2 Edited in response to comments, to make it clearer what I'm asking

Let

In answering this MO question, the issue was raised of characterizing when a given endofunctor $R:C\to C$ be an endofunctor. When does $R$ have has the form $U\circ F$ where $F:C\to D$ is left adjoint to $U:D\to C$, for some $D$? i.e. which admit a monad structure. Is there an algebraic or purely categorical characterization of such $R$?Is there any way to recover $F$ and $U$ from $R$?

From this MO question we know that in order to be a composition of adjoints $R$ must be a homotopy equivalence on the nerve $N(C)$. According to this MO question, the converse fails, so this is not a characterization.

We know $F$ has to preserve colimits and $U$ has to preserve limits. I'd be happy with an answer saying $R$ is of the form $U\circ F$ iff $R$ preserves $\langle$ fill in the blank $\rangle$. This seems like it should be known classically (e.g. in Categories for the Working Mathematician), but I never learned it and feel like it would be useful to know.

I should mention that I came up with this question while answering

From this MO question. The motivation for we know that in order to be a composition of adjoints $R$ must be a homotopy equivalence on the nerve $N(C)$. According to this MO questioncame from thinking of monads as monoids in , the category of endofunctorsconverse fails, and trying to figure out which monads came from adjunctionsso this is not a characterization.

1

# Is there an algebraic characterization for which endofunctors are compositions of adjoints?

Let $R:C\to C$ be an endofunctor. When does $R$ have the form $U\circ F$ where $F:C\to D$ is left adjoint to $U:D\to C$, for some $D$? Is there an algebraic characterization of such $R$? Is there any way to recover $F$ and $U$ from $R$?

From this MO question we know that in order to be a composition of adjoints $R$ must be a homotopy equivalence on the nerve $N(C)$. According to this MO question, the converse fails, so this is not a characterization.

We know $F$ has to preserve colimits and $U$ has to preserve limits. I'd be happy with an answer saying $R$ is of the form $U\circ F$ iff $R$ preserves $\langle$ fill in the blank $\rangle$. This seems like it should be known classically (e.g. in Categories for the Working Mathematician), but I never learned it and feel like it would be useful to know.

I should mention that I came up with this question while answering this MO question. The motivation for this question came from thinking of monads as monoids in the category of endofunctors, and trying to figure out which monads came from adjunctions.