In the post of two weeks ago, we showed how Grothendieck topologies form a neat framework for the categorical aspects of the more familiar (at least to algebraic geometers) Grothendieck pretopologies. In this final post of the series, we take this one step further, to the notion of a Lawevere–Tierney topology on an arbitrary elementary topos .
Definition. An elementary topos is a category that has finite limits, is Cartesian closed, and has a subobject classifier (see last week’s post).
The only example we’re interested in today is the category of presheafs (of sets) on a small category
, which we saw last week has subobject classifiers. But in fact, any Grothendieck topos (sheaves of sets on a (small) site) is an example of an elementary topos, so one could even introduce further topologies on those.
Example. The category of finite sets is an elementary topos, but not a Grothendieck topos since it is neither complete nor cocomplete.
Before giving the definition, we need to define one more structure on : the meet. Recall that the intersection (or meet) of two monomorphisms
is the fibre product
The intersection of and
is the monomorphism
given by
, which is classified by a map
. Since
is the universal monomorphism, we see that
is the universal intersection of two subobjects, i.e. if
and
are classified by
and
respectively, then
is classified by the composition
(If we denote this simply by , then
is
.)
Definition. Let be an elementary topos with subobject classifier
. Then a Lawvere–Tierney topology on
is a morphism
such that the following diagrams commute:
We saw two weeks ago that a Grothendieck topology is a certain subpresheaf , and last week that
is a subobject classifier
on
. Thus a subpresheaf
is classified by a morphism
, which we saw last week is given by
.
Lemma. The subpresheaf is a Grothendieck topology on
if and only if
is a Lawvere–Tierney topology on
. In particular, Grothendieck topologies on
are in bijective correspondence with Lawvere–Tierney topologies on
.
Thus Lawvere–Tierney topologies are an internalisation of the notion of Grothendieck topology to an arbitrary elementary topos .
Proof of Lemma. By definition of the morphism , we have a pullback square
The first commutative diagram in the definition above means that the top arrow has a section such that the composition
is
, i.e.
as subobjects of
. Since
is the map taking
to the maximal sieve
for any
, this means exactly that
for all
, which is condition 1 of a Grothendieck topology. For the second, consider the pullback
The condition means that
as subobjects of
. We already saw that
for a Grothendieck or Lawvere–Tierney topology, so pulling back along
gives
. Thus the second diagram in the definition of a Lawvere–Tierney topology commutes if and only if
, i.e. if
with
, then
. But
is given by
, so this is exactly axiom 3 of a Grothendieck topology.
For the third diagram, we first claim that is monotone for all
if and only if
satisfies axiom 2 of a Grothendieck topology. Indeed, if
is monotone and
satisfy
and
, then the inclusion
shows
, so
by axiom 3. Conversely, if
satisfies axiom 2 and
satisfy
, then for any
we have
, so
, i.e.
.
The third diagram in the definition above says that the map given by
is a morphism of meet semilattices. This implies in particular that
is monotone, as
if and only if
, so the third diagram above implies axiom 2 of a Grothendieck topology.
Conversely, if is a Grothendieck topology, then axiom 2 implies that
is monotone. In particular,
for any
, since
. For the reverse implication, if
satisfies
, then
and
, so the remark of two weeks ago shows that
, i.e.
. We see that
, showing that
is a morphism of meet semilattices.