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August 1, 2007

Solving Transformations for One of the Two n-Functors

Posted by Urs Schreiber

Here is a supposedly basic n-categorical question which I keep running into, and which doesn’t seem to be addressed in the literature that I am aware of, nor by the people that I talk to:

fix some integer n and some notion of (n+1)-category of n-categories. Then

Given two parallel n-functors F and G, what is the minimum of conditions and extra structure S on a transformation t:FG such that the consistency equation for the component map of t may be solved for F in terms of the data given by G, t and S?

Under which conditions does a morphism of n-functors allow to “solve for” one of the two n-functors, in terms of the other?

To start with, a sufficient condition is certainly to demand that t is an equivalence, and to let S={t¯,i,e,} be a specified weak inverse t¯ of t with specified (weak) unit i and counit e, and, if n>2, further structure.

For instance, to give the simplest example, if two 1-functors F and G are related by an invertible natural transformation t:FG we can “solve F for G and t” in that for any morphisms afb in the domain, we have F(a)F(f)F(b)=F(a)t(a)G(a)G(f)G(b)t(b) 1F(b) with the right hand side not involving F itself.

But, while demanding t to be an equivalence is sufficient for doing this, it is far from necessary.

For 1-functors, the sufficient and necessary condition is that t has a right inverse.

But already for 2-functors, the situation becomes more interesting. While I am not quite sure about necessity, I think that a sufficient condition for 2-functors which is truly weaker than demanding an equivalence of 2-functors is to demand that t:FG fits into a special ambidextrous adjunction: a left and a right adjunction such that the counit of one is the right-inverse of the unit of the other. See definition 3 here.

This allows to “solve F for G”, as described on p. 50.

Now, I have come to begin wondering about the analogous question for n=3.

More concretely, I am encountering the following issue: I seem to have a morphism of 3-functors, which behaves a lot like one would expect a pseudoadjunction to behave. But there is one crucial difference:

where in the defintion of an adjunction one usually has an identity morphism, I want a “weak” identity morphism: a morphism which is the identity only lax-ly and op-lax-ly, hence one which is a monad and co-monad on its domain.

I went back to Aaron Lauda’s Frobenius algebras and ambidextrous adjunctions, where on pages 16 and following Verity’s notion of pseudoadjunction is recalled, but it seems I am looking for something even weaker than that: the 2-morphisms on p. 16 labeled “1” I would like to allow to be just monads and comonads instead of identities.

I’d be grateful for any comments on this.

Posted at August 1, 2007 5:24 PM UTC

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