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Given a monad, $(M, \mu, \eta)$, where $M: C \rightarrow C$ for some category $C$, there is a category of factorizations, $F\cdot G = M$ where $F: X \rightarrow C$, $G: C \rightarrow X$. Though this may be a rather open question, does anyone have any examples of calculations whose result is the computation of some particular factorization that has some desired property? What are the methods of calculation? The result of the computation should be a category and the pair $F, G$. What might be an algorithm given a monad on a finite category that computes these objects and functors?

I will assume that G⊣F, And that the maps between factorizations are functors that commute nicely with the units and counits.

As per a comment, I would like to focus on the Kleisli category. With that said, I believe the comment has suggested that there is no way to have a "means" or requirement that would cause you to calculate anything else in the category of factorizations, aside from the category of M-algebras (which is Eilenberg-Moore?). This seems like a question all its own.

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    $\begingroup$ There are two canonical factorizations, one with $X=$ the category of $M$-algebras and another with $X=$ the Kleisli category for $M$. They both additionally satisfy the condition that the monad is the one induced by the adjunction (i. e. $\eta$ is the unit of the adjunction and $\mu=F\varepsilon G$ where $\varepsilon$ is the counit of the adjunction). If you do not impose these requirements I doubt any pleasant classification is possible. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 12, 2018 at 17:11
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    $\begingroup$ Concerning subsequent edit - I only want to say that if in the classification you do not actually take into account $\eta$ and $\mu$ then you actually ask for factorizations of an endofunctor that happens to carry some monad structure. For general endofunctors I believe the classification will be difficult, and I do not see how it could help to know that the endofunctor admits some monad structure. And yes, by algebras I mean Eilenberg-Moore. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 12, 2018 at 17:21
  • $\begingroup$ On the afterthought - I would like to add to my previous comment something obvious that did not occur to me when I wrote it. In fact if an endofunctor admits such factorization then it also admits at least one monad structure (coming from the adjunction) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 12, 2018 at 18:54
  • $\begingroup$ Still not clear enough I'm afraid. A more correct thing to say would be that if you do not impose any conditions tying together unit and counit of the adjunction with the $\eta$ and $\mu$ of the monad, then the classification you want will (a) only depend on $M$ as an endofunctor and (b) will in particular subsume classification of all monad structures on the endofunctor $M$. I believe it would be methodologically simpler to split this into two (quite different in nature, I believe) subtasks - classify adjunctions inducing fixed $\eta$ and $\mu$ and classify monad structures on $M$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 14, 2018 at 5:43

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Your last question, the one you call "a question all its own," is essentially the subject of this paper. (Free version from the author's website here.)

Here is a relevant excerpt from the introduction. The notation is as follows: $T$ is a monad, $C^T$ is the Eilenberg-Moore category of that monad (i.e., the category of $T$-algebras), and a "homological presentation of $T$" is an adjunction whose associated monad is $T$ and which satisfies a certain condition defined in Definition 2.2 in the paper. The relevant excerpt:

In Theorem 2.20, when $T$ is coequalizable and $\mathcal{C}^T$ has a biproduct and is Krull-Schmidt, we actually construct a "coordinate system" on the natural equivalence classes of homological presentations of $T$! Any homological presentation is determined uniquely (up to natural equivalence) by specifying a suitable subcollection of the collection of isomorphism classes of indecomposable objects of $\mathcal{C}^T$. In some practical cases, $\mathcal{C}^T$ is the category of finitely generated modules over an algebra, and then the vertices of the Auslander-Reiten quiver of $\mathcal{C}^T$ act as "coordinates" for the collection of natural equivalence classes of homological presentations of $T$.

A fairly explicit example is given in section 4, computing all the "finitary" homological presentations of the free $A$-module monad on the category of sets, where $A$ is a Dedekind domain.

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