B enabou, J. 1967. Introduction to Bicategories. In Midwest Category Seminar, Volume 42 of Lecture Notes in Mathematics. Springer-Verlag, 1--77.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....satisfy the BeckChevalley condition (pullback stability) We now recall the details of the above constructions and characterise them universally. 2. 1 The bicategory of spans Spn(B) We start by recalling the de nition of the bicategory of spans on a category with pullbacks, introduced in [B en67]. 2.1. De nition. Given a category B with pullbacks, the bicategory of spans Spn(B) consists of For brevity, we may write (d R ; R; c R ) X Y for this span. the spans, commuting with the domain and codomain morphisms: A A The identity span on X is w ....

.... c)i htf; d ; T )i 2 cells a 2 cell between morphisms (f; d; R; c) and (g; d ) is simply a 2 cell between the spans (d; R; c) and (d ) in Spn(B) The evident homomorphism Sp : SP Spn(B) induces a functor between the associated classifying categories (in the sense of [B en67]) which is clearly a bration. The functor P : P SP acts as the identity on objects, while its action on morphism is as follows: given a morphism, consider a (vertical, cartesian) factorisation and map it to the pair consisting of the vertical part and the associated representable span of ....

J. Benabou. Introduction to bicategories. In Reports of the Midwest Category Seminar, volume 47 of Lecture Notes in Mathematics, pages 1-77. Springer Verlag, 1967.

....interfaces, our results introduce an expressive, two fold algebra that can serve as a speci cation formalism for rewriting systems and for composing software modules and open programs. Key words: Spans, multi relations, monoidal categories, system speci cations. Introduction The use of spans [1,6] (and of the dual notion of cospans) have been quite ubiquitous in recent years. They have been used as a syntactical device for system speci cation [4,16,17] as a foundational tool for graphs and graph rewriting [13,14] or as a semantical domain for the study of partial and multialgebras ....

....p 1 and p 2 are the projections associated to the pullback of g and h in C, if it exists (see Figure 1) Since pullbacks are unique only up to iso, it is essential to work on equivalence classes of spans. Otherwise, either a choice of pullbacks would be required, or suitable bicategories of spans [1] should be considered. Summarizing, by slightly abusing the notation we use the terminology spans to denote equivalence classes rather than concrete diagrams. Similarly, we will (almost) ignore the obvious enrichment over Span(Set) making it a 2 category [18] Each homset has a natural ....

J. Benabou. Introduction to bicategories. In Midwest Category Seminar, volume 47 of Lectures Notes in Mathematics, pages 1-77. Springer Verlag, 1967.

....the class of objects of a category C is denoted by C 0 ; the class of morphisms of C is denoted by C 1 . The notation (A; B) is used for all arrows B A, which allows us to write the composition of arrows conveniently. For a standard text on categories the reader is referred to [16] See also [2] and [16] for an overview of bicategories. In several situations where we have a bifunctor B ThetaB B (for a category B) this bifunctor is not associative. If it is, and has a unit element, our category B becomes a so called strict monoidal category. For a (relaxed) monoidal category, there ....

....a category equivalence from the tensor product Q Omega S P . 2.3 Morita theory with use of bicategories To state the Morita theory in terms of bicategories, we first need to show that rings, bimodules and bilinear maps indeed form a bicategory. This fact (without proof) was already stated in [2] and [16] Recall Section 1.1. 16 Proposition 2.3.1. For any two rings R,S, let (R; S) be the category of R S bimodules as objects, and R S linear maps as arrows. Then the collection of all rings as objects and bimodules as arrows forms a bicategory [Rings] in which the composition functor (R; ....

B'enabou, J. Introduction to bicategories, Lecture Notes in Mathematics 47 (1967) 1-77.

.... several notions of morphism between bicategories where proposed in the literature, but adjoints were not preserved by ( A two sided enrichment F : V W between two bicategories was proposed in [KLSS99] as a slight generalisation of monoidal functors and Benabou s lax functors [Ben67]. With these new morphisms one obtained a bicategory Base together with the expected pseudo functor ( Base 2 Cat. Introducing some 3 cells on Base, one eventually gets a tricategory Caten such that the 2 categories of enrichments are representable V Cat = Caten(1; V) 1 is the unit ....

