Question

I'm interested in the theory of additive monads, but I can't find much, so does exist a source?

My interest came in the sense of n-lab, that is a monad over an additive category such its endofunctor is an additive functor. I'm interested in the study of additive categories, particurarly, small abelian and Grothendieck categories.

Answer 1

First, a small remark: if $C, D$ are additive categories, then an additive functor $C \to D$ is the same as an $Ab$-enriched functor $C \to D$. So for search terms, one might consider "monads enriched in $Ab$" or the like.

I am not aware of any comprehensive survey article for $Ab$-enriched monads, although I can think of some general examples of such (which might suggest other search terms), for example various localizations (see also "torsion theories"), and also monads which arise from free-forgetful adjunctions between module categories, where the study of the category of algebras of the monad is closely related to descent conditions.

Notice that most of the usual examples of $Set$-enriched monads, such as the free monoid monad, do not carry over straightforwardly to the $Ab$-enriched case (which might partly explain why you don't hear more about additive monads). For example, while the analogous monad $X \mapsto \sum_n X^{\otimes n}$ gives the right notion of free monoid, it is not $Ab$-enriched. Indeed, the only tensor power $X \mapsto X^{\otimes n}$ that is additive is the case where $n = 1$.

(One could try instead $X \mapsto \sum_{n \geq 0} X^n$, where the $X^n$ is the ordinary cartesian power. This construction is additive, but it doesn't give anything interesting.)

As long as one is studying $Ab$-enriched monads, it might be worth considering $V$-enriched monads for symmetric monoidal categories $V$. One line of research was inaugurated by John Power in his paper Enriched Lawvere Theories, which works out a precise correspondence between finitary $V$-enriched monads on (nice) $V$ and a suitable enriched form of Lawvere theories. Specialized to $V = Ab$ (or some other tensor abelian category), this might be your best bet for further investigations (although this is a pure guess, since your question is pretty non-specific).