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 $C$ 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 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 $E$ be a topos, with subobject classifier $\Omega$.
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
such that
$j true = true$, equivalently $\id_\Omega \leq j: \Omega \to \Omega$ (‘if $p$ is true, then $p$ is locally true’)
$j j = j$ (‘$p$ is locally locally true iff $p$ is locally true’);
$j \circ \wedge = \wedge \circ (j \times j)$ (‘$p \wedge q$ is locally true iff $p$ and $q$ are each locally true’)
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.).
By the definition of subobject classifier $j$ is equivalently a subobject
satisfying three conditions. This perspective gives the direct relation to Grothendieck topologies, as discussed below.
Equivalently, the third axiom in def. 1 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 $X$.
The closure operator induced by $j$ is the transformation
on the subobject lattice of $X \in E$, natural in $X$, that is given by the commuting diagram
This means that for $U \hookrightarrow X$ a subobject, with characteristic morphism $char U : X \to \Omega$, its closure is the subobject classified by
This appears for instance as (MacLaneMoerdijk, p. 220).
A morphism $j : \Omega \to \Omega$ is a Lawvere-Tierney topology, def. 1 precisely if the corresponding closure operator, def. 2 satisfies for all $X, Y \in E$
$A \subset \overline{A}$;
$\overline{\overline{A}} = \overline{A}$;
$\overline{A \cap B} = \overline{A} \cap \overline{B}$.
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 $E$ be a topos with Lawvere-Tierney topology $j$, def. 1 and associated closure operator $\overline{(-)} : Sub(-) \to Sub(-)$, def. 2.
A subobject $U \in Sub(X)$ is called dense if $\overline{U} = X$.
The corresponding monomorphism $U \hookrightarrow X$ is called a dense monomorphism.
An object $A \in E$ is called a $j$-sheaf if it is a local object with respect to the dense monomorphisms.
This means that $A$ is a $j$-sheaf if for every dense $U \hookrightarrow A$ the induced morphism
is an isomorphism.
$F$ is a $j$-separated presheaf if this morphism is itself a monomorphism.
This is for instance in (MacLaneMoerdijk, p. 223).
For $E$ a topos and $j$ a Lawvere-Tierney topology on $E$, the inclusion
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).
For $C$ a small category and $E := [C^{op}, Set]$ its presheaf topos, Lawvere–Tierney topologies in $E$ are equivalent to Grothendieck topologies on $C$.
The subobject classifier in a presheaf topos is the presheaf that assigns to $U \in C$ the set of all sieves in $C$ on $U$
since we have
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 1) 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
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
In other words, we find that the functor $\Omega: C^{op} \to Set$ is defined by
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
and this inclusion is natural in $c$, 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 $j: \Omega \to \Omega$.
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 $j$.
For $C$ a small site and $j$ the Lawvere-Tierney topology on the presheaf topos $E = [C^{op}, Set]$ given by prop. 3 the j-sheaves are precisely the sheaves in the ordinary sense of Grothendieck topologies.
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 $L$ 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.
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 $X$
by sending any monomorphism $A \hookrightarrow X$ classified by a characteristic function $\chi_A : X \to \Omega$ to the pullback $c_L(A)$ in
where $X \to \sharp X$ is the adjunction unit.
This is well defined. Moreover, $c_L$ commutes with pullback (change of base).
This appears as (Johnstone, lemma A4.3.2).
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.
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
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
(which is again a monomorphism since $\sharp$ preserves pullbacks).
For $A \hookrightarrow X$ any subobject with characteristic function $\chi_A : X \to \Omega$, we need to show that we have a pullback diagram
The pullback along the rightmost morphism is by definition $# * \to \sharp \Omega$
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 $\sharp$ 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.
Equivalently, by the pasting law, we have that $j : \Omega \to \Omega$ is the characteristic function of the $L$-closure, def. 5, of the universal subobject $* \to \Omega$, because we have a pasting diagram of pullback squares
In this form the statement appears in the proof of (Johnstone, Theorem A4.3.9).
Francis Borceux, Algebraic localizations and elementary toposes , Cah. Top. Géom. Diff. Cat. 21 no.4 (1980) pp.393-401.(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 $Set^{T}$ where $T$ is a commutative algebraic theory. Then $[\mathcal{C}^{op},\text{Set}^{T}]$. A $T$-sieve as an enriched subfunctor of $\mathcal{C}(-,x)\colon\mathcal{C}^{op}\rightarrow\text{Set}^{T}$. A $\mathbf{T}$-topology is a set $J(x)$ of $\mathbf{T}$-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 $\Omega_{T}$ in $[\mathcal{C}^{op},\text{Set}]$ which classifies subobjects in $[\mathcal{C}^{op},\text{Set}^{T}]$. Moreover, there is a correspondence betwen
(1) localizations of $[\mathcal{C}^{\text{op}},\text{Set}^{T}]$
(2) $T$-topologies on $\mathcal{C}$
(3) morphisms $j\colon\Omega_{\mathbf{T}}\rightarrow\Omega_{T}$ 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
Discussion in homotopy type theory is in