Eilenberg, S., G.M. Kelly, Closed categories, Proceedings of La Jolla Conference on Categorical Algebra, Springer Verlag 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(where # The author acknowledges partial financial assistance by Centro de Matematica da Universidade de Coimbra FCT. Partial financial assistance by NSERC is acknowledged. b means a b, and in which the tensor product is given by addition) V categories in the sense of Eilenberg and Kelly [11] are nothing but pairs (X, d) satisfying the basic laws 0 d(x, x) d(x, y) d(y, x) d(x, z) For a general V category A (with object set X) these are instances of the operations A(x, x) A(x, z) with# , I denoting the monoidal structure of V) which must satisfy the obvious ....

S. Eilenberg and G.M. Kelly, Closed categories, in: Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965.

....analysis that shows that one might at least want to restrict oneself to categories of ultra metric spaces, in order to obtain Cartesian closed categories. As the unifying concept of pre orders and metric spaces we follow Lawvere and use enriched categories. Enriched categories were introduced by Eilenberg and Kelly in 1966 ( Eilenberg Kelly 66] and popularized in the best sense of the word by Lawvere in 1973 ( Lawvere 73] In this latter paper Lawvere showed how to use essentially categorical tech niques on for instance pre orders and metric spaces by softening the requirements on what constitutes a category, ....

....might at least want to restrict oneself to categories of ultra metric spaces, in order to obtain Cartesian closed categories. As the unifying concept of pre orders and metric spaces we follow Lawvere and use enriched categories. Enriched categories were introduced by Eilenberg and Kelly in 1966 ( Eilenberg Kelly 66] and popularized in the best sense of the word by Lawvere in 1973 ( Lawvere 73] In this latter paper Lawvere showed how to use essentially categorical tech niques on for instance pre orders and metric spaces by softening the requirements on what constitutes a category, such that the resulting ....

[Article contains additional citation context not shown here]

Eilenberg, S., G.M. Kelly, Closed categories, Proceedings of La Jolla Conference on Categorical Algebra, Springer Verlag 1966.

....#W Cat . This question was largely answered in [KLSS99] with the introduction of the so called two sided enrichments. To explain partly these results, we should start from the definition of MonCat , the category of monoidal functors between monoidal categories [Ben63] and enrichments over them [EiKe66], Law73] A monoidal functor F : induces a 2 functor F : MonCat is equipped with a 2 categorical structure by defining 2 cells in it as monoidal natural transformations ( EiKe66] The process ( of sending to and F to F extends to a 2 functor from MonCat to 2 Cat . Adjunctions in ....

.... of MonCat , the category of monoidal functors between monoidal categories [Ben63] and enrichments over them [EiKe66] Law73] A monoidal functor F : induces a 2 functor F : MonCat is equipped with a 2 categorical structure by defining 2 cells in it as monoidal natural transformations ([EiKe66]) The process ( of sending to and F to F extends to a 2 functor from MonCat to 2 Cat . Adjunctions in MonCat were characterised in [Kel74] Moving to the case of enrichments over bicategories, several notions of morphism between bicategories were proposed in the literature, but no ....

S. Eilenberg, G.M. Kelly, Closed Categories, Proceedings of the conference on Categorical Algebra at La Jolla, Springer 1966, 421-562.

....both on spaces and on chain complexes, is intrinsically 2 categorical: homotopies are maps between maps. Like categories themselves, 2 categories are easy to understand. However, there is a glimpse of di#culties to come. There are strict 2 categories, defined by Ehresmann and Eilenberg and Kelly [45, 46] and there are weak 2 categories, defined by Benabou [14] under the name of bicategories. Intuitively, in strict 2 categories, coherence diagrams are required to commute; in weak ones, they are required to commute only up to natural isomorphism. The di#erence is only technical, since there is a ....

Samuel Eilenberg, G. Max Kelly, Closed categories, in Proceedings of Conference on Categorical Algebra (La Jolla, California,

....the advantage of being easily applied and understood also in other categories. The latter is useful for proving the existence of such products. Tensor Products as Left Adjoints. One way to define tensor products is to use closed categories and adjunctions. Informally (for a formal definition see [EK66]) a category C is closed if its hom sets are themselves objects of the category. That is, for every two objects C 1 and C 2 in C, the set C(C 1 ; C 2 ) of arrows in C can be regarded as an object of C. This is always the case for categories of algebras, like SL. See, e.g. Jac92] For every ....

