Contents

topos theory

# Contents

## Definition

### For maps between topological spaces

A function $f : X \to Y$ between topological spaces is called an open map if the image of every open set in $X$ is also open in $Y$.

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

#### Examples

• For any two topological spaces $X$, $Y$, the projection map $\pi \colon X \times Y \to Y$ out of the product topological space is open.

• If $G$ is a topological group and $H$ is a subgroup, then the projection to the coset space $p \colon G \to G/H$, where $G/H$ is provided with the quotient topology (making $p$ a quotient map), is open. This follows easily from the observation that if $U$ is open in $G$, then so is

$p^{-1}(p(U)) = U H = \bigcup_{h \in H} U h$
• If $p \colon A \to B$ and $q \colon C \to D$ are open maps, then their product $p \times q \colon A \times C \to B \times D$ is also an open map.

### For morphisms between locales

A continuous map $f\colon X \to Y$ of topological spaces defines a homomorphism $f^*\colon Op(Y) \to Op(X)$ between the frames of open sets of $X$ and $Y$. If $f$ is open, then this frame homomorphism is also a complete Heyting algebra homomorphism; the converse holds if $Y$ is a T-D space. Accordingly, we define a map $f\colon X \to Y$ of locales to be open if it is, as a frame homomorphism $f^*\colon Op(Y) \to Op(X)$, a complete Heyting algebra homomorphism, i.e. it preserves arbitrary meets and the Heyting implication.

This is equivalent to saying that $f^*\colon Op(Y) \to Op(X)$ has a left adjoint $f_!$ (by the adjoint functor theorem for posets) which satisfies the Frobenius reciprocity condition that $f_!(U \cap f^* V) = f_!(U) \cap V$.

In the special case that $Y$ is the one-point locale, the Frobenius reciprocity condition is automatically satisfied.

### For geometric morphisms of toposes

Categorifying, a geometric morphism $f\colon X \to Y$ of toposes is an open geometric morphism if its inverse image functor $f^*\colon Y \to X$ is a Heyting functor.

### For morphisms in a topos

A class $R \subset Mor(\mathcal{E})$ of morphisms in a topos $\mathcal{E}$ is called a class of open maps if it satisfies the following axioms.

1. Every isomorphism belongs to $R$;

2. The pullback of a morphism in $R$ belongs to $R$.

3. If the pullback of a morphism $f$ along an epimorphism lands in $R$, then $f$ is also in $R$.

4. For every set $S$ the canonical morphism $(\coprod_{s \in S} *) \to *$ from the $S$-fold coproduct of the terminal object to the terminal object is in $R$.

5. For $\{X_i \stackrel{f_i}{\to} Y_i\}_{i \in I} \subset R$ then also the coproduct $\coprod_i X_i \to \coprod_i Y_i$ is in $R$.

6. If in a diagram of the form

$\array{ Y &&\stackrel{p}{\to}&& X \\ & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}} \\ && B }$

we have that $p$ is an epimorphism and $g$ is in $R$, then $f$ is in $R$.

The class $R$ is called a class of étale maps if in addition to the axioms 1-5 above it satisfies

1. for $f : X \to Y$ in $R$ also the diagonal $Y \to Y \times_X Y$ is in $R$.

2. If in

$\array{ Y &&\stackrel{p}{\to}&& X \\ & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}} \\ && B }$

we have that $p$ is an epimorphism, and $p, g \in R$, then $f\in R$.

For instance (JoyalMoerdijk, section 1).

Application to transition systems and bisimulations