Timeline for Representation of Groupoids
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 17, 2015 at 1:32 | answer | added | Rohit D. Holkar | timeline score: 3 | |
| Apr 20, 2012 at 23:30 | comment | added | Omar Antolín-Camarena | Qiaochu (cont'd): ... particularly useful, for example, I don't think people do that for group representations! I've never heard someone say a representation of G is just a representation of the free monoid on the set of elements of G such that ρ(f∘g)=ρ(f)∘ρ(g); it doesn't seem to be a useful observation. Not even considering represenations of G to be special representation of the free group on the set G seems to be useful, as far as I know. | |
| Apr 20, 2012 at 23:27 | comment | added | Omar Antolín-Camarena | Qiaochu: A groupoid representation is a special kind of representation of the underlying quiver. A groupoid representation is a functor from the groupoid to the category Vect of vector spaces, but a quiver representation is a functor from the free category on the quiver to Vect. So the difference is that if you have two composable morphisms $f$ and $g$, a groupoid representation must satisfy $\rho(f \circ g) = \rho(f) \circ \rho(g)$, but a representation of the underlying quiver need not satisfy that. I don't think that viewing groupoid representations as special quiver representations is ... | |
| Jun 26, 2011 at 3:18 | answer | added | Benjamin Steinberg | timeline score: 5 | |
| Jun 7, 2010 at 0:29 | answer | added | David Carchedi | timeline score: 7 | |
| Feb 27, 2010 at 10:31 | comment | added | Shizhuo Zhang | then comonad $\mathfrak{G}$=$M\bigotimes_{A_{U}} -$,where $M$ is a bimodule ($A_{U}\bigotimes_{k} A_{U}^{op}-mod$). In other word, $(M,\delta)$ is a coalgebra in the monoidal category of $A_{U}\bigotimes_{k} A_{U}^{op}-mod$,where $\delta$: $M\rightarrow M\bigotimes_{A_{U}} M$. | |
| Feb 27, 2010 at 10:31 | comment | added | Shizhuo Zhang | @Tyler: I wonder whether you are talking about the following general formalism: Suppose $C_{X}$ is category of quasi coherent sheaves on stack $X$(or more general space). Further assumptions that $X$ is quasi compact and quasi separated. We have a collection of finite affine morphisms $U_{i}\rightarrow X$. Then we have $U=\coprod U_{i}\rightarrow X$. Then $Qcoh_{U}=\coprod Qcoh_{U_{i}}=A_{U}-mod$,where $A_{U}=\prod O_{U}(U_{i})$. And then we use the Beck's theorem for comonad. We get: $Qcoh_{X}=\mathfrak{G}-Comod$,if we further require the stack $X$(or general "space") is semiseparated | |
| Feb 27, 2010 at 2:44 | comment | added | Yuhao Huang | @Taler: Can you elaborate on 2? | |
| Feb 26, 2010 at 19:17 | answer | added | AFK | timeline score: 4 | |
| Feb 26, 2010 at 17:07 | answer | added | Theo Johnson-Freyd | timeline score: 9 | |
| Feb 26, 2010 at 15:13 | comment | added | Shizhuo Zhang | @Tyler: For 2, I think this description is special case of Barr-Beck's theorem right? And if you further require semiseparated condition, you can talk about comodule over coalgebra. | |
| Feb 26, 2010 at 13:56 | answer | added | Chris Schommer-Pries | timeline score: 6 | |
| Feb 26, 2010 at 13:21 | comment | added | Tyler Lawson | Yes, these are studied. For 1: Just as a every groupoid can be decomposed up to natural equivalence as a collection of components with one isomorphism class of group per component, a representation of same is equivalent to one representation per component. It is, yes, a handy formalism for naturally-occuring representations of $\pi_1$. For 2: Given a presentation of a stack by affines, quasicoherent sheaves on the stack can be described in terms of what are called comodules, and certain flavours of algebraic topologist study those all the time. | |
| Feb 26, 2010 at 7:48 | comment | added | Steven Sam | Qiaochu: Thinking of a groupoid representation as a quiver representation means forcing a bunch of the linear maps to be isomorphisms. This eliminates a lot of the interesting behavior of quiver representations, and wouldn't make use of the fact that we have a bunch of invertible linear maps. So I would think that this language wouldn't buy a whole lot. | |
| Feb 26, 2010 at 6:50 | comment | added | Qiaochu Yuan | Isn't a representation of a groupoid the same thing as a representation of the corresponding quiver? | |
| Feb 26, 2010 at 6:34 | history | asked | Yuhao Huang | CC BY-SA 2.5 |