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(dense,closed)-factorization

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

In point set topology, every subspace XX of a space AA has a unique largest subspace in which XX is dense, namely simply the closure X¯\overline{X}. Using maps, this amounts to say that XAX\hookrightarrow A factors as XX¯X\hookrightarrow\overline{X} followed by X¯A\overline{X}\hookrightarrow A.

The (dense,closed)-factorization generalizes this idea from topology to topos theory. It can be viewed as a way to associate to every subtopos Sh j()Sh_j(\mathcal{E})\hookrightarrow\mathcal{E} a closure Sh j()¯\overline{Sh_{j}(\mathcal{E})}.

Statement

A geometric embedding of elementary toposes

Sh j() Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}

factors as

Sh j()Sh c(ext(j))() Sh_j(\mathcal{E}) \hookrightarrow Sh_{c(ext(j))}(\mathcal{E}) \hookrightarrow \mathcal{E}

where ext(j)ext(j) (the “exterior” of jj) denotes the jj-closure of 1\emptyset \rightarrowtail 1 and

j¯c(ext(j)) \bar j \coloneqq c(ext(j))

the closed topology that corresponds to the subterminal object ext(j)ext(j).

Here the first inclusion exhibits a dense subtopos and the second a closed subtopos.

Remark

Sh c(ext(j))()Sh_{c(ext(j))}(\mathcal{E}) can be viewed as the ‘best approximation’ of Sh j()Sh_j(\mathcal{E}) by a closed subtopos and therefore might be called the closure Cl(Sh j())Cl(Sh_j(\mathcal{E})) of Sh j()Sh_j(\mathcal{E}).1

Its complement, the open subtopos Ext(Sh j())Ext(Sh_j(\mathcal{E})) corresponding to the subterminal object ext(j)ext(j) deserves in turn to be called the exterior of Sh j()Sh_j(\mathcal{E}).

The (dominant,closed)-factorization

The (dense,closed)-factorization is a special case for inclusions of a slightly more general factorization which attaches to a general geometric morphism the closure of its image.

Recall that an inclusion is dense precisely if it is a dominant geometric morphism, hence the following is pertinent for the (dense,closed)-factorization as well.

Proposition

Let i:Sh c(U)()i:Sh_{c(U)}(\mathcal{E})\hookrightarrow\mathcal{E} be dominant and a closed inclusion at the same time. Then ii is an isomorphism.

Proof: Recall that XX\in\mathcal{E} are in the closed subtopos precisely when they satisfy X×UUX\times U\cong U with UU the subterminal object associated to ii. But ii is dominant, or what comes to the same for inclusions: dense, hence \emptyset is in Sh c(U)()Sh_{c(U)}(\mathcal{E}) and therefore ×UU\emptyset\times U\cong U . From this follows UU\cong\emptyset, which in turn implies that all XX\in\mathcal{E} are in Sh c(U)()Sh_{c(U)}(\mathcal{E}) . \qed

Proposition

Let f:f:\mathcal{F}\to\mathcal{E} be a geometric morphism. Then ff factors as a dominant geometric morphism dd followed by a closed inclusion cc.

Proof: Let id 1i\circ d_1 be the surjection-inclusion factorization of ff. Since d 1d_1 is surjective, it is dominant (cf. this proposition). Then we use the (dense,closed)-factorization to factor ii into cd 2c\circ d_2. Since both d id_i are dominant, so is d:=d 2d 1d:=d_2\circ d_1 and cdc\circ d yields the demanded factorization of ff. \qed

References


  1. (SGA4, p.461) uses the term ‘l’adhérence’ for it.

Last revised on August 14, 2015 at 04:23:17. See the history of this page for a list of all contributions to it.