nLab (dense,closed)-factorization



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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})}.


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.


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.


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


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


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

Last revised on May 31, 2022 at 12:02:27. See the history of this page for a list of all contributions to it.