nLab (dense,closed)-factorization

Idea

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

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

Statement

A geometric embedding of elementary toposes

$$Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}$$

factors as

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

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

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

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

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

Remark

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

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

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

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

Proposition

Let $f:\mathcal{F}\to\mathcal{E}$ be a geometric morphism. Then $f$ factors as a dominant geometric morphism $d$ followed by a closed inclusion $c$.

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

Related entries

• dense

• dense subtopos

• dominant geometric morphism

• (geometric surjection, embedding) factorization system

References

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (Exposé IV 9.3.4-9.4., pp.456ff)

• Peter Johnstone, Sketches of an Elephant vol.I , Oxford UP 2002. (around Lemma A 4.5.19, p. 219)

• Olivia Caramello, Lattices of theories , arXiv:0905.0299 (2009). (section 8)

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