) (g( ) + h( )) = (24) 2 2 2 b − a u(b) − u(a) g(b − a) = h(b − a) = (25) b − a Beweis. Wendet man Lemma 3.2 auf die Funktionen g und h im Intervall [0, x] an, so erh¨alt man zun¨achst die Absch¨atzungen x g(x) ≥ u0(a + ) (26) 2 und x h(x) ≥ u0(b − ) (27) 2 Die Funktionen g und h sind differenzierbar im Intervall [0, b − a]. Wegen (26) und dem Satz von Rolle gilt deshalb 1 1 x 1 g0(x) = (u0(a + x) − g(x)) ≤ (u0(a + x) − u0(a + )) = u00(a + x − ) (28) x x 2 2 x fur¨ ein geeignetes im Intervall [0, 2 ]. Eine analoge Argumentation liefert die Absch¨atzung 1 1 x 1 h0(x) = (u0(b − x) − h(x)) ≤ (u0(b − x) − u0(b − )) = u00(b − x + δ) (29) x x 2 2 x 00 fur¨ ein geeignetes δ im Intervall [0, 2 ]. Fur¨ die zweite Ableitung g von g erh¨alt man damit die Absch¨atzung 1 1 g00(x) = (u00(a + x) − g0(x)) − (u0(a + x) − g(x)) = (30) x x2 1 = (u00(a + x) − 2g0(x)) ≥ (31) x 1 ≥ (u00(a + x) − u00(a + x − )) ≥ 0 (32) x
6 - 149 -
Die letzte Ungleichung folgt aus der Tatsache, dass u0 als konvexe Funktion vorausgesetzt wurde, und deshalb u00 monoton wachsend sein muß (siehe z.B. [4]). Mit der Nichtnegativit¨at von g00 ist gezeigt, dass g konvex ist. Analoge uberlegungen¨ fur¨ die Funktion h00 liefern die Absch¨atzung 1 1 h00(x) = (−u00(b − x) − h0(x)) − (u0(b − x) − h(x)) = (33) x x2 1 = (−u00(b − x) − 2h0(x)) ≥ (34) x 1 ≥ (−u00(b − x) + u00(b − x − δ)) ≥ 0 (35) x Folglich ist auch h eine konvexe Funktion. Die behauptete Ungleichung (23) ist eine Konsequenz aus Lemma 3.1. Die beiden anderen Gleichungen (24) und (25) erh¨alt man unmittelbar aus der Definition von g bzw. h. Mit Lemma 3.3 ist eine genauere Analyse des Verhaltens der durch f(x) := 1 2 (g(x) + h(x)) gegebenen Funktion m¨oglich. Folgerung 3.1. Sei u0 konvex und differenzierbar . Dann ist die Funktion f := 1 2 (g + h) konvex und im Intervall [0, (b − a)/2] monoton fallend. Wegen b − a u(b) − u(a) f( ) = f(b − a) = (36) 2 b − a gilt u(b) − u(a) f(x) ≤ = min f(y) (37) b − a b−a y∈[0, 2 ] b−a fur¨ alle x im Intervall [ 2 , b − a]. Mit diesen Vorbereitungen kann das Paradoxon der Großkundenrabatte for- muliert werden. Theorem 3.1. Das vorliegende Preisgestaltungsmodell fuhrt¨ zu gunstigeren¨ Ta- rifen fur¨ Abnehmer mit gr¨oßeren Kontingenten, sobald die Ableitung u0 der Kos- tenfunktion u konvex und differenzierbar ist. Beweis. Wir greifen auf die in Folgerung 3.1 definierte Funktion f und die im Abschnitt 2 eingefuhrte¨ Tariffunktion t zuruck.¨ Offenbar gilt
tx(a) + tx(b − x) = 2f(x) (38) fur¨ alle x im Intervall [0, b − a]. Wir legen nun die Verbrauchsmengen p und q mit p ≤ q zweier Marktteilnehmer zugrunde, die ihren Bedarf im Intervall [a, b] decken, wobei a der Gesamtbedarf vor Eintritt des ersten Teilnehmers und b der Bedarf nach Eintritt des zweiten Teilnehmers ist. Die Quantit¨aten p und q genugen¨ folglich der Ungleichung p + q ≤ b − a. Im Fall q ≥ (b − a)/2 muss deshalb p ≤ (b − a)/2 gelten. Nach Folgerung 3.1 ergibt sich in diesem Fall
