Let me start with 3 examples to illustrate my question (probably vague, I apologize in advance).
- $\mathbf{Man}$, the category of closed (compact without boundary) topological manifold . For any $M, N\in \mathbf{Man}$ there is the following theorem (Whittaker) which says that
$$\mathrm{Homeo}(M)\cong_{\text{as groups}} \mathrm{Homeo}(N) \, \textit{ if and only if } \, M\cong_{\text{as manifold}} N$$
- $\mathbf{NFields}$, the category of Number fields. For any $F, K\in \mathbf{NFields}$ there is a theorem (Neukirch-Uchida) which says that
$$\mathrm{Gal}(\overline{K}/K)\cong_{\text{as progroups}} \mathrm{Gal}(\overline{F}/F) \textit{ if and only if } K\cong_{\text{as fields}} F $$
- $\mathbf{Vect}$, the category of finite dimensional vector spaces. For any $V, W\in \mathbf{Vect}$ we have that
$$\mathrm{GL}(V)\cong \mathrm{GL}(W) \textit{ if and only if } V\cong W $$
In the first and the third examples we see that the automorphism group of an object determines the object, the second example seems to be similar in some sense however it does not admit the same naive interpretation.
$\textbf{Question:}$ What are other non trivial examples of interesting categories where the automorphism group of an object determines the object it self ? Is there a name for such categories? Is there a way to compare and characterize these kind of categories?
