nLab closed morphism

This entry is about the general concept. For the concept in topology see at closed map.

Definition

Between topological spaces

A continuous function $f \colon X \longrightarrow Y$ between topological spaces is called a closed map if the image of every closed subset in $X$ is also closed in $Y$.

Recall that $f$ is a continuous map if the preimage of every closed set in $Y$ is closed in $X$. For defining closed maps typically one restricts attention to closed continuous maps, although it also makes sense to speak of closed functions that are not continuous.

Between locales

Definition

A map $f : X \to Y$ of locales is closed iff for any $u \in O(X)$ and $v \in O(Y)$ the reciprocity relation

$$f_*(u \vee f^*v) = f_*(u) \vee v$$

holds. (“$\geq$” is trivially always satisfied.) The map $f_* : O(X) \to O(Y)$ is the monotone right adjoint to $f^* : O(Y) \to O(X)$, explicitly given by

$$f_*(u) = \sup\{ v \in O(Y) \,|\, f^*(v) \leq u \}.$$

Between toposes

Definition

A geometric morphism $f \;\colon\; \mathcal{F} \longrightarrow \mathcal{E}$ of toposes is closed iff for any object $A \in \mathcal{E}$, the induced locale homomorphism

$$\mathrm{Sub}_{\mathcal{F}/f^*A}(1) \to \mathrm{Sub}_{\mathcal{E}}(1)$$

between the spaces of subobjects of the corresponding terminal objects is closed in the sense of def. .

Example

Let $A$ be an object of a topos $\mathcal{E}$. Then the canonical etale geometric morphism $\mathcal{E}/A \to A$ is closed iff $A$ fulfills the following condition, formulated in the internal language:

$$\mathcal{E} \models \forall U \subseteq A{:} \forall p \in \Omega{:} \quad A \subseteq (U \cup \{ x \in A \,|\, p \}) \quad\Rightarrow\quad (A \subseteq U) \vee p.$$

Note that this condition is satisfied for any $A$ whatsoever if the internal language of $\mathcal{E}$ is boolean.

Example

Let $f : A \to B$ be a morphism in a topos $\mathcal{E}$. Then the induced geometric morphism $\mathcal{E}/A \to \mathcal{E}/B$ is closed iff, in the internal language of $\mathcal{E}$, the fibers of $f$ fulfill the condition displayed at the previous example.

Remark

A geometric morphism $f : \mathcal{F} \to \mathcal{E}$ is closed iff, in the internal language of $\mathcal{E}$, the unique locale map $f_* \Omega_{\mathcal{F}} \to \mathrm{pt}$ into the one-point space (given by the frame $\Omega_{\mathcal{E}}$) is closed.

Related concepts

• open morphism, open geometric morphism

• proper morphism

• étale morphism

• separated morphism

• covert space

References

Closed maps of locales and toposes are discussed in section C3.2 of

• Peter Johnstone, Sketches of an Elephant.