Timeline for "2-Sheafification" with Values in non $Cat$ categories?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| S Aug 10, 2016 at 6:29 | history | bounty ended | CommunityBot | ||
| S Aug 10, 2016 at 6:29 | history | notice removed | CommunityBot | ||
| Aug 2, 2016 at 23:11 | comment | added | David Roberts♦ | @TimCampion it's a common misconception that people who study (1-)stacks mostly consider the case where the 1-cells are invertible. This is the case for things like algebraic stacks, where people consider the stack of groupoids (though this is not actually a necessary restriction), but stacks of coherent sheaves, vector bundles, curves, etc work fine with the non-invertible 2-cells, and these are definitely considered. | |
| Aug 2, 2016 at 16:02 | history | edited | user84563 | CC BY-SA 3.0 |
added 510 characters in body
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| S Aug 2, 2016 at 4:35 | history | bounty started | user84563 | ||
| S Aug 2, 2016 at 4:35 | history | notice added | user84563 | Authoritative reference needed | |
| Aug 1, 2016 at 2:48 | comment | added | user84563 | Yes, I'm aware of the terminology but thought it was used in the context of (2,1)-sheaves. I'm interested in stacks with values in categories with non-invertible 2-cells but $X$ being locally discrete is fine. | |
| Jul 31, 2016 at 19:42 | comment | added | Tim Campion | [Can some expert correct this guess?] In the $(n-1)-Cat$-valued case, the $n$-sheafification should be obtained by applying $L$ $(n+1)$ times, where $L(F)(x) = \varinjlim F(y)$; the colimit is over coverings of $x$. An $n$-sheaf on $X$ valued in an $n$-cateogory $C$ should just be an $n$-functor $X^{op} \to C$ such that $C(c,F-): X^{op} \to (n-1)-Cat$ is an $n$-sheaf for each $c \in C$, and (hence?) iteratively applying the same colimit formula should work for $C$-valued $n$-sheafification as long as $C$ is locally finitely presentable, but I think(?) the number of iterations depends on $C$. | |
| Jul 31, 2016 at 19:42 | comment | added | Tim Campion | Are you aware that most people call a "2-sheaf" a "stack"? Most people study higher sheaves / stacks where the higher cells are all equivalences, and there is a lot of literature in this case. Are you specifically interested in 2-sheaves with values in 2-categories with non-invertible 2-cells? And where $X$ is not locally discrete? If so, what is a motivating example for you? For some discussion of the general case of $Cat$-valued 2-sheaves, see here and here. | |
| Jul 31, 2016 at 3:52 | history | asked | user84563 | CC BY-SA 3.0 |