R. Street. Fibrations in Bicategories. Cahiers de Topologie et Geometrie Differentielle, vol. XXI-2, pages 111-160, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and Cat ( Cat 0 ) the objects are small categories with colimits of chains (and an initial object) the morphisms are functors that preserve these colimits, and the 2 cells are natural transformations. Concerning exactness properties in 2 categories we will focus on bicategorical (co)limits [29]. We exemplify this notion with the most basic example. Bicategorical (or pseudo) initial object. An object 0 in a 2 category is said to be pseudo initial if, for every object C, there exists a morphism : 0 C such that for every morphism c : 0 C, we have that = c via a unique pseudo ....

....and (2) We will indicate these cones as the canonical cones hg l;n f n;l i l = fl n n and h n fl n i = 1A , respectively. Bicategorical colimits. A pseudo cone h Phi n : n 1 f n = n : A n Ai for an chain hf n : A n A n 1 i is said to be a bicategorical (pseudo) colimit [29] if it satisfies the following universal property: 1. For every pseudo cone h Psi n : n 1 f n = n : A n Xi there exists an arrow u : A X and an indexed family of pseudo cells h n : u n = n i such that n Delta (u Phi n ) Psi n Delta ( n 1 f n ) u n 1 f n ) n . 2. ....

R. Street. Fibrations in Bicategories. Cahiers de Topologie et Geometrie Differentielle, vol. XXI-2, pages 111-160, 1980.

.... F (M) commutes; the functor P F sends an arrow X Y of H to [X Y Y Omega I Y Omega F (I) I] X ffl Y ; if N is an object of G ; the component at N of F is [F (N) I Omega F (N) N ] F (N) ffl I : The kernel and the cokernel are special instances of bilimits (see [17]) and are determined, up to monoidal equivalences, by their universal property, which is discussed in detail in [24] In section 2 we use the explicit description of the kernel and of the cokernel, but in section 4 we essentially use their universal property. For this we recall here the universal ....

R. Street, Fibrations in bicategories, Cahiers Topologie G'eom'etrie Diff'erentielle 21 (1980) 111-160.

....that set ( Gamma) op is a monad on CAT (the putative unit, the Yoneda embedding, exists only for locally small arguments) it has the Kock Lawvere property and thus arises, roughly speaking, from a homomorphism with domain Delta, explicitly considered as a 2 category. We refer the reader to [19] for details. The ideas of Proposition 6.1 also apply to completely distributive lattices, as in Example 5. For suppose that Y : B Gamma C is the down segment embedding of an ordered set, B, into its lattice of down closed subsets and that we have an adjoint string of length 5, say U a V a W a X ....

....is also to be found in [10] In [6] Kock observed that, as a monoidal 2 category, Delta is generated by Lawvere s data and equations and the transformation 0 1 : 1 Gamma 2 subject to the two equations saying that this transformation is identified by both 0 Gamma 1 and 2 Gamma 1. In [19] Street pointed out that Delta, regarded as a cosimplicial complex in CAT, is generated by adjunction and pushout from the unique functors 0 Gamma 1 Gamma 2. That is to say Theory and Applications of Categories, Vol. 1, No. 6 144 n n 1 0 n Gamma 1 n 0 n Gamma1 n ....

R. Street. Fibrations in bicategories. Cahiers de topologie et g'eometrie diff'erentielle, XXI:111--160, 1980.

....spans in Auto have certain special properties pertaining to inputs and outputs, and then attempting to identify categorical properties that characterize the dataflow like spans. I found that the dataflow like spans in Auto could be described in terms of Street s notion of 0 fibration [12, 13], which adapts to more general 2 categories the notion of opfibration in Cat [2] Street s theory characterizes fibrations in a 2 category as being the algebras of a certain kind of 2 monad called a KZ doctrine. For dataflow networks, the endo 2 functor underlying this 2 monad corresponds to ....

....0 there corresponds a functor b : F Gamma1 (x) F Gamma1 (x 0 ) and a natural transformation b : J x Delta Gamma J x 0 b . The components of the natural transformations b are called cartesian morphisms and the components of b are called opcartesian morphisms. Street [12, 13] has developed an abstract theory of fibrations, so that the notion can be applied, not just in Cat, but more generally to any bicategory with suitable completeness properties. Here we summarize the basics of Street s theory as it applies to the 2 category Cat. Let Spn(A; B) denotes the 2 category ....

R. H. Street. Fibrations in bicategories. Cahier de Topologie et G'eometrie Diff'erentielle, XXI-2:111--159, 1980.

....k j T l; T k , T ( k) T k 0 (T :Tk) Tk 0 T ;k :T k 0 T ( k)k 0 ) T ( kk 0 ) T i T k;k 0 T : T k:T k 0 ) i T :T (kk 0 ) T ;kk 0 . Theory and Applications of Categories, Vol. 4, No. 5 97 2.15. The right fibrations from L to K are described in [ST2] as the algebras for a KZdoctrine RL;K on spnCAT(L;K) It takes but a simple extension of that account to show that those objects in [ CAT] which are right fibrations are the algebras for a single KZdoctrine R on [ CAT] with R restricting to the RL;K . Left fibrations are algebras for a ....

