Cohomology and homotopy
In higher category theory
A local isomorphism in a presheaf category is a morphism that becomes an isomorphism after passing to sheaves with respect to a given Grothendieck topology on .
The collection of all local isomorphisms not only determines the Grothendieck topology but is precisely the collection of morphisms that are inverted when passing to sheaves. Hence local isomorphisms serve to understand sheaves and sheafification in terms of the passage to a homotopy category of .
This is a particular case of the notion of reflective factorization system, applied to the sheafification reflector. It is discussed in more detail at category of sheaves:
In terms of the discussion at geometric embedding, local isomorphisms in are precisely the multiplicative system that is sent to isomorphisms by the sheafification functor
which is left exact left adjoint to the full and faithful inclusion
A system of local isomorphisms on is any collection of morphisms satisfying
local isomorphisms are a system of weak equivalences (i.e. every isomorphism is a local isomorphism and they satisfy 2-out-of-3);
a morphism is a local isomorphism if and only if its pullback
along any morphism , where is representable, is a local isomorphism.
Relation to Grothendieck topologies
Systems of local isomorphisms on are equivalent to Grothendieck topologies on .
The following indicates how choices of systems of local isomorphisms are equivalent to choices of systems of local epimorphisms. The claim follows by the discussion at local epimorphism.
Local epimorphisms from local isomorphisms
A system of local epimorphisms is defined from a system of local isomorphisms by declaring that is a local epimorphism precisely if is a local isomorphism.
Local isomorphisms from local epimorphisms
Given a Grothendieck topology in terms of a system of local epimorphisms, a system of local isomorphisms is constructed as follows.
A local monomorphism with respect to this topology is a morphism in such that the canonical morphism is a local epimorphism.
A local isomorphism with respect to a Grothendieck topology is a morphism in that is both a local epimorphism as well as a local monomorphism in the above sense.
Relation to Lawvere-Tierney topologies
Recall that Grothendieck topologies on a small category are in bijection with Lawvere-Tierney-topologies on and that sheafification with respect to a Lawvere-Tierney topology is encoded in terms of monomorphisms in which are dense with respect to the Lawvere-Tierney topology.
the dense monomorphisms are precisely the local isomorphisms which are also ordinary monomorphisms.
The sheafification functor which sends a presheaf to its weakly equivalent sheaf can be realized using a colimit over local isomorphisms. See there.
Characterization and relation to sieves
Often one concentrates on the local isomorphisms whose codomain is a representable presheaf, i.e. those of the form
where is an object in and is the Yoneda embedding. These come from covering sieves of a Grothendieck topology on : for and a covering sieve on , the coresponding local isomorphism is the presheaf which is the image of the joint injection map
Using the fact that morphisms in a presheaf category are strict morphisms, so that image and coimage coincide, it is useful, with an eye towards generalizations from sheaves to stacks and ∞-stacks (see in particular descent for simplicial presheaves), to say this equivalently in terms of the coimage: the local isomorphism corresponding to the covering sieve is
Notice that in general these are not all the local isomorphism with representable codomain (more generally these are hypercovers, where is replaced in turn by one of its covers).
Notice that local isomorphism with codomain a representable already induce general local isomorphisms using the fact that every presheaf is a colimit of representables (the co-Yoneda lemma) and that local isomorphisms/sieves are stable under pullback:
If is a local object with respect to local isomorphisms whose codomain is a representable, then every morphism of presheaves such that for every representble and every morphism the pullback is a local isomorphism, the canonical morphism
is an isomorphism.
We may first rewrite trivially
and then use the co-Yoneda lemma to write (suppressing notationally the Yoneda embedding)
and hence rewrite as
Then using that colimits of presheaves are stable under base change this is
Recall that by assumption the components of this are local isomorphisms. Hence
is a limit over isomorphisms, hence an isomorphism.
This is in section 16.2 of
See in particular exercise 16.5 there for the characterization of Grothendieck topologies in terms of local isomorphisms.