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\begin{document}

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\section*{Lawvere-Tierney topology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

\hypertarget{lawveretierney_topologies}{}\section*{{Lawvere--Tierney topologies}}\label{lawveretierney_topologies}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_closure_operator}{The closure operator}\dotfill \pageref*{the_closure_operator} \linebreak
\noindent\hyperlink{sheaves}{Sheaves}\dotfill \pageref*{sheaves} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SheafSubtoposes}{$j$-Sheaf subtoposes}\dotfill \pageref*{SheafSubtoposes} \linebreak
\noindent\hyperlink{RelationToGrothendieckTopology}{Equivalence with Grothendieck topologies}\dotfill \pageref*{RelationToGrothendieckTopology} \linebreak
\noindent\hyperlink{ClosureOperation}{Relation to lex reflectors}\dotfill \pageref*{ClosureOperation} \linebreak
\noindent\hyperlink{enriched_generalization}{Enriched generalization}\dotfill \pageref*{enriched_generalization} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{Lawvere--Tierney topology} (or \emph{operator} , or \emph{[[modal logic|modality]]} , also called \emph{\href{modal+type+theory#GeometricModality}{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 \emph{[[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 [[coverage|(Grothendieck) coverage]]; see [[Grothendieck topology]] for some possible replacements for ``Lawvere--Tierney topology.''

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\hypertarget{the_closure_operator}{}\subsubsection*{{The closure operator}}\label{the_closure_operator}

\begin{defn}
\label{LTTopologyDef}\hypertarget{LTTopologyDef}{}
A \textbf{Lawvere--Tierney topology} in $E$ is ([[internalization|internally]]) a [[closure operator]] given by a [[exact functor|left exact]] [[idempotent monad]] on the internal meet-[[semilattice]] $\Omega$.

This means that: a Lawvere--Tierney topology in $E$ is a [[morphism]]

\begin{displaymath}
j: \Omega \to \Omega
\end{displaymath}
such that

\begin{enumerate}%
\item $j true = true$, equivalently $\id_\Omega \leq j: \Omega \to \Omega$ (`if $p$ is true, then $p$ is locally true')

\begin{displaymath}
\itexarray{
    * &\stackrel{true}{\to}& \Omega
    \\
    & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}}
    \\
    && \Omega
  }
\end{displaymath}

\item $j j = j$ (`$p$ is locally locally true iff $p$ is locally true');

\begin{displaymath}
\itexarray{
    \Omega &\stackrel{j}{\to}& \Omega
    \\
    & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}}
    \\
    && \Omega
  }
\end{displaymath}

\item $j \circ \wedge = \wedge \circ (j \times j)$ (`$p \wedge q$ is locally true iff $p$ and $q$ are each locally true')

\begin{displaymath}
\itexarray{
    \Omega \times \Omega &\stackrel{\wedge}{\to}& \Omega
    \\
    {}^{\mathllap{j \times j}}\downarrow && \downarrow^{\mathrlap{j}}
    \\
    \Omega \times \Omega &\underset{\wedge}{\to}& \Omega
  }
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
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 (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk, V 1.}).

\begin{remark}
\label{LTTopologyAsSubobject}\hypertarget{LTTopologyAsSubobject}{}
By the definition of [[subobject classifier]] $j$ is equivalently a [[subobject]]

\begin{displaymath}
J \hookrightarrow \Omega
\end{displaymath}
satisfying three conditions. This perspective gives the direct relation to [[Grothendieck topologies]], as discussed \href{RelationToGrothendieckTopology}{below}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Equivalently, the third axiom in def. \ref{LTTopologyDef} 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 monad|strong]] (a special case of the fact that, for an [[endofunctor]] on some [[closed monoidal category|monoidal closed]] $V$, [[tensorial strength|tensorial strengths]] are the same as $V$-[[enriched category|enrichments]], as described in the article on the former), and therefore \emph{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 $X$.