J. B enabou, Introduction to bicategories, Lecture Notes in Math. 47 (Springer-Verlag), 67, 1-77.

....for Mealy automata. 2.3 Behaviour In order to study bicategories of ifo systems, we often consider structure preserving homomorphisms to other bicategories. Often such homomorphisms deserve the name behaviour. Given any finitely complete category E, recall that the bicategory Span(E) of spans ([4]) in E is a self dual compact closed bicategory (a symmetric monoidal autonomous bicategory, in the language of [13] the tensor product Omega being the obvious one inherited from the Cartesian structure of E. It thus admits a feedback operation or a trace. It fails to be a traced monoidal ....

....way. We call this category the quotient category of B, and we denote it by cBb. This construction defines a functor c Gamma b: Bicat Cat; where Bicat is the category of bicategories and homomorphisms and Cat is the category of categories and functors. 11 This construction can be found in [4]. It is also used in the definition of the Grothendieck group of a monoidal category: the Grothendieck group of a monoidal category (C; Omega ) is formed by inverting the arrows of c Sigma(C; Omega )b. Many structures that a bicategory may bear can be passed on to its quotient category. For ....

B'enabou J, Introduction to bicategories, in: Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 47, pages 1--77, Springer-Verlag, 1967.

....f(s) in B there is a corresponding unique transition from the state s in A. Furthermore, strong bisimulation of objects can be described by a span of 9 R maps in a fiber over a fixed alphabet Sigma: A oe R B. From any category X with pullbacks one can form the bicategory of spans (see [B en67] or [Bor94] in X : ffl the objects (or 0 cells) are those of X ffl the morphisms (or 1 cells) are spans (f; g) X Y in X: A X oe f Y g ffl the 2 cells are given by maps h 2 X such that P A oe f B g P 0 h g 0 oe f 0 ffl the span (1 X ; 1X ) is the identity morphism ....

J. B'enabou. Introduction to bicategories. Springer LNM, 40:1--77, 1967.

....relatively simple interface, and this is relevant from a practical point of view, as argued in the introduction. 4 On some structures for bi categories Appendix A. 2 recalls the basic definitions regarding monoidal bi categories: Most of them are standard, and can be found in classical references [2] (even if our presentation follows closely the recent survey [24] except for monoidality, for which we refer to [17] In Section 4.1 we first introduce pseudo monoids [12] then presenting our personal addition to the bi categorical folklore, namely, dgs monoidal bi categories, and spelling out ....

J. B'enabou. Introduction to bicategories. In Midwest Category Seminar, volume 47 of Lectures Notes in Mathematics, pages 1--77. Springer Verlag, 1967.

....certain sense be completed to yield a semantics for nondeterministic networks, so that the new semantics embeds the original one via an embedding that preserves important structure of the deterministic case. More precisely, we ask whether there is a way of embedding the locally posetal bicategory [B en67] Dom of Scott domains and continuous maps, into a larger bicategory Sys whose 1 cells (arrows) can serve This research was supported in part by NSF Grant CCR 9320846. as interpretations for nondeterministic dataflow networks, via a homomorphism of bicategories Dom Sys that respects the ....

.... an object of Sys and each arrow of Dom to a 1 cell of Sys, but also in the stronger sense that ordering relationships between arrows in Dom manifest themselves as unique 2 cells between the corresponding 1 cells of Sys, so that the embedding of Dom into Sys becomes a homomorphism of bicategories [B en67]. The relationship between Sys and Dom is analogous to, but more complicated than, the relationship between the category Set of sets and functions and the bicategory Rel of sets, binary relations between sets (1 cells) with inclusion relationships between binary relations as (2 cells) or more ....

J. B'enabou. Introduction to bicategories. In Reports of the Midwest Category Seminar, volume 47 of Lecture Notes in Mathematics, pages 1--77. SpringerVerlag, 1967.

....On the other hand, we can impose them only up to natural isomorphism, with these natural isomorphisms satisfying the coherence laws discussed in the previous section. This is clearly more compatible with the spirit of categorification. If we do this, we obtain the definition of weak 2 category [12]. We warn the reader that strict 2 categories are traditionally known as 2 categories , while weak 2 categories are known as bicategories . The present style of terminology, introduced by Kapranov and Voevodsky [40] has the advantage of generalizing easily to n categories for arbitrary n. ....

