Dear all, As a remark, this kind of result arguably grows ever more important as one scales the higher-categorical ladder. The next level up, namely that every internal biequivalence in a tricategory can be promoted into a "coherent" biadjoint biequivalence, has been proved by Gurski, Nick Gurski, Biequivalences in tricategories, http://arxiv.org/abs/1102.0979 One needs this for instance to transport structures (monoidal, braided-monoidal, ...) from a bicategory A to a bicategory B, given only that they are equivalent. The nature of the coherence laws is naturally closely related to topology (see eg. pg 140 of http://arxiv.org/abs/1112.1000 and Appendix C of http://arxiv.org/abs/1102.0979). Regards, Bruce Bartlett On Mon, Apr 8, 2013 at 11:09 AM, Jean Bénabou <jean.benabou@wanadoo.fr>wrote:
Dear Jonathan,
I think you misread Mac Lane. If the 2-category is Cat , suppose functors f: A --> B and u: B --> A are such that fu and uf are isomorphic to id(A) and id(B) by isomorphisms eta: id(A) -->uf and epsilon: fu --> id(B). He does not say that one can modify eta and epsilon can be to satisfy the the triangle identities. What he says is: You can keep f and modify u, eta and epsilon to get a functor u': B ---> A and isomorphisms eta': id(A) ---> u'f and epsilon': fu' ---> id(B) satisfying the triangle identities.
Under this form the result is true for any 2- Category C. Sketch of proof: Suppose f: A --> B and u: B ---> A are 1-cells and eta and epsilon are isomorphism 2-cells. Then for each object X of C, f,u,eta and epsilon define an equivalence between the categories C(X,A) and C(X,B). Take X=B, then f defines an equivalence C(B,f): C(B,A) --> C(B,B). By Mac Lane's result we can find an adjoint to C(B,f), which is en equivalence, U: C(B,B) ---> C(B,A). Take u' = U(id(B)): B --->A . By a tedious but straightforward computation one verifies that the pair (f, u') is an adjoint equivalence.
Bien amicalement,
Jean
Le 6 avr. 2013 à 22:22, Jonathan CHICHE 齊正航 a écrit :
Dear all,
This is a standard fact, proved for instance in details in "Categories for the Working Mathematician" (Chapter 4, Section 4, Theorem 1 in the second edition), that a functor is an equivalence of categories if and only if it is part of an adjoint equivalence. I would like to use the fact that this is true in an arbitrary 2-category, i.e. that given an equivalence in a 2-category the invertible 2-cells can be required to satisfy the triangle identities. Is there a standard reference for this fact? Well-known books such as classical introductions to category theory are particularly welcome.
Thanks,
Jonathan
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