Godement resolution and operad sheaf homotopy theory

Abstract

Abstract. We show how to induce products in sheaf cohomology for a wide variety of coefficients: sheaves of dg commutative and Lie algebras, symmetric Ω-spectra, filtered dg algebras, operads and operad algebras.

Prismatic cohomology and $p$-adic homotopy theory

“…We are now ready to discuss the Godement resolution, following [16]. In fact, we have the following proposition ( [16], Proposition 3.3.1):…”

“…If C is some category, then ∆C denotes the category of cosimplicial objects of C. We avoid any ∞-categorical language because the author does not know it. many of the arguments can be found in, for example, [5], [14], and [15], with the main conceptual ideas originating from Godement [6]. The aim of this section is to cover the following well known theorem.…”

“…For the purposes of this paper, we will mainly be concerned with D = Ch(k-mod) the category of cochain complexes of k-modules. In this situation, we have the following classical theorem, proven for far more general D by Rodríguez-Gonález and Roig (see [15] Proposition 3.4.5, Corollary 3.4.6, Proposition 3.4.7) though definitely known to Godement [6] albeit without modern language. See also Chataur and Cirici ([5] Definition 2.5).…”

“…We introduce two necessary definitions from [15] and [16] Remark. Recall that the symmetric monoidal structure on ∆Sh(C, D) is induced degree-wise by the symmetric monoidal structure on Sh(C, D), so the definitions above make sense for cosimplicial sheaves as well.…”

“…The equivalence above in definition 2.8 follows from the existence of the sheafification functor for presheaves with values in D = Ch(k-mod). Part of the work of Rodríguez-Gonález and Roig in [15] concerns the conditions on the coefficient category D for which a sheafification functor is available. For the purposes of this paper, we will actually mainly be concerned with the underlying presheaf structure than the sheaf structure per se.…”