Question

Coproducts of modules over an algebraic monad $\Sigma$ are described in Section 4.16.14/15 in Durov's thesis. 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?

Answer 1

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.