This post is the first goal in a series on sieves (subobjects of representable presheaves); I will give another generalisation in the next two posts. In the first post of the series, I defined sieves and gave basic examples, and last week I showed how the sheaf condition on a site can be stated in terms of sieves:
Corollary. Let be a (small) site. For a set of morphisms
with the same target, write
for the presheaf image of
. Then a presheaf
is a sheaf if and only if for every covering
in
, the inclusion
induces an isomorphism
Thus, if is a site (a small category with a Grothendieck pretopology), we should be able to obtain the category
of sheaves purely in terms of sieves. This is the notion of a Grothendieck topology that we describe at the end of this post.
Before giving the definition, note that any morphism in
gives a pullback
taking
to its inverse image under
(I avoid the word ‘pullback’ here to make sure this is truly a subpresheaf and not a presheaf with a monomorphism to
defined uniquely up to unique isomorphism). Thus,
is itself a presheaf
(it takes values in
since
is small).
Also note the following method for producing sieves: if is a presheaf,
a subpresheaf, and
a section over some
, we get a sieve
by
Definition. Let be a small category. Then a Grothendieck topology on
consists of a subpresheaf
such that
- For all
, the maximal sieve
is in
.
- If
and
with
, then
.
- If
is a sieve such that
, then
(equivalently, then
is the maximal sieve
).
The sieves are called covering sieves. Since
is a presheaf, we see that for any
and any covering sieve
, the pullback
is covering. Condition 2 says that any sieve containing a covering sieve is covering. In the presence of condition 1, conditions 2 and 3 together are equivalent to the local character found in SGA IV
, Exp. II, Def. 1.1:
- If
with
, such that for every morphism
the inverse image of
along
is in
, then
.
Indeed, applying this criterion when immedately shows
if
, since the inverse image of
along
is the maximal sieve
. Thus the local character implies criterion 2. The local character says that if
contains a covering sieve
, then
is covering. Assuming criterion 2, the sieve
contains a covering sieve if and only if
is itself covering, so the local character is equivalent to criterion 3.
Remark. One property that follows from the axioms is that is closed under binary intersection, i.e. if
then
. Indeed, if
for some
, then
Example. Let be a pretopology on the (small) category
; see Tag 00VH for a list of axioms. For each
, define the subset
as those
that contain a sieve of the form
for some covering
in
. (See the corollary at the top for the definition of
.) Concretely, this means that there exists a covering
such that
for all
, i.e.
is covered by morphisms
that are in the given sieve
.
Lemma. The association is a topology. It is the coarsest topology on
for which each
for
is a covering sieve.
Proof. We will use the criteria of Tag 00VH. If , then there exists
with
. If
is any morphism in
, then
by criterion 3 of Tag 00VH. But
, because a morphism
factors through
if and only if
factors through
. Thus,
, so
, and
is a subpresheaf of
.
Condition 1 follows immediately from criterion 1 in Tag 00VH, and condition 2 is satisfied by definition. For condition 3, suppose satisfies
. Then there exists
with
. This means that
for all
, i.e.
for all
. Thus, for each
there exists
in
such that
, i.e.
for all
and all
. Thus, if
denotes
, then we get
. But
is a covering by criterion 2 of Tag 00VH, so
.
If is any other Grothendieck topology for which each
for
is covering, then
contains
by criterion 2.
To state the obvious (hopefully), the notion of sheaf can therefore be defined on a Grothendieck topology in a way that coincides with the usual notion for a Grothendieck pretopology:
Definition. Let be a small category, and let
be a Grothendieck topology. Then a presheaf
is a sheaf if for any
and any
, the map
induces an isomorphism
Thus, a Grothendieck topology is an internal characterisation (inside ) of which morphisms
one needs to localise to get
. In the last two posts, we will generalise this even further to a Lawvere–Tierney topology on an arbitrary topos.