Timeline for $BG$ the stack, $BG$ the simplicial presheaf
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 20, 2018 at 7:20 | comment | added | Denis Nardin | @user51223 Sorry if I misunderstood, but to be clear $U(n)$ and $U$ are most certainly not $K(G,1)$ (well, except $n=0$ and $n=1$ for silly reasons) | |
| Oct 20, 2018 at 5:50 | vote | accept | HuynA | ||
| Oct 20, 2018 at 3:42 | comment | added | user51223 | @DenisNardin My first comment was rather about a phrase in the question. I think the part that ``$BG$ has no nontrivial homotopy groups other than $\pi_1BG$'' implies that $BG$ is taken to be $K(G,1)$ in this sentence. Now, I am not sure either $U(n)$ or $U$ as in my previous comment are examples of $K(G,1)$, right! | |
| Oct 19, 2018 at 17:37 | comment | added | Denis Nardin | @user51223 That has nothing to do with $U$ not being abelian and everything about $G$ being discrete | |
| Oct 19, 2018 at 17:25 | history | edited | HuynA | CC BY-SA 4.0 |
added 62 characters in body
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| Oct 18, 2018 at 15:40 | comment | added | mme | @Horstenson Rather $\text{colim } U(n)$. The individual groups $U(n)$ have well-understood homotopy only up to about degree $2n$ by comparison to this colimit, and after that it becomes more complicated. | |
| Oct 18, 2018 at 15:24 | answer | added | Marc Hoyois | timeline score: 16 | |
| Oct 18, 2018 at 14:47 | comment | added | Horstenson | @Qfwfq I think that's supposed to be the n-th unitary group U(n). | |
| Oct 18, 2018 at 13:08 | comment | added | David Roberts♦ | The stack BG in groupoids is presented by the groupoid in schemes with one object and G as the group of automorphisms. You can take the nerve of this internal groupoid to get a simplicial object in schemes, which is the other form of BG. The simplicial scheme presents a higher stack that is the image under the inclusion of ordinary stacks of the first BG. | |
| Oct 18, 2018 at 10:27 | comment | added | Jason Starr | The "simplicial" name for a "Cech hypercovering" is the "0-coskeleton". More generally, given the first n stages in a simplicial sheaf, there is a "n-coskeleton" functor that is adjoint to the forgetful functor that remembers only the first n stages of a given simplicial object. | |
| Oct 18, 2018 at 8:42 | comment | added | Qfwfq | @user51223: what is the $U$ in your comment? | |
| Oct 18, 2018 at 5:56 | comment | added | R. van Dobben de Bruyn | Any presentation of an algebraic stack $\mathscr X$ as a quotient of a scheme $U$ by an étale equivalence relation $R \rightrightarrows U$ gives rise to a simplicial scheme by taking its "Čech hypercovering": let $X_i$ be the $i$-fold fibre product of $U$ over $\mathscr X$. Any algebraic stack has such a presentation by [Tag 04T3]. It's not clear to me how well-defined this is, let alone whether this is an equivalence onto some subcategory of simplicial schemes. | |
| Oct 18, 2018 at 4:35 | comment | added | user51223 | For the main question, perhaps the nerve of the category is homotopy equivalent to the geometric realisation of your simplicial object. | |
| Oct 18, 2018 at 4:33 | comment | added | user51223 | What you heard is true in the case of $G$ being Abelian. In general, $BG$ can be nontrivial in more than one dimension. For instance, $\pi_nU\simeq\mathbb{Z}$ for $n$ even and trivial otherwise. | |
| Oct 18, 2018 at 4:06 | history | asked | HuynA | CC BY-SA 4.0 |