Timeline for Motivation for equivariant sheaves?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Apr 8, 2010 at 10:20 | vote | accept | Gerrit Begher | ||
| Apr 7, 2010 at 21:15 | answer | added | Zoran Skoda | timeline score: 16 | |
| Apr 7, 2010 at 20:52 | comment | added | Gerrit Begher | @Reid: So in the model case $Sh:Top^{op} \to Cat$ of topological spaces and ordinary set valued sheaves the colimit in question would be the quotient $X/G$ of a space - colimit in the sense that it is a universal, that is initial object with trivial action under $X$ ("$X/Triv^G \subset X/Top^G$ )? Do we hope for $Sh_G(X)$ to be the limit of $Sh\circ For$ i.e.(?) fiber of $Sh$ of the forgetful functor $For:X/Triv^G\to Top$ ? | |
| Apr 7, 2010 at 20:45 | comment | added | Reid Barton | I will make my comment into an answer in a little while, if nobody beats me to it. | |
| Apr 7, 2010 at 20:03 | history | edited | Gerrit Begher |
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| Apr 7, 2010 at 19:59 | comment | added | Bill Kronholm | maybe add equivariant-homotopy tag? | |
| Apr 7, 2010 at 18:46 | comment | added | Reid Barton | To elaborate on Emerton's comment, one would like the (pseudo)functor assigning to a space its category of sheaves to take colimits to limits. However, depending on your notion of space and sheaf, this may not be the case for colimits given by taking the quotient by a group action. The category of G-equivariant sheaves is what we would like the category of sheaves on "X/G" to be, if we had a nice enough object X/G. | |
| Apr 7, 2010 at 18:34 | comment | added | Reid Barton | @Emerton: that is enough motivation for me! Maybe you should make that into an answer. | |
| Apr 7, 2010 at 18:01 | comment | added | Emerton | Equivariant sheaves are more like fixed points then a quotient. If $G$ acts on $X$ then we can pull-back a sheaf $\mathcal F$ on $X$ via an element $g \in G$, giving a $G$-action on the category of sheaves on $X$. To give a $G$-equivariant structure is then to give isomorphisms $g^*\mathcal F \cong \mathcal F$ for all $g\in G$, satisfying a suitable compatibility. This can be thought of as a categorification of the set-theoretic notion of a fixed point of the $G$-action on the sheaves on $X$. | |
| Apr 7, 2010 at 17:54 | history | asked | Gerrit Begher | CC BY-SA 2.5 |