Timeline for On realizing a topos of sheaves as a topos of equivariant sheaves
Current License: CC BY-SA 4.0
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| Nov 4, 2021 at 17:28 | history | edited | Jens Hemelaer | CC BY-SA 4.0 |
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| Nov 4, 2021 at 13:16 | comment | added | Jens Hemelaer | About $G$-equivariant sheaves on topological spaces: there is an equivalence of categories between sheaves on $X$ and local homeomorphisms $Y \to X$. A $G$-equivariant sheaf then corresponds by definition to a local homeomorphism $\pi : Y \to X$ together with a continuous $G$-action on $Y$ such that $\pi(g \cdot x) = g \cdot \pi(x)$. The category of $G$-equivariant sheaves is then a Grothendieck topos. If $G$ is discrete, then an explicit site of definition is given in Johnstone's Elephant, Example 2.1.11(c), p. 76. | |
| Nov 4, 2021 at 13:11 | comment | added | Jens Hemelaer | Yes, it is indeed the pseudopullback in the 2-category of topoi. If the two morphisms defining the pullback come from morphisms of sites between sites that have finite limits (I believe this is the case in your setting), then you can compute a presenting site for the pullback as well, see here. I don't know how practical this calculation is in your case. | |
| Nov 4, 2021 at 12:43 | comment | added | Adrien MORIN | This is awesome ! I have a few questions and remarks. 1) By pullback of topoi do you mean the pullback in the 2-category of topoi I presume ? 2) Can we compute an explicit presenting site for the pullback ? 3) I'm not sure I understand what a G-equivariant sheaf is with your definition, do you have some reference I could read ? | |
| Nov 4, 2021 at 11:14 | history | edited | Jens Hemelaer | CC BY-SA 4.0 |
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| Nov 4, 2021 at 10:55 | history | answered | Jens Hemelaer | CC BY-SA 4.0 |