nLab
Lawvere-Tierney topology

Lawvere–Tierney topologies

Idea

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 CC is a small category, then choosing a Grothendieck topology on CC is equivalent to choosing a Lawvere–Tierney topology in the presheaf topos Set C op\Set^{C^\op} on CC.

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.”

Definition

Let EE be a topos, with subobject classifier Ω\Omega.

The closure operator

Definition

A Lawvere–Tierney topology in EE is (internally) a closure operator given by a left exact idempotent monad on the internal meet-semilattice Ω\Omega.

This means that: a Lawvere–Tierney topology in EE is a morphism

j:ΩΩ j: \Omega \to \Omega

such that

  1. jtrue=truej true = true, equivalently id Ωj:ΩΩ\id_\Omega \leq j: \Omega \to \Omega (‘if pp is true, then pp is locally true’)

    * true Ω true j Ω \array{ * &\stackrel{true}{\to}& \Omega \\ & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }
  2. jj=jj j = j (‘pp is locally locally true iff pp is locally true’);

    Ω j Ω j j Ω \array{ \Omega &\stackrel{j}{\to}& \Omega \\ & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }
  3. j=(j×j)j \circ \wedge = \wedge \circ (j \times j) (‘pqp \wedge q is locally true iff pp and qq are each locally true’)

    Ω×Ω Ω j×j j Ω×Ω Ω. \array{ \Omega \times \Omega &\stackrel{\wedge}{\to}& \Omega \\ {}^{\mathllap{j \times j}}\downarrow && \downarrow^{\mathrlap{j}} \\ \Omega \times \Omega &\underset{\wedge}{\to}& \Omega } \,.

Here \leq is the internal partial order on Ω\Omega, and :Ω×ΩΩ\wedge: \Omega \times \Omega \to \Omega is the internal meet.

This appears for instance as (MacLaneMoerdijk, V 1.).

Remark

By the definition of subobject classifier jj is equivalently a subobject

JΩ J \hookrightarrow \Omega

satisfying three conditions. This perspective gives the direct relation to Grothendieck topologies, as discussed below.

Remark

Equivalently, the third axiom in def. 1 can be replaced with the (internal) statement that jj 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 VV, tensorial strengths are the same as VV-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 Ω\Omega (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 XYX \hookrightarrow Y in EE, consider its characteristic morphism χ X:YΩ\chi_X: Y \to \Omega. Then jχ Xj \circ \chi_X is another morphism YΩY \to \Omega, which defines another subobject j *(X)j_*(X) of YY, taken as the closure of our original subobject. The elements of j *(X)j_*(X) are those elements of YY that are ‘locally’ in XX.

Definition

The closure operator induced by jj is the transformation

()¯ X:Sub(X)Sub(X) \overline{(-)}_X : Sub(X) \to Sub(X)

on the subobject lattice of XEX \in E, natural in XX, that is given by the commuting diagram

Hom(X,Ω) Sub(X) Hom(X,j) ()¯ Hom(X,Ω) Sub(X). \array{ Hom(X, \Omega) &\stackrel{\simeq}{\to}& Sub(X) \\ {}^{\mathllap{Hom(X,j)}}\downarrow && \downarrow^{\mathrlap{\overline{(-)}}} \\ Hom(X,\Omega) &\stackrel{\simeq}{\to}& Sub(X) } \,.
Remark

This means that for UXU \hookrightarrow X a subobject, with characteristic morphism charU:XΩchar U : X \to \Omega, its closure is the subobject classified by

charU¯:XcharUΩjΩ. char \overline{U} : X \stackrel{char U}{\to} \Omega \stackrel{j}{\to} \Omega \,.

This appears for instance as (MacLaneMoerdijk, p. 220).

Proposition

A morphism j:ΩΩj : \Omega \to \Omega is a Lawvere-Tierney topology, def. 1 precisely if the corresponding closure operator, def. 2 satisfies for all X,YEX, Y \in E

  1. AA¯A \subset \overline{A};

  2. A¯¯=A\overline{\overline{A}} = A;

  3. AB¯=A¯B¯\overline{A \cap B} = \overline{A} \cap \overline{B}.

This appears as (MacLaneMoerdijk, V 1., prop 1).

