Coproduct in an alternative category of groups

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).

• Sorry, yes you are right, that is what I meant. I'll edit the question. – Dave Sep 20 at 2:56
• Presumably you mean the morphisms are isomorphism classes of $G\times H^{op}$-sets. – Eric Wofsey Sep 20 at 3:04
• Well, composition would not be associative, since $(X\circ Y)\circ Z$ is only canonically isomorphic to $X\circ (Y\circ Z)$, not literally equal. – Eric Wofsey Sep 20 at 3:23
• Sorry, I just deleted my comment as you commented because I realised that is exactly the reason for only considering isomorphism classes. – Dave Sep 20 at 3:25

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.

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.