G.M. Kelly, A. Labella, V.Schmitt, R.Street, Categories enriched on two sides, Communication at Category Theory 99 - Coimbra 99. J. Pure Appl. Algebra 168 (2002), 53-98.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....generalisation in terms of enrichments of the later adjunction. Our first problem was to define a good notion of base morphism F : We had in mind that such a morphism should induce a 2 adjunction between 2 categories of enrichments, say F #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 ....

....on. We introduce the super two sided enrichments. They generalise the super monoidal functors used by one of the authors to code uniformly continuous maps from enriched functors [Sc01] They are more general than left adjoint two sided enrichments and enjoy nice properties. It was known from [KLSS99] that a two sided enrichment F induces a normal lax functor between bicategories of modules F # : For any super F , F # : preserves adjoints and F : preserves the Morita equivalence. Eventually we extend the notion of reversibility defined for enrichments in [Wal82] to two sided ....

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G.M. Kelly, A. Labella, V.Schmitt, R.Street, Categories enriched on two sides, Communication at Category Theory 99 - Coimbra 99. J. Pure Appl. Algebra 168 (2002), 53-98.

....113, 00198 Roma, Italy, e mail: labella dsi.uniroma1.it y Department of Mathematics and Computer science, University 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 ....

.... 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 adjoints were not preserved by ( A two sided enrichment F : V W between two bicategories was proposed in [KLSS99] as a slight generalisation of monoidal functors and Benabou s lax functors [Ben67] With these new morphisms one obtained a bicategory Base together with the expected pseudo functor ( Base 2 Cat. Introducing some 3 cells on Base, one eventually gets a tricategory Caten such that the ....

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G.M. Kelly, A. Labella, V.Schmitt, R.Street, Categories enriched on two sides, Communication at Category theory 99 - Coimbra 99. To appear in Journal of Pure and Applied Algebra.

....when one the following equivalent assertions is satis ed: i) They are isomorphic 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: ....

G.M. Kelly, A. Labella, V.Schmitt, R.Street, Categories enriched on two sides, Communication at Category theory 99 - Coimbra 99. 23

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