Coproducts of modules over an algebraic monad $\Sigma$ are described in Section 4.16.14/15 in Durov's [thesis](http://arxiv.org/abs/0704.2030). It is claimed there that for $\Sigma$-modules $M,N$, the set $M \coprod N$ generates $M \oplus N$ and that this implies that every element of $M \oplus N$ may be written as $t(x\_1,...,x\_n,y\_1,...,y\_n)$ for some $m,n \geq 0$, $t \in \Sigma(n)$ (surely he means $t \in \Sigma(n+m)$), $x\_i \in M$ and $y\_j \in N$.


Is this correct? What about taking the algebraic monad $\Sigma$ of groups (or even just monoids) so that $\mathrm{Mod}(\Sigma)$ is the category of groups. Here $G \oplus H$ is the free coproduct, it consists of the elements of the form $g\_1 h\_1 g\_2 h\_2 \dotsc g\_n h\_n$ with $g\_i \in G$ and $h\_i \in H$. We cannot write it as $g h$.


It seems to me that we can only find an expression as above if $\Sigma$ is commutative (5.1.1. loc.cit), so that in particular the example $\mathbb{Z}\_{\infty}$ should work, but in the general case Durov wanted to write $t(x\_1,...,x\_n)$ with $x\_i \in M$ or $x\_i \in N$. Is this true?



Coproducts of modules over an algebraic monad
I think the point is that an operation in an algebraic theory (even a noncommutative one) need not preserve the order of its inputs. There is a binary operation in the theory of groups which takes the input $(g,h)$ to the product $h g$. More generally, the symmetric group on $n$ letters acts on the set of $n$-ary operations, although in general the action may not have many fixed points. Thus, you can always find an operation which permutes all the $M$-inputs to the front and all the $N$-inputs to the rear.


