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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".

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