S. Mac Lane and R. Pare, Coherence in bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), 59-80.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this end, we shall examine here for morphisms of bundles the notion of descent equivalence, which was introduced in the rst author s Ph.D. thesis [3] and study its properties. We formulate this notion in the (essentially equivalent) language of internal categories and of indexed categories (see [5,6,7]) rather than that of brations, making extensive use of some of the results of [5] which we recall here in sucient detail. After some preliminary observations concerning descent equivalences and their comparison with e ective descent morphisms, in Theorem 1 we give a somewhat surprising ....

....constructs the internal functor q: Eq(p) Eq( where q 0 = q; q 1 = q q. Then, for a xed object B of C, the assignments: E; p) 7 Eq(p) and q 7 q; de ne the functor Eq B : C=B cat(C) 2. Indexed categories. A C indexed category A or a pseudo functor A : C CAT (cf. [5,7,8]) consists of the following data: for every object E of C a category A for every morphism f : E D of C a functor f for every f : E D; g : D B in C, two natural isomorphisms: 1 A D (1 D ) j : f (gf) which make the diagrams (1 E ) 1 D ) ....

S. Mac Lane and R. Pare, Coherence in bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), 59-80.

....made essential contributions, and Todd Trimble and Carsten Butz provided helpful advice and comments on an earlier draft of this paper. 1 Slices, stacks, and sheaves Throughout this section, let E be a fixed small topos. We begin by defining the E indexed category E (for indexed categories, see [20], 23] Recall that an E indexed category A is essentially the same thing as a pseudofunctor A : E op CAT, i.e. a contravariant functor up to isomorphism on E with values in the category CAT of (possibly large) categories. Since the only indexed categories to be considered here are ....

....functor Sigma ff : E=J E=I along ff. Thus for any composable pair of morphisms K fi J ff I, there is a canonical natural isomorphism OE ff;fi : fi ff Gamma (fffi) 1) Furthermore, these OE ff;fi for each composable pair ff; fi then satisfy the required coherence conditions (cf. [20]) making E= an indexed category. Observe also that since E is small, each E=I is a small category, and so E= is a small indexed category. An indexed category is called strict if all of its canonical natural isomorphisms (1) are identities. Thus a small, strict indexed category is the same thing ....

S. Mac Lane and R. Par'e. Coherence for bicategories and indexed categories. Journal of Pure and Applied Algebra, 37:59--80, 1985.

....this end, we shall examine here for morphisms of bundles the notion of descent equivalence, which was introduced in the first author s Ph.D. thesis [3] and study its properties. We formulate this notion in the (essentially equivalent) language of internal categories and of indexed categories (see [5,6,7]) rather than that of fibrations, making extensive use of some of the results of [5] which we recall here in sufficient detail. After some preliminary observations concerning descent equivalences and their comparison with effective descent morphisms, in Theorem 1 we give a somewhat surprising ....

....constructs the internal functor q: Eq(p) Eq( where q 0 = q; q 1 = q Theta B q. Then, for a fixed object B of C, the assignments: E; p) 7 Eq(p) and q 7 q; define the functor Eq B : C=B cat(C) 2. Indexed categories. A C indexed category A or a pseudo functor A : C op CAT (cf. [5,7,8]) consists of the following data: Delta for every object E of C a category A E Delta for every morphism f : E D of C a functor f : A D A E , Delta for every f : E D; g : D B in C, two natural isomorphisms: i D : 1 A D (1 D ) j f;g : f g (gf) which make the ....

S. Mac Lane and R. Par'e, Coherence in bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), 59-80.

....B (A; f) The condition FE = 1 C 2 translates into each F B preserving the terminal object 1B of C=B. How can F be recovered from its slices Assuming C to be finitely complete, B 7 C=B) becomes a C indexed category, with v : C=D C=B for v : B D in C given by pullback (cf. PS] [MLP]) also called the basic fibration of C. Now F B : C=B C=B becomes a lax C indexed functor, with the needed natural transformation ff v : F B v v F D given, as follows: for every object (C; g) in C=D, let (2) be the defining pullback for (A; f) v (C; g) and put v (C;g) ....

S. Mac Lane and R. Par'e, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985) 59-80.

....S. E. CRANS ON BRAIDINGS, SYLLEPSES, AND SYMMETRIES tures appropriate for a coherence theorem for weak n categories. In this section I will give precise definitions up to dimension 4, and a heuristic approach for higher dimensions. 2. 1 Dimension 3 Because of the coherence theorems for bi [29] and tricategories [15] lowdimensional teisi are familiar categorical structures. Definition 2.1 A 0 dimensional tas is a set. A 1 dimensional tas is a category. A 2 dimensional tas is a 2 category. 3 Recall that Gray is the monoidal category of 2 categories and 2 functors with tensor product ....

S. Mac Lane and R. Pare, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), 59--80.

....of these two calculi. A corollary is that all proofs of an equality in the substitution calculus are equal. We thus solve a coherence problem similar to the coherence problems studied in category theory. Compare for example the proof of coherence for indexed categories by MacLane and Par e [9]. The present development is closely related to the work by Curien 2 on Substitution up to isomorphism [5] There he introduces an explicit syntax for dependent types (including Pi types) and interprets it in a locally cartesian closed category. He shows several coherence properties, for ....

S. Mac Lane and R. Par'e. Coherence for bicategories and indexed categories. Journal of Pure and Applied Algebra, 37:59--80, 1985.

....would be between (homo)morphisms instead of between 2 functors, and one would probably get tricategories via some theory of weak enrichment . And in category theory, equality often is important, as can be seen from Kelly s body of work [24, 23, 4] and from the abundance of coherence theorems [27, 29, 30, 15]. No, the conceptual difference lies in the treatment of dimension. The cartesian product of 2 categories, and of categories, is basically set theoretical: C Theta D has as basic ingredient pairs (x; y) of dimension p for x 2 C p and y 2 D p , and functoriality then gives, more generally, pairs ....

S. Mac Lane and R. Par'e, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), 59--80.

....the coherence theorem is equivalent to the statement that all the diagrams in the free monoidal category commute. This is the approach we are employing in this formalisation. The coherence theorem has since been generalised to other kinds of categories. In some cases (for example, bicategories [23]) the statement was similar: there was some canonical category, which totally commuted. In other cases (as in biclosed monoidal categories) this did not work, so Lambek [20] restated the coherence problem as a decision problem whether two arrows in a certain free category are equal. In this ....

Saunders Mac Lane and Robert Par#. Coherence for bicategories and indexed categories. Journal of Pure and Applied Algebra, 37:59 80, 1985.

....prove that derivations yielding the same judgement are given the same meaning. This idea has also appeared in the context of category theory and our use of the term coherence is partially inspired by its use there, where it means the uniqueness of certain canonical morphisms (see e.g. KL71] and [LP85]) Although we have not attempted a rigorous connection in this paper, the possibility of unifying coherence results for a variety of different calculi offers an interesting direction of investigation. In the case of Fun, we show the coherence of our semantic approach by proving that ....

S. Mac Lane and R. Pare. Coherence for bicategories and indexed categories. Journal of Pure and Appled Algebra, 37:59--80, 1985.

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