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% «.3.13._universal_closure»	(to "3.13._universal_closure")

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{\setlength{\parindent}{0em}
\footnotesize

Notes on Peter Johnstone's ``Topos Theory'' (1977).

\ssk

These notes are at:

\url{http://angg.twu.net/LATEX/2020topostheory.pdf}

See:

\url{http://angg.twu.net/LATEX/2020favorite-conventions.pdf}

\url{http://angg.twu.net/math-b.html\#favorite-conventions}


}

\bsk
\bsk



% (find-topostheorypage (+ 24  76) "3.1. Topologies")
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\def\scrE{\mathscr{E}}
\def\scrS{\mathscr{S}}
\def\scrSCop{\mathscr{S}}
\def\Shv{\mathrm{Shv}}




\section*{3. Topologies and Sheaves}

\subsection*{3.1. Topologies}

We recall that the concept of Grothendieck topos was built up in two
distinct steps: from $\scrS$, the "pre-existing" category of sets, we
passed to the category $\scrS^{\catC^\op}$ of presheaves on a small
category $\catC$, and then to the category $\Shv(\catC, J)$ of sheaves
for a Grothendieck topology $J$. In chapter 2, we saw how the first
step may be done with $\scrS$ replaced by any elementary topos
$\scrE$; our objective in the present chapter is to give a similar
generalization of the second step.

{\bf 3.11 Definition} (Lawvere-Tierney [LH]). Let be a topos. A
topology in $\scrE$ is a morphism $j:Ω→Ω$ such that the diagrams
%
%D diagram ??
%D 2Dx     100 +20 +20 +20 +50 +20
%D 2D  100 A0  A1  B0  B1  C0  C1
%D 2D
%D 2D  +20     A2      B2  C2  C3
%D 2D
%D ren A0 A1 A2 ==> Ω Ω Ω
%D ren B0 B1 B2 ==> Ω Ω Ω
%D ren C0 C1 C2 C3 ==> Ω×Ω Ω Ω×Ω Ω
%D
%D (( A0 A1 -> .plabel= a t
%D    A0 A2 -> .plabel= l t
%D    A1 A2 -> .plabel= r j
%D    
%D    B0 B1 -> .plabel= a j
%D    B0 B2 -> .plabel= l j
%D    B1 B2 -> .plabel= r j
%D    
%D    C0 C1 -> .plabel= a ∧
%D    C0 C2 -> .plabel= r j×j
%D    C1 C3 -> .plabel= r j
%D    C2 C3 -> .plabel= a ∧
%D
%D    B1 relplace 20 0 \text{and}
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$
%
commute, where $∧$ is the morphism defined in 1.49(ii). If $j$ is a
topology, we write $J \monicto Ω$ for the subobject classified by j,
and $Ω_j \monicto Ω$ for the equalizer of $j$ and $1_Ω$ (equivalently,
the image of $j$, since $j$ is idempotent).

{\bf 3.12 Example.} Let $(\catC, J)$ be a site (cf. 0.32). On
comparing 0.32(ii) with 1.12(iii), we see that $J$ is a subobject of
$Ω$ in $\scrSCop$; so it has a classifying map $Ω \ton{j} Ω$ in
$\scrSCop$. Moreover, it is not hard to prove that a sub-presheaf $J$
of satisfies 0.32(i) and (iii) iff its classifying map $j$ satisfies
the conditions of 3.11; so we have a bijection between topologies (in
the sense of 3.11) in and Grothendieck topologies on C.

Although the overwhelming advantage of the elementary definition of a
topology is its conciseness, there is another, more explicitly
descriptive, way of defining a topology which has the advantage that
it can be interpreted in categories more general than toposes.

% «3.13._universal_closure»  (to ".3.13._universal_closure")
% (tptp 2 "3.13._universal_closure")
% (tpt    "3.13._universal_closure")

{\bf 3.13 Definition.} Let $\scrE$ be any category with pullbacks. A
{\sl universal closure operation} on $\scrE$ is defined by specifying,
for each $X∈\scrE$, a closure operation (i.e. an increasing,
order-preserving, idempotent map) on the poset of subobjects of $X$
--- we denote the closure of $X' \monicto X$ by $\ovl{X'} \monicto X$
--- in such a way that closure commutes with pullback along morphisms
of $\scrE$; i.e. given $Y \ton{f} X$, we have $f^*(\ovl{X'}) =
\ovl{f^*(X')}$ as subobjects of $Y$.

We shall use the words dense and closed with their usual meanings
relative to a universal closure operation; i.e. $X' \monicto X$ is
dense if $\ovl{X'} ≅ X$, and closed if $\ovl{X'} ≅ X'$.

\msk

{\bf 3.14 Proposition.} Let be a topos. Then there is a bijection
between topologies in $\scrE$ and universal closure operations on
$\scrE$.

{\sl Proof.} Let $j$ be a topology in $\scrE$. We define the
associated ``$j$-closure'' operation as follows; if $X' \monicto X$
has classifying map $X \ton{ϕ} Ω$, then $\ovl{X'}$ is the subobject
classified by $jϕ$. The first two diagrams of 3.11 trivially imply
that this operation is increasing (i.e. $X'≤\ovl{X'}$) and idempotent;
the fact that it preserves order is an easy consequence of the third
diagram, since we have $X'≤X''$ iff $X'∩X''≅X'$; and universality is
obvious from the form of the definition.

Conversely, suppose we are given a universal closure operation on
$\scrE$. By applying it to the generic subobject $j: 1 \monicto Ω$, we
obtain a subobject $J \monicto Ω$ with classifying map $Ω \ton{j} Ω$;
and universality then says that the entire closure operation is
induced by $j$ in the above manner. To show that $j$ is a topology, we
need to show that any universal closure operation commutes with
intersection of subobjects, which will imply the third condition of
3.11; the first two are immediate.

But it is clear that $\ovl{X'}$ may be characterized as the unique
subobject of $X$ such that is $X' \monicto \ovl{X'}$ dense and is
$\ovl{X'} \monicto X$ closed. Now in the diagram
%
%D diagram ??
%D 2Dx     100 +30 +30
%D 2D  100 A0  A1
%D 2D
%D 2D  +25 A2  A3  A4
%D 2D
%D 2D  +25     A5  A6
%D 2D
%D ren A0 A1    ==>      X'        X'∩X''
%D ren A2 A3 A4 ==> \ovl{X'} \ovl{X'}∩X''            X''
%D ren    A5 A6 ==>          \ovl{X'}∩\ovl{X''} \ovl{X''}
%D
%D (( A0 A1 <-<
%D    A0 A2 >->
%D    A1 A3 >->
%D    A2 A3 <->
%D
%D    A3 A4 >->
%D    A3 A5 >->
%D    A4 A6 >->
%D    A5 A6 >->
%D ))
%D enddiagram
%D
$$\pu
  \diag{??}
$$
%
both squares are pullbacks, from which it follows easily that
%
$$X'∩X'' \monicto \ovl{X'}∩\ovl{X''}$$
%
is dense; and similarly $\ovl{X'}∩\ovl{X''} \monicto X$ is closed. So
$\ovl{X'}∩\ovl{X''} ≅ \ovl{X'∩X''}$, as required.



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