Hermida, C. Representable multicategories. Adv. in Math. 151 (2000), 164225. 27  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... mathematical structures may be generalized quite dramatically, so as to include geometric structures like topological spaces and the much lesser known approach spaces (see [21] but also Lambek s [19] multicategories which enjoy renewed interest in higher dimensional category theory (see [13], 14] Indeed, it is well known that a topological space may be completely described by a convergence relation, i.e. by a function 2, where UX is the set of ultrafilters on X satisfying the two basic axioms . m(X) z) Here x x means that the ultrafilter x on X ....

.... (pre)metric space multi metric space approach space fuzzy T space Set category multicategory ultracategory T category V V category V multicategory V ultracategory (T , V) category Here M denotes the free monoid monad on Set, which was used also by Burroni [7] Leinster [20] and Hermida [13] to describe multicategories. While their approach (working with the bicategory Span T (B) for a cartesian monad T on a category B with pullbacks) allows for a good definition of internal multicategories, ours (working with Mat(V) instead) leads to an easy V enrichment, thus automatically ....

C. Hermida, Representable multicategories, Adv. Math. 151 (2000) 164-225.

....is conceptually analogous to Street s definition, except that it works one n at a time. Their n opetopic sets are given by a presheaf category, and their n categories are suitably restricted n opetopic sets. Variants of their definition are given and studied by Hermida, Makkai, and Power [70, 71, 72, 73, 73, 75], Makkai and Zawadowski [106, 108, 156] Leinster [94, 95] and Cheng, who also gives comparisons among definitions within this family [32, 33, 34, 35] Another conceptually similar definition, using an alternative diagram scheme, has recently been given by Higuchi, Miyada and 15 Tsujishita [76, ....

Claudio Hermida, Representable multicategories, Advances in Mathematics 151 (2000), no. 2, 164--225.

....requirement is easily derived. Composing the empty string of multi 2 cells with an input free multi 2 cell leaves unchanged. Advances in higher dimensional category theory led to renewed interest in T categories in the mid 1990 s, notably by Tom Leinster [13] and by Claudio Hermida [9]. More recently, Maria Manuel Clementino and Walter Tholen have opted to start from a bicategory of V valued relations or matrices for symmetric monoidal closed V instead of from a bicategory of spans [4] This leads to a notion of V enriched T category . A priori there is no provision for ....

Hermida, C., Representable multicategories, Adv. in Math. 151 (2000), pp. 164{ 225.

....provides the local composition of poly 2 cells . In contrast to the situation for multi bicategories, poly bicategories provide a natural setting for a notion of adjunction. A corresponding calculus of Australian mates is available as well. In Section 2, we adapt an insight of Claudio Hermida [13], who had, in particular, noticed that the coherence requirements for bicategories can usefully be expressed in terms of the universal multi properties expected of composite 1 cells. In the same manner the coherence issues for linear bicategories can alternately be expressed in terms of the ....

.... such functions by the tensor: then one characterizes them as homomorphisms of Abelian groups equalizing the m homomorphisms from R 0 Rm 2 #Mm 1 induced by the left and right actions, as well as module homomorphisms with respect to the left action of 0 and the right action , cf. [13]. Note then that the multi bicategory structure is more natural than the bicategory structure with the tensor; in the present context, the latter amounts to the representability of the multi bicategory (Section 2) Separating these notions (multi linear functions and tensors, or more generally, ....

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Hermida, C. Representable multicategories. University of Sydney, June 1999.

....operation in (1) is to compose i and j simultaneously with ; the sequential compositions can be simulated by arti cially inserting identity 2 cells, but yield the same result. The notion of (internal) T category was rediscovered independently by Tom Leinster [11] and by Claudio Hermida [8] in the mid 1990 s with the intension to advance higher dimensional category theory. More recently, Maria Manuel Clementino and Walter Tholen have explored the option of starting from a bicategory of V valued relations or matrices for symmetric monoidal closed V instead of from a bicategory of ....

.... V enriched T category . Notice that a priori there is no provision for horizontally composing multi2 cells, not even 1 cells. Such an operation , called tensor product in the untyped case, only becomes available in the context of what Hermida called representable multicategories , cf. [8]. Manfred E. Szabo in 1975 generalized multicategories to polycategories by allowing strings of 1 cells also as output [13] In terms of diagrams, one now is concerned with the compositional properties of poly 2 cells of the form V V V V V hm 1 : ....

