Suppose that $F:\mathscr{C}^{op} \to Set$ is a presheaf. For an object $C$, denote by $Aut(C)$ the group of isomorphisms $f:C \to C.$ There is a canonical action $Aut(C)$ of on $F(C).$ Is there a special name for those $F$ for which all of these actions are transitive? Has this situation been studied? Does this imply any nice categorical consequences? If $F$ is also a sheaf for a Grothendiek topology on $\mathscr{C},$ does $F$ satisfy any nice conditions in the associated topos of sheaves?
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2 Answers
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At least for fiber functors from Galois categories to finite sets this is the well-known notion of Galois objects. See Lenstra's notes, section 3.14. For example, if $L/K$ is a finite separable field extension, then it is Galois iff $\mathrm{Aut}_K(L)$ acts transitively on $\hom(L,\overline{K})$.
answered Jun 17, 2013 at 16:37
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I am not well-versed in these matters, but I believe Olivia Caramello uses something like this in her toposical formulation of Galois theory. See her paper and, if you can understand French, this brief lecture series expounding it.
