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If G is an abelian Lie group, we get the Deligne complex, used to define bundle n-gerbes with connection, alias Deligne cohomology. …

There are numerous expositions of simplicial homology in the literature. Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes. Hatcher in “Algebraic Topol …

One can define an analogue of the crystalline topos for smooth manifolds. This is known as the de Rham stack of $M$. One of the easiest constructions of the de Rham stack embeds smooth manifolds fully …

Suppose M is an arbitrary smooth manifold and D is its bundle of 1-densities. On the category of finite-dimensional vector bundles over M and linear differential operators between them there is a cont …

The nth Deligne cohomology is defined as cohomology with coefficients in a truncated chain complex of sheaves of U(1)-valued differential forms: U(1)→Ω^1→Ω^2→⋯→Ω^n for some n≥0. … This defines a nonabelian analog of the Deligne cohomology for any choice of the underlying site: smooth, holomorphic, algebraic. …

"in what way is cohomology a sheaf" leads one to notions like ∞-topoi etc. …

Chain complexes arise from simplicial abelian groups via the Dold–Kan correspondence: the normalized chain complex functor establishes an equivalence of categories from simplicial abelian groups to (n …

In “Chern–Weil forms and abstract homotopy theory”, Freed and Hopkins compute the de Rham cohomology of $\def\B{{\sf B}}\B_∇ G$, the moduli stack (in groupoids) of principal $G$-bundles with connections … The de Rham cohomology of $\B_∇ G$ turns out to be isomorphic to the graded algebra of $G$-invariant polynomials (placed in even degrees) on the Lie algebra of $G$, with the zero differential. …

One rich source of examples is that every combinatorial symmetric monoidal model category gives rise to such a category V. In particular, this covers all the examples in the main post, including cdga …

In derived functor cohomology one takes cochain resolutions of A, which again by co-Dold-Kan is the same thing as cosimplicial resolutions. Can one also do this the nonabelian setting? …

Cup products in sheaf (and presheaf) cohomology are often easy to compute by resolving the source (in the projective model structure, say), not the target. …