Locality and descent
Could not include topos theory - contents
Generally, for an object we think of as a space, a cover of is some other object together with a morphism , usually an epimorphism demanded to be well behaved in certain way.
The idea is that provides a “locally resolved” picture of in that and are “locally the same” but that is “more flexible” than .
The archetypical example are ordinary covers of topological spaces by open subsets : here is their disjoint union .
More generally, you might need a cover to be family of maps ; if the category has coproducts that get along well with the covers, then you can replace these families with single maps as above—see superextensive site.
In the context of sheaf and topos theory a cover on an object in a category is a collection of morphisms .
A specification of a collection of covers for each object of the category, subject to some compatibility condition, makes a coverage on . If the collection of covers in a coverage is being closed under some operations, the result is called a Grothendieck topology. Equipped with a coverage/Grothendieck topology, the category is called a site. See there for more details.
Covering families in have incarnations as single morphisms in the category of presheaves over , and these are also sometimes called covers :
the Cech nerve of the morphism in is a simplicial object in
Its colimit is the local epimorphism on that is the incarnation of the covering family in , now in .
In higher category theory, when we do not restrict to presheaves, for instance when we use simplicial presheaves, the full Cech nerve itself is the “local epimorphism”, the covering map.
More generally, given a coverage one can form hypercovers in the category of simplicial presheaves, by starting with a Cech nerve and then iteratively refining it in each degree further and further by more covers.
In the category =Top of topological spaces or = Diff of smooth manifolds or similar, one has the notions
open cover – for a space, an open cover is a collection of open subsets, that cover in the obvious naive sense of the word, i.e. which are such that their union equals ;
good cover – a cover is called a good cover (or good open cover if in addition it is an open cover) if all of the and all their finite intersections are contractible as topological spaces.
A parameterized version of this is a stacked cover.
There is also the notion of
of a topological space or manifold. This is a priori an independent notion of cover, but for the standard Grothendieck topologies on Top, Diff, etc. the projection from a covering space is a covering family.