Locality and descent
Cohomology and homotopy
In higher category theory
A presheaf on a site is a sheaf if its value on any object of the site is given by its compatible values on any covering of that object.
There are several equivalent ways to characterize sheaves. We start with the general but explicit componentwise definition and then discuss more general abstract equivalent reformulations. Finally we give special discussion applicable in various common special cases.
General definition in components
The following is an explicit component-wise definition of sheaves that is fully general (for instance not assuming that the site has pullbacks).
Let be a site in the form of a small category equipped with a coverage .
A presheaf is a sheaf with respect to if
for every covering family in
and for every compatible family of elements given by tuples such that for all morphisms in we have for all
- there is a unique element such that for all .
In this form the definition appears for instance in (Johnstone, def. C2.1.2).-
General definition abstractly
We now reformulate the above component-wise definition in general abstract terms.
for the Yoneda embedding.
Given a covering family in , its sieve is the presheaf defined as the coequalizer
Here the coproduct on the left is over the pullbacks
in , and the two morphisms between the coproducts are those induced componentwise by the two projections in this pullback diagram.
A sheaf on is a presheaf that is a local object with respect to all : an object such that for all covering families in we have that the hom-functor sends the canonical morphisms to isomorphisms.
Equivalently, using the Yoneda lemma and the fact that the hom-functor sends colimits to limits, this says that the diagram
is an equalizer diagram for each covering family.
This is also called the descent condition for descent along the covering family.
We may express the set of natural transformations (as described there) by the end
Using this in the expression of the equalizer
as a subset of the product set on the left manifestly yields the componenwise definition above.
Characterizations over special sites
We discuss equivalent characterizations of sheaves that are applicable if the underlying site enjoys certain special properties.
Characterizations over sites of opens
An important special case of sheaves is those over a (0,1)-site such as a category of open subsets of a topological space . We consider some equivalent ways of characterizing sheaves among presheaves in such a situation.
(The following was mentioned in Peter LeFanu Lumsdaine’s comment here).
Suppose is the category of open subsets of some topological space regarded as a site with the canonical coverage where is covering if the union in .
Then a presheaf on is a sheaf precisely if for every complete full subcategory , takes the colimit in over to a limit:
A complete full subcategory is a collection of open subsets that is closed under forming intersections of subsets. The colimit
is the union of all these open subsets. Notice that by construction the component maps of the colimit are a covering family of .
Inspection then shows that the limit is the corresponding set of matching families (use the description of limits in terms of products and equalizers ). Hence the statement follows with def. 1.
Characterization over canonical topologies
The above prop. 2 shows that often sheaves are characterized as contravariant functors that take some colimits to limits. This is true in full generality for the following case
Sheaves and localization
We now describe the derivation and the detailed description of various aspects of sheaves, the descent condition for sheaves and sheafification, relating it to all the related notions
We start by assuming that a geometric embedding into a presheaf category is given and derive the consequences.
So let be a small category and write for the corresponding topos of presheaves.
Assume then that another topos is given together with a geometric embedding
i.e. with a full and faithful functor
and a left exact functor
Such that both form a pair of adjoint functors
with left adjoint to .
Write for the category
consisting of all those morphisms in that are sent to isomorphisms under .
From the discussion at geometric embedding we know that is equivalent to the full subcategory of on all -local objects.
Recall that an object is called a -local object if for all in the morphism
is an isomorphism. This we call the descent condition on presheaves (saying that a presheaf “descends” along from “down to” ). Our task is therefore to identify the category , show how it determines and is determed by a Grothendieck topology on – equipping with the structure of a site – and characterize the -local objects. These are (up to equivalence of categories) the objects of , i.e. the sheaves with respect to the given Grothendieck topology.
A morphism is in if and only if for every representable presheaf and every morphism the pullback is in
Since is stable under pullback (as described at geometric embedding: simply because preserves finite limits) it is clear that is in if is.
To get the other direction, use the co-Yoneda lemma to write as a colimit of representables over the comma category (with the Yoneda embedding):
Then pull back over the entire colimiting cone, so that over each component we have
Using that in colimits are stable under base change we get
But since the right hand is , which is just . So and we find that is a morphism of colimits. But under the two respective diagrams become isomorphic, since is in . That means that the corresponding morphism of colimits (since preserves colimits) is an isomorphism, which finally means that is in .
A presheaf is a local object with respect to all of already if it is local with respect to those morphisms in whose codomain is representable
Rewriting the morphism in in terms of colimits as in the above proof
we find that equals
If is local with respect to morphisms with representable codomain, then by the above if is in all the morphisms in the limit here are isomorphisms, hence
Every morphism in factors as an epimorphism followed by a monomorphism in with both being morphisms in .
Use factorization through image and coimage, use exactness of to deduce that the factorization exists not only in but even in .
More in detail, given we get the diagram
Because is exact, the pullbacks and pushouts in this diagram remain such under . But since is an isomorphism by assumption, the all these are pullbacks and pushouts along isomorphisms in , so all morphisms in the above diagram map to isomorphisms in , hence the entire diagram in is in .
Since the morphism out of the coimage is at the same time the equalizing morphism into the image , it is a monomorphism.
The monomorphisms in which are in are called dense monomorphisms.
Every monomorphism with representable is of the form
for a disjoint union of representables
This is a direct consequence of the standard fact that subfunctors are in bijection with sieves.
If a presheaf is local with respect to all dense monomorphisms, then it is already local with respect to all morphisms of the form
with the left vertical morphism a dense monomorphism
(and with the disjoint union (of representable presheaves) over a covering family of objects.)
The morphisms in with representable codomain
of the form as above are covers:
of the form (with a cover of ) as above are hypercovers
of the representable .
Urs: the above shows this almost. I am not sure yet how to see the remaining bit directly, without making recourse to the full machinery leading up to section VII, 4, corollary 7 in Sheaves in Geometry and Logic.
So we finally conclude:
From the assumption that is a geometric embedding follows at once the following explicit description of the sheafification functor .
For a presheaf, its sheafification is the presheaf given by
By the discussion at geometric embedding the category is equivalent to the localization , which in turn is the category with the same objects as and with morphisms given by spans out of hypercovers in
So we have
by Yoneda that ;
by the hom-adjunction this is ;
by the equivalence just mentioned this is .
For a presheaf, the plus-construction on is the presheaf
where the colimit is over all dense monomorphisms (instead of over all local isomorphisms as for sheafification ).
The archetypical example of sheaves are sheaves of functions:
for a topological space, a topological space and the site of open subsets of , the assignment of continuous functions from to for every open subset is a sheaf on .
for a complex manifold and a complex manifold, the assignment of holomorphic functions in a sheaf.
Section C2 in
The book by Kashiwara and Shapira discusses sheaves with motivation from homological algebra, abelian sheaf cohomology and homotopy theory, leading over in the last chapter to the notion of stack.
A quick pedagogical introduction with an eye towards the generalization to (∞,1)-sheaves is in