J.W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics 391 (Springer-Verlag, 1974).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....where n 3. We allow n to go to infinity and include the study of # categories (or # categories) The theory of 3 categories worked out in [66] is explicit and combinatorial. While not every 3 category is equivalent to a strict 3 category, there is an intermediate notion of a Gray category [68], or semi strict 3 category, such that every 3 category is equivalent to a Gray category. The idea is that some but not all coherence diagrams can be arranged to commute strictly without essential loss of information. Trimble [152] has defined 4 categories in a similarly concrete vein, but it is ....

John W. Gray, Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Mathematics 391, Springer, 1974.

....in P, so this is really only suitable for linear bicategories (rather than for poly bicategories) If P also has all left and right adjoints, linear natural transformations give rise to certain cyclic adjoint poly modules, thus placing them in a context which allows both compositions. A fact [1, 12] that seems sometimes to be forgotten is that whiskering of lax natural transformations with a lax functor on the codomain side does not, in general, produce a lax natural transformation. We are grateful to an (anonymous) referee for pointing this out to us. The prospect of restricting attention ....

Gray, J. W. Formal Category Theory: Adjointness for 2--Categories. vol. 391 of LNM, Springer-Verlag, Berlin -- New York, 1970.

....within categorical models of reduction [48,70,75] where types are represented as objects, terms as morphisms and reductions as 2 cells. In such models the introduction and elimination rules for the exponential type constructor form an adjoint pair of functors whose local unit and counit [33,45] correspond to j expansion and fi contraction respectively. These rewrite rules are linked by local triangle laws and the restrictions on the applicability of the expansionary j rewrite rule prevent exactly those expansions which occur in these triangle laws. This thesis demonstrates how these ....

....categorical structures based on term equality to categorical structures based on term reduction. Just as the natural constructions on traditional models may be used to derive an equational theory for the terms of the calculus, so the natural categorical constructions on these ordered categories [33,39,48,47,75,72] implicitly contain rewrite relations whose equational theory matches that suggested by the traditional semantics. Although these structures, and the more general enriched categories [24,50,51, 56] are beginning to gain wider acceptance, their use is still limited enough to warrant formal ....

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J. W. Gray. Formal Category Theory: Adjointness for 2-Categories. Number 391 in Lecture Notes in Mathematics. Springer Verlag, 1974.

....is given with respect to all pullback diagrams in the base categories and some other times (as for us) only with respect to pullbacks of monomorphisms. Kelly s book [65] provides an excellent introduction to enriched categories, while Kelly and Street paper [67] and Gray s monography [47] deal with many 2 , pseudo and bi categorical concepts. Bicategories were introduced by Benabou [9] Street s paper [127] contains definitions and results about pseudo limits (bilimits in that paper s jargon) in bicategories. Borceux s monography [17] covers some material about enriched concepts ....

....the functor H to be defined from P to the Cauchy completion Q of Q. Some people use the word distributor or bimodule [10, 72] Adjoint pairs in bicategories are defined in analogy with the 2 categorical case, but for some extra care needed to take account of the coherence isomorphisms [47]. 68 CHAPTER 4. PROFUNCTORS In terms of Kan extensions, the theorem above is saying that Lan y P (F ) H and yQ (D) H # . As we shall see later, just like Rel, also Prof (or better Cocont as far as our definitions will be concerned) is compact closed [35, 36] though in a ....

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John W. Gray. Formal Category Theory: Adjointness for 2-Categories, volume 391 of Lecture Notes in Mathematics. Springer-Verlag, 1974.

....# categories) The theory of 3 categories worked out in [62] is explicit and combinatorial. While not every 3 category is equivalent to a strict 3 category, there is an intermediate notion of a Gray category, or semi strict 3 category, such that every 3 category is equivalent to a Gray category [64]. The idea is that some but not all coherence diagrams can be arranged to commute strictly without essential loss of information. Trimble [144] has defined 4 categories in a similarly concrete vein, but it is apparent that more conceptual encapsulations are essential to a coherent theory of ....

