Locality and descent

Topos Theory

topos theory



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Topos morphisms

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Cohomology and homotopy

In higher category theory


under construction



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

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

The archetypical example are ordinary covers of topological spaces XX by open subsets {U i}\{U_i\}: here YY is their disjoint union Y:= iU iY := \coprod_i U_i.

More generally, you might need a cover to be family of maps (π i:Y iX) i(\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.


In the context of sheaf and topos theory a cover on an object UU in a category CC is a collection of morphisms {U iU} iI\{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 CC. 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 iU}\{U_i \to U\} in CC have incarnations as single morphisms in the category of presheaves PSh(C)PSh(C) over CC, and these are also sometimes called covers :

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

C({U i})=( ijU i× UU j iU i)U. 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 UU that is the incarnation of the covering family in CC, now in PSh(C)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})UC(\{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.


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

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

  • good cover – a cover {U iX}\{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 iU_i and all their finite intersections U i 1× XU i 2× X× XU i nU_{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 {X^X}\{\hat X \to X\} from a covering space is a covering family.

Revised on May 9, 2017 12:37:12 by Urs Schreiber (