It’s a curious (though well known) phenomenon that an equivalence in a bicategory can always be converted into an *adjoint* equivalence by tweaking one of the 2-cells. There are two ways to prove it, that I know of. The elementary way is just to write down a (rather complicated) definition and laboriously prove that it has the desired properties. I always found this rather baffling, so I preferred the more abstract approach: in essence, it’s easy to show that equivalences in **Cat** have this property by using the Yoneda lemma, then one can use the bicategorical Yoneda lemma to transfer the result to an arbitrary bicategory. But recently I learnt something that makes the direct approach a lot easier for me to understand.

At the CMS meeting in Calgary I gave a talk about monoidal categories in which I described how, under certain circumstances, the coherence conditions for a monoidal category are redundant. What I meant is that if you give me any natural isomorphisms

*α _{A,B,C}: A⊗(B⊗C) → (A⊗B)⊗C*

*λ*

_{A}: I⊗A → A*ρ*

_{A}: A⊗I → Athen I can use them to construct new ones that satisfy the coherence conditions. This doesn’t always work – my argument assumes a technical condition called *tensor generation* – but a key part of the construction is perfectly general. Here it is: suppose we have a category ** C** equipped with a functor

*⊗:*and a specified object

**C**×**C**→**C***I*. If you give me natural isomorphisms

*λ _{A}: I⊗A → A*

*ρ*

_{A}: A⊗I → Athen I can construct a new natural isomorphism *λ’* with the property that *λ’ _{I} = ρ_{I}*. I simply define

*λ’*to be the composite

_{A}(The only trick needed to see that *λ’ _{I} = ρ_{I}* is the old observation of Kelly that, since

*ρ*is natural, the square

commutes, and since it’s invertible that implies *ρ _{I}⊗I = ρ_{I⊗I}*. Just use that and the naturality of

*λ*.)

After the talk, Robert Paré asked me whether turning an equivalence into an adjoint equivalence was an instance of this construction. Not as far as I know, I said. But his hunch was right, as he showed me the next morning after he’d figured it out. Given an equivalence

in some 2-category ** B**, consider the hom category

*, define a tensor on it by*

**B**(X, Y)*a⊗b := bga*, and take the ‘unit’ object to be

*f*. The

*λ*and

*ρ*are given by the 2-cells

*fg ⇒ 1*and

*gf ⇒ 1*from the equivalence. Then you construct

*λ’*, and lo and behold! the equation

*λ’*corresponds exactly to one of the triangle equations for an adjunction. (The other can then be derived using the fact that the 2-cells are invertible.) It also works when

_{I}= ρ_{I}**B**is a bicategory: the only real difference is you have to choose one of the bracketings for

*bga*, say

*b(ga)*, and put in a couple of associators.

I am currently concerned with a certain application, where it seems to turn out that the right notion of “equality” in a bicategorical setup is not quite that of an (adjoint) equivalence, but something slightly weaker.

From any adjunction one obtains a monad, from an “ambidextrous adjunction” one obtains a Frobenius algebra like monad. These may have a property called “specialness”. I called the adjunctions that lead to special Frobenius algebras “special ambidextrous adjunctions”.

The details are described here.

I wonder if this makes sense to anyone else, and if so, if it has been considered by anyone else. I expect it has, but I haven’t seen anything like that discussed in the literature.

What’s an “adjoint equivalence”?

An adjoint equivalence is an adjunction whose unit and counit are invertible. So it’s an equivalence and an adjunction at the same time.

Sorry your comments took so long to come through, Urs. (WordPress misidentified them as spam, and I only just spotted them.)

I haven’t come across these “special ambidextrous adjunctions” before, but they look intriguing. Do you have a general result about transport of structure across such a thing, or is it just in one special case?

Nerver mind. I thought the first one got lost. I wish you would erase it, now that there are two of them…

No.

The way I encountered them was as follows:

Say you have two 2-functors F, G : A –> B and a pseudonatural transformation F –a-> G.

Under which conditions can one recover F from knowing just G and the data implicit in the transformation “a” (i.e the map a: Mor_1(A)–>Mor_2(B) ).

This arises in local trivialization of surface transport. F would be a globally defined surface transport which locally can be trivialized to G. For that to make sense, you need to be able to reconstruct F from G and the trivialization data.

Consider it done.

I have the vague idea that this paper of Kelly and Lack may be relevant, but I haven’t looked into the details. Do you know it?

I always liked the string diagram proof that you can turn an equivalence into an adjoint equivalence. You can see a version of it here:

John Baez and Aaron Lauda

Higher-Dimensional Algebra V: 2-Groups

starting on page 19, in the section on “Improvement”. We take the cryptic formula for “improving” an equivalence to an adjoint equivalence, write it out using string diagrams, and use string diagrams to check that it works.

We’re doing this in a specific context: we’ve got a “weak 2-group”, i.e. a monoidal category that’s a groupoid where every object x is part of an equivalence:

e: x* tensor x -> 1

i: 1 -> x tensor x*

and we’re improving e to make this adjunction into an adjoint equivalence. We call the result a “coherent 2-group”. We wind up showing that the 2-category of weak 2-groups is equivalent to the category of coherent 2-groups.

But just look at the pictures!!!

I wrote:

and we’re improving e to make this adjunction into an adjoint equivalence.Ugh – I meant we’re making this equivalence into an adjoint equivalence. Note that the data I wrote down is enough to specify an equivalence, because i and e are invertible.