Contents
Context
Topos Theory
topos theory
Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
Contents
Idea
For a cover of a space , the corresponding Čech groupoid is the internal groupoid
whose set of objects is the disjoint union of the covering patches, and the set of morphisms is the disjoint union of the intersections of these patches.
This is the -coskeleton of the full Čech nerve. See there for more details.
If we speak about generalized points of the (which are often just ordinary points, in applications), then
-
an object of is a pair where is a point in ;
-
there is a unique morphism for all pairs of objects labeled by the same such that ;
-
hence the composition of morphism is of the form
Definition
Definition
(Cech groupoid)
Let be a site, and an object of that site. For each covering family of in the given coverage, the Cech groupoid is the presheaf of groupoids
which, regarded as an internal category in the category of presheaves over , has as presheaf of objects the coproduct
of the presheaves represented (under the Yoneda embedding) by the covering objects , and as presheaf of morphisms the coproduct over all fiber products of these:
This means that for any the groupoid assigned by has as set of objects pairs consisting of an index and a morphism in , and there is a unique morphism between two such objects
precisely if
(1)
Properties
Codescent
We discuss (Prop. below) how the Cech groupoid co-represents matching families as they appear in the definition of sheaves.
For reference, we first recall that definition:
Definition
(matching family – descent object)
Let be a small category equipped with a coverage, hence a site and consider a presheaf (Example ) over .
Given an object and a covering of it we say that a matching family (of probes of ) is a tuple such that for all and pairs of morphisms satisfying
(2)
we have
(3)
We write
(4)
for the set of matching families for the given presheaf and covering.
This is also called the descent object of for descent along the covering .
Proposition
(Cech groupoid co-represents matching families – codescent)
For Grpd regarded as a cosmos for enriched category theory, via its cartesian closed category-structure, and a site, let
be a presheaf on , regarded as a Grpd-enriched presheaf, let be any object and a covering family with induced Cech groupoid (Def.).
Then there is an isomorphism
between the hom-groupoid of Grpd-enriched presheaves and the set of matching families (Def. ).
Since therefore the Cech-groupoid co-represents the descent object, it is sometimes called the codescent object along the given covering.
Moreover, under this identification the canonical morphism induces the comparison morphism
In conclusion, this means that the presheaf is a sheaf (Def. ) precisely if homming Cech groupoid projections into it produces an isomorphism.
Proof
The hom-groupoid is computed as the end
where the “integrand” is the functor category (here: a groupoid) from the Cech groupoid at a given to the set (regarded as a groupoid) assigned by to .
Since is just a set, that functor groupoid, too, is just a set, regarded as a groupoid. Its elements are the functors , which are equivalently those functions on sets of objects
which respect the equivalence relation induced by the morphisms in the Cech groupoid at .
Hence the hom-groupoid is a subset of the end of these function sets:
Here we used: first that the internal hom-functor turns colimits in its first argument into limits (see at internal hom-functor preserves limits), then that limits commute with limits, hence that in particular ends commute with products , and finally the enriched Yoneda lemma, which here comes down to just the plain Yoneda lemma . The end result is hence the same Cartesian product set that also the set of matching families is defined to be a subset of, in (4).
This shows that an element in is a tuple , subject to some condition. This condition is that for each the assignment
constitutes a functor of groupoids.
By definition of the Cech groupoid, and since the codomain is a just set regarded as a groupoid, this is the case precisely if
which is exactly the condition (3) that makes a matching family.
Examples
For a smooth manifold and an atlas by coordinate charts, the Čech groupoid is a Lie groupoid which is equivalent to as a Lie groupoid:
For the Lie groupoid with one object coming from a Lie group morphisms of Lie groupoids of the form
are also called anafunctors from to . They correspond to smooth -principal bundles on .
References
For instance