TITLE: Where is there a treatment of "exponential monads"?
QUESTION [13 upvotes]: 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: $\mathrm{Sym}=\bigoplus \mathrm{Sym}^q$.

REPLY [11 votes]: I've just spent some time working out for myself the details of what an exponential monad is supposed to be, so I might as well post what I learned here.  (I don't know that I'll ever have a reason to write it up more formally.)
Here is the correct definition.  Given symmetric monoidal $(C,\otimes, 1)$, with finite coproducts $+$ and initial object $0$, an exponential structure on a monad $T$ on $C$ should be 

the structure of strong symmetric monoidal functor on $T: (C,+,0)\to (C,\otimes, 1)$, consisting of natural isomorphisms $\nu\colon 1\to T0$ and $\alpha\colon TX\otimes TY\to T(X+Y)$ satisfying a bunch of coherence properties.

There is one additional condition you need to impose.  To state this condition, let $\gamma: T(TX\otimes TY)\to TX\otimes TY$ be the composite
$$ T(TX\otimes TY) \xrightarrow{T\alpha} TT(X+Y) \xrightarrow{\mu} T(X+Y) \xrightarrow{\alpha^{-1}} TX\otimes TY,
$$
where $\mu: TT\to T$ is part of the monad structure.  The additional condition is 

for all $X$ and $Y$, we have $\gamma\circ T(\mu\otimes \mu)=(\mu\otimes \mu)\circ \gamma$ as maps $T(TTX\otimes TTY)\to TX\otimes TY$.

The map $\gamma$ defines a $T$-algebra structure on $TX\otimes TY$ (it is the free algebra structure on $T(X+Y)$, transported along the isomorphism $\alpha$); the additional property says that $\mu\otimes \mu$ is itself a map of $T$-algebras.  Note also that the map $\nu$ identifies $1$ with the initial $T$-algebra $T0$.
Given this, you can prove (with no more difficulty than you would expect) that for every pair of $T$-algebras $A$ and $B$, we can put a canonical $T$-algebra structure on $A\otimes B$, which exhibits it as the coproduct of $A$ and $B$ in the category of $T$-algebras.  In particular, the forgetful functor from $T$-algebras to $C$ becomes strong symmetric monoidal, using coproduct as the monoidal structure for $T$-algebras.  
I would have expected that you would need another condition, relating $\gamma$ to the unit map $\eta\colon I\to T$, or perhaps a condition on $\nu$, but it doesn't seem that this is necessary as far as I can tell.
You don't need the hypothesis that $\otimes$ distribute over coproduct, as I suggested in my question.  It might seem surprising, but apparently you don't even need the monoidal structure on $C$ to be symmetric or associative; being unital appears to be enough to exhibit coproducts of $T$-algebras using $\otimes$.  I suspect that for such a "unital monoidal" category $C$, you might be able to show that the existence of an exponential monad implies that $C$ is symmetric monoidal.  (This does not seem so crazy in light of the way the symmetric monoidal smash product of  EKMM spectra comes about).