Coproduct in an alternative category of groups

Question

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).

Answer 1

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.

Answer 2

This is a standard category to consider, but as the other answer shows, it has some deficiencies. A standard way to repair them is to consider instead the category whose objects are small categories, where a left module is generalized to a covariant functor into sets and dually.

Talking about isomorphism classes is not ideal; this is really a bicategory, with natural transformations as 2-morphisms. It should be clear how this specializes to the case of groups: bimodule homomorphisms. This bicategory does have colimits in the appropriate sense, which are given by the corresponding colimits of categories. Thus the closest thing to a coproduct of two groups $G,H$ in your category is probably the disjoint union of $G$ and $H$, viewed as one-object categories.

Answer 3

My five cents here.

After linearization (taking free abelian groups on morphism sets and identifying $2m$ with $m \sqcup m$) it is a very useful object and known as a biset category. Endomorphism of an object in this category was known even before categories were defined and evidently a Burnside ring of a group.

There is a surprising construction stemming from it. If you take in consideration only finite $p$-groups, and linearize everything over rationals, then resulting category will have an explicitly defined — by some universal relations — abelian (!) quotient into which category of finite $p$-groups embeds. In some sense this category completely describes rational representations and natural operations on them. Tag word is Roquette category.