Cohomology and homotopy
In higher category theory
Locality and descent
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.
Let be a site: a (small) category equipped with a coverage.
This appears for instance as (Johnstone, corollary 2.1.11). See also Lawvere-Tierney topology.
The category of sheaves is equivalent to the homotopy category of the category with weak equivalences with the weak equivalences given by local isomorphisms
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
a reflective subcategory, hence a full subcategory with a left adjoint , such that moreover preserves finite limits.
Write for the class of morphisms in that are sent to isomorphisms by .
A presheaf is in (meaning: in the essential image of ) precisely if for all in the induced function
is a bijection.
If , then by the -adjunction isomorphism we have
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.
By the assumption that is a full and faithful functor and basic properties of adjoint functors we habe that the counit
is a natural isomorphism. By the zig-zag law the composite
is the identity and therefore is an isomorphism and so is in , under our assumption on .
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 .
A morphism is in precisely if for every morphism with representable domain, the pullback in
is in .
Assume first that is in . Since by assumption preserves finite limits, it follows that
is still a pullback diagram in and hence that is the pullback of the isomorphism and thus itself an isomorphism. Therefore is in .
Conversely, suppose that all these pullbacks are in . Then use the “co-Yoneda lemma” to write the presheaf as a colimit over all representables mapping into it
Forming the pullback along , using that in a topos (such as our presheaf topos) colimits are preserved by pullbacks, we get
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 .
The maximal sieve is covering. Clear: applied to an isomorphism is an isomorphism.
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.
By prop. 5 and prop. 3 we are reduced to showing that an object is in already if for all monomorphisms in the function is a bijection.
Categories of sheaves are examples of categories that are toposes: they are the Grothendieck toposes characterized among all toposes as those satisfying Giraud's axioms.
This appears for instance as (Johnstone, corollary 2.1.11).
As accessible reflective subcategories
See at reflective sub-(∞,1)-category.
Dependence on the site
For a category of sheaves and a site such that we have an equivalence of categories we say that is a site of definition for the topos .
For a site with coverage and any subcategory, the induced coverage on has as covering sieves the intersections of the covering sieves of with the morphisms in .
Let be a site (possibly large). A subcategory (not necessarily full) is called a dense sub-site with the induced coverage if
every object has a covering in with all in ;
for every morphism in with there is a covering family such that the composites are in .
Let be a (possibly large) site with a locally small category and let be a small dense sub-site. The pair of adjoint functors
with given by precomposition with and given by right Kan extension induces an equivalence of categories between the categories of sheaves
This appears as (Johnstone, theorm C2.2.3).
Epi- mono- and isomorphisms
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 monomorphism or isomorphism precisely if it is so globally in that for each object in the site, then the component is an injection or bijection of sets, respectively.
is an epimorphism precisely if it is so locally, in that: for all there is a covering such that for all and every element the element is in the image of .
But if is a set of points of a topos for such that these are enough points (def.) then the morphism is epi/mono/iso precisely it is is so an all stalks, hence precisely if
is a surjection/injection/bijection of sets, respectively, for all .
Every sheaf topos satisfies the following exactness properties. it is an
Locally presentable categories: Large categories whose objects arise from small generators under small relations.
|(n,r)-categories…||satisfying Giraud's axioms||inclusion of left exact localizations||generated under colimits from small objects||localization of free cocompletion||generated under filtered colimits from small objects|
|(0,1)-category theory||(0,1)-toposes||algebraic lattices|| Porst’s theorem||subobject lattices in accessible reflective subcategories of presheaf categories|
|category theory||toposes||locally presentable categories|| Adámek-Rosický’s theorem||accessible reflective subcategories of presheaf categories||accessible categories|
|model category theory||model toposes||combinatorial model categories|| Dugger’s theorem||left Bousfield localization of global model structures on simplicial presheaves|
|(∞,1)-topos theory||(∞,1)-toposes||locally presentable (∞,1)-categories|| |
|accessible reflective sub-(∞,1)-categories of (∞,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: