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    • CommentRowNumber1.
    • CommentAuthorRichard Williamson
    • CommentTimeJul 6th 2020
    • (edited Jul 6th 2020)

    Given an (adjoint) equivalence in a 2-category, does anyone know if it is possible to replace the objects and/or 1-arrows in some reasonable way (up to equivalence/isomorphism) so that either the unit or the co-unit becomes an identity, not just a natural isomorphism? I don’t have time to think about it just now, and maybe someone knows something off the top of their head.

    • CommentRowNumber2.
    • CommentAuthorRichard Williamson
    • CommentTimeJul 6th 2020
    • (edited Jul 6th 2020)

    I guess this is fairly obvious actually. I’ll drop adjointness for simplicitly. Suppose that we have F:ABF: A \rightarrow B and G:BAG: B \rightarrow A, a natural isomorphism ϕ:GFid\phi: GF \rightarrow id, and a natural isomorphism ψ:FGid\psi: FG \rightarrow id. Let aa be an object of AA. Since FF is an equivalence, there must be an isomorphism f af_{a} in BB such that G(f a)G(f_{a}) is equal (on the nose) to the isomorphism ϕ(a):GF(a)a\phi(a): GF(a) \rightarrow a. We can then replace FF by FF', where F(a)F'(a) is the target of f f(A)f_{f(A)}, and F(g)F'(g) for g:aag: a \rightarrow a' is f f(a)F(g)f f(a) 1f_{f(a')} \circ F(g) \circ f_{f(a)}^{-1}. Keeping GG the same, we then have that GFGF' is on the nose equal to id(A)id(A), and FGF'G is still naturally isomorphic to idid.

    If people agree, I’ll add this to equivalence or some such page. If I have not made a mistake in the above, it must be in the literature somewhere; does anybody know of an explicit reference? Or if there is some canonical/abstract formalism which recovers the above, that would be very good to add too.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJul 6th 2020

    That would only be true if you already knew there was an object bb such that G(b)=aG(b)=a on the nose.

  1. Yes, thanks, good point! The functor GG would have to be assumed to be surjective-on-objects, which is probably fine in practise in most cases. I’ll add something about this a little later to the nLab.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJul 7th 2020

    Note that surjective-on-objects equivalences are the acyclic fibrations in the canonical model structure on Cat.

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