John W. Gray, Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Mathematics 391, Springer, 1974.

....in connection with physics and geometry ( BN] C] FY] JS] KV] and of particular interest to these applications have been the notions of a braiding for a tensor and a dual for an object. In this paper, we prove a theorem which demonstrates how an old construction (the bicategory of squares [G]) can be used to define braided compact closed bicategories which arise in the study of topological quantum field theories. Specifically, the main theorem of this paper states the following. Suppose Sq(B) is the monoidal bicategory of squares in a monoidal bicategory B. Then an object f : A B ....

Gray JW, Formal Category Theory: Adjointness for 2-Categories. Number 391 Lecture Notes in Mathematics, Springer-Verlag, New York, 1974.

....X (B X) A for the functor X 7 (B X) A . Similarly, a function with storage A X B X can be viewed as a Kleisli morphism A X X B for the comonad A 7 A X X , or as a coalgebra X B A X for the functor X 7 B A X . In both cases, there is a double category [Gra74] displaying the structure and behaviour as the horizontal and the vertical arrows respectively. It has pairs hA; Xi as objects, where the component A is to be thought of as a data type, and X as a state space. The horizontal arrows represent computations, captured as transducers (functions with ....

J. W. Gray. Formal Category Theory: Adjointness for 2-categories. Springer, 1974.

....on normalized algebras. We believe also that the approach via the adjoint string gives us a better insight into the nature of ffi : Dd Gamma dD. We work 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 ....

J. W. Gray, Formal category theory: Adjointness for 2-categories. Lecture Notes in Mathematics 391, Springer-Verlag, 1974

....QC, Canada H3A 2K6. e mail:hermida triples.math.mcgill.ca. The reader may consult [KS74, Str72b] for the relevant 2 categorical concepts involved in the present paper. Relevant background material for bred categories can be found in [Gra66, Jac91, Pav90] A warning should be made about [Gra74], which refers to a bration as a bration in the 2 category 2 Cat. This concept is weaker than the one we consider here. 2 Fibred 2 categories We introduce the relevant notions of cartesian 1 cells and 2 cells appropriate to characterise (in elementary terms) brations. 2.1. De nition ....

....was given in elementary 2 categorical terms, it is possible to give a more abstract formulation of brations. The notion of bration in a 2 category can be made completely internal to it, i.e. without relying on representability. This requires the presence of comma objects in the 2 category, cf. [Gra74, Str73]. The situation for brations is much more delicate, as they are not simply brations in the 3 category 2 Cat . The appropriate setting for brations is actually weakly tricategorical : the formal de nition of a bration in the above style takes place in (a mild variant of) Gray s 2 Cat ....

J.W. Gray. Formal Category Theory: Adjointness for 2-categories, volume 391 of Lecture Notes in Mathematics. Springer Verlag, 1974.

....Thus we have a locally ordered category Lax(A; B) given by locally ordered functors and lax transformations. Our monoidal biclosed category V consists of the category of small locally ordered categories and locally ordered functors, together with that monoidal structure (described in Gray s book [8], Theorem I,4.9. for which, for each object A of V, Gamma Omega A has right adjoint Lax(A; Gamma) We call this tensor product the Gray tensor product. To give an explicit definition of it, let A and B be small locally ordered categories. Then we define A Omega B to be the locally ordered ....

J.W. Gray. Formal category theory---adjointness for 2-categories, volume 391 of Springer Lecture Notes in Mathematics. Spriger-Verlag, 1974.

....A B giving the additive exponent. Cat also has another closed structure, where A B is the category whose objects are functors and whose morphisms are transformations, i.e. families of maps but without naturality constraints. The symmetric monoidal structure is given by Gray s tensor product [18] with the one object category as unit. So Cat is an ane, bicartesian dcc. These are the only symmetric monoidal closed structures on Cat [14] A number of other naturally occurring examples arise from Day s construction, including a model of higher dimensional automata [17] and complexity models ....

