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Stream: theory: algebraic topology

Topic: Čech cohomology & sheaf cohomology

Sophie El Agami (May 16 2022 at 18:46):

Hi, I posted something similar in learning: questions already.
I am trying to follow Bredon's proof, that Cech and Sheaf cohomology coincide, but I am struggling to see, where the space $X$ being Hausdorff is used. I think it is needed to prove that the stalks of the Čech cohomology of a presheaf $P$ are zero if the stalks of $P$ are zero everywhere.

To be precise, let $P$ be a presheaf of abelian groups on a Hausdorff paracompact space $X$ with zero stalks $P_y =0$ for all $y\in X$. If $U\subset X$ is an open subset and $x\in X$ is a point with $x\in \overline{U}\backslash U$, then is the colimit $\lim_{W\ni x}P(W\cap U)$ zero? I think the stalk of the Cech cohomology (seen as the presheaf $U\mapsto H(X,P_{|U})$) is a colimit of finite products of such colimits?

Sorry, if this is actually obvious...

Tim Hosgood (May 16 2022 at 21:30):

so this hypothesis of paracompactness and Hausdorff is pretty nice because it tells you that Čech and sheaf cohomologies (aka "Grothendieck cohomology) agree for all sheaves, whereas lots of other theorems about cohomology theories agreeing place weaker conditions on the space itself and instead consider conditions on the sheaves in question

Tim Hosgood (May 16 2022 at 21:33):

(I know this doesn't answer your question but I'm just thinking about it, and I though I would mention this in the mean time!)

Tim Hosgood (May 16 2022 at 21:41):

one reason it's hard to know where exactly the Hausdorff assumption is being used is because Godement (who wrote the proof that I once read, a few years ago now), amongst others, defines a paracompact topological space to be, in particular, Hausdorff!

Peter Arndt (May 16 2022 at 23:19):

I don't know about your precise technical question with that boundary point... But a proof that a presheaf with zero stalks on a paracompact Hausdorff space has zero Cech cohomology is given in Brylinski," Loop spaces, characteristic classes and geometric quantization", Thm. 1.3.13 (3):
Brylinski.png
(for Brylinski "paracompact" means paracompact & Hausdorff)

Peter Arndt (May 17 2022 at 02:22):

For that proof he uses that for an open covering $\{U_i\}$ of a paracompact Hausdorff space, there exists an open covering $\{V_i\}$ such that $\bar{V}_i \subseteq U_i$ for all $i$ (his Prop. 1.3.12):
Brylinski2.png

Peter Arndt (May 17 2022 at 02:22):

This statement (his Prop. 1.3.12) is proven as Lemma 41.6 (page 258) of Munkres, Topology and uses regularity of paracompact Hausdorff spaces.
The proof that paracompact Hausdorff spaces are regular is embedded in the proof of Munkres, Thm. 41.1 (page 253) and explicitly uses the Hausdorff property.
(for Munkres "paracompact" means paracompact and not necessarily Hausdorff)

Sophie El Agami (May 18 2022 at 22:17):

Peter Arndt said:

I don't know about your precise technical question with that boundary point... But a proof that a presheaf with zero stalks on a paracompact Hausdorff space has zero Cech cohomology is given in Brylinski," Loop spaces, characteristic classes and geometric quantization", Thm. 1.3.13 (3):
Brylinski.png
(for Brylinski "paracompact" means paracompact & Hausdorff)

Thanks a lot - This is exactly what I was searching for! I somehow did not think about checking Brylinski's Book. I'm more interested in loop spaces now
I read that part and now actually think that the colimit is not zero...

Sophie El Agami (May 18 2022 at 22:21):

Tim Hosgood said:

one reason it's hard to know where exactly the Hausdorff assumption is being used is because Godement (who wrote the proof that I once read, a few years ago now), amongst others, defines a paracompact topological space to be, in particular, Hausdorff!

Yeah right, this is exactly what puzzled me :upside_down:

Tim Hosgood (May 19 2022 at 07:37):

I’m very out of practice with this sort of stuff, but I also don’t think the colimit is zero, and I think you could cook up some counterexample showing this on something like the Sierpinski space

Sophie El Agami (May 21 2022 at 20:31):

Tim Hosgood said:

I’m very out of practice with this sort of stuff, but I also don’t think the colimit is zero, and I think you could cook up some counterexample showing this on something like the Sierpinski space

I will look that that... though I have no idea about Siepinski spaces tbh :)

John Baez (May 21 2022 at 22:39):

The Sierpinski space is the most interesting topological space with 2 points.

John Baez (May 21 2022 at 22:40):

A continuous map from a topological space X to the Sierpinski space is the same as an open subset of X, so we say the Sierpinski space is the 'classifying space' for open sets.

Sophie El Agami (May 22 2022 at 17:29):

John Baez said:

A continuous map from a topological space X to the Sierpinski space is the same as an open subset of X, so we say the Sierpinski space is the 'classifying space' for open sets.

Well, that's a good primer!

Patrick Nicodemus (Jun 06 2022 at 10:23):

Is there a well known theory of Cech cohomology with coefficients in a profinite group?

Patrick Nicodemus (Jun 06 2022 at 10:25):

I have a cohomology theory for CH spaces and it naturally takes coefficients in a profinite group. I am curious about what the standard cohomology theories are that take coefficients in a profinite group as I have personally only worked with ordinary Abelian groups as coefficient groups.

Patrick Nicodemus (Jun 06 2022 at 10:25):

I think I should compare my cohomology theory to other cohomology theories to see what it agrees with but I don't know what to compare it to!

John Baez (Sep 07 2022 at 12:06):

https://twitter.com/CihanPostsThms/status/1567223117054971904

The paracompactness assumption in the common comparison theorems of sheaf vs singular cohomology can be dropped.

- Some theorems (@CihanPostsThms)

John Baez (Sep 07 2022 at 12:07):

Click on the tweet for some more details.