S. Eilenberg and G.M. Kelly. Closed categories. In S. Eilenberg et al., editor, Proc. of La Jolla Conf. on Categorical Algebra, pages 421--562. Springer-Verlag, 1966.

....of Leicester, University road, Leicester LE1 7RH, England e mail: vs27 mcs.le.ac.uk 1 objects respectively of V Cat and W Cat. The rst point of our problem was answered in [KLSS99] Let us start from the de nition of monoidal functor between monoidal categories [Ben63] and enrichments over them [EiKe66], Law73] A monoidal functor F : V W induces a 2 functor F : V Cat W Cat. MonCat is equipped with a 2 categorical structure by de ning 2 cells in it as monoidal natural transformations ( EiKe66] The process ( of sending V to V Cat and F to F extends to a 2 functor from MonCat to 2 ....

....the de nition of monoidal functor between monoidal categories [Ben63] and enrichments over them [EiKe66] Law73] A monoidal functor F : V W induces a 2 functor F : V Cat W Cat. MonCat is equipped with a 2 categorical structure by de ning 2 cells in it as monoidal natural transformations ([EiKe66]) The process ( of sending V to V Cat and F to F extends to a 2 functor from MonCat to 2 Cat. Adjunctions in MonCat were characterised in [Kel74] Moving to the case of enrichments over bicategories, several notions of morphism between bicategories where proposed in the literature, but ....

S. Eilenberg, G.M Kelly, Closed Categories, Proceedings of the conference on Categorical Algebra at La Jolla, Springer, 66, 421-562.

....in V Mod; ii) Their Cauchy completions are isomorphic in V Cat. 4 More on enrichments 4. 1 About the change of base The change of base for enrichments (over bicategories) has recently known some new developments [KLSS99] Nevertheless we shall only use an old formulation of a classical result [Eil Kel66]. Restricting ourselves to the case where bases are partial orders, we get: Fact 4.1 There is a 2 functor ( Mon 2 Cat where Mon is the 2 category with: objects: monoidal partial orders, arrows: monoidal functors, 2 cells: natural transformations 2 , and 2 Cat is the 2 category of ....

S. Eilenberg, G. M. Kelly, Closed categories, Proceedings of the conference on Categorical Algebra at La Jolla Springer-Verlag 66, 421-562

....if is the object A 1 A k , then 2 will be represented by 2(A 1 A k ) but clearly we mean 2A 1 2A k . Thus we shall make the further simplifying assumption that 2 is a symmetric monoidal functor, 2; m A;B ; m 1 ) This notion is originally due to Eilenberg and Kelly [14]. In essence this provides a natural transformation m A;B : 2A 2B 2(A B) and morphism m 1 : 1 21 which satisfy a number of conditions which are detailed below. DEFINITION 2. A monoidal functor, 2; m A;B ; m 1 ) on a CCC C satisfies the four following equations. 1. id 2A m 1 ; m A;1 ....

S. Eilenberg and G.M. Kelly. Closed categories. In Proceedings of Conference on Categorical Algebra, La Jolla, 1966.

.... Gamma) defined above) because for all r and s in [0; 1] t s r ( s r Delta Gamma t: Generalized metric spaces can be viewed as categories enriched in [0; 1] or [0; 1] categories for short. By taking this view, we follow Lawvere s [Law73] conception of metric spaces as V categories [EK66, Kel82]. The main advantage of this approach is that many results from enriched category theory can be applied to metric spaces. We just saw that any preorder induces a generalized metric space. There is also the reverse construction: any generalized metric space X induces a preordered space hX; X i ....

S. Eilenberg and G.M. Kelly. Closed Categories. In S. Eilenberg, D.K. Harrison, S. Mac Lane, and H. Rohrl, editors, Proceedings of the Conference on Categorical Algebra, pages 421--562. Springer-Verlag, 1966.

....1 morphisms, and so on up to n morphisms. There should be various ways of composing j morphisms for 1 j n, and these should satisfy various laws. As with 2 categories, we can try to impose these laws either strictly or weakly. Strict n categories have been understood for quite some time now [23, 28], but more interesting for us are the weak ones. Various definitions of weak n category are currently under active study [5, 10, 36, 57, 58, 61, 62, 63] and we discuss our own in Section 5. Here, however, we wish to sketch the main challenges any theory of weak n categories must face, and some of ....

S. Eilenberg and G. M. Kelly, Closed categories, in Proceedings of the Conference on Categorical Algebra, eds. S. Eilenberg et al, Springer Verlag, New York, 1966.