J. W. Gray. Formal Category Theory { Adjointness for 2-Categories, volume 391 of Lecture Notes in Math. Springer, 1974.

....of Gray categories, as in [6] continuing the development of the formal theory of pseudomonads started in [9] This paper is organized as follows: In section 2 we provide a brief description of the framework that we use, namely that of Gray categories. For more details we refer the reader to [6, 5]. In section 3 we recall the definition and some properties of pseudomonads given in [9] the definition uses the definition of pseudomonoid given in [3] We also define the change of base 2 functors, change of base strong transformations and the change of base modifications that we will need in ....

J. W. Gray, Formal category theory: Adjointness for 2-categories. Lecture Notes in Mathematics 391, Springer-Verlag, 1974.

.... [2, 1] This presentation of type constructor via introduction and elimination rules opens the way to applying some of the ideas of adjoint rewriting as developed in [7] In this analysis, the introduction and elimination rules of a type constructor form a pair of (locally) adjoint functors [11, 13] whose unit and counit are respectively an expansionary (not contractive) j rewrite rule and contractive fi rewrite rule. The equational theory generated by these rewrite rules is automatically sound and complete for a semantics which interprets the introduction and elimination rules as being ....

J. W. Gray. Formal Category Theory: Adjointness for 2-Categories. Number 391 in Lecture Notes in Mathematics. Springer Verlag, 1974.

....suggests that Johnson s pasting schemes are not unnecessarily complicated. In chapter 3, I will use pasting presentations to give a detailed description of the tensor product on the collection of categories, which is defined by Steiner [16] which extends Gray s tensor product on 2 categories [6] and which is closely related to Brown Higgins s tensor product on groupoids [2] This description will lead to a better understanding of Gray categories [5] so that hopefully methods of chapter 1 can be used to show the homotopical relevance of Graygroupoids [9] and their higher dimensional ....

J. W. Gray. Formal category theory: adjointness in 2-categories, volume 391 of Lecture Notes in Math. Springer Verlag, New York, 1974.

....a preorder, each homset is a poset. Both these kinds of order enriched category can be thought of as a 2 dimensional category: the arrows of C represent the horizontal dimension and the those of the order relation the vertical dimension. This property is formalized by the notion of a 2 category [12], of which preorder enriched categories are a special case. Functors between such categories must preserve both the horizontal and vertical structure, which in this case means that functors must be monotonic. Therefore when we refer to a functor between order enriched categories, we will always ....

Gray, J.W.: Formal Category Theory: Adjointness for 2-categories. Springer-Verlag Lecture Notes in Mathematics 391 (1974)

....category of denotations. Just as in the denotational semantics of terms, models of reduction should be cartesian closed categories, where cartesian closure and, more generally, adjunctions are re interpreted to accommodate 2 cells. Exactly how this should be done is an area of active research (J.W. Gray, 1974; S. Kasangian et al. 1983; C.B. Jay, 1988; C.B. Jay, 1990; A. Carboni et al. 1990; C.B. Jay, 1991a; C.A.R. Hoare et al. 1989) but most developments share the following properties. There is a local counit and a local unit which are linked by local triangle laws. In our examples the local ....

J.W. Gray, Formal category theory: Adjointness for 2-categories, Lecture Notes in Mathematics 391, (Springer, 1974).

....[KV] and of particular interest to these applications have been the notions of a braiding for a tensor and a dual for an object. In this paper we prove a theorem which deepens the connection between duals and adjoints, and which demonstrates how an old construction (the bicategory of squares [G]) can be used to define new braided compact closed bicategories. Specifically, the main theorem of this paper states the following. Suppose Sq(B) is the monoidal bicategory of squares in a monoidal bicategory B. Then an object f : A B of Sq(B) has a right bidual if and only if (i) both A and ....

