Definition (Grothendieck topology):
Let be a category. Then a Grothendieck topology on is a class , whose elements are sets of maps in , that satisfies the following three axioms:
- For each object , the singleton set is contained within
- If is in and is a morphism of , then is in (here, we assume that all the required fibered products exist, and that is the morphism implied by the definition of a fibered product)
- If is in and for each , the set is in , then the set is in
Example (model example of a Grothendieck topology):
For a topological space , we construct a category as follows: The objects are precisely the open subsets of , and for each inclusion of open sets, there is a morphism given by the inclusion, and these are all morphisms of .
A site is a category together with a Grothendieck topology on it.
Many of the statements below hold true for general functor categories. We loose no generality by imposing the existence of a Grothendieck topology, since we may equip each category with the trivial Grothendieck topology consisting of the trivial covers () only.
Definition (category of open subsets):
Let be a topological space. The category of open subsets of , denoted by , is the category whose objects are precisely the open subsets of , and whose morphisms are precisely the inclusions of open sets, that is,
Let be a category, and let be a site. A presheaf on with values in is a contravariant functor .
Let be a concrete category (with functor ), a site, and a presheaf. A section of on is an element of .
Definition (morphism of presheaves):
Let be a category, and let be a site. Let be two presheaves on . Then a morphism of presheaves is a natural transformation .
Definition (category of presheaves):
Let be a category, and let be a site. Then the category of presheaves on with values in shall be the category whose objcts are presheaves on with valus in and whose morphisms are morphisms of presheaves. That is, the category of presheaves on with values in is the functor category from to .
Example (constant presheaf):
Let be a category possessing a terminal object, and a topological space. The constant presheaf associated to , denoted , is the functor that associates to every nonempty open subset of the object , to the empty set the terminal object of , and to every inclusion the identity, lest it is the inclusion of the empty set, in which case we associate to it the unique morphism to the terminal object.
Let be a category, a site, and a presheaf. We order the inverse systems in as follows: One inverse system is less than or equal to the other if and only if there exists an injective natural transformation from the supposedly smaller one to the supposedly larger one. Then a point in is a maximal directed system with respect to that order.
this definition does not yet capture all points; improvement is under way.
Let be a category, a site, a point in , and a presheaf. Then the stalk of at is defined as
where the morphisms implied in the direct limit are given by ().
Remark (points exist in small sites):
If the underlying category of is small, then it is easily seen that the inverse systems on form a set, so that Zorn's lemma implies the existence of points.
Proposition (morphism of presheaves induces morphisms of stalks):
Let be two presheaves on a site with values in a category . If is a morphism of presheaves and a point in , then there is a functor
which on objects is given by
Proof: If is a morphism of presheaves, then the cocone implied in the definition of may be precomposed with , which yields a cocone over the same diagram that is used in the definition of . Thus, the universal property of yields a morphism , and we use this assignment as the definition of the desired functor on morphisms. The uniqueness of this assignment ensures that whenever and are morphisms of presheaves. Also, the functor thus constructed maps all identities of to identities of .
Definition (stalk functor):
The functor , described in the above proposition shall be called stalk functor at .
Proposition (existence of limits and colimits in a category of presheaves):
Let be a site, and suppose that is a category that admits any limit. Then admits that limit. The same statement for colimits is true as well.
Proof: This follows since is nothing but a functor category, and for functor categories that statement is true.
Proposition (image and kernel of presheaf morphisms commute with the stalk functor):
Let be two presheaves on a site with values in a category . If is a morphism of presheaves, then
- and .
Proof: Since the stalk functor maps the zero map to the zero map (as is readily seen from the universal property defining this functor), factors over .
Definition (continuous functor):
Let and be sites. A continuous functor from to is a covariant functor in the other direction such that the following three conditions are satisfied:
- The final object of is mapped to the final object of
- preserves fibered products
- The image of a covering under is again a covering
Definition (pushforward presheaf):
Let be a continuous functor from the site to the site , and let be a presheaf on . Then the pushforward presheaf is defined to equal .
Definition (pullback presheaf):
Let be a continuous functor from the site to the site , and let be a presheaf on . Then the pullback presheaf is defined by
where the morphisms are given by the universal property of the inverse limit, which is supposed to range over all maps from to any object in the image of .
Proposition (pullback and pushforward of presheaves are adjoint functors):
Let be a functor (which may, for instance, be a continuous functor from to ). Then there exists an isomorphism
that is natural in and ; that is, and are adjoint functors.
Proof: On an object , a morphism will yield a morphism
Let now and suppose that . Then from the above, we obtain a morphism
since then the direct limit on the right hand side of the first morphism is simply , as this will be a maximal element. From the functoriality of and the fact that the morphism was a morphism of presheaves, one easily sees that one may use that the restriction maps of the pullback are simply the ones of on the image of in order to prove that a morphism of presheaves is thus defined.
Conversely, on an object , a morphism will yield a morphism
If now is given, we obtain from this a morphism
for every morphism in . The universal property of the inverse limit turns this into a morphism
This is a morphism of presheaves, as is seen from applying the universal property of to the cone with tip for a morphism in .
Since by uniqueness a morphism
may just as well arise from the universal property of the latter inverse limit and morphisms
- (there is a morphism in from to )
and for , the map from presheaf morphisms on to presheaf morphisms on does not change the morphism at all, the two maps are inverses of each other. Thus, we have shown
Naturality in follows from the commutativity of the triangl
- triangle.svg (F'(F(V)) to F(F(V)) to G(V))
and naturality in