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Why is it that when it comes to sheaves all of the attention is put on cohomology and almost nothing on homology? I have barely found any articles on sheaf homology.
That’s a very good question.
The reason is that:
cohomology is homotopy groups of derived hom-spaces $Hom(-,A)$
homology is homotopy groups of derived tensors $(-) \otimes A$.
and hom-spaces always exist as soon as there is a category,
but tensors are (1.) extra structure and (2.) tend to be hard to construct/control in homotopy theory.
That’s also the reason why there is a lot about “non-abelian cohomology” but almost no “non-abelian homology”:
For defining cohomology we need no assumption on the coefficient object $A$
(maybe except that it be pointed, but not that it be abelian/stable).
I see. But surely this is not a problem in sufficiently nice settings such as quasicoherent sheaves?
edit: for what it’s worth, this comes from a recent discussion I had with Eric Sharpe where he talked about how (under the hypothesis of quantization by derived categories of sheaves) open string states between D-branes are counted by Ext groups of the corresponding coherent sheaves. However, curiously, it seems Tor never appeared in his work. This would obviously appear if considering homology instead (the analogous case in Hypothesis K would be studying cycles in K-homology, as has been done elsewhere to analyze where D-branes can wrap), yet I couldn’t even find work on homology of sheaves, let alone how it is relevant in physics.
It’s not a problem of principle, but it explains the phenomenon of the literature that you observe, I think.
Regarding flux quantization: This is intrinsically an issue in cohomology, rooted in the fact that flux densities of higher gauge fields are differential forms, hence define classes in (de Rham) cohomology instead of in homology.
If one gets to the bottom of why this is, it has to do with the same asymmetry as in #1: A flux density is like an observable which as such is a map from chains in spacetime to the real numbers, assigning the field flux through that chain.
I’d say the relation of K-homology to D-brane worldvolumes is more speculative than that of K-cohomology to RR-flux. If one assumes Hypothesis K then the primary object is K-cohomology, and K-homology plays a role as a secondary tool for understanding K-cohomology. But this is a little vague and maybe we should look at more concrete situations.
Another response to 2.
I would like to add to Urs’s (good) answer 3 by quoting more old-fashioned answer. In topology, cycles are compact and to have good properties one needs to deal with the finiteness and compactness problems somehow. Skljarenko’s classical paper
shows e.g. how to deal with compactifying families of supports to obtain good homology theories in sheaf theoretic generality. Bredon’s book tries something with less success (Sklyarenko wrote around 250 remarks to his book when redacting the Russian translation). One can write naively Čech homology complex, but because passing to the colimit over refinements is not well behaved it fails to satisfy the long exact sequence. Compactifying tools try to avoid such problems.
One standard case of well behaved homology theory for rather general class of spaces are Borel-Moore homology and Steenrod-Sitnikov homology theory and there is a more general construction called strong homology by (Lisica and) Mardešić.
Strong homology trick is not to do the limits (on refinements) prematurely and essentially to work with coherent homotopy theory at the chain level, not to break the coherences; if we correct Čech homology with these tools we indeed obtain a homology theory, and Mardešić calls this theory a strong homology because it is closely related to strong shape theory (when Lurie talks about shape theory, or Simpson/Toen, they mean strong shape, of course).
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