I've been thinking about the following category $\mathbb{G}$. Objects of $\mathbb{G}$ are groups and a morphism from $G$ to $H$ is a set $X$ equipped with commuting left, right actions of $G, H$; equivalently, a left action of $G \oplus H^{op}$ on $X$. The identity morphism on $G$ is $G$ itself, with the actions just given by left and right multiplication. If $X \in \mathbb{G}(G,H)$ and $Y \in \mathbb{G}(H,K)$, the composition $X\circ Y$ is the cartesian product $X \times Y$, modulo the relation $(x\cdot h, y) \sim (x,h\cdot y)$. This admits a well-defined $G \oplus K^{op}$ action.
Is this a standard category to consider? It is similar to the category of rings, with bimodules as morphisms.
I am interested in whether this category has a coproduct, and if it is different from the coproduct in the standard category of groups (free product).
Best Answer
This category does not have coproducts. The simplest example is that it has no initial object: an initial object would be a group $G$ such that for any group $H$ there is exactly one $(G,H)$-bimodule (up to isomorphism), but this is impossible since there is always a proper class of different $(G,H)$-bimodules. (Here by $(G,H)$-bimodule of course I mean set with commuting left $G$-action and right $H$-action.)
Or, consider a coproduct of two copies of the trivial group. That would be a group $G$ together with two right $G$-modules $A$ and $B$ such that for any group $H$ with two right $H$-modules $C$ and $D$, there is a unique $(G,H)$-bimodule $X$ such that $A\circ X\cong C$ and $B\circ X\cong D$. But, since $A\circ X$ is a quotient of $A\times X$, it is empty iff either $A$ or $X$ is empty. So for instance, if $C$ is empty and $D$ is nonempty, then we find that $B$ must be empty and $A$ must be nonempty, but then we get a contradiction if we swap the roles of $C$ and $D$. A similar argument (with messier notation) shows that actually no coproducts at all exist besides unary coproducts.