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?

  • 7
    $\begingroup$ 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. $\endgroup$
    – tomasz
    Commented Nov 26, 2021 at 17:43
  • $\begingroup$ Maybe you could also do something with compact connected groups via Lie algebras, but I'm not sure. $\endgroup$
    – tomasz
    Commented Nov 26, 2021 at 17:54
  • 5
    $\begingroup$ 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. $\endgroup$ Commented Nov 27, 2021 at 21:59
  • $\begingroup$ My best regards from TeX.SE. :-) $\endgroup$
    – Sebastiano
    Commented Mar 8, 2022 at 20:02
  • $\begingroup$ 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 ... $\endgroup$ Commented Aug 29, 2022 at 18:15


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