DOCSLIB.ORG

Introduction and Preliminaries 1. Coproduct Preservation Properties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction and Preliminaries 1. Coproduct Preservation Properties Journal of Pure and Applied Algebra 46 (1987) 35-47 35 North-Holland COPRODUCTS AND ULTRAFILTERS Reinhard BORGER Fachbereich Mathematik und lnformatik, Fernuniversitiit, D-5800 Hagen 1, Fed. Rep. Germany Communicated by G.M. Kelly Received 28 February 1985 Revised 1 November 1985 Introduction and preliminaries In this paper we study connections between ultrafilters and coproduct preserva- tion properties of set-valued functors. It turns out that the problem can be reduced to coproducts indexed by measurable cardinals, if all needed coproducts exist. In particular, preservation of countable coproducts implies preservation of arbitrary ones provided that there are no uncountable measurable cardinals. This result has been proved for endofunctors of Set by Trnkovfi [9] and for arbitrary functors it is contained in [2]. Furthermore, we see that the ultrafilter functor is terminal in the (meta-)category of all endofunctors (see [1]). Then we prove some generalisations for functors that are not set-valued. Finally, we show by a counterexample that our results cannot be generalized to colimits of chains. I am indebted to the referee for valuable suggestions, notably for the simplifica- tion of the proof of Theorem 4.1. We always assume the axiom of choice. For all set-theoretical terminology see [4]. Particularly, if a is an infinite cardinal, an ultrafilter is called a-complete iff it is closed under intersections of less than a members. A cardinal a is called measurable iff there is a non-principal a-complete ultrafilter on a. (By von Neumann's conven- tion, a is identified with the set of all ordinals < a.) In particular, the countable car- dinal ~0 is measurable by our definition. Note that any a+-closed ultrafilter on a cardinal a is principal (a + is the least cardinal >a). If a is a cardinal, then a coproduct indexed by a set of cardinality < a is called an a-coproduct. The image of a map f is denoted by im f. The cardinality of a set M is called #M. 1. Coproduct preservation properties Our further investigation will be based on 0022-4049/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland) 36 R. Bdrger Lemma 1.1. Let a, ,8, be regular cardinals with fl <_ a. Let o~ be a category, in which all a +-coproducts exist. Furthermore, let F : O~-* Set be a functor that preserves all fl-coproducts and let (A¢)¢<a be a family of ;Y{-objects. For any subset MCa define B M := H ~EMA¢, and let "~M : BM ~ Ba be the canonically induced morphism. Then the following assertions hold: (i) For any MC a, F(,,M) is injective; (ii) For any ~ e F(Ba), the collection ~:= {MCa[x6im(F(,,M)) } is a fl-complete ultrafilter on ct; (iii) The coproduct B a = H ~<~ A~ is preserved by F iff for all ~ ~ F(Ba) the ultra- filter q~, is principal. Proof. (i) follows immediately from the fact that B~ is the coproduct of B M and B.\M with the injections "M and "%\M- (ii) By the above argument, F(B,~) is the disjoint union of imF(,,M) and im F(.%\M), hence exactly one of the two sets M, a\M belongs to ~. As ~eF(Ba), we obtain 0~. If McNCa are sets with Me~, consider the induced map : BM'-~BN; then we have ~e im F(UM) = im F(,,NO*)Cim F(~N), hence Ne D=. Now we have to show that ~ is closed under ),-indexed intersections for all y < fl (in particular for ),=2). So let (M¢)~< r be a family of members of ~. Define Ny'=~ {M~ t(<y}, and N,7:= ~ {M~ I ~<r/} \M r, for all r/<y. Then (N,),<y+l is a partition of a, and hence Ba is the coproduct of all BN,, r/<y+ 1 (with injections "N,)- Now the maps F(..%), rl<_y form a coproduct sink in Set, hence (im F(..N,),I<y is a partition of a, therefore ~e im F(..N,) for a unique r/_< y. But for all/7 < (, N,r e ~. would contradict M,1 e ~.~. Thus we have (iii) For any ~<a we have B{~} =A~, and B o is the coproduct of the B{¢} with the injections ,,{¢1. By (i) all F(,,{~I) are injective, and for any ~o<~1<a we have im(F(,,{~0l)) n im(F(,,{~}))= 0, since the contrary would lead to the contradiction 0= {~0} n {~1} ff~x for some xeF(Ba). Thus the given coproduct is preserved iff we have U {im(F("{¢}))l~<(} =F(B,~). This holds iff all ~.