\end{remark}
\begin{defn}
\label{TheClosureOperator}\hypertarget{TheClosureOperator}{}
The \textbf{[[closure operator]]} induced by $j$ is the [[natural transformation|transformation]]

\begin{displaymath}
\overline{(-)}_X : Sub(X) \to Sub(X)
\end{displaymath}
on the [[subobject]] lattice of $X \in E$, [[natural transformation|natural]] in $X$, that is given by the [[commuting diagram]]

\begin{displaymath}
\itexarray{
    Hom(X, \Omega) &\stackrel{\simeq}{\to}& Sub(X)
    \\
    {}^{\mathllap{Hom(1,j)}}\downarrow && \downarrow^{\mathrlap{\overline{(-)}}}
    \\
    Hom(X,\Omega) &\stackrel{\simeq}{\to}& Sub(X)
  }
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This means that for $U \hookrightarrow X$ a [[subobject]], with characteristic morphism $char U : X \to \Omega$, its closure is the subobject classified by

\begin{displaymath}
char \overline{U} : X \stackrel{char U}{\to} \Omega \stackrel{j}{\to} \Omega
  \,.
\end{displaymath}
\end{remark}
This appears for instance as (\href{
MacLaneMoerdijk}{MacLaneMoerdijk, p. 220}).

\begin{prop}
\label{}\hypertarget{}{}
A morphism $j : \Omega \to \Omega$ is a Lawvere-Tierney topology, def. \ref{LTTopologyDef} precisely if the corresponding closure operator, def. \ref{TheClosureOperator} satisfies for all $X, Y \in E$

\begin{enumerate}%
\item $A \subset \overline{A}$;


\item $\overline{\overline{A}} = \overline{A}$;


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



\end{enumerate}
\end{prop}
This appears as (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk, V 1., prop 1}).

\hypertarget{sheaves}{}\subsubsection*{{Sheaves}}\label{sheaves}

Using Lawvere--Tierney topologies, the notion of [[sheaf]] and [[sheafification]] generalizes from [[Grothendieck topos|Grothendieck topoi]] to arbitrary topoi.

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

\begin{defn}
\label{DenseMonos}\hypertarget{DenseMonos}{}
A subobject $U \in Sub(X)$ is called \textbf{dense} if $\overline{U} = X$.

The corresponding [[monomorphism]] $U \hookrightarrow X$ is called a \textbf{[[dense monomorphism]]}.

\end{defn}
\begin{defn}
\label{JSheaf}\hypertarget{JSheaf}{}
An object $F \in E$ is called a \textbf{$j$-[[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

\begin{displaymath}
Hom(X,F) \to Hom(U,F)
\end{displaymath}
is an [[isomorphism]].

$F$ is a \textbf{$j$-[[separated presheaf]]} if this morphism is itself a [[monomorphism]].

\end{defn}
This is for instance in (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk, p. 223}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{SheafSubtoposes}{}\subsubsection*{{$j$-Sheaf subtoposes}}\label{SheafSubtoposes}

\begin{prop}
\label{RelationToCoveringSieves}\hypertarget{RelationToCoveringSieves}{}
For $E$ a [[topos]] and $j$ a Lawvere-Tierney topology on $E$, the inclusion

\begin{displaymath}
Sh_j(E) \hookrightarrow E
\end{displaymath}
of \hyperlink{JSheaf}{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 [[exact functor|left exact]] [[left adjoint]] functor $E \to Sh_j(E)$: this is called the \textbf{[[sheafification]]} functor.

\end{prop}
This appears for instance as (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk V 3., theorem 1}).

\hypertarget{RelationToGrothendieckTopology}{}\subsubsection*{{Equivalence with Grothendieck topologies}}\label{RelationToGrothendieckTopology}

\begin{prop}
\label{RelationToCoveringSieves}\hypertarget{RelationToCoveringSieves}{}
For $C$ a [[small category]] and $E := [C^{op}, Set]$ its [[presheaf topos]], Lawvere--Tierney topologies in $E$ are equivalent to [[Grothendieck topology|Grothendieck topologies]] on $C$.

\end{prop}
\begin{proof}
The [[subobject classifier]] in a [[presheaf topos]] is the presheaf that assigns to $U \in C$ the set of all [[sieve]]s in $C$ on $U$

\begin{displaymath}
\Omega : U \mapsto Sieves_C(U)
  \,.
\end{displaymath}
since we have

\begin{displaymath}
Sieves_C (U)=Sub_C(y(U))=hom(y(U),\Omega)=\Omega(U)
\end{displaymath}
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 \ref{LTTopologyAsSubobject}) is a Lawvere-Tierney topology.

