Cohomology with coefficients in a sheaf

[feuxfollets]

I just want to know if I'm understanding this right. I haven't really seen homology/cohomology outside of Z-related coefficients before, so this still seems kind of weird. I also haven't actually learned sheaf theory, so this might just be totally wrong.


So if I have a top space and a sheaf mapping from it to something, say free modules:
First if this is a cellular space, then the sheaf maps each cell to a free module. So is the cochain essentially a direct sum of free modules, one assigned to each cell?

And for singular cohomology, the cochains would map the singular simplices to the free modules which are mapped to by the sheaf from the interiors of the geometric realizations of the simplices? I'm kind of just guessing this, not sure what else it would be.


Also could someone give me an elementary example of where taking cohomology coefficients in a sheaf would be useful? preferably one not involving too much category or manifold theory.


Thanks a lot for the help.

 

[mathwonk]

this is rather big question. you seem a bit confused. in sheaf cohomology the coefficients tend to be functions on open sets rather than integers assigned to a single cell. I.e. a cochain is not an assignment of one integer to each cell, but an assignment oif one function to each open set.

It is possible to look at a simplicial complex and replace each simplex by an opens set, the open star of that simplex. So instead of a vertex we look at the union of all open faces and open edges touching that vertex.

then we can assign as a cochain, not just a single integer to that vertex, i.e. a constant function, but any continuous function on that open set.

Notice that an "edge" woulkd be the union of all open faces touiching that edge,a nd would be the intersection of the opoen stars of the two vertices which are extremities of that edge.

so using this "Cech" approach we could recover simplicial cohomology by using only constant unctions, but we have the flexibility to obtain much richer cohomology, one that encodes also information about the continuous functions, or even holomorphic functions,This allows a very great and useful expansion of cohomology so that it is no longer a homotopy invariant but a holomorphic isomorphism invariant, and which sometimes can distinguish between different holomorphic structures on the same topological surface.

here are some notes, but its a long slog.

http://math.arizona.edu/~jschettler/sheafcohomology.pdf

 

[homeomorphic]

One use for cohomology with sheaf coefficients is to prove the de Rham theorem that de Rham cohomology is isomorphic to singular cohomology with real coefficients. Bott and Tu cover this in their Differential Forms in Algebraic Topology book, but they only mention presheaves, not sheaves. I'm not too familiar with this subject myself, but I'm assuming it's essentially the same thing. Anyway, they give some good motivation for being interested in presheaves, which leads you to be interested in sheaves (although, not as much as I'd personally like). Anyway, a good example of a presheaf is given by differential forms on each open set of a manifold. I wouldn't recommend shying away from manifolds if you want to have good motivation for this stuff.

Here's a paper that tries to apply sheaf theory to networks (doesn't assume too much background knowledge):

http://www.math.upenn.edu/~ghrist/preprints/networkcodingshort.pdf

Maybe the reason I mention this is that somehow sheaf theory is supposed to be patching together local information to get global information, so those networks are a good real-world example of that.

 

[mathwonk]

a presheaf assigns to each open set an abelian group, and to each inclusion a restriction map. a presheaf is a sheaf if essentially the abelian group is a collection of functions defined by a local property. thus the presheaf of continuos functions or smooth functions on open sets is automatically a sheaf. i agree the example of the de rham complex is a great one to appreciate sheaf cohomology.

 

[blue_raver22]

I can confirm that your understanding of cohomology with coefficients in a sheaf is correct. In general, cohomology with coefficients in a sheaf is a way of measuring the "twisted" or "localized" cohomology of a space, where the coefficients are not just integers but can be more complicated objects, such as modules or sheaves.

In the case of a cellular space, the cochains do indeed correspond to direct sums of free modules, one for each cell. This is because each cell can be thought of as a "local" space, and the sheaf assigns a module to each local space.

For singular cohomology, your understanding is also correct. The cochains map the singular simplices to the free modules, which are then mapped to by the sheaf from the interiors of the geometric realizations of the simplices. This way, we are able to capture more information about the space by considering not just the topology, but also the sheaf structure.

An elementary example where cohomology with coefficients in a sheaf is useful is in the study of vector bundles. Vector bundles can be thought of as "twisted" or "localized" versions of the trivial bundle, and cohomology with coefficients in the sheaf of sections of a vector bundle can help us understand the global properties of the bundle. This is important in applications such as differential geometry and physics.