R. Street, Fibrations and Yoneda's Lemma in a 2-category, in: Sydney Category Seminar, LNM  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and give several characterizations of them. Among others, connected functors will be proved to be exactly the representably fully faithful functors (Condition 4, Proposition 3. 6) Such morphism in general 2 categories have already been studied by John Gray, Ross Street and other authors, see [12] or [4] In [5] Brian Day calls such functors Cauchy dense. 2. Preliminaries We use the notation and terminology of [10] We fix a locally small symmetric monoidal closed category V o , which is complete and cocomplete. The tensor product in V o is denoted #, its unit by I. The internal hom ....

R. Street, Fibrations and Yoneda's Lemma in a 2-category, in: Sydney Category Seminar, LNM

....F ) ParCat(X; G) naturally in (the paracategory) X, which means that the lax square above is the universal one with respect to F and G. When G = id , we write F=C for F=G and similarly when F = id . An analogous construction of the comma object (in the spirit of 2 category theory [Str73]) can be carried out in Cat P : given functors F : A ; A) C ; C) and G : B ; B) C ; C) their comma object F=G in Cat P is the category C (F=G) with objects triples (x; a : Fx Gy; y) with a 2 C morphisms (f; g) x; a : Fx Gy; y) x ) given by f : x x such ....

....do compose whenever their images in the base category do, and their composite is p cartesian as well. For a functor of paracategories p : with a category, there is a simple adjoint characterisation of the property of being a bration, analogous to the one for brations of categories [Str73]. 7.3. Proposition. Given a functor p : with a category, the following are equivalent: 1. p is a bration of paracategories 2. The functor p : p induced as depicted below p # =p :B B id B p : x 7 (px; id : px px ; x) has a right adjoint right ....

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R. Street. Fibrations and Yoneda's lemma in a 2-category. In Category Seminar, volume 420 of Lecture Notes in Mathematics. Springer Verlag, 1973.

.... pre algebra F : K 2 K, a factorization algebra structure on F is an isomorphism # : FF 2 # ## FCK which satisfies: #I K 2 = 1 F (1) #(I K ) 2 = 1 F (2) #C K 2 #(F 2 ) 2 = #(CK ) 2 F# 2 (3) these equations being the specialization of the equations in 2 of [STR] to the case at hand. 136 ROBERT ROSEBRUGH AND R.J. WOOD 1.5. Remark. In the absence of normality, a ( 2 pseudo algebra structure further requires an isomorphism # : 1 K # ## F I K and equations (1) and (2) above must then be replaced by: #I K 2 #F = 1 # ) 1 F #(I K ) 2 F ....

R. Street. Fibrations and Yoneda's lemma in a 2-category. In Lecture Notes in Math. 420, 104--133, Springer-Verlag, 1974.

....is a lax epimorphism. We show a simple example demonstrating that this sucient condition is not necessary. Lax monomorphisms and epimorphisms (the latter also called full and faithful morphisms) in general 2 categories have already been studied by John Gray and Ross Street in the early 1970 s, see [11], and the latter by other authors, see, e.g. 3] and [4] In the latter reference Brian J. Day also calls lax epimorphisms in CAT Cauchy dense functors. However, the explicit characterization of lax epimorphisms in CAT we provide below is new. How is the concept of lax epimorphism related to other ....

R. Street, Fibrations and Yoneda's Lemma in a 2-category, in: Sydney Category Seminar, LNM 420, Springer-Verlag 1974, 104-133

....brations for them so that the Yoneda object on M does correspond to the object of discrete brations on it. Without indulging in details here, let us point out that this provides a natural example of a 2 category where the appropriate notion of bration is not the representable one advocated in [Str73]. In fact the second and third approaches above can be formally related (and shown equivalent) in the context of internal category theory (see x8) The second part of the paper is devoted to three basic examples in this context: the one already mentioned of monoidal categories, monoidal globular ....

....their monoid classi ers. Part I General Theory 2 Preliminaries on bimodules Since there is no comprehensive account of the elementary properties of bicategories of bimodules internal to a 2 category, we collect some basic facts in this section. Background material on bimodules can be found in [Str73, Str80, Woo82, Woo85, CJSV94] and [BC80, x1] In order to build a bicategory of bimodules internal to a given 2 category K, we assume the following: 1. K admits pullbacks of brations and co brations 2. K admits comma objects 3. K admits coequalizers, stable under pullback. 2.1. Remark. If K admits pullbacks, it also admits ....

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R. Street. Fibrations and Yoneda's lemma in a 2-category. In Category Seminar, volume 420 of Lecture Notes in Mathematics. Springer Verlag, 1973. 54

....to the converse of the above corollary. 4. 2 Comma objects in Fib As a preliminary to our treatment of brations over brations in x4.3 below, and also as an illustration of limits in a bred 2 category, we show that Fib and its bre 2 categories Fib(B ) admit comma objects in the sense of [Str73]. The construction is entirely analogous to that of ordinary limits in bred categories in Corollary 4.9. That is, we rst build comma objects in Fib, and then obtain them in the bres by restriction along the diagonal. We make the details explicit in the following proposition, as we need them in ....

....That is, in K # B H we only consider the vertical morphisms of the comma category K #H . It is clear then that it enjoys the required universal property within Fib(B ) 2 4.15. Remark. The existence of the appropriate 2 categorical limits in Fib(B ) follows from its algebraic treatment in [Str73]. The construction above shows the special case of comma objects as an instance of a limit in a bre of a bration, thereby making explicit its relation to the corresponding limit in the total 2 category, which is essential for our algebraic proof of B enabou s characterisation of brations over ....

