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Modalities, Closure and Reflection
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.”
Let be a topos, with subobject classifier .
The closure operator
A Lawvere–Tierney topology in is (internally) a closure operator given by a left exact idempotent monad on the internal meet-semilattice .
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’)
Here is the internal partial order on , and is the internal meet.
This appears for instance as (MacLaneMoerdijk, V 1.).
The closure operator induced by is the transformation
on the subobject lattice of , natural in , that is given by the commuting diagram
This appears for instance as (MacLaneMoerdijk, p. 220).
A morphism is a Lawvere-Tierney topology, def. 1 precisely if the corresponding closure operator, def. 2 satisfies for all
This appears as (MacLaneMoerdijk, V 1., prop 1).
Using Lawvere–Tierney topologies, the notion of sheaf and sheafification generalizes from Grothendieck topoi to arbitrary topoi.
Let be a topos with Lawvere-Tierney topology , def. 1 and associated closure operator , def. 2.
A subobject is called dense if .
The corresponding monomorphism is called a dense monomorphism.
An object is called a -sheaf if it is a local object with respect to the dense monomorphisms.
This means that is a -sheaf if for every dense the induced morphism
is an isomorphism.
is a -separated presheaf if this morphism is itself a monomorphism.
This is for instance in (MacLaneMoerdijk, p. 223).
This appears for instance as (MacLaneMoerdijk V 3., theorem 1).
Equivalence with Grothendieck topologies
The subobject classifier in a presheaf topos is the presheaf that assigns to the set of all sieves in on
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 .
For a small site and the Lawvere-Tierney topology on the presheaf topos given by prop. 3 the j-sheaves are precisely the sheaves in the ordinary sense of Grothendieck topologies.
Relation to lex reflectors
As discussed there, categories of sheaves are also characterized as being reflective subcategories of the given ambient topos
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.
Given a reflector , define for each object a closure operator, being a functor on the poset of subobjects of
by sending any monomorphism classified by a characteristic function to the pullback in
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
is the adjunction unit;
is the characteristic function of the result of applying to the universal subobject
(which is again a monomorphism since preserves pullbacks).
For any subobject with characteristic function , we need to show that we have a pullback diagram
The pullback along the rightmost morphism is by definition
Moreover, by the naturality of the adjunction unit we have a commuting diagram
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
is a pullback. But then also the left square is a pullback, by def. 5. This finally means, by the pasting law, that also the total rectangle is a pullback.
- 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
- Bill Lawvere, Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)
Detailed discussion of Lawvere-Tierney operators as geometric modalities is in
- Robert Goldblatt, Grothendieck topology as geometric modality, Mathematical Logic Quarterly, Volume 27, Issue 31-35, pages 495–529, (1981)
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