Sheaves

Using Lawvere–Tierney topologies, the notion of sheaf and sheafification generalizes from Grothendieck topoi to arbitrary topoi.

Let EE be a topos with Lawvere-Tierney topology jj, def. 1 and associated closure operator ()¯:Sub()Sub()\overline{(-)} : Sub(-) \to Sub(-), def. 2.

Definition

A subobject USub(X)U \in Sub(X) is called dense if U¯=X\overline{U} = X.

The corresponding monomorphism UXU \hookrightarrow X is called a dense monomorphism.

Definition

An object AEA \in E is called a jj-sheaf if it is a local object with respect to the dense monomorphisms.

This means that AA is a jj-sheaf if for every dense UXU \hookrightarrow X the induced morphism

Hom(X,F)Hom(U,F) Hom(X,F) \to Hom(U,F)

is an isomorphism.

FF is a jj-separated presheaf if this morphism is itself a monomorphism.

This is for instance in (MacLaneMoerdijk, p. 223).

Properties

jj-Sheaf subtoposes

Proposition

For EE a topos and jj a Lawvere-Tierney topology on EE, the inclusion

Sh j(E)E Sh_j(E) \hookrightarrow E

of j-sheaves is a geometric embedding.

So in particular Sh j(E)Sh_j(E) is itself a topos and the embedding is a full and faithful functor which has a left exact left adjoint functor ESh j(E)E \to Sh_j(E): this is called the sheafification functor.

This appears for instance as (MacLaneMoerdijk V 3., theorem 1).

Equivalence with Grothendieck topologies

Proposition

For CC a small category and E:=[C op,Set]E := [C^{op}, Set] its presheaf topos, Lawvere–Tierney topologies in EE are equivalent to Grothendieck topologies on CC.

Proof

The subobject classifier in a presheaf topos is the presheaf that assigns to UCU \in C the set of all sieves in CC on UU

Ω:USieves C(U). \Omega : U \mapsto Sieves_C(U) \,.

since we have

Sieves C(U)=Sub C(y(U))=hom(y(U),Ω)=Ω(U)Sieves_C (U)=Sub_C(y(U))=hom(y(U),\Omega)=\Omega(U)

A subobject JΩJ \hookrightarrow \Omega 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 CC is equivalent to the statement that the characteristic map j:ΩΩj : \Omega \to \Omega of JΩJ \hookrightarrow \Omega (see remark 1) is a Lawvere-Tierney topology.

Here is more discussion of this point:

Suppose that CC is a small site. Then given a subpresheaf inclusion FGF \hookrightarrow G in Set C op\Set^{C^\op}, an object XX of CC, and an element ff of G(X)G(X), we say ff is locally in FF (that is, fj *(F)(X)f \in j_*(F)(X)) if and only if, for some covering family c=(c i:U iX) ic = (c_i: U_i \to X)_i on XX, the restriction c *(f)c^*(f) of ff to cc is in FF (that is, each c i *(f)F(U i)c_i^*(f) \in F(U_i)). This intuitively defines the “local” modality that is the Lawvere–Tierney topology corresponding to the given Grothendieck topology on CC.

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) G:X[X,R]G: X \mapsto [X,R] on Top\Top; the constant functions form a subpresheaf FF of GG. A real-valued function f:XRf: X \to R belongs to j *(F)j_*(F) iff it is locally constant; that is, for some open cover (U i) i(U_i)_i of the domain XX, each restriction f|U if|U_i is constant.