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R. Street. Fibrations and Yoneda's lemma in a 2-category. In Category Seminar, volume 420 of Lecture Notes in Mathematics. Springer

....network forming operations of series composition, parallel composition, and feedback. The main result of the paper is that this can be done, resulting in a bicategory Sys that is in a sense equivalent to a certain bicategory of two sided fibrations in Dom, with cartesian arrows of spans as 2 cells [Gra66, Str74, Str80]. In more explicit detail, our construction starts with Dom and produces a new structure Sys, which has as its objects the objects of Dom (i.e. the Scott domains) as its 1 cells (arrows) certain systems of inequalities, which are syntactic objects denoting nondeterministic networks, and as its ....

....of relating a and b may have some additional structure (e.g. that of a Scott domain) Finally, we come to the relationship with fibrations. We show that, for each pair of domains A and B, the ordered category Sys(A; B) is equivalent to a full subcategory of the category of two sided fibrations [Str74] from A to B in Dom, with cartesian arrows of spans as morphisms. We use the term systemic fibration to refer to fibrations that correspond to systems, and we obtain a characterization of the systemic fibrations. We can then show that the syntactic operation of series composition of systems of ....

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R. H. Street. Fibrations and Yoneda's lemma in a 2-category. In Lecture Notes in Mathematics 420, pages 104--133. Springer-Verlag, 1974.

....to verify that M : T 2 M TM is such a left adjoint: given a sequence of sequences h x 1 ; x n i of objects of M , id x 1 : x n : h x 1 ; x n i x 1 : x n is a universal arrow in TM . So T is a 2 monad whose pseudo algebras are (left) adjoints to the units [Str73, Koc95]. We define RepMulticat to be the locally full (i.e. all 2 cells) sub 2 category of Multicat consisting of representable multicategories and morphisms between such which preserve universal arrows. We write Ps T alg for the 2 category of pseudo algebras, pseudo morphisms and all 2 cells between ....

R. Street. Fibrations and Yoneda's lemma in a 2-category. In Category Seminar, volume 420 of Lecture Notes in Mathematics. Springer Verlag, 1973.

....brations for them so that the Yoneda object on M does correspond to the object of discrete brations on it. Without indulging in details here, let us point out that this provides a natural example of a 2 category where the appropriate notion of bration is not the representable one advocated in [Str73]. 4 In fact the second and third approaches above can be formally related (and shown equivalent) in the context of internal category theory (see x8) The second part of the paper is devoted to three basic examples in this context: the one already mentioned of monoidal categories, monoidal ....

....theory (Remark 11.2) Part I General Theory 2 Preliminaries on bimodules Since there is no comprehensive account of the elementary properties of bicategories of bimodules internal to a 2 category, we collect some basic facts in this section. Background material on bimodules can be found in [Str73, Str80, Woo82, Woo85, CJSV94] and [BC80, x1] In order to build a bicategory of 6 bimodules internal to a given 2 category K, we assume the following: 1. K admits pullbacks of brations and co brations 2. K admits comma objects 3. K admits coidenti ers, stable under pullback along brations. 2.1. Remark. If K admits ....

[Article contains additional citation context not shown here]

R. Street. Fibrations and Yoneda's lemma in a 2-category. In Category Seminar, volume 420 of Lecture Notes in Mathematics. Springer Verlag, 1973.

....by the usual adjunctions but in the 2 category ICat(B) instead of Cat. This is not quite right, because function spaces (and dependent products) are given via an adjunction with parameter. A 2 categorical reformulation may have to rely on brations in a 2 category, which is far from simple (see [Str73]) Polymorphic types are modelled like universal quanti ers (in Hyperdoctrines) while abstract data types are modelled like existential quanti ers (see [MP88] Remark 3.1 The requirement Obj(C[X] B(X; for the kind of all types is not always justi ed in relation to programming ....

....types, embedding between categories with attributes. Moreover, it introduces some operations on categories with attributes (see De nition 6.14) Remark 6.1 In the introduction we advocate formulating concepts 2 categorically. Since the concept of bration can be formulated 2 categorically (see [Str73]) it is possible to formulate categories with attributes and other concepts introduced below 2 categorically, too. However, we have not done so, because it seemed too complicated and unintelligible. Nevertheless, a 2 categorical formulation is essential to de ne indexed categories with attributes ....

R. Street. Fibrations and Yoneda's lemma in a 2-category. In Category Seminar, volume 420 of Lecture Notes in Mathematics. Springer Verlag, 1973.

....: T 2 M TM is such 32 a left adjoint: given a sequence of sequences h x 1 ; x n i of objects of M , id x 1 Delta: Delta x n : h x 1 ; x n i x 1 Delta : Delta x n is a universal arrow in TM . So T is a 2monad whose pseudo algebras are (left) adjoints to the units [Str73, Koc95]. We define RepMulticat to be the locally full (i.e. all 2 cells) sub 2category of Multicat consisting of representable multicategories and morphisms between such which preserve universal arrows. We write Ps GammaT Gammaalg for the 2 category of pseudo algebras, pseudo morphisms and all 2 cells ....

R. Street. Fibrations and Yoneda's lemma in a 2-category. In Category Seminar, volume 420 of Lecture Notes in Mathematics. Springer Verlag, 1973.

....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 and Yoneda's lemma in a 2-category. In Lecture Notes in Mathematics 420, pages 104--133. Springer-Verlag, 1974.

....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 and Yoneda's lemma in a 2-category. In Lecture Notes in Mathematics 420, pages 104--133, Springer-Verlag, 1974.

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