Reference: equivalences can be made into adjunctions

Reference: equivalences can be made into adjunctions

[Jonathan CHICHE 齊正航]

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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Re: Reference: equivalences can be made into adjunctions

[Paul Levy]

On 6 Apr 2013, at 21:22, Jonathan CHICHE 齊正航 wrote:

> 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?

Dear Jonathan,

A reference is "Two-dimensional monad theory" by Blackwell, Kelly and  
Power.  Journal of Pure and Applied Algebra 59, 1989.

I asked a similar question in 2001:

http://facultypages.ecc.edu/alsani/ct01%289-12%29/msg00071.html

which led to some very interesting responses.

Paul



--
Paul Blain Levy
School of Computer Science, University of Birmingham
+44 121 414 4792
http://www.cs.bham.ac.uk/~pbl












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Re: Reference: equivalences can be made into adjunctions

[Jean Bénabou]

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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