# nLab cover

topos theory

## Theorems

under construction

# Contents

## 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 coproducts that get along well with the covers, then you can replace these families with single maps as above—see superextensive site.

## Definitions

In the context of sheaf and topos theory a cover on an object $U$ in a category $C$ is a collection of morphisms $\{U_i \to U\}_{i \in I}$.

A specification of a collection of covers for each object of the category, subject to some compatibility condition, makes a coverage on $C$. 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 $\{U_i \to U\}$ in $C$ have incarnations as single morphisms in the category of presheaves $PSh(C)$ over $C$, and these are also sometimes called covers :

the Cech nerve of the morphism $\coprod_i U_i \to U$ in $PSh(C)$ is a simplicial object in $PSh(C)$

$C(\{U_i\}) = \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} \coprod_{i j} U_i \times_U U_j \stackrel{\to}{\to} \coprod_i U_i \right) \to U \,.$

Its colimit is the local epimorphism on $U$ that is the incarnation of the covering family in $C$, now in $PSh(C)$.

In higher category theory, when we do not restrict to presheaves, for instance when we use simplicial presheaves, the full Cech nerve itself $C(\{U_i\}) \to U$ 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.

## Examples

In the category $C$ =Top of topological spaces or $C$ = Diff of smooth manifolds or similar, one has the notions

• open cover – for $X \in C$ a space, an open cover is a collection $\{U_i \subset X\}$ of open subsets, that cover $X$ in the obvious naive sense of the word, i.e. which are such that their union equals $X$;

• good cover – a cover $\{U_i \to X\}$ is called a good cover (or good open cover if in addition it is an open cover) if all of the $U_i$ and all their finite intersections $U_{i_1} \times_X U_{i_2} \times_X \cdots \times_X U_{i_n}$ 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 $\{\hat X \to X\}$ from a covering space is a covering family.

Revised on July 17, 2014 22:57:22 by Urs Schreiber (82.136.246.44)