J. B'enabou, Introduction to bicategories, Lecture Notes in Mathematics 47, Springer Verlag, Berlin, 1967, pp. 1-77. 47

....and the second defines sylleptic monoidal bicategories. Ordinary and enriched category theory are assumed throughout this article; as usual, Mac71] and [Kel82] are references for these subjects respectively. Familiarity with 2dimensional algebra is assumed and the reader may wish to consult [B en67] or [KS74] for general theory on this subject; in particular we shall use the theory of mates [KS74] We use the string calculus of [JS93] and the definition of a monoidal bicategory [GPS95] The approach taken, and terminology used in this article follows [DS97] and [McC99b] the latter being ....

J. B'enabou. Introduction to bicategories, in Reports of the Midwest Categories Seminar, Lecture Notes in Math., vol. 47, 1--77, Springer-Verlag, 1967.

....certain kind of allegories, called unitary pretabular allegories [18] and cartesian bicategories. It would then be important to identify a natural categorical structure that could be used as a semantic domain, still retaining suitable properties of freeness. In our opinion, categories of spans [2,7] o er such a domain. Our view is supported by their use as models for circuits and predicate transformers [23,30,31] and as a syntax for nets and graphs [5,10,22] More generally, it seems to us that these (co)span categories arise naturally in the description of distributed systems with ....

....category. These correspondence results are, to the best of our knowledge, new. Our work is still preliminary, since on the one side our results could be generalised to (co)spans over complete and (co)complete categories; on the other side, we could consider the bicategorical enrichment [2,6] over (co)spans. For the time being, we conclude the paper by presenting a table that summarizes the equational properties of the structured categories we presented, and relating our results of to the ongoing work on spans over the category of graphs [10,11,20] 2 1 Preliminaries We introduce ....

[Article contains additional citation context not shown here]

J. Benabou. Introduction to bicategories. In Midwest Category Seminar, volume 47 of Lectures Notes in Mathematics, pages 1-77. Springer Verlag, 1967.

....we characterise them in terms of nite trees and their grafting composition (Prop. 5.2) Having achieved the basic identi cation of Def. 4.3, we proceed to set up the 2 category of multicategories. In order to do so, we continue the reformulation of internal category theory (along the lines of [B en67] mentioned above) giving an algebraic interpretation of natural transformations in terms of morphisms of bimodules (Prop. 6.1) as well as an analysis of their horizontal and vertical compositions. We then instantiate this algebraic formulation of transformations in the context of ....

.... I I I I I C 1 d 1 C 1 c 1 B B B B B B C 0 C 1 d1 oo c 1 C 0 C 0 id iiR R R R R R R R R R R R R R R OO id 55 l l l l l l l l l l l l l l l 9 This point of view of internal categories as monoid structures on graphs was pioneered by B enabou in [B en67]; it is an essential insight which lies at the heart of our present work. Having identi ed categories with monads, we could expect the rest of the structure (namely functors and natural transformations) to follow from this identi cation. This is not quite as straightforward: Street s original ....

[Article contains additional citation context not shown here]

J. Benabou. Introduction to bicategories. In Reports of the Midwest Category Seminar, volume 47 of Lecture Notes in Mathematics, pages 1-77. Springer Verlag, 1967.

....if X is a homotopy 1 type. However, if X is a homotopy 1 type, we can classify it up to homotopy equivalence using 1 (X) There is a suggestive pattern here. To see how it continues, we need to categorify the notion of category itself This gives the concept of a 2category or bicategory [10, 20]. We will not give the full de nitions here, but the basic idea is that a 2 category has objects, morphisms between objects, and also 2 morphisms between morphisms, like this: Just as a category allows us to distinguish between equality and isomorphism for objects, a 2 category allows us to make ....

J. Benabou, Introduction to bicategories, Lecture Notes in Mathematics 47, Springer Verlag, Berlin, 1967, pp. 1-77. 28

....we characterise them in terms of finite trees and their grafting composition (Prop. 5.2) Having achieved the basic identification of Def. 4.3, we proceed to set up the 2 category of multicategories. In order to do so, we continue the reformulation of internal category theory (along the lines of [B en67] mentioned above) giving an algebraic interpretation of natural transformations in terms of morphisms of bimodules (Prop. 6.1) as well as an analysis of their horizontal and vertical compositions. We then instantiate this algebraic for4 mulation of transformations in the context of ....