\end{proof}
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 [[cover|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 \emph{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

\begin{displaymath}
Sub: E^{op} \to Set
\end{displaymath}
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

\begin{displaymath}
Sub(\hom_C(-, c)) \cong \hom_{Set^{C^{op}}}(\hom_C(-, c), \Omega) \stackrel{Yoneda}{\cong} \Omega(c)
\end{displaymath}
In other words, we find that the functor $\Omega: C^{op} \to Set$ is defined by

\begin{displaymath}
\Omega(c) = \{sieves\,on\,c\}
\end{displaymath}
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

\begin{displaymath}
J(c) \hookrightarrow \Omega(c)
\end{displaymath}
and this inclusion is natural in $c$, by virtue of the first axiom on covering sieves. Thus we have a subobject

\begin{displaymath}
J \hookrightarrow \Omega
\end{displaymath}
and again, by definition of subobject classifier, this subobject corresponds to a uniquely determined element

\begin{displaymath}
j \in \hom_E(\Omega, \Omega)
\end{displaymath}
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 $j$.

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a [[small site]] and $j$ the Lawvere-Tierney topology on the [[presheaf topos]] $E = [C^{op}, Set]$ given by prop. \ref{RelationToCoveringSieves} the \hyperlink{SheafSubtoposes}{j-sheaves} are precisely the [[sheaves]] in the ordinary sense of Grothendieck topologies.

\end{prop}
\hypertarget{ClosureOperation}{}\subsubsection*{{Relation to lex reflectors}}\label{ClosureOperation}

As discussed there, [[categories of sheaves]] are also characterized as being [[reflective subcategories]] of the given ambient topos

\begin{displaymath}
Sh_j(\mathcal{E})
   \stackrel{\overset{L}{\leftarrow}}{\underset{}{\hookrightarrow}}
  \mathcal{E}
 \,.
\end{displaymath}
Here we discuss explicit translations between the structure given by the [[localization|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.

\begin{defn}
\label{ClosureOperatorOfReflection}\hypertarget{ClosureOperatorOfReflection}{}
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 \textbf{closure operator}, being a [[functor]] on the [[poset of subobjects]] of $X$

\begin{displaymath}
c_L : Sub(X) \to Sub(X)
  \,,
\end{displaymath}
by sending any [[monomorphism]] $A \hookrightarrow X$ classified by a [[characteristic function]] $\chi_A : X \to \Omega$ to the [[pullback]] $c_L(A)$ in

\begin{displaymath}
\itexarray{
    c_L(A) &\to& \sharp A
    \\
    \downarrow && \downarrow
    \\
    X &\to& \sharp X
  }
  \,,
\end{displaymath}
where $X \to \sharp X$ is the [[unit of an adjunction|adjunction unit]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
This is well defined. Moreover, $c_L$ commutes with [[pullback]] ([[change of base]]).

\end{prop}
This appears as (\hyperlink{Johnstone}{Johnstone, lemma A4.3.2}).

\begin{defn}
\label{}\hypertarget{}{}
A family of functors $Sub(X) \to Sub(X)$ for all objects $X$ that commutes with [[change of base]] is called a \textbf{[[universal closure operation]]}.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Given a [[left exact functor|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

\begin{displaymath}
j : \Omega \to \sharp \Omega \stackrel{\chi_{\sharp true}}{\to} \Omega  
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $\Omega \to \sharp \Omega$ is the [[unit of an adjunction|adjunction unit]];


\item $\chi_{\sharp true} : \sharp \Omega \to \Omega$ is the [[characteristic function]] of the result of applying $\sharp$ to the [[subobject classifier|universal subobject]]

\begin{displaymath}
(* \stackrel{\sharp true}{\hookrightarrow} \sharp \Omega)
   :=
  \sharp (* \stackrel{true}{\hookrightarrow} \Omega)
\end{displaymath}
(which is again a [[monomorphism]] since $\sharp$ preserves [[pullbacks]]).



\end{itemize}
\end{prop}
\begin{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]]

\begin{displaymath}
\itexarray{
    c_L(A) &\to& &\to& &\to& *
    \\
    \downarrow && && && \downarrow
    \\
    X &\stackrel{\chi_A}{\to}& \Omega
    &\stackrel{}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
  \,.
\end{displaymath}
The pullback along the rightmost morphism is by definition $# * \to \sharp \Omega$

\begin{displaymath}
\itexarray{
    c_L(A) &\to& &\to& # * = * &\to& *
    \\
    \downarrow && && \downarrow && \downarrow
    \\
    X &\stackrel{\chi_A}{\to}& \Omega
    &\stackrel{}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
  \,.
\end{displaymath}
Moreover, by the [[natural transformation|naturality]] of the [[unit of an adjunction|adjunction unit]] we have a [[commuting diagram]]

