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Stream: theory: category theory

Topic: Is my category a Kleisli category over C

Ralph Sarkis (Feb 23 2023 at 13:21):

The monadicity theorems and many other results about categories of Eilenberg-Moore algebras allow us to answer questions like: is $\mathbf{D}$ a category of algebras over $\mathbf{C}$? I am wondering if there has been similar results for Kleisli categories to answer questions like: is $\mathbf{D}$ a category of free algebras over $\mathbf{C}$?

Nathanael Arkor (Feb 23 2023 at 14:47):

A left adjoint is (up to isomorphism) the inclusion of a category into a Kleisli category if and only if it is bijective-on-objects.

Nathanael Arkor (Feb 23 2023 at 14:53):

This observation is due to Linton (e.g. "Triples vs theories"), but is stated more explicitly in Schumacher's Minimale und maximale Tripelerzeugende und eine Bemerkung zur Tripelbarkeit.

Matteo Capucci (he/him) (Feb 23 2023 at 19:32):

:O wow!

John Baez (Feb 23 2023 at 21:06):

:hushed:

dusko (Feb 23 2023 at 22:13):

looking at this question from the point of view of "categories as a language" it becomes a nice piece of evidence that we still didn't learn to speak the language of adjunctions.

if $T:{\bf C}\to {\bf C}$ is a monad and $L\dashv R:{\bf X }\to{\bf C}$ any resolution of $T$ (i.e. an adjunction with $RL=T$). call $\bf K$ the kleisli category of $T$. then ${\bf K}(a,b) \cong {\bf C}(a, Tb) \cong {\bf X}(La, Lb)$. so the kleisli category of $T$ can be defined as the category with the objects of $\bf C$ and the hom-sets of $\bf X$ determined along $L$ --- for any resolution of $T$ whatsoever. in other words, the kleisli category is obtained by taking the essentially-surjective/full-and-faithful factorization of any resolution of $T$. (when you are given a concrete functor, the essentially-surjective boils down to identity-on-the-objects-functor, aka ioof...)

but then you say "doh, if i take $\bf X$ to be the category of algebras, all that this says is that the kleisli category is the essential image of the free functor". bien-sur. the only amazing thing about this fact is that it is not one of the first exercises in every textbook that includes monads.

but we are pretty much blind to adjunctions.

(plug: read dominic hughes' and my papers about the nucleus and the tight completions to see how much of the iceberg surfaces when the ocean of adjunctions is thickened just a little.)

Nathanael Arkor (Feb 23 2023 at 22:56):

the only amazing thing about this fact is that it is not one of the first exercises in every textbook that includes monads.

I completely agree. In fact, I have been slowly building up a collection of nice exercises for an introduction to category theory, and this was the very first one I added :)

dusko (Feb 24 2023 at 01:19):

keep us posted about the exercises, whenever they get released :)

JS PL (he/him) (Feb 24 2023 at 07:34):

Ralph Sarkis said:

The monadicity theorems and many other results about categories of Eilenberg-Moore algebras allow us to answer questions like: is $\mathbf{D}$ a category of algebras over $\mathbf{C}$? I am wondering if there has been similar results for Kleisli categories to answer questions like: is $\mathbf{D}$ a category of free algebras over $\mathbf{C}$?

A possible related notion is "abstract Kleisli categories" or "thunk-force categories" https://ncatlab.org/nlab/show/thunk-force+category

JS PL (he/him) (Feb 24 2023 at 07:36):

Essentially, if a category is an "thunk-force category", then theres a special class of maps called the "thunkable maps". And on this subcategory of thunkable maps, there's an induced monad and the Kleisli category of the induced monad is equivalent to the starting "thunk-force category"

JS PL (he/him) (Feb 24 2023 at 07:39):

Conversely, every Kleisli category of a monad is a "thunk-force category". But the subcategory of thunkable maps may not be equivalent the starting base category, but the resulting Kleisli category of the monad on thunkable maps is still equivalent to the first Kleisli category anyways.

JS PL (he/him) (Feb 24 2023 at 07:40):

So "thunk-force categories" are precisely the Kleisli categories of monads, or alternatively a "thunk-force category" is a way of writing down a Kleisli category without a starting monad.

Nathanael Arkor (Feb 24 2023 at 10:40):

JS PL (he/him) said:

So "thunk-force categories" are precisely the Kleisli categories of monads, or alternatively a "thunk-force category" is a way of writing down a Kleisli category without a starting monad.

Although this is only true for a specific subclass of monads, according to the nLab page.

Matteo Capucci (he/him) (Feb 24 2023 at 13:56):

dusko said:

in other words, the kleisli category is obtained by taking the essentially-surjective/full-and-faithful factorization of any resolution of $T$. (when you are given a concrete functor, the essentially-surjective boils down to identity-on-the-objects-functor, aka ioof...)

what do you mean by this? you factor the adjunction or just the left adjoint?

dusko said:

but we are pretty much blind to adjunctions.

alas...

Nathanael Arkor (Feb 24 2023 at 13:57):

what do you mean by this? you factor the adjunction or just the left adjoint?

You factor the left adjoint.

JS PL (he/him) (Feb 25 2023 at 22:36):

Nathanael Arkor said:

JS PL (he/him) said:

So "thunk-force categories" are precisely the Kleisli categories of monads, or alternatively a "thunk-force category" is a way of writing down a Kleisli category without a starting monad.

Although this is only true for a specific subclass of monads, according to the nLab page.

What the nlab page is saying is that the construction monad ---> thunk-force category ---> monad gets you back to original monad when the monad is exact. I would still say thunk-force categories are indeed just Kleisli categories since:

JS PL (he/him) (Feb 25 2023 at 22:37):

For any monad, its Kleisli category is a thunk-force category. Covnersely, every thunk-force category is the Kleisli category of an (exact) monad.

JS PL (he/him) (Feb 25 2023 at 22:38):

Anyone curious about thunk-force/abstract categories, we talk about the coKleisli version in this paper of ours: https://arxiv.org/pdf/2108.04304.pdf (see Section 4, its about differential categories but if you're not interested in that, you can ignore those parts)