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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
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