.... fflffl I I I I I I I C 1 d 1 C 1 c 1 B B B B B B C 0 C 1 d 1 oo c 1 C 0 C 0 id ii R R R R R R R R R R R R R R R OO id 55 l l l l l l l l l l l l l l l This point of view of internal categories as monoid structures on graphs was pioneered by B enabou in [B en67]; it is an essential insight which lies at the heart of our present work. Having identified categories with monads, we could expect the rest of the structure (namely functors and natural transformations) to follow from this identification. This is not quite straightforward: Street s original ....

[Article contains additional citation context not shown here]

J. B'enabou. Introduction to bicategories. In Reports of the Midwest Category Seminar, volume 47 of Lecture Notes in Mathematics, pages 1--77. Springer Verlag, 1967.

....of a linear category. For illustration we consider Abramsky s category SP roc, which arises from the deterministic transition systems and cover system for bisimulation of the previous section. 3. 1 The process construction Given a category X with pullbacks, the bicategory Span(X) of spans in X [B en67] is given as follows: 14 Delta the 0 cells A are those of X; Delta the hom category Span(X) A; B) has as objects f : A ; B spans B A . f 1 f 0 in X, and as arrows f ) g the morphisms x of X for which . B A x g 1 g 0 f 1 f 0 commutes in X; Delta horizontal identities i A : A ; A ....

J. B'enabou. Introduction to bicategories. Lecture Notes in Mathematics, 40:1--77, 1967.

....left factor closed. The cover system for weak bisimulation is obtained by constructing a stable functor from the Kleisli category TranD back to Tran and taking the preimage of 9 R . 2. 4 Process categories From any category X with pullbacks one can form the bicategory of spans in X (see B enabou [Ben67]) the objects are those of X, 1 cells A Gamma B are spans (f; g) in X, and 2 cells (f; g) Gamma (f 0 ; g 0 ) are maps h of X such that P B A P h g f g f commutes in X. Span composition is given by pullback i.e. f; g) h; k) is (p; f; q; k) where: C B A Q P R q p k h g f A cover ....

B'enabou J., Introduction to bicategories. Lecture Notes in Mathematics 47, Springer Verlag, 1967.

....comments on an earlier draft. A Spans, twists and oplaxness The axioms for bicategories are similar to those for 2 categories except that identity and associativity axioms hold up to isomorphism rather than equality and are subject to coherence conditions. For details see Borceux [5] or B enabou [4]. One can quotient the 1 cells of a bicategory to obtain a category. Following B enabou, the category obtained by identifying all 1 cells which are 2 isomorphic is the classifying category . We write B [ for the classifying category for a bicategory B. Given a category C with pullbacks, the ....

Jean B'enabou. Introduction to bicategories. Lecture Notes in Mathematics, 47, 1967.

....3. 4 Every fibration F : D C with the finite limits in C and the finite fibrewise products in D induces a free regular fibration Cod : F=C C : hR; r : F(R) Ai 7 A: The observation that the comma category F=C freely adds the stable direct images to the fibred category D is due to B enabou [2]. These direct images are obtained simply by composing: f hR; ri : hR; r; f)i Since the finite fibrewise products can be freely added to any fibration, this construction actually yields a left adjoint to the inclusion of regular fibrations in the general ones, over a finitely complete base. ....

....questions, let us describe in more detail the setting in which the maps are defined. Bicategory of predicates. A regular fibration P induces a bicategory of predicates, just like a regular category induces an allegory [12] For a formal introduction to bicategories, the reader is referred to [2]. By definition, the objects of the bicategory of predicates R = R P , associated with the regular fibration P : C, are the objects of C. The hom category from A to B consists of the predicates on A Theta B: R(A;B) A Theta B) A predicate R(x; y) 2 (A Theta B) thus appears as the ....

J. B'enabou, Introduction to bicategories, in: Reports of the Midwest Category Seminar I, Lecture Notes in Mathematics 47 (Springer, 1967) 1--77

....transformations. A natural transformation : h h 0 is said to be lax monoidal if j ; j 0 and ( AB ; A Omega B ) Gamma ( A Theta B ) 0 AB Delta . Proposition 2.2 R=Bij [R; Set] lax Omega 2.2 Specifying types Definition 2. 3 Let R be a category and B a bicategory [7]. A lax functor P : R B is an assignment for each object A of R of an object PA in B and for each arrow f : A B of a 1 cell Pf : PA PB in B. Furthermore, P comes equipped with the 2 cells fg : Pf ; Pg Gamma P(f ; g) for every composable f and g, and jA : id PA Gamma P(id A ) for ....

