I have a category $C$, which is equipped with a symmetric monoidal structure (tensor product $\otimes$, unit object $1$). My category also has finite coproducts (I'll write them using $\oplus$, and write $0$ for the initial object), and $\otimes$ distributes over $\oplus$.

By an *exponential monad*, I mean a monad $(T,\eta,\mu)$ on $C$, where the functor $T:C\to C$ is equipped with some structure maps of the form
$$\nu \colon 1 \to T(0)$$
and
$$\alpha\colon T(X)\otimes T(Y) \to T(X\oplus Y).$$
The structure maps are isomorphisms, and are suitably "coherent" with respect to the two monoidal structures $\otimes$ and $\oplus$.

The simplest example is: $C$ is the category of $k$-vector spaces, and $T=\mathrm{Sym}$ is the commutative $k$-algebra monad (i.e., $\mathrm{Sym}(X)$ is the symmetric algebra $\bigoplus \mathrm{Sym}^q(X)$).

Now, I'm sure I can work out all the formalism that I need for this, if I have to. My question is: is there a convenient place in the literature I can refer to for this? Alternately, is there suitable categorical language which makes this concept easy to talk about?

I'd also like to have a good formalism for talking about a "grading" on $T$. This means a decomposition of the functor $T=\bigoplus T^q$, where $T^q\colon C\to C$ are functors, which have "nice" properties (for instance, $T^m(X\oplus Y)$ is a sum of $T^p(X)\otimes T^{m-p}(Y)$). The motivating example again comes from the symmetric algebra: $\mathrm{Sym}=\bigoplus \mathrm{Sym}^q$.

ifthe two monoidal structures otimes and + are actually the same. I'm interested in a case where they are not. – Charles Rezk Feb 19 '13 at 14:04