In the first post of this series, I explained how subobjects of the constant presheaf (resp. constant sheaf) on a small category (resp. small site) with terminal object
correspond to left closed (resp. local) properties on
. In this post, I will explain the main examples that intervene in setting up topoi, and show how to define the sheaf condition using sieves (instead of coverings).
For simplicity, assume is a small category with fibre products.
Definition. Given a set of morphisms with the same target
, define the sieve
generated by
as the sieve on
of those morphisms
that factor through some
.
It is in a sense the right ideal in generated by the
. What does this look like as a subobject of
?
Example. If has one element, i.e.
, then
is the image of the morphism of representable presheaves
. In the case where
is already a monomorphism (this is always the case when
is a poset, such as
for some topological space
), then
is itself injective (this is the definition of a monomorphism!), so
is just
.
In general, is the image of the map
This shows that, in fact, every sieve is of this form for some set
: take as index set (the objects of) the slice category
, which as in the previous post gives a surjection
. This corresponds to generating an ideal by all its elements.
But we can also characterise without using the word ‘image’ (which somehow computes its first syzygy):
Lemma. Let be a set of morphisms with common target, and
the sieve generated by
. Then
is the coequaliser of the diagram
.
We will give two proofs, one using the description of coequalisers of sets, and the other using that presheaves are colimits of representable presheaves, as discussed in the previous post.
Proof 1. The diagram
Proof 2. By the previous post, the presheaf is the colimit over
of
(see post for precise statement). Let
be the diagram
of the two projections, and let
be the category of elements of
, as in this post. There is a natural functor
taking
to
and
to
, taking the morphisms
in
to the projections
. We claim that
is cofinal, hence the colimit can be computed over
instead (see Tag 04E7).
To verify this, we use the criteria of Tag 04E6. If , then by definition the composition
is given by a morphism
that is contained in
. Since
is generated by the
, this factors through some
over
, giving a map
.
If and
are two such maps, they factor uniquely through
. The general result for
and
for
(either of the form
or of the form
) follows since elements of the form
always map to the elements
and
, showing that the category
is weakly connected.
Corollary. Let as above, and let
be a presheaf on
. Then
Proof. By the lemma above, we compute
Corollary. Let be a (small) site. Then a presheaf
is a sheaf if and only if for every object
and every covering
in the site, the inclusion
induces an isomorphism
Proof. Immediate from the previous corollary.
Thus, the category of sheaves on can be recovered from
if we know at which subobjects
we should localise (make the inclusion invertible). Next week, we will use this to give a definition of a Grothendieck topology, abstracting and generalising the notion of a site (i.e. Grothendieck pretopology).