Could not include topos theory - contents
A Lawvere–Tierney topology (or operator , or modality , also called geometric modality) on a topos is a way of saying that something is ‘locally’ true. Unlike a Grothendieck topology, this is done directly at the stage of logic, defining a geometric logic. In fact, it is a generalisation of Grothendieck topology in this sense: If is a small category, then choosing a Grothendieck topology on is equivalent to choosing a Lawvere–Tierney topology in the presheaf topos on .
The use of “topology” for this and the related Grothendieck concept is regarded by some people as unfortunate; see historical note on Grothendieck topology for some reasons why. A proposed replacement for “Grothendieck topology” is (Grothendieck) coverage; see Grothendieck topology for some possible replacements for “Lawvere–Tierney topology.”
This means that: a Lawvere–Tierney topology in is a morphism
, equivalently (‘if is true, then is locally true’)
(‘ is locally locally true iff is locally true’);
(‘ is locally true iff and are each locally true’)
This appears for instance as (MacLaneMoerdijk, V 1.).
Equivalently, the third axiom in def. 1 can be replaced with the (internal) statement that is order-preserving.
The equivalence amounts to the fact that, within the internal logic of topoi, one can demonstrate that every monad on the preorder of truth values is in fact strong (a special case of the fact that, for an endofunctor on some monoidal closed , tensorial strengths are the same as -enrichments, as described in the article on the former), and therefore automatically preserves finite meets.
Thus, a Lawvere-Tierney topology is the same thing as an internal closure operator on the preorder (aka, a Moore closure on the one-element set), which amounts to the same thing as a natural closure operator on subobjects.
Specifically, given any subobject inclusion in , consider its characteristic morphism . Then is another morphism , which defines another subobject of , taken as the closure of our original subobject. The elements of are those elements of that are ‘locally’ in .
This means that for a subobject, with characteristic morphism , its closure is the subobject classified by
This appears for instance as (MacLaneMoerdijk, p. 220).
This appears as (MacLaneMoerdijk, V 1., prop 1).
A subobject is called dense if .
This means that is a -sheaf if for every dense the induced morphism
is an isomorphism.
This is for instance in (MacLaneMoerdijk, p. 223).
For a topos and a Lawvere-Tierney topology on , the inclusion
This appears for instance as (MacLaneMoerdijk V 3., theorem 1).
since we have
A subobject is therefore precisely a choice of a collection of sieves on each object, which is closed under pullback. The proof therefore amounts to checking that the condition that such a collection of sieves is a Grothendieck topology on is equivalent to the statement that the characteristic map of (see remark 1) is a Lawvere-Tierney topology.
Here is more discussion of this point:
Suppose that is a small site. Then given a subpresheaf inclusion in , an object of , and an element of , we say is locally in (that is, ) if and only if, for some covering family on , the restriction of to is in (that is, each ). This intuitively defines the “local” modality that is the Lawvere–Tierney topology corresponding to the given Grothendieck topology on .
As a specific example, take the usual Grothendieck topology on Top, given by the usual notion of open cover. Taking real-valued functions on a space defines a presheaf (in fact a sheaf) on ; the constant functions form a subpresheaf of . A real-valued function belongs to iff it is locally constant; that is, for some open cover of the domain , each restriction is constant.
To make this precise in terms of the above definition, we need to understand the subobject classifier in . But according to the definition, is simply the representing object for the functor
which takes an object of to the collection of subobjects of , . In other words, . Applied to , we have then
In other words, we find that the functor is defined by
Next, if is a Grothendieck topology on , then the collection of -covering sieves on [which we denote by ] is a subcollection of all sieves on , and so we have an inclusion
and this inclusion is natural in , by virtue of the first axiom on covering sieves. Thus we have a subobject
and again, by definition of subobject classifier, this subobject corresponds to a uniquely determined element
which is just the Lawvere–Tierney operator .
Conversely, any morphism determines a subobject of , which therefore associates to every object a set of sieves on . It is easy to check that the axioms for covering sieves in a Grothendieck topology correspond exactly to the required properties of the operator .
Here we discuss explicit translations between the structure given by the reflector and the corresponding Lawvere-Tierney topology in a way that makes the relation to modal type theory and monads (in computer science) most manifest.
where is the adjunction unit.
This appears as (Johnstone, lemma A4.3.2).
Given a left exact reflector as above with induced closure operation , the corresponding Lawvere-Tierney operator is given as the composite
The pullback along the rightmost morphism is by definition
Using this in the remaining bottom morphism of our would-be pullback square we find that equivalently
needs to be a pullback diagram. Since preserves pullbacks we have that also the middle square in
In this form the statement appears in the proof of (Johnstone, Theorem A4.3.9).
Francis Borceux, Sheaves of algebras for a commutative theory, Ann. Soc. Sci. Bruxelles Sér. I 95 (1981), no. 1, 3–19, MR83c:18006
Let be a small category enriched over where is a commutative algebraic theory. Then . A -sieve as an enriched subfunctor of . A -topology is a set of -sieves for every , satisfying some axioms. Borceux defines the notion of a sheaf over such enriched site and proves the existence and exactness of the associated sheaf functor.
He proves that there is an object in which classifies subobjects in . Moreover, there is a correspondence betwen
(1) localizations of
(2) -topologies on
(3) morphisms satisfying the Lawvere-Tierney axioms for a topology
The notion is introduced as a geometric modality on p. 3 of
Detailed discussion of Lawvere-Tierney operators as geometric modalities is in
Textbook accounts include section V.1 of
(the notion of sheaves in section V.3, the sheafification functor in section V.3 and the relation to Grothendieck topologies in section V.4);
and section A4.4 of