Dear Jean & others, I don't have Mac Lane in front of me, so I don't know what he says. But it is true that given functors f and and u, and isomorphims eta:1->uf and epsilon:fu->1 then you can modify just one (either one) of eta and epsilon and obtain an adjoint equivalence. It's useful to note also that for equivalences, either triangle equation implies the other. There's a higher-dimensional version for internal biadjunctions in a Gray-category, which can be found for example in my paper "A Quillen model structure for Gray-categories" (Proposition 3.1.) Bien amicalement, Steve. On 08/04/2013, at 8:09 PM, Jean Bénabou 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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