nLab
Lawvere-Tierney topology

Idea

A Lawvere–Tierney topology (or operator, or modality, also called geometric modality) 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 topos ${Set}^{C^{op}}$ of presheaves on $C$.

The use of “topology” for this and the related Grothendieck concept is regarded by some people as unfortunate; see 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 $E$ be a topos, with subobject classifier $\Omega$. A Lawvere–Tierney topology in $E$ is a map $j:\Omega \to \Omega$ that satisfies certain axioms.

The axioms say that $j$ is (internally) a left exact monad on the internal meet-semilattice $\Omega$. To be explicit:

• ${id}_{\Omega }\le j:\Omega \to \Omega$ (‘if $p$ is true, then $p$ is locally true’);
• $j\circ j=j:\Omega \to \Omega$ (‘$p$ is locally locally true iff $p$ is locally true’);
• $j\circ \wedge =\wedge \circ j\times j:\Omega \times \Omega \to \Omega$ (‘$p\wedge q$ is locally true iff $p$ and $q$ are each locally true’).

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

Equivalently, the third axiom above 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 $X$.

Equivalence with Grothendieck topologies

As mentioned above, a Lawvere–Tierney topology on ${Set}^{C^{op}}$ is equivalent to a Grothendieck topology on $C$. 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 $f\mid U_i$ is constant.

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

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

which takes an object $F$ of $E$ to the collection of subobjects of $F$, $\mathrm{Sub}(F)$. In other words, $\mathrm{Sub}(F)\cong {\mathrm{hom}}_E(F,\Omega )$. Applied to $F={\mathrm{hom}}_C(-,c)$, we have then

$$\mathrm{Sub}({\mathrm{hom}}_C(-,c))\cong {\mathrm{hom}}_{{\mathrm{Set}}^{C^{\mathrm{op}}}}({\mathrm{hom}}_C(-,c),\Omega )\stackrel{\mathrm{Yoneda}}{\cong }\Omega (c)$$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 $\Omega :C^{\mathrm{op}}\to \mathrm{Set}$ is defined by

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

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

$$J\hookrightarrow \Omega$$J \hookrightarrow \Omega

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

$$j\in {\mathrm{hom}}_E(\Omega ,\Omega )$$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 $j$.

Sheaves

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

Technically this is achieved essentially by replaxing local isomorphisms everywhere with dense monomorphisms.

If $E=\mathrm{PSh}(S)$ is a presheaf Grothendieck topos for a site $S$ regarded as a topos with Lawvere–Tierney topology and hence equipped with a notion of dense monomorphisms, then a presheaf $F\in \mathrm{PSh}(S)$ is a sheaf with respect to the given topology precisely if

$${\mathrm{Hom}}_{\mathrm{PSh}(S)}(-,F):\mathrm{PSh}(S{)}^{\mathrm{op}}\to \mathrm{Set}$$Hom_{PSh(S)}(-, F): PSh(S)^{op} \to Set

sends all dense monomorphisms to isomorphisms.

This condition clearly makes sense for every topos with Lawvere–Tierney topology.

Sheafification

By using dense monomorphisms in place of local isomorphisms, this induces a notion of sheafification on an arbitrary topos $E$ with a Lawvere–Tierney topology.

This sheafification is a functor

$$\overline{(-)}:E\to \mathrm{Sh}(E)$$\bar {(-)} : E \to Sh(E)

to the subcategory $i:\mathrm{Sh}(E)\hookrightarrow E$ of objects local with respect to dense monomorphisms which is

• left exact (commutes with all small limits);

• left adjoint to $i$.

In the case that $E=\mathrm{PSh}(S)$ and the Lawvere–Tierney topology is that corresponding to a Grothendieck topology on $S$, the two notions of sheafification coincide.

References

Lawvere–Tierney topologies are discussed in section V.1 of

• MacLane, 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.