Some forgetful functors of algebraic categories preserve colimits, but most do not. In order to understand this phenomen in general, I have classified cocontinuous monadic functors whose target is an algebraic category $C$; they correspond to monoid objects in the monoidal category of "internal objects" of $C$ (i.e. $x$ such that $\hom(x,-)$ has values in $C$; what is the common name for this?).

The equivalence is given by module categories. So for example, the cocontinuous monadic functors to $\mathsf{Set}$ are given by $M\mathsf{-Set}$ for monoids $M$, and the cocontinuous monadic functors to $\mathsf{Ab}$ are given by $\mathsf{Mod}(R)$ for rings $R$. For $C=\mathsf{Grp}$ Kan has shown in *On monoids and their dual* that the category of internal objects is equivalent to the category of pointed sets (they are free groups), the monoids are then pointed monoids. A module over a pointed monoid $M$ is then a group $G$ together with a morphism $M \to \mathrm{End}(G)$ of pointed monoids.

Probably this result is not new, but I would like to know where I can find it in the literature. So my question is: Has there any work been done towards the classification of cocontinuous monadic functors?

**EDIT.** Yes there is some literature. For example:

- Paragraphs 8-9 in: Wraith,
*Algebraic theories* - Paragraphs 63-64 in: George M. Bergman, Adam O. Hausknecht, \textit{Cogroups and co-rings in categories of associative rings}, American Mathematical Society, Mathematical Surveys and Monographs # 45, 1996.

Internal monoids are called TW-monads. I vote to close as for "no longer relevant".