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Stream: theory: category theory

Topic: Composition of adjunctions

Tomáš Jakl (Oct 30 2023 at 11:32):

Hello, I have what seems like a beginner's question about adjunctions. Assume we have three adjunctions as shown below:

image.png

Assume further that $L_3 = L_1 \circ L_2$ and $R_3 = R_2 \circ R_1$.

Recall that we have transposition/conjugation operations, i.e. natural bijections

• $b_i : \hom(L_i(-), -) \to \hom(-, R_i(-))$
  for $i=1,2,3$.

I might be overlooking something but is it automatic that $b_3 = b_2 \circ b_1$? If this is not the case, is there a name for this scenario? I've come across this in adjunctions induced by (co)monad morphisms where this is always the case. Maybe it's always true for composition of (co)monadic adjunctions. Any pointers would be appreciated.

Nathanael Arkor (Oct 30 2023 at 11:39):

The condition that $b_3 = b_2 \circ b_1$ is precisely the condition remaining for the adjunction between $L_3$ and $R_3$ to be the composite of the other two adjunctions. It's not a necessary condition for, given any $b_3$, we could form a ${b_3}'$ exhibiting $L_3 \dashv R_3$ by, for instance, composing an automorphism to $b_3$.

Nathanael Arkor (Oct 30 2023 at 11:42):

In other words, the condition for an adjunction to be the composite of two others requires this condition, in addition to having the left and right adjoints be composite functors.

Tomáš Jakl (Oct 30 2023 at 11:48):

I see, thanks.

Tomáš Jakl (Oct 31 2023 at 13:34):

A followup question: Suppose we have monads $T$ and $S$ on a category $\mathscr C$ such that $\mathscr C^S$ has coequalizers of reflexive pairs. Then a monad morphism $\lambda : T \Rightarrow S$ induces an adjoint pair $L \dashv R$ between the categories of EM algebras (cf. Johnstone 1974):

image.png

Is it written up somewhere that this adjoint situation is in fact a composition of two adjunctions (in the same sense as discussed above)?

Nathanael Arkor (Oct 31 2023 at 14:57):

I don't personally know of a reference, but I also don't see why it should be true: the left adjoint $L$ is defined using the reflexive coequalisers in $\mathscr C^S$, which are defined only up to isomorphism, so I don't see why the triangle in that diagram involving the left adjoints should commute.

Mike Shulman (Oct 31 2023 at 16:05):

If a diagram of right adjoints commutes, then the corresponding diagram of left adjoints will automatically commute, by uniqueness of adjunctions.

Morgan Rogers (he/him) (Oct 31 2023 at 16:07):

But only up to isomorphism, so that doesn't conflict with what Nathanael said.

Mike Shulman (Oct 31 2023 at 16:43):

I assumed when Tomas said "is" he meant up to isomorphism. Why would you ask for more than that?

Tomáš Jakl (Oct 31 2023 at 16:44):

Yes, there's some choice involved but as we showed in our technical report the equaliser for free algebras can be given by an explicit formula so that the diagram commutes on the nose.

Tomáš Jakl (Oct 31 2023 at 16:45):

Separately to that I needed the fact that the composite of the two adjunctions gives the third one so my question is if this fact has been written up somewhere. I was surprised that it is not so trivial and needed the techniques of bimorphisms that we also discuss in the above mentioned technical report.

Nathanael Arkor (Oct 31 2023 at 17:38):

Mike Shulman said:

I assumed when Tomas said "is" he meant up to isomorphism. Why would you ask for more than that?

If you wanted the assignment from monad morphisms to categories of algebras to be (covariantly) functorial rather than pseudofunctorial, which makes everything simpler. (To be clear, Tomáš asked for composition "in the same sense as discussed above", which involves strict composition of both adjoints.)

Tomáš Jakl (Oct 31 2023 at 18:36):

Nathanael Arkor said:

If you wanted the assignment from monad morphisms to categories of algebras to be (covariantly) functorial rather than pseudofunctorial, which makes everything simpler.

I'm not considering the category of monad morphisms so I don't understand why this would make any difference :upside_down:.

Either way, I would be glad for any reference, whether that would be in the strict or weak sense. Is there any?

Nathanael Arkor (Oct 31 2023 at 19:25):

I think I'm slightly confused by what you're asking, which is leading me to not give helpful answers. If you are happy for the adjoints to compose only up to isomorphism, then the additional condition involving the strict equality of the natural isomorphisms defining the adjunctions is unnecessary (because the choices of natural isomorphism are not important if you are reasoning up to isomorphism). As far as I can see, the only reason to require the additional condition is if you want everything to commute strictly. So I expect there is not a reference in the literature, because people are generally happy to reason about adjoints up to isomorphism.

Nathanael Arkor (Oct 31 2023 at 19:25):

Presumably the condition you are asking for is that the isomorphism $\mathscr C^S(L F^T {-}, {-}) \cong \mathscr C^S(F^S{-}, {-}) \cong \mathscr C({-}, U^S{-}) = \mathscr C({-}, U^T R{-})$ is equal to the isomorphism defining the composite adjunction? Is there a reason you want this to hold?

Nathanael Arkor (Oct 31 2023 at 19:40):

(However, given that you can define $L$ in such a way as to make the triangle commute, you should be able to phrase your result as giving a functor from the category of monads whose categories of algebras have reflexive coequalisers, to the coslice category under $\mathscr C$, which will furthermore form something like a local adjunction with the usual (contravariant) assignment of categories of algebras to monads. This seems like a nice formulation that I have not seen elsewhere.)

Tomáš Jakl (Nov 01 2023 at 09:01):

Nathanael Arkor said:

Presumably the condition you are asking for is that the isomorphism $\mathscr C^S(L F^T {-}, {-}) \cong \mathscr C^S(F^S{-}, {-}) \cong \mathscr C({-}, U^S{-}) = \mathscr C({-}, U^T R{-})$ is equal to the isomorphism defining the composite adjunction? Is there a reason you want this to hold?

Yes, this is precisely all I care about and I don't care all that much about the strictness of the composition (although, as I said, I have it for free). I need this for applications, when proving locality theorems (such as Gaifman locality) in the setting of game comonads. It would be nice if I could just have a reference that proves the above property for monad morphisms (or even better lax monad morphisms/Eilenberg-Moore laws).

Nathanael Arkor (Nov 01 2023 at 11:27):

(Perhaps someone else can provide a reference, but I expect there is not a reference; in the only contexts I have seen this theorem, the authors are satisfied simply with the existence of the left adjoint.)

Tomáš Jakl (Nov 01 2023 at 16:22):

Great, thanks a lot!