nLab
cover

Idea

Generally, for $X$ an object we think of as a space, a cover of $X$ is some other object $Y$ together with a morphism $\pi :Y\to X$, usually an epimorphism demanded to be well behaved in certain way.

The idea is that $Y$ provides a “locally resolved” picture of $X$ in that $X$ and $Y$ are “locally the same” but that $Y$ is “more flexible” than $X$.

The archetypical example are ordinary covers of topological spaces $X$ by open subsets $\{U_i\}$: here $Y$ is their disjoint union $Y:={\coprod }_i{U}_i$.

More generally, you might need a cover to be family of maps $({\pi }_i:Y_i\to X{)}_i$; if the category has a coproducts that get along well with the covers, then you can replace these families with single maps as above.

Formalizations

There are several different but related formalizations of the notion of cover.

• In the context of sheaf and topos theory a cover is a sieve. A collection of sieves for all objects in a category equips that category with a Grothendieck topology and hence makes it a site.

• In the context of homotopy theory and in particular in the corresponding homotopical cohomology theory a cover is an acyclic fibration. In the homotopy theory of simplicial objects this is usually called a hypercover.