The Stacks project

Remark 59.22.2. In the statement of Lemma 59.22.1 the covering $\mathcal{U}$ is a refinement of $\mathcal{V}$ but not the other way around. Coverings of the form $\{ V \to U\}$ do not form an initial subcategory of the category of all coverings of $U$. Yet it is still true that we can compute Čech cohomology $\check H^ n(U, \mathcal{F})$ (which is defined as the colimit over the opposite of the category of coverings $\mathcal{U}$ of $U$ of the Čech cohomology groups of $\mathcal{F}$ with respect to $\mathcal{U}$) in terms of the coverings $\{ V \to U\}$. We will formulate a precise lemma (it only works for sheaves) and add it here if we ever need it.