....p 6 q. By a slight abuse of language, any gum stemming from a preorder in this way will itself be called a preorder. 3. The set [0; 1] with distance, for r and s in [0; 1] 0; 1] r; s) ae 0 if r s s if r s. We briefly review Lawvere s [Law73] conception of metric spaces as V categories [EK65, Kel82] Then we shall follow and further elaborate his approach for the special case of generalized ultrametric spaces, which will be shown to be [0; 1] categories. The main point is that, in general, many properties of V categories derive from the structure on the underlying category V . In our ....

S. Eilenberg and G.M. Kelly. Closed Categories. In S. Eilenberg, D.K. Harrison, S. Mac Lane, and H. Rohrl, editors, Proceedings of the Conference on Categorical Algebra, pages 421--562, La Jolla, June 1965. Springer-Verlag.

....analysis that shows that one might at least want to restrict oneself to categories of ultra metric spaces, in order to obtain Cartesian closed categories. As the unifying concept of pre orders and metric spaces we follow Lawvere and use enriched categories. Enriched categories were introduced by Eilenberg and Kelly in 1966 ( Eilenberg Kelly 66] and popularized in the best sense of the word by Lawvere in 1973 ( Lawvere 73] In this latter paper Lawvere showed how to use essentially categorical techniques on for instance pre orders and metric spaces by softening the requirements on what constitutes a category, ....

....might at least want to restrict oneself to categories of ultra metric spaces, in order to obtain Cartesian closed categories. As the unifying concept of pre orders and metric spaces we follow Lawvere and use enriched categories. Enriched categories were introduced by Eilenberg and Kelly in 1966 ( Eilenberg Kelly 66] and popularized in the best sense of the word by Lawvere in 1973 ( Lawvere 73] In this latter paper Lawvere showed how to use essentially categorical techniques on for instance pre orders and metric spaces by softening the requirements on what constitutes a category, such that the resulting ....

[Article contains additional citation context not shown here]

Eilenberg, S., G.M. Kelly, Closed categories, Proceedings of La Jolla Conference on Categorical Algebra, Springer Verlag 1966.

....6 q. By a slight abuse of language, any gms stemming from a preorder in this way will itself be called a preorder. 3. The set [0; 1] with distance, for r and s in [0; 1] 0; 1] r; s) ae 0 if r s s Gamma r if r s. Generalized metric spaces are [0; 1] enriched categories in the sense of [EK66, Law73, Kel82] As shown in [Law73] 0; 1] is a complete and cocomplete symmetric monoidal closed category. It is a category because it is a preorder (objects are the non negative real numbers including infinity; and for r and s in [0; 1] there is a morphism from r to s if and only if r s) It ....

S. Eilenberg and G.M. Kelly. Closed Categories. In S. Eilenberg, D.K. Harrison, S. Mac Lane, and H. Rohrl, editors, Proceedings of the Conference on Categorical Algebra, pages 421--562. Springer-Verlag, 1966.

....geometric surjection, we get the result for equivariant extension structures, as claimed. In the context of SDG, there arise further variants on this theme, cf. 13] or our forthcoming [14] 3. Enrichment strength of fractional exponents We recall some notions from enriched category theory, cf. [4] or [7] Recall that a cartesian closed category E is enriched in itself (i.e. is made into an E category) by means of Y X as the object of maps from X to Y . Then an E enrichment of an endofunctor G : E E consists of a family of maps GX;Y : Y X G(Y ) G(X) 6 natural in X and Y , ....

....V Theta GY G(U Theta V Theta Y ) 2) respectively, for all U; V; Y , under the evident identifications like 1 Theta GY = GY etc. We also recall that there is a notion for a natural transformation X : G 1 (X) G 2 (X) between two E functors to be E natural, or strongly natural, see [4] 1.10; in terms of the tensorial form of enrichments (for endofunctors G 1 and G 2 on E) this may be expressed simply as commutativity of all squares of the form X Theta G 1 Y t (1) X;Y G 1 (X Theta Y ) X Theta G 2 Y 1 Theta Y t (2) X;Y G 2 (X Theta Y ) X ThetaY ....