To make this precise in terms of the above definition, we need to understand the subobject classifier in E=Set C opE = Set^{C^{op}}. But according to the definition, Ω\Omega is simply the representing object for the functor

Sub:E opSetSub: E^{op} \to Set

which takes an object FF of EE to the collection of subobjects of FF, Sub(F)Sub(F). In other words, Sub(F)hom E(F,Ω)Sub(F) \cong \hom_E(F, \Omega). Applied to F=hom C(,c)F = \hom_C(-, c), we have then

Sub(hom C(,c))hom Set C op(hom C(,c),Ω)YonedaΩ(c)Sub(\hom_C(-, c)) \cong \hom_{Set^{C^{op}}}(\hom_C(-, c), \Omega) \stackrel{Yoneda}{\cong} \Omega(c)

In other words, we find that the functor Ω:C opSet\Omega: C^{op} \to Set is defined by

Ω(c)={sievesonc}\Omega(c) = \{sieves on c\}

Next, if JJ is a Grothendieck topology on CC, then the collection of JJ-covering sieves on cc [which we denote by ] is a subcollection of all sieves on cc, and so we have an inclusion

J(c)Ω(c)J(c) \hookrightarrow \Omega(c)

and this inclusion is natural in cc, by virtue of the first axiom on covering sieves. Thus we have a subobject

JΩJ \hookrightarrow \Omega

and again, by definition of subobject classifier, this subobject corresponds to a uniquely determined element

jhom E(Ω,Ω)j \in \hom_E(\Omega, \Omega)

which is just the Lawvere–Tierney operator j:ΩΩj: \Omega \to \Omega.

Conversely, any morphism j:ΩΩj:\Omega\to\Omega determines a subobject JJ of Ω\Omega, which therefore associates to every object cc a set of sieves on cc. It is easy to check that the axioms for covering sieves in a Grothendieck topology correspond exactly to the required properties of the operator jj.

Observation

For CC a small site and jj the Lawvere-Tierney topology on the presheaf topos E=[C op,Set]E = [C^{op}, Set] 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

Sh j()L. Sh_j(\mathcal{E}) \stackrel{\overset{L}{\leftarrow}}{\underset{}{\hookrightarrow}} \mathcal{E} \,.

Here we discuss explicit translations between the structure given by the reflector LL and the corresponding Lawvere-Tierney topology j:ΩΩj : \Omega \to \Omega in a way that makes the relation to modal type theory and monads (in computer science) most manifest.

Definition

Given a reflector :LSh j()\sharp : \mathcal{E} \stackrel{L}{\to} Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}, define for each object XX \in \mathcal{E} a closure operator, being a functor on the poset of subobjects of XX

c L:Sub(X)Sub(X), c_L : Sub(X) \to Sub(X) \,,

by sending any monomorphism AXA \hookrightarrow X classified by a characteristic function χ A:XΩ\chi_A : X \to \Omega to the pullback c L(A)c_L(A) in

c L(A) A X X, \array{ c_L(A) &\to& \sharp A \\ \downarrow && \downarrow \\ X &\to& \sharp X } \,,

where XXX \to \sharp X is the adjunction unit.

Proposition

This is well defined. Moreover, c Lc_L commutes with pullback (change of base).

This appears as (Johnstone, lemma A4.3.2).

Definition

A family of functors Sub(X)Sub(X)Sub(X) \to Sub(X) for all objects XX that commutes with change of base is called a universal closure operation.

Proposition

Given a left exact reflector \sharp as above with induced closure operation c Lc_L, the corresponding Lawvere-Tierney operator j:ΩΩj : \Omega \to \Omega is given as the composite

j:ΩΩχ trueΩ, j : \Omega \to \sharp \Omega \stackrel{\chi_{\sharp true}}{\to} \Omega \,,

where

  • ΩΩ\Omega \to \sharp \Omega is the adjunction unit;

  • χ true:ΩΩ\chi_{\sharp true} : \sharp \Omega \to \Omega is the characteristic function of the result of applying \sharp to the universal subobject

    (*trueΩ):=(*trueΩ) (* \stackrel{\sharp true}{\hookrightarrow} \sharp \Omega) := \sharp (* \stackrel{true}{\hookrightarrow} \Omega)

    (which is again a monomorphism since \sharp preserves pullbacks).