R. Street. Fibrations in bicategories. Cahiers de Topologie et G'eometrie Diff'erentielle, 21:111--160, 1980.

....i.e. for every small category C , the functor C : T 2 C Gamma T C is left adjoint to the functor j TC : T C Gamma T 2 C , and the adjunctions are preserved by functors H : C Gamma D . It is equivalent to ask that be right adjoint to Tj, with the identity being the unit (see [13, 21]) The notion of KZ monad was introduced to study particular features of 2monads given by free completions under classes of colimits [13] But they do not characterise such free completions, as the following example shows. Example 12. Consider the 2 monad on Cat that sends every category to the ....

.... ) T D ; T (C D ) have coproducts that are preserved by composition and the coprojections T (i) T C Gamma T (C D ) Gamma T (D ) T (j) have right adjoints (that become projections for the product) Such a result allows us to conclude that T (C D ) is a product in a bicategorical sense [21] but not in the strict sense we are asserting; that is, from the limit colimit coincidence we could only deduce (in principle) an equivalence T (C D ) T C Theta TD but not an isomorphism as we do in the proof of Theorem 25. Nonetheless the link of Theorem 25 to the more general question of ....

R. Street. Fibrations in Bicategories. Cahiers de Topologie et G'eom'etrie Diff'erentielle, XXI(2):111-160, 1980.

....appropriate conditions involving an adjunction between structure and unit as here. In [22] Kock considered a 2 functor T : K K and 2 natural transformations m : T 2 T and i : 1 T satisfying the unit conditions strictly, but with m being associative only up to coherent isomorphism; in [27], Street considered the general bicategorical notion of monad on a bicategory, calling them doctrines ; in [25] Marmolejo considered (the formal theory of) pseudo monads. All three authors had notions of a monad with structure adjoint to unit in a suitable sense, and in all three cases the ....

R.H. Street, Fibrations in bicategories, Cahiers de Topologie et G'eometrie Differentielle, XXI2: 111--160, 1980.

....situation, this doctrine corresponds to the construction compose with an input buffer. Thus, the dataflow like spans are those spans that are algebras of the input buffering doctrine. The theory of fibrations was first developed in terms of concrete constructions on categories [3] Then, Street [14, 15], building on work of Gray [2] showed that this theory has a bicategorical formulation, which can be applied not only to the 2 category Cat, but to any bicategory with sufficient completeness properties. Here, we examine how the theory applies to the category Auto of automata and the category ....

....outputs. We show that these monotone automata are in fact the algebras of an input buffering monad on a category of spans in AutoWk. This result prepares the connection, made in Section 3, with fibrations in AutoWk. For this, the use of 2categories is necessary, and the reader is referred to [6, 14, 15] for the basic terminology and notations. In Section 4, we apply the theory to the 2 category EvOrdWk of domains and obtain a similar characterization of the dataflow like spans in EvOrdWk. Finally, a comment on notation. In this paper, fx or f(x) denotes the application of a mapping f to its ....

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R. H. Street. Fibrations in bicategories. Cahier de Topologie et G'eometrie Diff'erentielle, XXI-2:111--159, 1980.

.... in the framework of enriched category theory [2] where the category V is equal to the category Gray with strict tensor product [5] see [4] as well) By working in the context of Gray categories we are developing the formal theroy of KZ doctrines in the way that, by working in a 2 category, [13] develops the formal theory of monads . Notice that this is a very general setting since every tricategory is equivalent to a Gray category [5] The idea of defining KZ doctrines in an enriched setting is also suggested in [9] We adopt the definition of a pseudomonoid given in [1] We show that ....

.... a pseudomonoid given in [1] We show that every KZdoctrine is a pseudomonad (pseudomonoid in the Gray monoid determined by an object of the Gray category) and that the 2 categories of algebras defined as adjunctions coincide with the classical algebras for a pseudomonad (Theorem 10.7) We follow [13] in defining the algebras for a pseudomonad and the algebras for a KZ doctrine with arbitrary objects of the Gray category as domains. R. Street [13] gives a conceptual global account of KZ doctrines in terms of the simplicial category Delta. Recall that in that context a doctrine on a bicategory ....

[Article contains additional citation context not shown here]

Ross Street, Fibrations in bicategories, Cahiers de topologie et g'eom'etrie diff'erentielle, vol. XXI-2, 1980, pp. 111-159.

.... in the framework of enriched category theory [2] where the category V is equal to the category Gray with strict tensor product [5] see [4] as well) By working in the context of Gray categories we are developing the formal theroy of KZ doctrines in the way that, by working in a 2 category, [13] develops the formal theory of monads . Notice that this is a very general setting since every tricategory is equivalent to a Gray category [5] The idea of defining KZ doctrines in an enriched setting is also suggested in [9] We adopt the definition of a pseudomonoid given in [1] We show that ....

.... a pseudomonoid given in [1] We show that every KZdoctrine is a pseudomonad (pseudomonoid in the Gray monoid determined by an object of the Gray category) and that the 2 categories of algebras defined as adjunctions coincide with the classical algebras for a pseudomonad (Theorem 10.7) We follow [13] in defining the algebras for a pseudomonad and the algebras for a KZ doctrine with arbitrary objects of the Gray category as domains. R. Street [13] gives a conceptual global account of KZ doctrines in terms of the simplicial category Delta. Recall that in that context a doctrine on a bicategory ....

[Article contains additional citation context not shown here]

Ross Street, Fibrations in bicategories, Cahiers de topologie et g'eom'etrie diff'erentielle, vol. XXI-2, 1980, pp. 111-159.

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