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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?

Carla only proves trivial prop
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    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. – tomasz Nov 26 '21 at 17:43
  • Maybe you could also do something with compact connected groups via Lie algebras, but I'm not sure. – tomasz Nov 26 '21 at 17:54
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    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. – Chris Grossack Nov 27 '21 at 21:59
  • My best regards from TeX.SE. :-) – Sebastiano Mar 08 '22 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 https://ncatlab.org/nlab/show/total+category ... – Dabouliplop Aug 29 '22 at 18:15
  • @Dabouliplop An answer for a category of groups of cardinality (strictly) bounded by some cardinal $\kappa$ would also be interesting. – Carla only proves trivial prop Aug 29 '22 at 22:38
  • Ok, but in that case, we might as well allow the "set of points" to be large. Usually, in a duality you have a notion of "set of points" defined as the set of morphisms to another object (possibly outside of the category). But here, an obstacle is that there are lots of simple groups. You could say "my set of points is just the collection of morphisms to all the groups" but then you recover the Yoneda embedding, and from a point of view the whole point of interesting dualities is to improve this embedding. – Dabouliplop Aug 30 '22 at 11:18
  • Maybe something is possible for groups but I'm giving a possible reason for why nobody gave an answer. – Dabouliplop Aug 30 '22 at 11:18

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