Bsrgvty

I've been thinking about the following category $mathbbG$. Objects of $mathbbG$ 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 $X in mathbbG(G,H)$ and $Y in mathbbG(H,K)$, the composition $Xcirc Y$ is the cartesian product $X times Y$, modulo the relation $(xcdot h, y) sim (x,hcdot y)$. 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).

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 $Acirc Xcong C$ and $Bcirc Xcong D$. But, since $Acirc X$ is a quotient of $Atimes X$, 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.

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.

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.