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.