This entry is about the properties and the characterization of the category of (set-valued) sheaves on a (small) site , which is a Grothendieck topos. Among other things it gives a definition and a characterization of the notion of sheaf itself, but for more details on sheaves themselves see there.
on those presheaves which are sheaves with respect to .
Every category of sheaves is a reflective subcategory
hence a subtopos of the presheaf topos. Moreover, every such subtopos arises in this way: there is a bijection between Grothendieck topologies on and equivalence classes of geometric embeddings into .
Details on the first statement are at sheafification. A full proof for the second statement is at (∞,1)-category of (∞,1)-sheaves (there proven in (∞,1)-category theory, but the proof is verbatim the same in category theory).
The converse is also true: for every left exact functor (preserving finite limits) which is left adjoint to the inclusion of its image, there is a Grothendieck topology on such that the image of is the category of sheaves on with respect to that topology.
We spell out proofs of some of the above claims.
Let be a small category and
is a bijection.
But by assumption is an isomorphism, so the claim is immediate.
Conversely, if for all the function is a bijection, define and let be the -unit.
Using this it follows that
is an isomorphism. Write for the preimage of under this isomorphism, which is therefore a left inverse of . This immediately implies that also is in , and so we can enter the same argument with to find that it has a left inverse itself. But this means that is in fact an isomorphism and hence so is , which thus exhibits as being in the essential image of .
is in .
Assume first that is in . Since by assumption preserves finite limits, it follows that
Since preserves all colimits and finite limits, we also get
Since by assumption now all are isomorphisms, also is an isomorphism and hence three sides of the above square are isomorphisms. Therefore also is and hence is in .
We check the list of axioms, given at Grothendieck topology:
Pullbacks of covering sieves are covering :
First of all, the pullback of a sieve along a morphism of representables is still a sieve, because monomorphisms are (as discussed there) stable under pullback.
Next, since preserves finite limits, applied to the pullback sieve is the pullback of an isomorphism, hence is an isomorphism, hence the pullback sieve is in .
Two sieves cover precisely if their intersection covers. This is again due to the pullback-stability of elements of , due to the preservation of finite limits by .
If all pullbacks of a sieve along morphisms of a covering sieve are covering, then the original sieve was covering .
This is the same argument as in the second part of the proof of prop. 4.
is a Grothendieck topos.
This appears for instance as (Johnstone, corollary 2.1.11).
See at reflective sub-(∞,1)-category.
There are always different sites whose categories of sheaves are equivalent. First of all for fixed and given a coverage , the category of sheaves depends only on the Grothendieck topology generated by . But there may be site structures also on inequivalent categories that have equivalent categories of sheaves.
If is a full subcategory then the second condition is automatic.
This appears as (Johnstone, theorm C2.2.3).
Let be a locale with frame regarded as a site with the canonical coverage ( covers if the join of the is ). Let be a basis for the topology of : a complete join-semilattice such that every object of is the join of objects of . Then is a dense sub-site.
For the category of all paracompact topological manifolds equipped with the open cover coverage, the category CartSp is a dense sub-site: every paracompact manifold has a good open cover by open balls homeomorphic to a Cartesian space.
Let be a small site and let be its category of sheaves. Let be a homomorphism of sheaves, hence a morphism in . Then:
is a surjection/injection/bijection of sets, respectively, for all .
Every sheaf topos satisfies the following exactness properties. it is an
category of sheaves
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
|(n,r)-categories||toposes||locally presentable||loc finitely pres||localization theorem||free cocompletion||accessible|
|(0,1)-category theory||locales||suplattice||algebraic lattices||Porst’s theorem||powerset||poset|
|category theory||toposes||locally presentable categories||locally finitely presentable categories||Adámek-Rosický’s theorem||presheaf category||accessible categories|
|model category theory||model toposes||combinatorial model categories||Dugger’s theorem||global model structures on simplicial presheaves||n/a|
|(∞,1)-topos theory||(∞,1)-toposes||locally presentable (∞,1)-categories||Simpson’s theorem||(∞,1)-presheaf (∞,1)-categories||accessible (∞,1)-categories|
The characterization of sheaf toposes and Grothendieck topologies in terms of left exact reflective subcategories of a presheaf category is also in
where it is implied by the combination of Corollary VII 4.7 and theorem V.4.1.
The characterization of as the homotopy category of with respect to local isomorphisms is emphasized at the beginning of the text
Details are in
It’s a bit odd that the full final statement does not seem to be stated there explicitly, but all the ingredients are discussed: