Reference: equivalences can be made into adjunctions
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Bruce Bartlett -
Jean Bénabou -
Jonathan CHICHE 齊正航 -
Paul Levy -
Steve Lack