Hermida, C., Representable multicategories, Adv. in Math. 151 (2000), pp. 164{ 225.

....may be composed with any multi 2 cell with no inputs, leaving unchanged. The substitution operation above then arises as a special case where all but possibly one i are identity 2 cells. The notion of T category was rediscovered independently by Tom Leinster [8] and by Claudio Hermida [6] in the mid 1990 s. Notice that there is no provision for horizontally composing multi 2 cells, not even 1 cells. These operations become available in the context of Hermida s representable multicategories , cf. 6] Manfred E. Szabo in 1975 generalized multi categories to poly categories by ....

....was rediscovered independently by Tom Leinster [8] and by Claudio Hermida [6] in the mid 1990 s. Notice that there is no provision for horizontally composing multi 2 cells, not even 1 cells. These operations become available in the context of Hermida s representable multicategories , cf. [6]. Manfred E. Szabo in 1975 generalized multi categories to poly categories by allowing strings of 1 cells also as output [9] Geometrically, one now is concerned with the compositional properties of poly 2 cells of the form f0 f1 : fn 1 g0 g1 : gm 1 (0 02) Here composition corresponds ....

Hermida, C. Representable multicategories. Adv. in Math. 151 (2000), 164-225. 16

....is conceptually analogous to Street s definition, except that it works one n at a time. Their n opetopic sets are given by a presheaf category, and their n categories are suitably restricted n opetopic sets. Variants of their definition are given and studied by Hermida, Makkai, and Power [66, 67, 68, 69, 69, 71], Makkai and Zawadowski [102, 104, 148] Leinster [89, 90] and Cheng, who also gives comparisons among definitions within this family [28, 29, 30, 31] Another conceptually similar definition, using an alternative diagram scheme, has recently been given by Higuchi, Miyada and Tsujishita [72, 114] ....

Claudio Hermida, Representable multicategories, Advances in Mathematics 151 (2000), no. 2, 164--225.

....the local composition of poly 2 cells . In contrast to the situation for multi bicategories, poly2 bicategories provide a natural setting for the notion of adjunction. A corresponding calculus of Australian mates is available as well. In Section 2, we adapt an insight of Claudio Hermida [10], who had, in particular, noticed that the coherence requirements for bicategories can usefully be expressed in terms of the universal multi properties expected of composite 1 cells. In the same manner the coherence issues for linear bicategories can be alternately be expressed in terms of the ....

.... Mm 1 to N . These are homomorphisms of Abelian groups equalizing the m homomorphisms from M 0 R 0 M 1 Rm 2 Mm 1 to M 0 M 1 Mm 1 induced by the left and right actions, as well as module homomorphisms with respect to the left action of M 0 and the right action of Mm 1 , cf. [10]. b) Since rings with unit are just monads in ab , the monoidal category of abelian groups, and monoidal categories are just one object bicategories, the previous example admits a generalization to monads in any bicategory B and modules between such. c) The previous examples can be generalized ....

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Hermida, C. Representable multicategories. University of Sydney, June 1999.

....of Graph, given the importance of the resulting categories in the modeling of the operational behaviour of rewriting systems and of automata, as pointed out in [9,11,20] and [30,31] respectively. In this sense, we consider interesting the results on monads and pseudo algebras presented in [25]. ....

C. Hermida. Representable multicategories. Submitted.

....intended example of the cartesian closed paracategory of bivariant functors T : with a cartesian closed category) and dinatural transformations between these (Example 5.6) In [Mat00] the combination of probabilistic automata involves a certain product without diagonals . Following [Her00], we introduce partial multicategories (which is an instance of the abstract notion of paramonoid or saturated partial algebra from [HM02] and de ne the appropriately restricted notion of representability . This framework allows us to exhibit the above product without diagonals as the tensor ....