~ are principal. [] Lemma 1.1 enables us to prove the following surprising theorem, which was proved by Trnkovfi [9] for endofunctors of Set. Theorem 1.2. Let a be an uncountable cardinal, and let o~ be a category, in which all a-coproducts exist. Let F: or'--, Set be a functor. Then the following statements are equivalent: (i) F preserves all a-coproducts; (ii) F preserves all fl+-coproducts for all measurable cardinals fl<a. Coproducts and ultrafilters 37 Proof. (i) = (ii) is trivial. (ii) = (i). Assume (ii) is true and (i) is false. Application of (ii) to fl= 1~0 yields that Fpreserves all countable coproducts. Now there is a smallest a such that F does not preserve all fl+-coproducts. (Then, of course, F preserves fl-coproducts.) Let (A¢)¢< B be a family, whose coproduct is not preserved by F. By Lemma 1.1(iii), there is an ~eF(II¢<#A~) such that the ultrafilter ~, (as above) is not principal. On the other hand, by Lemma 1. l(ii), ~ is fl-complete. Hence fl is measurable, which contradicts (ii). [] Application of Theorem 1.2 to contravariant hom-functors from the category of unital rings to Set gives the following result: If R is an unital ring and a is a non- Ulam-measurable cardinal, then every unital ring homomorphism from a product of a rings to R depends on only one component provided that this is true for count- able products. This and related results can be found in [3]. Theorem 1.2 reduces the question of a-coproduct preservation to the case where a =fl+ and fl is measurable. But the assumption that g0 is the only measurable car- dinal is relatively consistent with ZFC (= Zermelo-Fraenkel set theory + axiom of choice). On the other hand, uncountable measurable cardinals have to be very large [8, 10], and adding the assumption of their existence leads to a theory strictly stronger than ZFC. If ~0 is the only measurable cardinal, then Theorem 1.2 says that preservation of countable coproducts implies preservation of arbitrary ones. Proposition 1.3. Let a be an infinite cardinal, and let ~5" be a category with a- coproducts, let F, G : ~'--, Set be the functors, and let (o : F~ G be a natural trans- formation. If G preserves a-coproducts and if F preserves finite coproducts, then F preserves a-coproducts. Proof. For any family (A¢)¢<# of ~'-objects with fl<a and for x~F(L[~<aA), the ultrafilter ~ is contained in the ultrafilter @,(x)~), which is constructed like ~, but with F being replaced by G. As both are ultrafilters, ~ = @~,(x)(~) is u-complete by Lemma 1.1, and this proves that the coproduct is preserved by F. [] 2. The ultrafilter functor In this section we want to apply Lemma 1.1 and Theorem 1.2 to endofunctors of Set. Every set is a disjoint union (= Set-coproduct) of singletons, and we can apply Lemma 1.1 for this canonical coproduct. Then the assignment ~--* ~, even becomes natural in a suitable way. In order to show this, we have to define the ultrafilter functor T: Set-,Set: For every set Xlet T(X) be the set of all ultrafilters on Xand for any set map/:X~ Yand any ~ T(X) define T(/)(~):= {BC rl/-l[B] ~.~}. Obviously T is a functor. 38 R. B6rger Theorem 2.1. (i) T preserves finite coproducts. (ii) If F: Set ~ Set /s an endofunctor preserving finite coproducts, then there exists a unique natural transformation x : F--* T. (iii) lf F and x are as in (ii) and if fl is a cardinal, then F preserves all fl-coproducts iff for all sets X and all ~ ~ F(X) the ultrafilter x(X)(x) is fl-complete. Proof. (i) is trivial. (ii) Let X be a set, and for any subset A CX let "A : A ---'X be the inclusion map. Applying Lemma 1.1(ii) to the representation of X as a coproduct of singletons yields that x(X)(x) := {A CX [~imF(,~,4)} is an ultrafilter on X. x :F~ T turns out to be a natural transformation. Now we have to show the uniqueness of x. Let x': F~ T be an arbitrary natural transformation. Then it suffices to show that x(X)(~)Cx'(X)(~) (since both are ultrafilters) for all sets X and all ~ e F(X). For any A ~ x(X)(~) we get ~, = F(,, A )(a) for some aeF(A), hence ,~,~1[A] =A ¢x'(A)(,~) and x' (X)(a:) = (x' (X)) o (F(a A ))(a) = ( T(~ A )) o (x'(A))(a), yields A e x'(X)(,~). (iii) The 'only if' part is a direct translation of Lemma 1.1 (ii) to this special case, and the 'if'-part follows from Lemma 1. l(iii). [] In particular Theorem 2.1 means that the functor that assigns to every set X the set of fl-complete ultrafilters on X is terminal in the meta-category of fl-coproduct preserving endofunctors.