Proof

For AXA \hookrightarrow X any subobject with characteristic function χ A:XΩ\chi_A : X \to \Omega, we need to show that we have a pullback diagram

c L(A) * X χ A Ω Ω Ω. \array{ c_L(A) &\to& &\to& &\to& * \\ \downarrow && && && \downarrow \\ X &\stackrel{\chi_A}{\to}& \Omega &\stackrel{}{\to}& \sharp \Omega &\stackrel{}{\to}& \Omega } \,.

The pullback along the rightmost morphism is by definition #*Ω# * \to \sharp \Omega

c L(A) #*=* * X χ A Ω Ω Ω. \array{ c_L(A) &\to& &\to& # * = * &\to& * \\ \downarrow && && \downarrow && \downarrow \\ X &\stackrel{\chi_A}{\to}& \Omega &\stackrel{}{\to}& \sharp \Omega &\stackrel{}{\to}& \Omega } \,.

Moreover, by the naturality of the adjunction unit we have a commuting diagram

X X χ A χ A Ω Ω. \array{ X &\to& \sharp X \\ {}^{\mathllap{\chi_A}}\downarrow && \downarrow^{\mathrlap{\sharp \chi_A}} \\ \Omega &\to& \sharp \Omega } \,.

Using this in the remaining bottom morphism of our would-be pullback square we find that equivalently

c L(A) #*=* * X X χ A Ω Ω \array{ c_L(A) &\to& &\to& # * = * &\to& * \\ \downarrow && && \downarrow && \downarrow \\ X &\stackrel{}{\to}& \sharp X &\stackrel{\sharp \chi_A}{\to}& \sharp \Omega &\stackrel{}{\to}& \Omega }

needs to be a pullback diagram. Since \sharp preserves pullbacks we have that also the middle square in

c L(A) A #*=* * X X χ A Ω Ω \array{ c_L(A) &\to& \sharp A &\to& # * = * &\to& * \\ \downarrow && \downarrow && \downarrow && \downarrow \\ X &\stackrel{}{\to}& \sharp X &\stackrel{\sharp \chi_A}{\to}& \sharp \Omega &\stackrel{}{\to}& \Omega }

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.

Remark

Equivalently, by the pasting law, we have that j:ΩΩj : \Omega \to \Omega is the characteristic function of the LL-closure, def. 5, of the universal subobject *Ω* \to \Omega, because we have a pasting diagram of pullback squares

c L(*) *=* * Ω Ω χ true Ω. \array{ c_L(*) &\to& \sharp * = * &\to & * \\ \downarrow && \downarrow && \downarrow \\ \Omega &\to& \sharp \Omega &\stackrel{\chi_{\sharp true}}{\to} & \Omega } \,.

In this form the statement appears in the proof of (Johnstone, Theorem A4.3.9).

Enriched generalization

  • Francis Borceux, Sheaves of algebras for a commutative theory, Ann. Soc. Sci. Bruxelles Sér. I 95 (1981), no. 1, 3–19, MR83c:18006

Let 𝒞\mathcal{C} be a small category enriched over Set TSet^{T} where TT is a commutative algebraic theory. Then [𝒞 op,Set T][\mathcal{C}^{op},\text{Set}^{T}]. A TT-sieve as an enriched subfunctor of 𝒞(,x):𝒞 opSet T\mathcal{C}(-,x)\colon\mathcal{C}^{op}\rightarrow\text{Set}^{T}. A T\mathbf{T}-topology is a set J(x)J(x) of T\mathbf{T}-sieves for every xx, 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 Ω T\Omega_{T} in [𝒞 op,Set][\mathcal{C}^{op},\text{Set}] which classifies subobjects in [𝒞 op,Set T][\mathcal{C}^{op},\text{Set}^{T}]. Moreover, there is a correspondence betwen

(1) localizations of [𝒞 op,Set T][\mathcal{C}^{\text{op}},\text{Set}^{T}]

(2) TT-topologies on 𝒞\mathcal{C}

(3) morphisms j:Ω TΩ Tj\colon\Omega_{\mathbf{T}}\rightarrow\Omega_{T} satisfying the Lawvere-Tierney axioms for a topology

References

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

Revised on November 4, 2013 12:02:58 by Urs Schreiber (77.251.114.72)