# Is there a known simple characterization of the category $\mathbf{Grp}^{\text{op}}$?

For some common categories, there are nice characterizations of their opposite category: ie any group is isomorphic to it's opposite, $$\mathbf{Set}^{\text{op}}$$ is equivalent to the category of complete atomic boolean algebras and $$\mathbf{CRing}^{\text{op}}$$ is equivalent to the category of affine schemes. This question also asks about a description of the opposite category of $$\mathbf{Top}$$, also has some interesting answers.

Is there a nice characterization of $$\mathbf{Grp}^{\text{op}}$$ (ie as some simple concrete category)? Has this category been studied, and if yes, what are some references about its properties?

• Also, the category of finite abelian groups is self-opposite. The opposite of the category of (abstract or discrete) abelian groups is the category of compact abelian groups, by Pontryagin duality. The category of locally compact abelian group is self-opposite, for the same reason. With non-abelian groups, I have no idea, though. Commented Nov 26, 2021 at 17:43
• Maybe you could also do something with compact connected groups via Lie algebras, but I'm not sure. Commented Nov 26, 2021 at 17:54
• It's worth noting that we basically define the category of affine schemes to be $\mathsf{CRing}^\text{op}$, so this isn't necessarily enlightening. Similarly, we define the category of locales to be opposite the category of frames, etc. For a slightly more natural example, notice the category of stone spaces is equivalent to the opposite of the category of boolean algebras. Commented Nov 27, 2021 at 21:59
• My best regards from TeX.SE. :-) Commented Mar 8, 2022 at 20:02
• A problem with groups is that there are simple groups of arbitrarily large cardinality. The opposite of the category of groups is not total, so there is no topological functor from $\mathbf{Grp}^{\mathrm{op}}$ to sets, see ncatlab.org/nlab/show/total+category ... Commented Aug 29, 2022 at 18:15