Load more

Recommended publications
  • A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
    A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience.
    [Show full text]
  • Notes and Solutions to Exercises for Mac Lane's Categories for The
    Stefan Dawydiak Version 0.3 July 2, 2020 Notes and Exercises from Categories for the Working Mathematician Contents 0 Preface 2 1 Categories, Functors, and Natural Transformations 2 1.1 Functors . .2 1.2 Natural Transformations . .4 1.3 Monics, Epis, and Zeros . .5 2 Constructions on Categories 6 2.1 Products of Categories . .6 2.2 Functor categories . .6 2.2.1 The Interchange Law . .8 2.3 The Category of All Categories . .8 2.4 Comma Categories . 11 2.5 Graphs and Free Categories . 12 2.6 Quotient Categories . 13 3 Universals and Limits 13 3.1 Universal Arrows . 13 3.2 The Yoneda Lemma . 14 3.2.1 Proof of the Yoneda Lemma . 14 3.3 Coproducts and Colimits . 16 3.4 Products and Limits . 18 3.4.1 The p-adic integers . 20 3.5 Categories with Finite Products . 21 3.6 Groups in Categories . 22 4 Adjoints 23 4.1 Adjunctions . 23 4.2 Examples of Adjoints . 24 4.3 Reflective Subcategories . 28 4.4 Equivalence of Categories . 30 4.5 Adjoints for Preorders . 32 4.5.1 Examples of Galois Connections . 32 4.6 Cartesian Closed Categories . 33 5 Limits 33 5.1 Creation of Limits . 33 5.2 Limits by Products and Equalizers . 34 5.3 Preservation of Limits . 35 5.4 Adjoints on Limits . 35 5.5 Freyd's adjoint functor theorem . 36 1 6 Chapter 6 38 7 Chapter 7 38 8 Abelian Categories 38 8.1 Additive Categories . 38 8.2 Abelian Categories . 38 8.3 Diagram Lemmas . 39 9 Special Limits 41 9.1 Interchange of Limits .
    [Show full text]
  • Derived Functors and Homological Dimension (Pdf)
    DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic.
    [Show full text]
  • Generalized Inverse Limits and Topological Entropy
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of Faculty of Science, University of Zagreb FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Zagreb, 2016. FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Supervisors: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Mentori: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016. Acknowledgements First of all, i want to thank my supervisor professor Judy Kennedy for accept- ing a big responsibility of guiding a transatlantic student. Her enthusiasm and love for mathematics are contagious. I thank professor Vlasta Matijevi´c, not only my supervisor but also my role model as a professor of mathematics. It was privilege to be guided by her for master's and doctoral thesis. I want to thank all my math teachers, from elementary school onwards, who helped that my love for math rises more and more with each year. Special thanks to Jurica Cudina´ who showed me a glimpse of math theory already in the high school. I thank all members of the Topology seminar in Split who always knew to ask right questions at the right moment and to guide me in the right direction. I also thank Iztok Baniˇcand the rest of the Topology seminar in Maribor who welcomed me as their member and showed me the beauty of a teamwork.
    [Show full text]
  • Categories, Functors, and Natural Transformations I∗
    Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0.
