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Results tagged with grothendieck-topology
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user 402
9
votes
Accepted
Is there a notion of Čech groupoid of a cover of an object in a Grothendieck site?
Take $U=\coprod_{i∈I}Y(U_i)$, where $Y\colon C\to\mathop{\rm Presh}(C,{\rm Set})$ is the Yoneda embedding.
We have a canonical morphism $U→Y(X)$.
The Čech groupoid of $J_c$ can now be defined as
the …
- 37.8k
21
votes
Resources for topos theory
For a beginner, the more accessible textbooks seem to be the following two.
Francis Borceux, Handbook of Categorical Algebra, Volume 3.
Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and Logi …
- 37.8k
6
votes
How to construct cup-product in a general site?
Yes. This is treated in detail in Section 8.4
of Jardine's book “Local homotopy theory”.
See also the introduction to Chapter 8 there
for a historical comment on cup products and Godement resolutions …
- 37.8k
11
votes
Accepted
When is a basis of a topological space a Grothendieck pretopology?
This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies.
By a “base” in this answer I mean what appears to be the most common definition: a c …
- 37.8k
9
votes
1
answer
401
views
Reference for the Brown-Gersten property for smooth manifolds
A classical result by Brown and Gersten says that to verify the homotopy descent property for the Zariski topology
it suffices to verify it for Zariski squares and the empty cover of the empty scheme. …
- 37.8k
2
votes
Accepted
Reference for the Brown-Gersten property for smooth manifolds
I typed up a proof of this result:
Numerable open covers and representability of topological stacks.
The result is proved in greater generaility for arbitrary numerable open covers of topological sp …
- 37.8k
7
votes
1
answer
343
views
Which dense inclusions of sites are ∞-dense?
An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D.
Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ∞-sheav …
- 37.8k