3
$\begingroup$

Let $X$ be a 2-site and consider the category of 2-presheaves over $X$, which will be denoted as $Pshv(X;Cat)$. These are $Cat$-valued 2-functors, where $Cat$ is the 2-category of categories. There is a functor "2-Sheafification" $L:Pshv(X;Cat)\rightarrow Shv(X;Cat)$, which is left adjoint to the inclusion $i:Shv(X;Cat)\rightarrow Pshv(X;Cat)$.

My question is: How does 2-Sheafification work with 2-presheaves valued in 2-categories other than $Cat$? I have in mind 2-categories with "extra structure", for example 2-functors valued in the 2-category of monoidal categories.

$\mathbf{Edit}$: The following is from the nLab's entry on sheafification:

If a category A satisfies the following assumptions, sheafification of presheaves in $[S^{op},A]$ exists and is constructed analogously as for Set-valued sheaves.

A admits small limits;

A admits small colimits;

Small filtered colimits in A are exact;

A satisfies the IPC-property.

Maybe the answer to my question is related to the target 2-category satisfying 2-categorical versions of the above conditions?

Improve this question
$\endgroup$
4
  • $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Commented Jul 31, 2016 at 19:42
  • $\begingroup$ [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 $C$. $\endgroup$
    – Tim Campion
    Commented Jul 31, 2016 at 19:42
  • $\begingroup$ 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. $\endgroup$
    – user84563
    Commented Aug 1, 2016 at 2:48
  • 1
    $\begingroup$ @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. $\endgroup$
    – David Roberts ♦
    Commented Aug 2, 2016 at 23:11

0

Reset to default

You must log in to answer this question.