....saturated partial algebra. Here we present another useful instance of this latter notion, namely partial multicategories . Such structures allow us to capture certain tensor products which arise in the context of probabilistic automata (x8) Our reference for the theory of multicategories is [Her00]. As indicated there, an internal multicategory with object of objects C 0 is an internal monoid in Spn M (B ) C 0 ; C 0 ) Hence a partial multicategory is an internal paramonoid in Spn M (B ) C 0 ; C 0 ) We will not indulge here in this internal version of partial multicategories and give ....

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C. Hermida. Representable multicategories. Advances in Mathematics, 151:164-225, 2000. Available at http://www.cs.math.ist.utl.pt/s84.www/cs/claudio.html.

....map. The resulting structure is then an instance of the general notion of partial algebra for a monad which we proceed to examine. 13 5. 1 Partial algebras for a monad Given B with finite limits and a class of monos M, consider a monad hT ; i on B , it the monad is cartesian in the sense of [Her00], we get an induced monad Ptl M (T ) Ptl M (B ) Ptl M (B ) applying T to a partial map span yields another such) with unit (id ; and multiplication (id ; analogous to the Spn T construction in the Appendix of ibid. Now we can consider lax algebras for Ptl M (T ) which we call partial ....

.... Kleene morphism) factorisation of the unit x : x UE(x) of the adjunction of Theorem 5.8 An interesting instance of the notion of partial algebra appears in our follow up article [HM02] partial multicategories. For a partial multicategory we formulate representability , in the sense of [Her00] (for a restricted class of morphisms) A representable partial multicategory provides a suitable axiomatisation for certain partial tensor products which arise in the context of probabilistic automata [Mat00] We conclude our brief incursion into partial algebras by pointing out that they ....

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C. Hermida. Representable multicategories. Advances in Mathematics, 151:164-225, 2000. Available at http://www.cs.math.ist.utl.pt/s84.www/cs/claudio.html.

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C. Hermida. Representable multicategories. Advances in Mathematics, 151:164-225, 2000. Available at http://www.cs.math.ist.utl.pt/s84.www/cs/claudio.html.

....ordinary monos in Set) this span is itself a partial map. Thus, in the presence of the free monoid monad T , the data for a paramonoid can be organised into a single partial map. Given B with nite limits and a class of monos M, consider a monad hT ; i on B which is cartesian in the sense of [Her00], i.e. 1. The functor T preserves pullbacks 2. The unit : id ) T and multiplication : T ) T are cartesian transformation, meaning that the naturality squares are pullbacks. and T (M) M. We have then an induced monad Ptl M (T ) Ptl M (B ) B ) applying T to a partial map span ....

C. Hermida. Representable multicategories. Advances in Mathematics, 151:164-225, 2000. Available at http://www.cs.math.ist.utl.pt/s84.www/cs/claudio.html.

....and therefore they must be provided as part of the data, and their coherence conditions veri ed. However, if we regard such monoidal structure as algebraic structure on a multicategory M (with underlying category C ) it becomes a universal property, namely that M be representable in the sense of [Her99b]. The developments in ibid. lead us to seek a systematic way of achieving such a transformation. That is, given the 2 monad T on Cat whose pseudoalgebras are monoidal categories (thus T C = freemonoidonC ) we would like to derive the 2 category Multicat of multicategories and the 2 monad T 0 on ....

....cf. x6. We thus get an e ective and simple description of the classi er of lax morphisms out of a (pseudo )T algebra, and we recover some important monoid classi ers, cf. x9.1 and x10.3. Our consideration of bimodules was motivated by the pursuit of the theory of representable multicategories in [Her99b]. In trying to understand how to deal with the hom of a multicategory M , we observed that the hom sets of (multi)arrows M ( x; y) could be organised into a bimodule M : TM 6 M , where M is the category of linear morphisms (those with a singleton domain) of M and TM is the free strict ....

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C. Hermida. Representable multicategories. available at http://www.maths.usyd.edu.au:8000/u//hermida/, June 1999.

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Hermida, C. Representable multicategories. Adv. in Math. 151 (2000), 164225. 27

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Hermida, C. Representable multicategories. Adv. in Math. 151 (2000) 164--225.

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C. Hermida. Representable multicategories. Advances in Mathematics, 151:164-- 225, 2000.

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Claudio Hermida, Representable multicategories (2000). Advances in Mathematics 151, pp. 164--225.

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