Multiversal Algebra A T U M D P F E P S Francisco Lobo School of Computer Science Contents Introduction Background . Structures . . Relations . . Quotients . Algebraic Structures . Nodularity . . Magmoids . . Categories . . Naturality . Adjoint Structures . Transpositions . . Sums and Products . . Adjunctions . . Internal Algebras . Sequential Scheduling . Introduction . . Global Scheduling . . Local Scheduling . . Scheduling Algebras . . Dialogue Games . Abstract This thesis discusses two ideas, multiversal algebra and algebraic enrichment, and one potential application for the latter, sequential scheduling. Multiversal algebra is a proposal for the reconsideration of semigroupoid and category theory within a framework that extends the approach of universal algebra. The idea is to introduce the notion of algebraic operation relative to a given binary relation, as an alternative to the notion of operation on a carrier class. It is shown that for a particular class of relations the derived notion of category coincides with that of standard category theory. Algebraic enrichment is the name given to a series of similar constructions translating between external and internal algebraic structure, which are studied as a first step towards generalizing the seminal results of Eckmann and Hilton, and for the application to sequential scheduling. This well-known combinatorial engine of game semantics is shown to form part of a double semigroupoid, and this new algebraic perspective on scheduling offers a new direction for the study of game models and their innocence condition. Declaration No portion of the work referred to in this dissertation has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 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(iv) Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see <http://www.campus.manchester.ac. uk/medialibrary/policies/intellectual-property.pdf>), in any relevant Thesis restriction declarations deposited in the University Library, TheUni- versity Library’s regulations (see <http://www.manchester.ac.uk/library/ aboutus/regulations>) and in The University’s policy on presentation of Theses. Introduction This thesis discusses two ideas, multiversal algebra and algebraic enrichment, and one potential application for the latter, sequential scheduling. Multiversal algebra is a proposal for the reconsideration of semigroupoid and category theory within a framework that extends the approach of universal algebra which is introduced in chapter . Algebraic enrichment is the name for a series of similar constructions translating between external and internal algebraic structure that are considered in chapter . Sequential scheduling is the well-known combinatorial engine of game semantics [Hyl; Abr] which is studied in chapter from an algebraic perspective. Chapter reviews background material which, disregarding terminology, the reader is expected to be familiar with. In particular, section . on relations is just a reminder of notions that are relevant in later chapters. On the other hand, some set theoretic aspects are discussed in detail in section . and section .. These are used to establish lemma .. about the existence of quotients for equivalence relations on arbitrarily large classes. The idea of multiversal algebra is to introduce the notion of algebraic opera- tion relative to a given binary relation, as an alternative to the notion of operation on a carrier class which is standard in universal algebra [Coh]. For clarity, it was decided to only consider unary and binary operations. This is done at the endof section ., after introducing a class of relations that were named nodular. It was at a very late stage that the author understood that nodular relations are simply the ∗-regular graphs for the ∗-semigroup [Law] given by the composition and con- verse operations. Section . puts the idea of multiversal algebra in action by studying binary operations on a nodular relation. These structures were called magmoids by analogy with the term magma that was introduced by Bourbaki [Bou]. Some of the basic universal algebra results for magmas are considered to contrast the two frameworks, and show the potential of magmoids. For example, corollary .. establishes that a magmoid supports at most one pair of left and right identities, just like a magma supports at most one left and right identity. Further evidence for the emphasis on nodular relations is given by lemma .. which establishes that these graphs can be given a 2-sorted quiver structure by quo- tient, and then recovered by fibre product. This fact is used in section . to demon- strate that the notion of category derived from that of magmoid coincides with the notion of category which is studied in the literature (lemma ..). The notion of naturality from category theory is reviewed in section . as a first example of the double magmoid structures which are amenable to algebraic enrichment. The latter is related to, but distinct from, the well established notion of enrichment in category theory [Kel], but it is outside the scope of this thesis to compare these two ideas. In enriched categories it is hom sets which are given additional structure, and not necessarily of an algebraic nature. In the cases which are discussed in section ., and more abstractly in section ., it is a segment of predecessors or successors to a given object that gets enriched. The background for algebraic enrichment is an outer double magmoid which is induced by a limit product or coproduct on a category or semigroupoid, say X. In this case there is a correspondence between algebraic structure on segments of X and internal representatives of this structure, which was first exposed by Eckmann and Hilton [EH]. The construction described in section . is a first step towards establishing that such a correspondence exists in the case of a fibred limit. In preparation for the future study of this correspondence when the outer mag- moid is just a semigroupoid, the basic results of the theory of adjunctions from cat- egory theory were reconsidered for magmoids. This is done in sections . and .. It is established that the construction of adjoint and coadjoint morphisms is always possible in the case of semigroupoids, but may not be for general magmoids. The product and composition of adjunctions is also treated. Chapter presents a new perspective for the sequential scheduling [HS; HHM] of game semantics. It is established in lemma .. that the scheduling op- eration will only be compatible with the quiver structure representing the inten- ded data flow of games for a particular binary language, in fact, the one postulated in game semantics to govern the interaction of players. The main result, given in lemma .., is that scheduling interchanges with a concatenation and forms part of a double semigroupoid. This offers a new direction for the study of gamemod- els and the innocence condition using algebraic enrichment, which is described in section .. Chapter Background . Structures In order to avoid logical paradoxes while developing the theory of structures which are supported by arbitrarily large collections of objects, it is convenient to frame set theoretic assertions in the context of a fixed Grothendieck universe [Gab]. This is a class U which satisfies the following axioms [cf. Bou]. (U.I) If u is in x and x is in U, then u is also in U. (U.II) If x is in U, then the power set xP containing all subsets of x is in U. (U.III) If y = (yu ∈ U | u ∈ x ∈ U) is a family of sets in U indexed by a set x ⋃ which is also in U, then its union y = {v ∈ yu | u ∈ x} is in U. These axioms fix the semantics of the membership predicate (∈) relative to U and, by extension, also determine the U-semantics of the inclusion predicate (⊆). They were designed to capture the idea of a class that is sufficiently large to contain the values of standard set theoretic constructions on its members [Gro]. The following are valid statements for any universe U. (U.IV) If u ⊆ x and x ∈ U, then u is in U —(U.II) and (U.I). (U.V) If x ∈ U, then the boolean false 0 = ∅ is in U —(U.IV). (U.VI) If x ∈ U, then the boolean true 1 = {0} is in U —(U.V) and (U.II). (U.VII) If x ∈ U, then the boolean classifier 2 = {0, 1} is in U —(U.VI) and (U.II). The logical semantics for 2 is standard whenever 0 and 1 are the values of some indexed family. If the boolean classifier is the index set of a family with two values x and y in U, then the sets 0 and 1 are instead interpreted as left and right, respectively.