    [Show full text]
  • N-Quasi-Abelian Categories Vs N-Tilting Torsion Pairs 3
    N-QUASI-ABELIAN CATEGORIES VS N-TILTING TORSION PAIRS WITH AN APPLICATION TO FLOPS OF HIGHER RELATIVE DIMENSION LUISA FIOROT Abstract. It is a well established fact that the notions of quasi-abelian cate- gories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of t-structures. Firstly, we extend this picture into a hierarchy of n-quasi-abelian categories and n-tilting torsion classes. We prove that any n-quasi-abelian category E admits a “derived” category D(E) endowed with a n-tilting pair of t-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t-structures as quotient categories of coherent functors, generalizing Auslander’s Formula. Thirdly, we apply our results to Bridgeland’s theory of perverse coherent sheaves for flop contractions. In Bridgeland’s work, the relative dimension 1 assumption guaranteed that f∗-acyclic coherent sheaves form a 1-tilting torsion class, whose associated heart is derived equivalent to D(Y ). We generalize this theorem to relative dimension 2. Contents Introduction 1 1. 1-tilting torsion classes 3 2. n-Tilting Theorem 7 3. 2-tilting torsion classes 9 4. Effaceable functors 14 5. n-coherent categories 17 6. n-tilting torsion classes for n> 2 18 7. Perverse coherent sheaves 28 8. Comparison between n-abelian and n + 1-quasi-abelian categories 32 Appendix A. Maximal Quillen exact structure 33 Appendix B. Freyd categories and coherent functors 34 Appendix C. t-structures 37 References 39 arXiv:1602.08253v3 [math.RT] 28 Dec 2019 Introduction In [6, 3.3.1] Beilinson, Bernstein and Deligne introduced the notion of a t- structure obtained by tilting the natural one on D(A) (derived category of an abelian category A) with respect to a torsion pair (X , Y).
    [Show full text]
  • Coreflective Subcategories
    transactions of the american mathematical society Volume 157, June 1971 COREFLECTIVE SUBCATEGORIES BY HORST HERRLICH AND GEORGE E. STRECKER Abstract. General morphism factorization criteria are used to investigate categorical reflections and coreflections, and in particular epi-reflections and mono- coreflections. It is shown that for most categories with "reasonable" smallness and completeness conditions, each coreflection can be "split" into the composition of two mono-coreflections and that under these conditions mono-coreflective subcategories can be characterized as those which are closed under the formation of coproducts and extremal quotient objects. The relationship of reflectivity to closure under limits is investigated as well as coreflections in categories which have "enough" constant morphisms. 1. Introduction. The concept of reflections in categories (and likewise the dual notion—coreflections) serves the purpose of unifying various fundamental con- structions in mathematics, via "universal" properties that each possesses. His- torically, the concept seems to have its roots in the fundamental construction of E. Cech [4] whereby (using the fact that the class of compact spaces is productive and closed-hereditary) each completely regular F2 space is densely embedded in a compact F2 space with a universal extension property. In [3, Appendice III; Sur les applications universelles] Bourbaki has shown the essential underlying similarity that the Cech-Stone compactification has with other mathematical extensions, such as the completion of uniform spaces and the embedding of integral domains in their fields of fractions. In doing so, he essentially defined the notion of reflections in categories. It was not until 1964, when Freyd [5] published the first book dealing exclusively with the theory of categories, that sufficient categorical machinery and insight were developed to allow for a very simple formulation of the concept of reflections and for a basic investigation of reflections as entities themselvesi1).
    [Show full text]
  • Coequalizers and Tensor Products for Continuous Idempotent Semirings
    Coequalizers and Tensor Products for Continuous Idempotent Semirings Mark Hopkins? and Hans Leiß?? [email protected] [email protected] y Abstract. We provide constructions of coproducts, free extensions, co- equalizers and tensor products for classes of idempotent semirings in which certain subsets have least upper bounds and the operations are sup-continuous. Among these classes are the ∗-continuous Kleene alge- bras, the µ-continuous Chomsky-algebras, and the unital quantales. 1 Introduction The theory of formal languages and automata has well-recognized connections to algebra, as shown by work of S. C. Kleene, H. Conway, S. Eilenberg, D. Kozen and many others. The core of the algebraic treatment of the field deals with the class of regular languages and finite automata/transducers. Here, right from the beginnings in the 1960s one finds \regular" operations + (union), · (elementwise concatenation), * (iteration, i.e. closure under 1 and ·) and equational reasoning, and eventually a consensus was reached that drew focus to so-called Kleene algebras to model regular languages and finite automata. An early effort to expand the scope of algebraization to context-free languages was made by Chomsky and Sch¨utzenberger [3]. Somewhat later, around 1970, Gruska [5], McWhirter [19], Yntema [20] suggested to add a least-fixed-point op- erator µ to the regular operations. But, according to [5], \one can hardly expect to get a characterization of CFL's so elegant and simple as the one we have developed for regular expressions." Neither approach found widespread use. After the appearance of a new axiomatization of Kleene algebras by Kozen [10] in 1990, a formalization of the theory of context-free languages within the al- gebra of idempotent semirings with a least-fixed-point operator was suggested in Leiß [13] and Esik´ et al.
