nLab Lawvere-Tierney topology

Idea

A Lawvere–Tierney topology (or (local) 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 $C$ is a small category, then choosing a Grothendieck topology on $C$ is equivalent to choosing a Lawvere–Tierney topology in the presheaf topos $\Set^{C^\op}$ on $C$.

The use of “topology” for this and the related Grothendieck concept is regarded by some people as unfortunate; see the historical note at Grothendieck topology for some reasons why. A proposed replacement for “Grothendieck topology” is (Grothendieck) coverage; see the afore-mentioned historical note for some possible replacements for “Lawvere–Tierney topology.”

Definition

Let $E$ be a topos, with subobject classifier $\Omega$.

The closure operator

Definition

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

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

$$j: \Omega \to \Omega$$

such that

1. $j true = true$, equivalently $\id_\Omega \leq j: \Omega \to \Omega$ (‘if $p$ is true, then $p$ is locally true’)

   $$\array{ * &\stackrel{true}{\to}& \Omega \\ & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }$$

2. $j j = j$ (’$p$ is locally locally true iff $p$ is locally true’);

   $$\array{ \Omega &\stackrel{j}{\to}& \Omega \\ & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }$$

3. $j \circ \wedge = \wedge \circ (j \times j)$ (’$p \wedge q$ is locally true iff $p$ and $q$ are each locally true’)

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

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

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

Remark

By the definition of subobject classifier $j$ is equivalently a subobject

$$J \hookrightarrow \Omega$$

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

Remark

Equivalently, the third axiom in def. can be replaced with the (internal) statement that $j$ 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 $V$, tensorial strengths are the same as $V$-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 $\Omega$ (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 $X \hookrightarrow Y$ in $E$, consider its characteristic morphism $\chi_X: Y \to \Omega$. Then $j \circ \chi_X$ is another morphism $Y \to \Omega$, which defines another subobject $j_*(X)$ of $Y$, taken as the closure of our original subobject. The elements of $j_*(X)$ are those elements of $Y$ that are ‘locally’ in $X$.

Definition

The closure operator induced by $j$ is the transformation

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

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

$$\array{ Hom(X, \Omega) &\stackrel{\simeq}{\to}& Sub(X) \\ {}^{\mathllap{Hom(1,j)}}\downarrow && \downarrow^{\mathrlap{\overline{(-)}}} \\ Hom(X,\Omega) &\stackrel{\simeq}{\to}& Sub(X) } \,.$$

Remark

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

$$char \overline{U} : X \stackrel{char U}{\to} \Omega \stackrel{j}{\to} \Omega \,.$$

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

Proposition

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

1. $A \subset \overline{A}$;

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

3. $\overline{A \cap B} = \overline{A} \cap \overline{B}$.

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 $E$ be a topos with Lawvere-Tierney topology $j$, def. and associated closure operator $\overline{(-)} : Sub(-) \to Sub(-)$, def. .

Definition

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

The corresponding monomorphism $U \hookrightarrow X$ is called a dense monomorphism.

Definition

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

This means that $F$ is a $j$-sheaf if for every dense $U \hookrightarrow X$ the induced morphism

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

is an isomorphism.

$F$ is a $j$-separated presheaf if this morphism is itself a monomorphism.

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

Properties

$j$-Sheaf subtoposes

Proposition

For $E$ a topos and $j$ a Lawvere-Tierney topology on $E$, the inclusion

$$Sh_j(E) \hookrightarrow E$$

of j-sheaves is a geometric embedding.

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

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

Equivalence with Grothendieck topologies

Proposition

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

Proof

The subobject classifier in a presheaf topos is the presheaf that assigns to $U \in C$ the set of all sieves in $C$ on $U$

$$\Omega : U \mapsto Sieves_C(U) \,.$$

since we have

$$Sieves_C (U)=Sub_E(y(U))=hom(y(U),\Omega)=\Omega(U)$$

A subobject $J \hookrightarrow \Omega$ 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 $C$ is equivalent to the statement that the characteristic map $j : \Omega \to \Omega$ of $J \hookrightarrow \Omega$ (see remark ) is a Lawvere-Tierney topology.