J. B'enabou, Introduction to bicategories, in: Reports of the Midwest Category Seminar I, Lecture Notes in Mathematics 47 (Springer, 1967) 1--77

....up to equivalence, and so on. For example, a weak 1 category is just an ordinary category, defined by Eilenberg and MacLane [15] in their 1945 paper. In a category, composition of 1 morphisms is associative on the nose : fg)h = f(gh) Weak 2 categories first appeared in the work of B enabou [9] in 1967, under the name of bicategories . In a bicategory, composition of 1 morphisms is associative only up to an invertible 2 morphism, the associator : A f;g;h : fg)h f(gh) The associator allows one to rebracket parenthesized composites of arbitrarily many 1 morphisms, but there may be ....

J. B'enabou, Introduction to bicategories, Lecture Notes in Mathematics 47, Berlin, 1967, pp. 1-77.

....3. 4 Every fibration F : D C with the finite limits in C and the finite fibrewise products in D induces a free regular fibration Cod : F=C C : hR; r : F(R) Ai 7 A: The observation that the comma category F=C freely adds the stable direct images to the fibred category D is due to B enabou [2]. These direct images are obtained simply by composing: f hR; ri : hR; r; f)i Since the finite fibrewise products can be freely added to any fibration, this construction actually yields a left adjoint to the inclusion of regular fibrations in the general ones, over a finitely complete base. ....

....questions, let us describe in more detail the setting in which the maps are defined. Bicategory of predicates. A regular fibration P induces a bicategory of predicates, just like a regular category induces an allegory [12] For a formal introduction to bicategories, the reader is referred to [2]. By definition, the objects of the bicategory of predicates R = R P , associated with the regular fibration P : C, are the objects of C. The hom category from A to B consists of the predicates on A Theta B: R(A;B) A Theta B) A predicate R(x; y) 2 (A Theta B) thus appears as the ....

J. B'enabou, Introduction to bicategories, in: Reports of the Midwest Category Seminar I, Lecture Notes in Mathematics 47 (Springer, 1967) 1--77

....corresponds to a choice of communication, the choice of the behaviour corresponds to a choice of an abstraction level. The semantic categories are very often of the type considered in iteration theories [Elg75] BE93] The abstract mathematical setting (strictly speaking the theory of bicategories [B en67], CW87] CKW87] is not emphasized in this paper, rather the application to concurrency. A more precise analysis of the mathematical setting is begun in [KSW94] It has come as some surprise to the second author that his earlier interest in bicategories has been reawakened by these ....

J. B'enabou. Introduction to bicategories. In Reports of the Midwest Category Seminar, Lecture Notes in Mathematics, number 47, (Springer-Verlag, Berlin, 1967) 1--77

....there is no cause for ambiguity, is also denoted by Omega . Let N denote the additive monoid of natural numbers, that is, the one object category generated by one arrow. If B is a bicategory, let Omega B denote the bicategory of functors, lax transformations and modifications from N to B. See [2] for more on these and other bicategorical notions. Explicitly, we have: ffl an object of Omega B is an endomorphism in B, X : a a; Katis, Sabadini and Walters 4 ffl an arrow from X : a a to Y : b b is a pair (U; ff) where U is an arrow and ff is a 2 cell in B of the form a X Y U U a ....

B'enabou J., Introduction to bicategories, in: Lecture Notes in Mathematics, Vol. 47 (Springer, Berlin, 1967) 1-77.

....functors (in some cases these are strong monoidal homomorphisms) to other (bi)categories. Often such functors deserve the name behaviour. For example, there is a strong monoidal homomorphism B : Omega ( Sigma(Set; Theta) Span(Set) Omega ) where Span(Set) is the bicategory of spans ([4]) in Set. It maps an object X to the set X N and, if (U; ff) X Y is a circuit, B(U; ff) is the span of sets X N W Y N where W = f(x i ; u i ; y i ) i2N j ff(x i ; u i ) u i 1 ; y i )g. So an element of B(U; ff) is a behaviour of (U; ff) as described in the previous subsection on ....

B'enabou J, Introduction to bicategories, in: Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 47, pages 1--77, Springer-Verlag, 1967.

No context found.

B enabou, J. 1967. Introduction to Bicategories. In Midwest Category Seminar, Volume 42 of Lecture Notes in Mathematics. Springer-Verlag, 1--77.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC