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Results tagged with sheaf-theory
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user 126667
For questions about sheaves on a topological space.
12
votes
Accepted
Taking the category of sheaves is symmetric monoidal
Provided at least one of $M$ and $N$ is locally compact, the $\infty$-topos $\mathrm{Sh}(M \times N)$ is the product of $\mathrm{Sh}(M)$ and $\mathrm{Sh}(N)$ in $\mathrm{RTop}$. This is HTT 7.3.1.11. …
- 25.2k
4
votes
Accepted
Sheaves on sites given by a (regular) cd-structure
In general sections $b \in F(B)$ and $c \in F(C)$ that agree on $F(A)$ don't induce a matching family on $\{B \to D, C \to D\}$ though. The sheaf condition for that family is that
$$
F(D) \to F(B) \ti …
- 25.2k
6
votes
Left Kan extension that preserves colimit
Usually it does not. We can take $C \to B$ to be the universal example of a functor to a cocomplete category, namely the Yoneda embedding $C \to PSh$. Then the left Kan extension of $y : C \to PSh$ al …
- 25.2k
10
votes
Accepted
Does sheafification preserve sheaves for a different topology?
I think my answer to this question provides a counterexample: Let C be the category a → b, and consider the topologies T1 generated by the single covering family {a → b} and T2 generated by declaring …
- 25.2k
6
votes
Accepted
Simplicial Sheaves?
If I understand correctly, these are constructible sheaves with respect to the stratification of your simplicial complex by its skeleta. I think by a theorem of MacPherson the category of such sheave …
- 25.2k
10
votes
Sheaf cohomology and injective resolutions
If you're willing to take for granted that (bounded-below) chain complexes and quasi-isomorphisms are good things to study, then left exact functors have the defect that they do not preserve quasi-iso …
- 25.2k