structures in a cohesive (∞,1)-topos
The cohomology of a topological space with values in a sheaf of abelian groups / abelian sheaf was originally defined as the value of the right derived functor of the global section functor, the derived direct image functor.
But by embedding sheaves with values in abelian groups as special cases of simplicial sheaves into the more general context of ∞-groupoid-valued sheaves via the Dold-Kan correspondence and thus the abelian sheaf cohomology into the more general context of the intrinsic nonabelian cohomology of an (∞,1)-topos , this definition becomes equivalent to a special case of the general notion of nonabelian cohomology defined simply as the set of homotopy classes of maps
from the space regarded a (“nonabelian”!) sheaf, to the Eilenberg-MacLane object in degree , defined by .
The relation of this more conceptual and more general point of view on abelian sheaf cohomology to the original definition was originally clarified in
(whose proof is reproduced below).
Brown constructed effectively the homotopy category of using a model of a category of fibrant objects paralleling the model structure on simplicial presheaves as a presentation of the (∞,1)-category of (∞,1)-sheaves. This says that ordinary abelian sheaf cohomology in fact computes the equivalence classes of the ∞-stackification of a sheaf with values in chain complexes of abelian groups.
The general (∞,1)-topos-theoreric perspective on cohomology is described in more detail at cohomology. The details on how to realize abelian sheaf cohomology as an example of this are discussed below.
Using the Dold-Kan correspondence in higher topos theory, complexes of abelian sheaves can be understood as a generalization of topological spaces, in a precise sense recalled below. This induces a notion of cohomology of complexes of abelian sheaves from the familiar notion of cohomology of spaces.
Which is a simple one: recall that the cohomology of one topological space with coefficients in another space is nothing but the space of morphisms (continuous maps) from to – or, in a more restrictive sense traditionally adopted, the set of connected path components of this space.
Similarly, when considering chain complexes of abelian sheaves in their natural higher topos theoretic home, the cohomology of a complex of sheaves on a space is nothing but the hom-space – where the space itself is regarded as a special case of a sheaf.
One reason this conceptually simple picture is not usually presented is that the space is typically not represented by an abelian complex of sheaves, so that the full simplicity of the story becomes manifest only in general nonabelian cohomology.
More precisely, via the Dold-Kan correspondence (non-negatively graded) complexes of abelian sheaves – which are equivalently sheaves with values in (non-negatively graded) categories of chain complexes – can be regarded as special cases of simplicial sheaves. But thanks to the model category structure on the category of simplicial sheaves, this in turn is a model for the (infinity,1)-topos of generalized spaces called infinity-stacks. The very point of -topoi is that they are (infintiy,1)-categories which behave in all structural aspects relevant for homotopy theory as the archetypical example Top does. In particular, as in Top, the notion of cohomology in any (infinity,1)-topos simply coincides with that of hom-spaces.
In the 1-categorical model theoretic models these hom-spaces are computed technically by derived functors. More precisely, the Hom-space for an ordinary space computes the global sections of the complex of abelian sheaves which is computed by the right derived functor of the global section of the global section functor , which does exist entirely within the abelian context.
This, then, is the definition of sheaf cohomology as usually presented: the cohomology of the complex .
Under the (stable) Dold-Kan correspondence we have the following identification of sheaves taking values in chain complexes with sheaves taking values in infinity-groupoids and spectra, crucial for a conceptual understanding of abelian sheaf cohomology:
let be a site
the category of non-negatively graded chain complexes of abelian sheaves is homotopically equivalent to the category of sheaves with values in simplicial abelian groups (i.e. Kan complexes with strict abelian group structure);
Write for the terminal sheaf of , i.e. for the sheaf that corresponds to the space itself.
Now write for the free abelianization of the sheaf . This is the sheaf constant on the abelian group of integers. Then the above cohomology set, which of course happens to be a cohomology group here, due to the abelianness of , is canonically isomorphic to the cohomology set
This, finally, is the same group as obtained by choosing any complex of abelian sheaves that is injective? and quasi-isomorphic to regarded as a complex concentrated in degree 0 and then computing the homology group of the complex of global sections of :
Let be a topological space, a sheaf on (the category of open subsets of) with values in abelian groups, and the image of the complex of abelian sheaves ( in degree , trivial elsewhere) under the Dold-Kan correspondence in sheaves with values in Kan complexes
Then we have the following natural isomorphism of cohomologies:
This is the first four steps in the proof of theorem 2 in BrownAHT.
The proof proceeds along the following four steps, which we describe in more detail below:
On the other hand, by the general formula for hom-sets in homtotopy categories obtained by localizing at the multiplicative system given by quasi-isomorphisms of complexes (e.g. def. 13.1.2 in CaS) we have
But due to the injectiveness of , the integrand on the right is constant (lemma 14.1.5 in CaS) and hence the colimit is isomorphic to , as desired.
The second step uses that the inclusion functor
is full and faithful. This in turn follows from
first observing that the inclusion of chain complexes concentrated in non-negative degree into all complexes of sheaves is full and faithful and has the obvious right adjoint obtained by truncating a complex.
But by the adjoint functor lemma for homotopical categories, is also left adjoint to (since both preserve weak equivalences). So that once again with the general properties of adjoint functors it follows that is full and faithful.
The fourth step finally uses that the forgetful functor that only remembers the Kan complex underlying a simplicial group has a left adjoint, the free abelian group functor (see Dold-Kan correspondence for details), and that preserves weak equivalences (see the discussion at simplicial group for more on that).
We have a commuting diagram
abelian sheaf cohomology
Chênevert, Kassaei, Sheaf Cohomology (pdf)
Cibotaru, Sheaf cohomology (pdf)
A discussion of the Čech cohomology description of sheaf cohomology along the above lines is in