tq(a) + tq(b − q) = 2f(q) ≤ 2f(p) = tp(a) + tp(b − p) (39)
7 - 150 -
Im Fall q ≤ (b − a)/2 gilt auch p ≤ (b − a)/2. Nach Folgerung 3.1 ist f monoton fallend im Intervall [0, (b − a)/2]. Deshalb erh¨alt man auch in diesem Fall
tq(a) + tq(b − q) = 2f(q) ≤ 2f(p) = tp(a) + tp(b − p) (40)
Zun¨achst liegt die Vermutung nahe, dass hoher Verbrauch bei sehr stark anwachsenden Kostenfunktionen (was bei konvexer erster Ableitung der Kos- tenfunktion der Fall ist) zu hohen Tarifen fuhrt.¨ Das Paradoxon besteht darin, dass genau in dieser Situation die Großverbraucher gunstigere¨ Tarife erhalten als die Kleinverbraucher. Die ublichen¨ Rabattsysteme beruhen darauf, dass bei ho- hem Mengenverbrauch die Produktionskosten niedrig gehalten werden k¨onnen. Im vorliegenden Fall ist aber genau dieser Effekt nicht zutreffend. Dass das Ph¨anomen der gunstigen¨ Tarife fur¨ Großkunden bei konkaver Ableitung der Kostenfunktionen nicht zu beobachten ist, zeigt folgendes Beispiel.
Beispiel 3.1: Wir bestimmen die Tarife einer Reihe von Verbrauchern fur¨ die durch u(x) = x1.5 gegebene Kostenfunktion. Verglichen wird das Ergebnis mit der Situation u(x) = x3.0. Die eingetragenen Werte sind jeweils die ermittelten Tarife nach der Shapley-Wert-Methode.
Abbildung 1: Kostenfunktionen u(x) = x1.5 und u(x) = x3.0
W¨ahrend im Fall u(x) = x1.5 (konkave erste Ableitung der Kostenfunktion) das Tarifsystem zugunsten der Kleinverbraucher ausf¨allt, ergibt sich fur¨ u(x) = x3.0 ein v¨ollig kontr¨ares Bild.
8 - 151 -
4 N¨aherungsverfahren zur Bestimmung des Shapley- Werts
Die exakte Bestimmung des Shapley-Werts st¨oßt sehr schnell auf numerische Grenzen. Da die Anzahl der Permutationen einer n-elementigen Menge durch n! gegeben ist, w¨achst die Zahl der erforderlichen Rechenoperationen mit der Gr¨oße des Marktes rapide an (20! = 2.43290200817664∗1018). Aus diesem Grund ist es sinnvoll, nach N¨aherungsverfahren zu suchen, die zu einer drastischen Re- duktion des Rechenaufwands beitragen k¨onnen. Allgemeine L¨osungen zu die- sem Problem sind abgesehen von Monte-Carlo-Verfahren noch nicht verfugbar.¨ Fur¨ manche Modelle vernetzter Akteure lassen sich einfach auszuwertende For- meln fur¨ den Shapley-Wert finden (vgl. z.B. [5]). Auf das vorliegende Modell eines Marktes sind diese Ergebnisse aber nicht anwendbar. Wir werden des- halb ein Sch¨atzverfahren vorschlagen, das sich auf die n¨aherungsweise Bestim- mung des Erwartungswerts der Marginalbeitr¨age der Akteure stutzt¨ und eine entsprechende Fehlerabsch¨atzung erlaubt. Die Ergebnisse werden wir mit den mittels Monte-Carlo-Simulation gewonnenen vergleichen. Als Simulationswerk- zeug dient hierbei die Entwicklungsumgebung NetLogo ([6]) fur¨ agentenbasierte Modelle.
