Timeline for How is a Stack the generalisation of a sheaf from a 2-category point of view?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2017 at 16:02 | vote | accept | HaroldF | ||
| Jun 14, 2017 at 21:06 | comment | added | user40276 | @HaroldF Ah! Ok. But there are no problems with Vistoli's presentation. Most of the cases that deals with moduli do not require 2-stacks (or 3-sheaves). Ah! And I forgot to write this. That cocycle condition guarantees that every possible way of restricting a local section gives you the same result. It's analogous to that theorem of Maclane which says that when the Maclane pentagon commutes then all the possible ways of associating an n-ary monoidal product gives you isomorphic objects. | |
| Jun 14, 2017 at 20:56 | comment | added | user40276 | @Ben Not exactly. I've written this in a rush. Maybe later when I have time, I may try to make an answer containing all the details. I hate incomplete answers. And I'm a little against the reputation point system of mathoverflow, so I usually avoid answering. Furthermore, I don't know how to draw diagrams properly here. | |
| Jun 14, 2017 at 20:09 | comment | added | HaroldF | @user40276 Yes, my reply was (more naively, without the $\infty$ case) this. Thank you | |
| Jun 14, 2017 at 6:29 | comment | added | Ben | Don't you think those four comments constitute an answer, @user40276? | |
| Jun 14, 2017 at 5:22 | answer | added | David Benjamin Lim | timeline score: 15 | |
| Jun 13, 2017 at 19:23 | comment | added | user40276 | And it should be $O(X)^{op}$ instead of $O(X)$. I cannot correct it anymore. | |
| Jun 13, 2017 at 19:18 | comment | added | user40276 | For the $n$-categorial case you can truncate this $\infty$-groupoid to an n-groupoid and you will get the same result up to homotopy. Furthermore, as it was already noticed in the comments, this is equivalent to the preservation of a homotopy limit. This process of requiring things to be isomorphic instead of equal is the usual procedure for vertical categorification. However when requiring things to be isomorphic, you must require these isomorphisms to be compatible and so on. | |
| Jun 13, 2017 at 19:17 | comment | added | user40276 | Now for the higher categorial case for $F: O(X) \rightarrow \infty-Grpd$, we require this big polyhedra to be commutative up to a 3-morphism and that these 3-morphisms must satisfy compatibilities (now a 4-dimensional polyhedra) up to 4-morphisms. This data of compatibilities can be packed up into a homotopy colimit. More precisely the groupoid $U \times_X U \xrightarrow{\rightarrow} U$ for $U = \coprod_{I} U_i$ can be extended to an $\infty$-groupoid $U$ such that $U_n = U \times_X ... \times_X U$ (n times) and a descent data will be the same thing as a section of $holim F(U_n)$. ... | |
| Jun 13, 2017 at 19:04 | comment | added | user40276 | Now for the 2-categorial case for $F: O(X) \rightarrow 2-Grpd$, a descent data will be the same thing as a descent data for the 1-categorial case except that we will have a 2-morphism $\lambda_{ijk}: \phi_{ij} \phi_{jk} \rightarrow \phi_{ik}$ instead of equality and this 2-morphism must satisfy a commutative diagram for each $i, j$ and $k$. This diagram will be a polyhedra with faces indexed by each $\lambda_{lmn}$ and edges given by the triangles with faces $\phi_{wv}$..... | |
| Jun 13, 2017 at 19:04 | comment | added | user40276 | I don't know if I've understood your reply. In a sheaf $F: O(X) \rightarrow Set$, for $s_i \in F (U_i)$, we have that the descent data $s_i|_{U_{ij}} = s_j|_{U_{ij}}$ for every $I$ and $j$ is equivalent to the data $s \in F(X)$. Now if we want a 1-categorial version of this for $F: O(X) \rightarrow Grp$, we can define analogously that a descent data $\phi_{ji} : s_i|_{U_{ij}} \rightarrow s_{j}|_{U_{ij}}$ such that $\phi_{ij}\phi_{jk} = \phi_{ik}$ on $F(U_{ijk})$ is equivalent to a global section. ..... | |
| Jun 13, 2017 at 8:58 | comment | added | მამუკა ჯიბლაძე | (where exact truncated simplicial, resp. cosimplicial object means that $X$, resp. $F(X)$, is the homotopy colimit, resp. homotopy limit, of this truncated simplicial, resp. cosimplicial object considered as a diagram) | |
| Jun 13, 2017 at 8:54 | comment | added | მამუკა ჯიბლაძე | If I understand correctly the comment by @user40276, it is about maximally general formulation: sheaf/stack/... condition is that certain kind of colimits are carried to the same kind of limits. In every case, coproducts must go to products; the rest is about some kind of coequalizers go to the same kind of equalizers. For sheaves it is that if $U\times_XU\rightrightarrows U\to X$ is a coequalizer then $F(X)\to F(U)\rightrightarrows F(U\times_XU)$ must be an equalizer. Higher and higher, you get $n$-truncated simplicial objects, which, if exact, must go to exact $n$-truncated cosimplicials,... | |
| Jun 12, 2017 at 16:41 | comment | added | HaroldF | Oh ok, maybe i see what is the problem. Usually (see Vistoli's notes) a descent data is given by sections, and arrows between section in the double intersection (isomorphism) such that on triple intersections they satisfy cocycle condition with equality$\phi_{ij}\phi_{jk}=\phi_{ik}$. But in a 2-categorical environment we have that this equality will be a 2-isomorphism with compatibility in quadruple intersections (what a pseudofunctor guarantee). Is this the point? | |
| Jun 12, 2017 at 14:50 | comment | added | user40276 | This diagram is not naive. It's actually correct once you use homotopy colimits and add all the intersections (not only the double intersections). In other words, in a sheaf you require that sections coinciding in a intersection must glue to a global section, while in a stack, sections connected by an arrow in each double intersection (that is coherent with other arrows) must glue to a global section. More generally, you could relax these commutative triangles to be only commutative up to a 2-morphism with coherences between these 2-morphisms and so on (this would be an $\infty$-stack) | |
| Jun 12, 2017 at 8:19 | comment | added | მამუკა ჯიბლაძე | Strictly speaking, this depends on how do you realize presheaves as particular prestacks. Since your context is 2-categorical, your definitions must be equivalence-invariant. One such invariant way is to consider presheaves as prestacks valued in categories that are equivalent to discrete categories (sometimes called setoids). In this context then, the cocycle condition is not entirely trivial. It is related to things like local equivalence relations, foliations, etc. | |
| Jun 12, 2017 at 7:28 | history | asked | HaroldF | CC BY-SA 3.0 |