rev 2020.10.9.37784, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. groupal model for universal principal ∞-bundles, fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos, Abstract Homotopy Theory and Generalized Sheaf Cohomology. How does Stockfish know if the king is in check? This implies that also Id→HoT∘HoSId \to Ho T \circ Ho S is an isomorphism. A much cleaner approach to the computation of some cohomology groups is the Borel–Bott–Weil theorem, which identifies the cohomology groups of some line bundles on flag manifolds with irreducible representations of Lie groups. It relates sections on open subsets of the space to cohomology classes on the space. What are the "18 rescue missions" on Apollo 11, and which 10 of them did Michael Collins not feel comfortable with? Now write ℤ[X]\mathbb{Z}[X] for the free abelianization of the sheaf XX. The sheaf $\mathcal f_*\Bbb Z$ has stalks $\Bbb Z^2$ and the two components are permuted by the monodromy, say $T$. Why does a left curly brace impact the flalign alignment? Let $X$ be a manifold, $\mathcal{F}$ be an abelian sheaf on $X$. We take the trivial local system. The left hand side is $\Bbb Z$ in degree $0$ and $\Bbb Z$ in degree $1$. from the space XX regarded a (“nonabelian”!) We now assume $\mathcal F$ has non trivial monodromy. where on the right we have the degree qq abelian sheaf cohomology group with coefficients in the given ℱ\mathcal{F}. But due to the injectiveness of I •I^\bullet, the integrand on the right is constant (lemma 14.1.5 in CaS) and hence the colimit is isomorphic to ⋯≃Hom K(Sh(X,Ab))(ℤ,I •)≃H 0(I •(X))\cdots \simeq Hom_{K(Sh(X,Ab))}(\mathbb{Z}, I^\bullet) \simeq H^0(I^\bullet(X)), as desired. sheaf, to the Eilenberg-MacLane object in degree nn, defined by FF. The fourth step finally uses that the forgetful functor Sh(X,SimpAbGrp)→Sh(X,∞Grpd)Sh(X, SimpAbGrp) \to Sh(X, \infty Grpd) that only remembers the Kan complex underlying a simplicial group has a left adjoint, the free abelian group functor ℤ:Sh(X,∞Grpd)→Sh(X,AbSimpGrp)\mathbb{Z} : Sh(X,\infty Grpd) \to Sh(X, AbSimpGrp) (see Dold-Kan correspondence for details), and that preserves weak equivalences (see the discussion at simplicial group for more on that). Similarly, when considering chain complexes of abelian sheaves in their natural higher topos theoretic home, the cohomology of a complex of sheaves AA on a space XX is nothing but the hom-space H(X,A)=[X,A]H(X,A) = [X,A] – where the space XX itself is regarded as a special case of a sheaf. In fact, comparing sheaf cohomology to de Rham cohomology and singular cohomology provides a proof of de Rham's theorem that the two cohomology theories are isomorphic.
Let f −1:Y→Xf^{-1} \colon Y \to X be a morphism of sites. and quasi-isomorphic to FF regarded as a complex concentrated in degree 0 and then computing the nn homology group of the complex Γ(X,I F)\Gamma(X,I_F) of global sections of FF: Historically the development of abelian sheaf cohomology was precisely in reverse order to this derivation from the general (∞,1)(\infty,1)-categorical cohomology. Asking for help, clarification, or responding to other answers. Unix Epoch in International Space Station. The traditional definition of sheaf cohomology in terms of the right derived functor of the global sections functor: Ugo Bruzzo, Derived Functors and Sheaf Cohomology, Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes: Volume 2 (doi:10.1142/11473), Chênevert, Kassaei, Sheaf Cohomology (pdf), Its discussion in the more general nonabelian cohomology and infinity-stack context emphasized above is due to, This uses homotopical structures of a category of fibrant objects on complexes of abelian sheaves. ), If $\mathcal F$ is a locally constant sheaf of stalk $\Bbb Z/\ell \Bbb Z$, the étalé space $F$ is a degree $\ell$ cover, and the monodromy of this cover $f : F \to X$ is given by $\mathcal F$. For example, the de Rham complex is a resolution of the constant sheaf on any smooth manifold, so the sheaf cohomology of is equal to its de Rham cohomology. 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. Then the above cohomology set, which of course happens to be a cohomology group here, due to the abelianness of FF, is canonically isomorphic to the cohomology set, which can be regarded as the hom-set in the derived category of complexes of abelian sheaves.
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