4.1 Vorbereitungen Zun¨achst stellen wir einige Berechnungen zu den Erwartungswerten und Va- rianzen der kumulierten Verbr¨auche der Konsumenten zusammen. Wir fassen dazu jede Permutation π einer gegebenen Spielermenge P := {1,...,N} als Realisierung einer Zufallsvariablen ρ = (ρ1, . . . , ρN ) auf, die auf der Menge Π aller Permutationen, versehen mit der Gleichverteilung, definiert ist, und angibt, welcher Spieler jeweils zum Zeitpunkt i = 1,...,N in den Markt eintritt. Die Zufallsvariable ρ sei folglich definiert durch
−1 ρi(π) := π (i) ∀i = 1, . . . , N, π ∈ Π. (41)
Jede der Zufallsvariablen ρi ist damit gleichverteilt auf {1,...,N}. Wir geben nun eine beliebige relle Funktion f : {1,...,N} → R vor und definieren die Zufallsvariablen Xi (i = 1,...,N) durch
Xi := f ◦ ρi ∀i = 1,...,N. (42)
Da die Zufallsvariablen ρi (i = 1,...,N) gleichverteilt sind, erh¨alt man fur¨ PN s := j=1 f(j) die Beziehung s E(X ) = ∀i = 1,...,N (43) i N und damit
k k X X ks E( X ) = E(X ) = ∀k = 1,...,N. (44) i i N i=1 i=1
9 - 152 -
PN 2 Ganz analog erh¨alt man mit sb := j=1 f(j) die Beziehung s E(X2) = b ∀i = 1,...,N. (45) i N
Fur¨ die Varianzen V ar(Xi) der Zufallsvariablen Xi gilt folglich Ns − s2 V ar(X ) = b = V ar(f) ∀i = 1,...,N, (46) i N 2 wobei auf {1,...,N} die Gleichverteilung zugrunde gelegt wird. Da die bedingte Verteilung der Zufallsvariablen ρj unter der Bedingung ρi = m fur¨ i 6= j und m ∈ {1,...,N} eine Gleichverteilung auf der N − 1-elementigen Menge {1,...,N}\ {m} ist, berechnet man
s − f(m) E(X X |ρ = m) = f(m) ∀i 6= j. (47) i j i N − 1 Somit gilt
s − X s2 − s E(X X ) = E(X i ) = b ∀i 6= j. (48) i j i N − 1 N(N − 1)
Zusammen mit (43) ergibt sich daraus fur¨ die Kovarianzen Cov(Xi,Xj) der Zufallsvariablen Xi und Xj die Gleichung
s2 s 1 Cov(X ,X ) = − b = − V ar(f) ∀i 6= j. (49) i j N 2(N − 1) N(N − 1) N − 1
Mit diesen Vorbereitungen k¨onnen wir die Varianzen der Summenvariablen Pk i=1 Xi fur¨ k = 1,...,N berechnen und erhalten die Formel
k k k k X X X X V ar( Xi) = V ar(Xi) + Cov(Xi,Xj) = (50) i=1 i=1 i=1 j=1,j6=i k(k − 1)V ar(f) k(N − k) = kV ar(f) − = V ar(f) (51) N − 1 N − 1 ∀k = 1,...,N (52)
4.2 N¨aherungsformel fur¨ den Shapley-Wert Ahnlich¨ wie in Abschnitt 2 greifen wir einen beliebigen, aber festen Konsu- menten m heraus, der zum Zeitpunkt t ∈ {1, . . . , n} in den Markt eintritt. Der Verbrauch der zu diesem Zeitpunkt bereits in den Markt eingetretenen Pt−1 Akteure summiert sich insgesamt zum Wert St−1 := i=1 qρi auf, wobei wir S0 := 0 setzen und die Nomenklatur von Abschnitt 4.1 mit der Spielermenge P := {1, . . . , n}\{m} unf f(i) := qi (i ∈ P ) zugrunde legen. Der zus¨atzliche Verbraucher m ben¨otigt die Energiemenge qm und verursacht damit Zusatzkos- ten in der H¨ohe von u(St−1 + qm) − u(St−1).