[Article contains additional citation context not shown here]

S. Eilenberg and M. Kelly, Closed Categories, in Proc. Conf. Catgeorical Alg., LaJolla, Springer Verlag 1966.

....The present paper intends to be an extended abstract; therefore, most of the proofs are omitted and those remaining are just sketched. Full expositions can be found in [15, 16] 1 Background T he notion of monoidal category dates back to [1] see [11] for an easy thorough introduction and [4] for advanced topics) In this paper we shall be concerned only with a particular kind of symmetric monoidal categories, name 1 We remark that the existence of a similar axiomatization was conjectured also in [6] ### ly those which are strict monoidal and whose objects form a free ....

S. Eilenberg, and G.M. Kelly. Closed Categories. In Proceedings of the Conference on Categorical Algebra, La Jolla, S. Eilenberg et al., Eds., pp. ###--###, Springer, ####.

....both on spaces and on chain complexes, is intrinsically 2 categorical: homotopies are maps between maps. Like categories themselves, 2 categories are easy to understand. However, there is a glimpse of di#culties to come. There are strict 2 categories, defined by Ehresmann and Eilenberg and Kelly [52, 53] and there are weak 2 categories, defined by Benabou [18] under the name of bicategories. Intuitively, in strict 2 categories, coherence diagrams are required to commute; in weak ones, they are required to commute only up to natural isomorphism. Here the di#erence is only technical, since there is ....

Samuel Eilenberg, G. Max Kelly, Closed categories, in Proceedings of Conference on Categorical Algebra (La Jolla, California,

....for some related right adjoints, when they exist; as well as another explicit expression for MonG as a large limit, which uses a new representation of any monad as a (large) limit of monads of two special kinds, and an analogous result for general endofunctors. 1. Introduction Recall from [EK] that a monoidal functor F : V V 0 between monoidal categories V = V; I) and V 0 = V 0 ; 0 ; I 0 ) consists of a triple F = f; e f ; f ) where f : V V 0 is an ordinary functor, e f is a natural transformation with components e f XY : fX 0 fY f(X Y ) and f : I 0 ....

....is the B component of the counit A of the adjunction (2.1) the unit operation j : I A(A; A) corresponds under (2.1) to : I A A; the associativity and unit axioms follow just as in the special case A = V, treated in Section 1. 6 of [KB] or, in greater detail, in Sections II.3 and II.4 of [EK]; and the isomorphism between A and A 0 is given by the isomorphisms A(A; B) A( 1) ## A(I A; B) ## V(I; A(A; B) A 0 (A; B) 2.4) It is moreover straightforward to verify that, if A is identi ed with A 0 by this isomorphism, the functor A : A op A V occurring in the adjunction ....

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S. Eilenberg and G.M. Kelly, Closed categories, in Proc. Conf. on Categorical Algebra (La Jolla,

....side condition involving the monoidal structure is satisfied. Many important properties of a monoidal category V are inherited by the category V Cat of small V categories. For instance, if V is symmetric monoidal, V Cat has a canonical symmetric monoidal structure, as was observed already in [4]. Much later [7, Remark 5.2] it was realized that if V is only braided monoidal then V Cat still has a canonical monoidal structure, although it need not have a braiding unless the braiding on V is in fact a symmetry. Similarly, it is straightforward to show that V Cat is monoidal closed when ....

S. Eilenberg and G.M. Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra (La Jolla,

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Eilenberg, S., G.M. Kelly, Closed categories, Proceedings of La Jolla Conference on Categorical Algebra, Springer Verlag 1966.

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Samuel Eilenberg and Gregory Maxwell Kelly. Closed categories. In S. Eilenberg, D.K. Harrison, S. MacLane, and H. Roehrl, editors, Proceedings of the La Jolla Conference in Categorical Algebra, pages 421--562. Springer, 1966.

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Samuel Eilenberg and Gregory Maxwell Kelly. Closed categories. In S. Eilenberg, D.K. Harrison, S. MacLane, and H. Roehrl, editors, Proceedings of the La Jolla Conference in Categorical Algebra, pages 421--562. Springer, 1966.

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S. Eilenberg, and G.M. Kelly (1966), Closed Categories, in Proceedings of the Conference on Categorical Algebra, S. Eilenberg et. al. (Eds.), 421--562, Springer-Verlag.

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S. Eilenberg and G. M. Kelly. Closed categories. In Proceedings of the Conference on Categorical Algebra, La Jolla 1965.

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S. Eilenberg, and G.M. Kelly (1966), Closed Categories, in Proceedings of the Conference on Categorical Algebra, S. Eilenberg et. al. (Eds.), 421--562, Springer-Verlag.

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S. Eilenberg - G.M. Kelly, Closed categories, in: Proceedings of the Conference on Categorical Algebra, La Jolla

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