Here is more discussion of this point:

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

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

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

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

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

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

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

$$\Omega(c) = \{sieves \, on \, c\}$$

Next, if $J$ is a Grothendieck topology on $C$, then the collection of $J$-covering sieves on $c$ (which we denote by $J(c)$( is a subcollection of all sieves on $c$, and so we have an inclusion

$$J(c) \hookrightarrow \Omega(c)$$

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

$$J \hookrightarrow \Omega$$

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

$$j \in \hom_E(\Omega, \Omega)$$

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

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

Observation

For $C$ a small site and $j$ the Lawvere-Tierney topology on the presheaf topos $E = [C^{op}, Set]$ given by prop. 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

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

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

Definition

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

$$c_L : Sub(X) \to Sub(X) \,,$$

by sending any monomorphism $A \hookrightarrow X$ classified by a characteristic function $\chi_A : X \to \Omega$ to the pullback $c_L(A)$ in

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

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

Proposition

This is well defined. Moreover, $c_L$ commutes with pullback (change of base).

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

Definition

A family of functors $Sub(X) \to Sub(X)$ for all objects $X$ 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 $c_L$, the corresponding Lawvere-Tierney operator $j : \Omega \to \Omega$ is given as the composite

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

where

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

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

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

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

Proof

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

$$\array{ c_L(A) &\to& &\to& &\to& * \\ \downarrow && && && \downarrow \\ X &\stackrel{\chi_A}{\to}& \Omega &\stackrel{}{\to}& \sharp \Omega &\stackrel{}{\to}& \Omega } \,.$$

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

$$\array{ c_L(A) &\to& &\to& \sharp * = * &\to& * \\ \downarrow && && \downarrow && \downarrow \\ X &\stackrel{\chi_A}{\to}& \Omega &\stackrel{}{\to}& \sharp \Omega &\stackrel{}{\to}& \Omega } \,.$$

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

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

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

$$\array{ c_L(A) &\to& &\to& \sharp * = * &\to& * \\ \downarrow && && \downarrow && \downarrow \\ X &\stackrel{}{\to}& \sharp X &\stackrel{\sharp \chi_A}{\to}& \sharp \Omega &\stackrel{}{\to}& \Omega }$$

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

$$\array{ c_L(A) &\to& \sharp A &\to& \sharp * = * &\to& * \\ \downarrow && \downarrow && \downarrow && \downarrow \\ X &\stackrel{}{\to}& \sharp X &\stackrel{\sharp \chi_A}{\to}& \sharp \Omega &\stackrel{}{\to}& \Omega }$$

is a pullback. But then also the left square is a pullback, by def. . This finally means, by the pasting law, that also the total rectangle is a pullback.

Remark

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

$$\array{ c_L(*) &\to& \sharp * = * &\to & * \\ \downarrow && \downarrow && \downarrow \\ \Omega &\to& \sharp \Omega &\stackrel{\chi_{\sharp true}}{\to} & \Omega } \,.$$

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

Enriched generalization

• Francis Borceux, Algebraic localizations and elementary toposes, Cah. Top. Géom. Diff. Cat. 21 (1980), no. 4, 393–401. (MR82g:18002, pdf)

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

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

1. localizations of $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$

2. $\mathbf{T}$-topologies on $\mathcal{C}$

3. morphisms $j\colon\Omega_{\mathbf{T}}\rightarrow\Omega_{\mathbf{T}}$ satisfying the Lawvere-Tierney axioms for a topology

Related concepts

• modality

• modal type theory

• monad (in programming theory)

References

The notion is introduced as a geometric modality on p. 3 of

• William 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

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic,

(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

• Peter Johnstone, Sketches of an Elephant

Discussion in homotopy type theory is in

• Kevin Quirin, Nicolas Tabareau, Lawvere-Tierney sheafification in Homotopy Type Theory, Journal of Formalized Reasoning, Vol 9, No 2, (2016) (web)

The additive version of Lawvere-Tierney modular closure operators on categories of modules (as an additive version of a presheaf topos) – equivalent to Gabriel topology (=$Ab$-enriched Grothendieck topology) has been studied (and the equivalence proven) several years before Lawvere and Tierney in

• Kim Lin Chew, Closure operations in the study of rings of quotients, Bull. Math. Soc. Nanyang Univ. (1965) 1-20.