\begin{displaymath}
\itexarray{
    X &\to& \sharp X
    \\
    {}^{\mathllap{\chi_A}}\downarrow && \downarrow^{\mathrlap{\sharp \chi_A}}
    \\
    \Omega &\to& \sharp \Omega
  }
  \,.
\end{displaymath}
Using this in the remaining bottom morphism of our would-be pullback square we find that equivalently

\begin{displaymath}
\itexarray{
    c_L(A) &\to& &\to& # * = * &\to& *
    \\
    \downarrow && && \downarrow && \downarrow
    \\
    X &\stackrel{}{\to}& \sharp X
    &\stackrel{\sharp \chi_A}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
\end{displaymath}
needs to be a pullback diagram. Since $\sharp$ preserves pullbacks we have that also the middle square in

\begin{displaymath}
\itexarray{
    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
  }
\end{displaymath}
is a pullback. But then also the left square is a pullback, by def. \ref{ClosureOperatorOfReflection}. This finally means, by the [[pasting law]], that also the total rectangle is a pullback.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Equivalently, by the [[pasting law]], we have that $j : \Omega \to \Omega$ is the [[characteristic function]] of the $L$-closure, def. \ref{ClosureOperatorOfReflection}, of the universal subobject $* \to \Omega$, because we have a [[pasting diagram]] of [[pullback]] squares

\begin{displaymath}
\itexarray{
    c_L(*) &\to& \sharp * = * &\to & * 
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \Omega &\to& \sharp \Omega &\stackrel{\chi_{\sharp true}}{\to} & \Omega
  }
  \,.
\end{displaymath}
In this form the statement appears in the proof of (\hyperlink{Johnstone}{Johnstone, Theorem A4.3.9}).

\end{remark}
\hypertarget{enriched_generalization}{}\subsection*{{Enriched generalization}}\label{enriched_generalization}

\begin{itemize}%
\item [[Francis Borceux]], \emph{Algebraic localizations and elementary toposes} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{21} no.4 (1980) pp.393-401.(\href{http://archive.numdam.org/article/CTGDC_1980__21_4_393_0.pdf}{pdf})


\item Francis Borceux, \emph{Sheaves of algebras for a commutative theory}, Ann. Soc. Sci. Bruxelles S\'e{}r. I \textbf{95} (1981), no. 1, 3--19, \href{http://www.ams.org/mathscinet-getitem?mr=628032}{MR83c:18006}



\end{itemize}
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

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[modality]] [[modal type theory]]


\item [[monad (in programming theory)]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

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

\begin{itemize}%
\item [[William Lawvere]], \emph{Quantifiers and sheaves}, Actes, Congr\`e{}s intern, math., 1970. Tome 1, p. 329 \`a{} 334 (\href{http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0329.0334.ocr.pdf}{pdf})

\end{itemize}
Detailed discussion of Lawvere-Tierney operators as [[geometric modalities]] is in

\begin{itemize}%
\item [[Robert Goldblatt]], \emph{Grothendieck topology as geometric modality}, Mathematical Logic Quarterly, Volume 27, Issue 31-35, pages 495--529, (1981)

\end{itemize}
Textbook accounts include section V.1 of

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]},

\end{itemize}
(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

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
Discussion in [[homotopy type theory]] is in

\begin{itemize}%
\item Kevin Quirin, Nicolas Tabareau, \emph{Lawvere-Tierney sheafification in Homotopy Type Theory}, Journal of Formalized Reasoning, Vol 9, No 2, (2016) (\href{https://jfr.unibo.it/article/view/6232}{web})

\end{itemize}
[[!redirects sheafification in a Lawvere-Tierney topos]] [[!redirects sheafification in a Lawvere–Tierney topos]] [[!redirects sheafification in a Lawvere--Tierney topos]] [[!redirects Lawvere-Tierney topology]] [[!redirects Lawvere–Tierney topology]] [[!redirects Lawvere--Tierney topology]] [[!redirects Lawvere-Tierney topologies]] [[!redirects Lawvere–Tierney topologies]] [[!redirects Lawvere--Tierney topologies]]

[[!redirects Lawvere-Tierney operator]] [[!redirects Lawvere–Tierney operator]] [[!redirects Lawvere--Tierney operator]] [[!redirects Lawvere-Tierney operators]] [[!redirects Lawvere–Tierney operators]] [[!redirects Lawvere--Tierney operators]]

[[!redirects local modality]] [[!redirects local modalities]]



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