Wiki. “层上同调” [层上同调]
Wiki. “层上同调” [层上同调]
Before arriving at the full picture of higher topos theory … people had a pretty good guess about some aspects of this story, and this aspect they called sheaf cohomology. It turns out that sheaf cohomology is precisely the abelian part of general cohomology.
Here is what “abelian part of general cohomology” means: For many sheaves that appear in practice, the topological space $X(U)$ assigned to any test space $U$ is not just a topological space, but happens to have the structure of an abelian group, too. In this case a cascade of simplifications kicks in: first of all, such topological spaces, by a special case of something called the homotopy hypothesis (which is now really a precise theorem) can be identified with simplicial abelian groups. Second, by another theorem called the Dold–Kan correspondence, simplicial abelian groups are equivalently encoded more simply in the collection of data given by (non-positively graded) chain complexes in abelian groups.
定义
拓扑空间 $X$ 上 Abel 群层的上同调 $H^i(X,-)$ 是整体截面 $\Gamma$ 的右导出函子, 也即 $X\to *$ 的导出前推.
空间 $X$ 上的层上同调可视为 $X$ 上的 ∞-层 ∞-意象中的映射同伦类: $$ H^n(X,F)\simeq \pi_0\operatorname{Hom}_{\operatorname{Sh}(X,\infty \mathsf {Grpd})}(X,B^nF). $$
具体地, 设有内射消解 $F\to I^\bullet$, 定义 $H^n(X,F) := H^n(\Gamma(X,I^\bullet))$.