8
$\begingroup$

The collection of all self-equivalences of a category $C$ constitutes a $2$-group, which is a categorification of the notion of a group. My question is about what happens when one replaces equivalences by general adjoint functors.

The n-lab page on adjunctions says:

a morphism in an adjunction need not be invertible, but it has in some sense a left inverse from below and a right inverse from above.

I would like to know if this can be formalised in an interesting way.

In order to ask a specific question, I want to take a single adjoint pair of functors $(L,R):C\to C$, and let $M$ be the monoid of endofunctors of $C$ generated by $L$ and $R$ (together with some suitably chosen family of natural transformations).

Q. What kind of ($2$-)algebraic structure is $M$?

I apologise that this is a somewhat vague question. An ideal answer would have the form "$M$ is a (lax/colax/...) $2$-blah", where "blah" is an interesting algebraic structure that arises in other contexts not a priori having anything to do with adjoint functors. If it makes more sense to change the setting a little---e.g., to consider instead of $M$ the collection of all functors on $C$ admitting a two-sided adjoint---then please go ahead and do so.

$\endgroup$

2 Answers 2

5
$\begingroup$

If everything has both a left and a right adjoint, then you're talking about a rigid monoidal category. (Here I've done the usual dimension shift where a 2-category with one object is the same as a monoidal category.)

If you only want adjoints on one side it's a bit more awkward because that's not left rigid (since the left adjoint will only have a right adjoint) nor right rigid. But for the single functor case it'd be a monoidal category generated by a right rigid object.

$\endgroup$
1
  • 4
    $\begingroup$ I am of the opinion that rigid monoidal categories (especially the ones which are $k$-linear for some field $k$) are an excellent generalization of the notion of group. $\endgroup$ Dec 21, 2017 at 21:48
3
$\begingroup$

The free adjunction, i.e. the 2-category generated by an adjunction $C^\to_\leftarrow D$ (with $C$ and $D$ distinct) is described by Schanuel and Street. In particular, if you restrict to the full sub-2-category on $C$, you get the free monad, and the free full sub-2-category on $D$ is the free comonad. The free monad itself is an interesting 2-category; it's the delooping of the strict mononoidal category $\Delta_a$ of augmented simplices, otherwise known as finite linear orders. Since a monad is just a monoid in the endomorphism category, this is equivalent to the fact that $\Delta_a$ is the free strict monoidal category on a monoid (see the last link).

The free adjunction with $C=D$ will be more complicated. I suspect it's not that much more complicated, but I don't know of a reference where it's written down.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.