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Wenn wir u als zweimal stetig differenzierbar im Intervall [0, 1] voraussetzen, dann ist die durch ∆q(x) := u(x+q)−u(x) definierte Funktion ∆q fur¨ alle q ∈ R+ Lipschitz-stetig. Nach dem Satz von Rolle gilt die Absch¨atzung
0 0 |∆q(x2) − ∆q(x1)| ≤ |x2 − x1| max |u (y + q) − u (y)| ≤ (53) 0≤y≤1−q 00 ≤ q|x2 − x1| max |u (y)| ∀x1, x2 ∈ [0, 1] . (54) 0≤y≤1
Fur¨ den Shapley-Wert Φm des Spielers m erhalten wir aufgrund der Defini- tion der Spielwertsfunktion V (siehe Gleichung (4)), und weil jede Position t des Spielers m innerhalb einer Permutation gleichwahrscheinlich ist, den Ausdruck
n 1 X Φ = E(∆ (S )). (55) m n qm t−1 t=1 Der Erwartungswert wird dabei bezuglich¨ der Gleichverteilung auf der n − 1- elementigen Menge {1, . . . , n}\{m} gebildet. Wir definieren den N¨aherungswert Φ˜m fur¨ Φm durch n 1 X Φ˜ := ∆ (E(S ))) (56) m n qm t−1 t=1 und geben eine Schranke fur¨ die Genauigkeit dieser N¨aherungsformel an. Wegen der Lipschitz-Stetigkeit von ∆qm (53) und (50) gilt fur¨ gegebenes t ∈ {1, . . . , n} die Absch¨atzung
2 (E(∆qm (St−1)) − ∆qm (E(St−1))) ≤ (57) 2 ≤ E (∆qm (St−1) − ∆qm (E(St−1))) ≤ (58) ≤ M 2 E((S − E(S ))2) = (59) qm t−1 t−1
2 (t − 1)(n − t) 2 = M V ar(St−1) = M V ar(q ), (60) qm n − 2 qm (−m)
00 wobei Mqm := qm max0≤y≤1 |u (y)| und q(−m) := (q1, . . . , qm−1, qm+1, . . . , qn) gesetzt wird. Aus diesen Uberlegungen¨ folgt zusammen mit bekannten Summen- formeln die Ungleichung
n 1 X (E(∆ (S )) − ∆ (E(S )))2 ≤ (61) n qm t−1 qm t−1 t=1 n 1 X ≤ M 2 V ar(q ) (t − 1)(n − t) = (62) n(n − 2) qm (−m) t=1 n − 1 = M 2 V ar(q ). (63) 6 qm (−m) Zusammenfassend ergibt sich damit der folgende Sachverhalt:
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Lemma 4.1. Fur¨ jede im Intervall [0, 1] zweimal stetig differenzierbare Kos- tenfunktion u und jeden Spieler m gilt die Absch¨atzung
2 n − 1 2 (Φm − Φ˜m) ≤ M V ar(q ), (64) 6 qm (−m) mit 00 Mq := qm max |u (y)|. (65) m 0≤y≤1 Wie sich Lemma 4.1 in bestimmten Spezialf¨allen auswirkt, zeigen die nach- folgenden Anmerkungen.
Bemerkungen 4.1:
1. Sofern die Verbrauchswerte qi aller Spieler i 6= m ubereinstimmen,¨ ergibt sich fur¨ den Spieler m aus Lemma 4.1
Φm = Φ˜m . (66)
In diesem Fall ist die angegebene Absch¨atzung scharf. Der Markt be- steht in der gegebenen Situation aus einem Großverbraucher und mehreren Kleinverbrauchern mit identischen Verbrauchswerten.
2. Fur¨ große“ M¨arkte, bei denen die Verbr¨auche qi der Spieler i fur¨ eine ” c passende Konstante c > 0 durch n beschr¨ankt sind, konvergiert die in Lemma 4.1 angegebene Schranke fur¨ n → ∞ gegen 0. Asymptotisch liefert folglich der N¨aherungswert Φ˜m im vorliegenden Fall den Shapley-Wert des Spielers m.
3. Fur¨ den Tarif Φm/qm des Spielers m liefert Lemma 4.1, dass die N¨aherungsformel Φ˜m/qm der Absch¨atzung ˜ Φm Φm 2 n − 1 00 2 ( − ) ≤ ( max |u (y)|) V ar(q(−m)), (67) qm qm 6 0≤y≤1 genugt.¨ Sofern die Varianz der Verbr¨auche der ubrigen¨ Spieler gering ist, kann demnach der Tarif des Spielers m mit der angegebenen N¨aherungsformel gut approximiert werden.