Gray JW, Formal Category Theory: Adjointness for 2-Categories. Number 391 Lecture Notes in Mathematics, Springer-Verlag, New York, 1974.

....are nearer to the well used 2 categories. On the other hand, their monoidal closed structure, which follows from the equivalence with the previous example, seems more difficult to describe than in either of the previous examples. The corresponding case of 2 categories is dealt with by Gray in [32, 33]. The equivalence of 2 categories with a form of double categories with connection is due to Spencer [53, 54] but seems not to have been otherwise exploited, except in recent work of Verity on cubical nerves of categories. The work of [54] shows the value of the connections and the ....

Gray, J.W., Formal category theory: adjointness for 2-categories, Lecture Notes in Math. 391, Springer-Verlag, New York, (1974).

....is a 3 category except that horizontal composition of 2arrows results in a 3 isomorphism between the two possible ways of composing these 2arrows vertically, rather than these two vertical composites being equal. More formally, a Gray category is a category enriched in the monoidal category Gray [14, 13]. A braided (strict) monoidal category can be reconsidered as a Gray category which has only one object and one arrow. This reconsideration has the form of a reindexing : for a braided (strict) monoidal category C , the corresponding Gray category, which I will denote by Sigma 2 (C ) has (one ....

....G C C , ii) an object I 2 C , such that the following equations hold: Gamma Omega ( Omega ) Gamma Omega ) Omega (2.1) I Omega Gamma = Gamma (2.2) Gamma Omega I = Gamma: 2. 3) 3 In this definition, Omega G is the pseudo version of the Gray tensor product of 2 categories [13, 14]. For a more extensive description of the data and axioms of a monoidal 2 category, see [3, Lemma 4] 2.2 Braided monoidal 2 categories Semistrict braided monoidal 2 categories have been defined, with a slightly incorrect and incomplete long list of axioms, by Kapranov and Voevodsky [19, 18] ....

J. W. Gray. Formal category theory: adjointness in 2-categories, volume 391 of Lecture Notes in Math. Springer Verlag, New York, 1974.

....all follow from their definition of a braided monoidal 2 category. In our framework there is only one Zamolodchikov equation. 2 Definitions We begin by defining semistrict monoidal 2 categories and semistrict braided monoidal 2 categories. Following traditional practice among category theorists [18, 23], we use 2 category to mean what Kapranov and Voevodsky [21] call a strict 2 category, and 2 functor to mean what Kapranov and Voevodsky call a strict 2 functor. Composition of 1 morphisms, the horizontal composition of a 1 morphism and a 2 morphism (in either order) and the horizontal ....

....of 2 morphisms is denoted by Delta. We use the ordering in which, for example, the composite of f : A B and g: B C is denoted f ffi g. We use C Omega G D to denote Gordon, Power and Street s [17] Gray tensor product of the 2 categories C and D. This differs from Gray s original version [18] in being the pseudo rather than the lax weakening of the Cartesian product. For readers unfamiliar with these distinctions, let us simply recall that given a 1 morphism f : A A 0 in C and a 1 morphism g: B B 0 in D, the Cartesian product C Theta D contains a commuting square (A; B ....

J. Gray, Formal Category Theory: Adjointness for 2-Categories, Springer Lecture Notes in Mathematics 391, Berlin, 1974.

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J.W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics 391 (Springer-Verlag, 1974).

No context found.

J. W. Gray. Formal Category Theory: Adjointness for 2-Categories. Number 391 in Lecture Notes in Mathematics. Springer Verlag, 1974.

No context found.

J. W. Gray. Formal Category Theory { Adjointness for 2-Categories, volume 391 of Lecture Notes in Math. Springer, 1974.

No context found.

J.W. Gray. Formal Category Theory: Adjointness for 2-categories. Lecture Notes in Mathematics, vol. 391. Springer 1974.

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J. Gray, Formal Category Theory: Adjointness for 2-Categories (1974). Springer LNM 391.

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