R. Betti, R.F.C. Walters, The symmetry of the Cauchy completion of a category, Lecture Notes in Math. 962 (1982), 7-12. Home/Search Document Not in Database Summary Related Articles Check |
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Change Of Base, Cauchy Completeness And Reversibility - Anna Labella And (Correct)
....reversibility defined for enrichments in [Wal82] to two sided enrichments and to the Cauchy completion. Most familiar examples of enrichments over locally partially ordered bicategories (sheaves and group actions) enjoy a nice property of symmetry, called reversibility. As pointed out by Kasangian [BeWal82] a Cauchy completion of a reversible enrichment fails in general to be reversible. This fact forces us to define a completion for reversible enrichments that preserves reversibility. This is the so called Cauchy reversible completion. Thus we get the desired generalisation: an adjoint pair in ....
....on both sides of the inequality that for any a, a # , b#Obj(A) b, a) a # , b) i.e. 1 . Similarly one gets from ( #) 1 that # . 1. There are reversible enrichments with non reversible Cauchy completions. A very simple example (due to S. Kasangian) may be found in [BeWal82]. This motivates the notion of Cauchy reversible completeness defined further. 4.6. Definition. Reversible modules] A # between reversible reversible when it has right adjoint # . 4.7. Remark. If f is a between reversible then the is reversible. 4.8. Definition. ....
R. Betti, R.F.C. Walters, The symmetry of the Cauchy completion of a category, Lecture Notes in Math. 962 (1982), 7-12.
Change of base, Cauchy-completness and reversibility - Anna Labella Vincent (2000) (Correct)
....to introduce the notion of reversibility for bicategories, two2 sided enrichments and Cauchy completion. Most familiar examples of enrichments over locally posetal bicategories (sheaves and group actions) enjoy a nice property of symmetry that we call reversibility. As pointed out by Kasangian [BeWal82] a Cauchy completion of a reversible enrichment fails in general to be reversible. This fact forces us to de ne a completion for reversible enrichments that preserves reversibility. This is the so called Cauchy reversible completion. Thus we get the desired generalisation: an adjoint pair in Caten ....
...., A(a 0 ; a) b2Obj(A) s (b; a) s (a 0 ; b) i.e. 1 s ( s . Similarly one gets from ( s s 1 that 1. There are reversible enrichments with non reversible Cauchy completions. A very simple example (due to S. Kasangian) may be found in [BeWal82]. This motivates the further notion of Cauchy reversible completion. De nition 4.5 (Reversible modules) A V module between reversible V categories is reversible when it has right adjoint s . Remark 4.6 If f is a V functor between reversible V categories then the V module f is ....
R. Betti, R.F.C. Walters, The symmetry of the Cauchy-completion of a category, LNM 962, 82, 7-12.
Applying Enriched Categories to Quasi-Uniform Spaces. - Schmitt (2000) (Correct)
....get an equivalence of categories IntR : AUnif = DAUnif . The functors ER : AUnif REnr and ERD : DAUnif REnr satisfy ER = ERD IntR . Since ERD is faithfull then also is ER . Recall that the Cauchy completion does not generally preserve reversibility [Be Wal82]. But according to 7.5, one sees that for any uniform 4 uple S, the Cauchy completion of any reversible enrichment over CS is also reversible. Thus from 7.11, one deduces Theorem 9.5 The inclusion functor SC AUnif , AUnif has a left adjoint. Precisely the restriction to the category ....
R. Betti, R.F.C. Walters, The symmetry of the Cauchy-completion of a category, LNM 962, 82, 7-12
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