4. Die Sch¨atzung Φ˜m(m = 1, . . . , n) fur¨ die Shapley-Werte der Marktteilneh- mer ist im Allgemeinen nicht effizient im Sinne der Spieltheorie. In der Regel ist folglich davon auszugehen, dass die Ungleichung
n X Φ˜m 6= V ({1, . . . , n}) (68) m=1 gilt. Will man die Effizienz herstellen, so mussen¨ dazu die Sch¨atzwerte nachtr¨aglich entsprechend normiert werden.
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5. Wenn die Ableitung u0 der Kostenfunktion u konvex ist, sind die Funktio- nen ∆q(q ∈ R+) ebenfalls konvex. In dieser Situation gilt
Φ˜m ≤ Φm . (69)
Folglich ist der N¨aherungswert Φ˜m in diesem Fall eine Unterschrankensch¨atzung fur¨ den Shapley-Wert.
5 Vergleich der verschiedenen Methoden zur Ap- proximation des Shapley-Werts
Wegen des nicht-polynomial anwachsenden Rechenaufwands bei der exakten Be- stimmung des Shapley-Werts gelingt der Vergleich zwischen den verschiedenen N¨aherungsverfahren und der exakten Methode nur fur¨ M¨arkte mit wenigen Teil- nehmern. Der Vergleich des in Abschnitt 4.2 vorgestellten N¨aherungsverfahrens mit den Ergebnissen von Monte-Carlo-Simulationen ist jedoch auch fur¨ umfang- reichere M¨arkte mit geringen Rechenaufwand zu bewerkstelligen.
Beispiel 5.1: Im ersten Beispiel eines Marktes mit 9 Verbrauchern und der durch u(x) = x4 definierten Kostenfunktion ergibt sich folgendes Bild.
Abbildung 2: Vergleich der Methoden fur¨ die Kostenfunktion u(x) = x4
Die Daten sind entsprechend den Verbrauchswerten der Marktteilnehmer in aufsteigender Reihung angeordnet. Die Datenreihen geben die nach den ver- schiedenen Methoden ermittelten Tarife an. Die mit dem Proxi-Verfahren ge- wonnenen N¨aherungen der Shapley-Werte wurden nachtr¨aglich normiert (vgl.
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Bemerkung 4.1(4)). Das Beispiel zeigt, dass das Proxi-Verfahren (SHV-Proxi) fur¨ die Tarife zumindest den Trend der exakten Werte wiederspiegelt, w¨ahrend die Monte-Carlo-Methode (mit 1000 zuf¨alligen Permutationen, drei Varianten MC1, MC2, MC3) sehr stark variierende Ergebnisse liefert.
Beispiel 5.2: Im Beispiel eines Marktes mit 100 Verbrauchern und der durch u(x) = x4 definierten Kostenfunktion erkennt man ebenfalls noch eine gr¨oßere Bandbreite der Ergebnisse bei der Monte-Carlo-Methode (1000 zuf¨allige Permu- tationen). Das Proxi-Verfahren gibt wiederum den Trend der Tarife an.
Abbildung 3: Vergleich des Proxi-Verfahrens mit der Monte-Carlo-Methode
Die Anordnung der Datenreihen entspricht dem vorigen Beispiel.
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Literatur
[1] D. Balog, 2010 , [email protected], Risk based capital allocation , Proceedings of FIKUSZ10´ Symposium for Young Researchers, 17-26.
[2] Eugen Grycko, Werner Kirsch, Tobias Muhlenbruch,¨ 2014, Eine Reminiszenz an Reinhard B¨orger, gleicher Band [3] L. S. Shapley, 1953, A Value for n-person Games, Contributions to the Theo- ry of Games, volume II, by H.W. Kuhn and A.W. Tucker, editors. Annals of Mathematical Studies v. 28, 307–317. Princeton University Press. [4] M. Strickmann, 2013, Konvexe Funktionen und wichtige Unglei- chungen, Seminar Analysis SoSe 2013, www.mathematik.uni − dortmund.de/.../Strickmann Konvexe F kt.pdf. [5] P. L. Szczepanski, T. Michalak and T. Rahwan, 2012, A New Approach to Betweenness Centrality Based on the Shapley Value, Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2012), Conitzer, Winikoff, Padgham, and van der Hoek (eds.), 4-8 June 2012, Valencia, Spain. [6] U. Wilensky, 1999, NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Uni- versity, Evanston, IL.
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