    [Show full text]
  • A Few Points in Topos Theory
    A few points in topos theory Sam Zoghaib∗ Abstract This paper deals with two problems in topos theory; the construction of finite pseudo-limits and pseudo-colimits in appropriate sub-2-categories of the 2-category of toposes, and the definition and construction of the fundamental groupoid of a topos, in the context of the Galois theory of coverings; we will take results on the fundamental group of étale coverings in [1] as a starting example for the latter. We work in the more general context of bounded toposes over Set (instead of starting with an effec- tive descent morphism of schemes). Questions regarding the existence of limits and colimits of diagram of toposes arise while studying this prob- lem, but their general relevance makes it worth to study them separately. We expose mainly known constructions, but give some new insight on the assumptions and work out an explicit description of a functor in a coequalizer diagram which was as far as the author is aware unknown, which we believe can be generalised. This is essentially an overview of study and research conducted at dpmms, University of Cambridge, Great Britain, between March and Au- gust 2006, under the supervision of Martin Hyland. Contents 1 Introduction 2 2 General knowledge 3 3 On (co)limits of toposes 6 3.1 The construction of finite limits in BTop/S ............ 7 3.2 The construction of finite colimits in BTop/S ........... 9 4 The fundamental groupoid of a topos 12 4.1 The fundamental group of an atomic topos with a point . 13 4.2 The fundamental groupoid of an unpointed locally connected topos 15 5 Conclusion and future work 17 References 17 ∗e-mail: [email protected] 1 1 Introduction Toposes were first conceived ([2]) as kinds of “generalised spaces” which could serve as frameworks for cohomology theories; that is, mapping topological or geometrical invariants with an algebraic structure to topological spaces.
    [Show full text]
  • The Hecke Bicategory
    Axioms 2012, 1, 291-323; doi:10.3390/axioms1030291 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Communication The Hecke Bicategory Alexander E. Hoffnung Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122, USA; E-Mail: [email protected]; Tel.: +215-204-7841; Fax: +215-204-6433 Received: 19 July 2012; in revised form: 4 September 2012 / Accepted: 5 September 2012 / Published: 9 October 2012 Abstract: We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail. Keywords: Hecke algebras; categorification; groupoidification; Yang–Baxter equations; Zamalodchikov tetrahedron equations; spans; enriched bicategories; buildings; incidence geometries 1. Introduction Categorification is, in part, the attempt to shed new light on familiar mathematical notions by replacing a set-theoretic interpretation with a category-theoretic analogue. Loosely speaking, categorification replaces sets, or more generally n-categories, with categories, or more generally (n + 1)-categories, and functions with functors. By replacing interesting equations by isomorphisms, or more generally equivalences, this process often brings to light a new layer of structure previously hidden from view.
    [Show full text]
  • Abelian Categories
    Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists.
    [Show full text]
  • Categorical Models of Type Theory
    Categorical models of type theory Michael Shulman February 28, 2012 1 / 43 Theories and models Example The theory of a group asserts an identity e, products x · y and inverses x−1 for any x; y, and equalities x · (y · z) = (x · y) · z and x · e = x = e · x and x · x−1 = e. I A model of this theory (in sets) is a particularparticular group, like Z or S3. I A model in spaces is a topological group. I A model in manifolds is a Lie group. I ... 3 / 43 Group objects in categories Definition A group object in a category with finite products is an object G with morphisms e : 1 ! G, m : G × G ! G, and i : G ! G, such that the following diagrams commute. m×1 (e;1) (1;e) G × G × G / G × G / G × G o G F G FF xx 1×m m FF xx FF m xx 1 F x 1 / F# x{ x G × G m G G ! / e / G 1 GO ∆ m G × G / G × G 1×i 4 / 43 Categorical semantics Categorical semantics is a general procedure to go from 1. the theory of a group to 2. the notion of group object in a category. A group object in a category is a model of the theory of a group. Then, anything we can prove formally in the theory of a group will be valid for group objects in any category. 5 / 43 Doctrines For each kind of type theory there is a corresponding kind of structured category in which we consider models.
    [Show full text]