nLab
open morphism

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Topology

Topos Theory

topos theory

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Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Definition

For maps between topological spaces

A function f:XYf : X \to Y between topological spaces is called open if the image of every open set in XX is also open in YY.

Recall that ff is a continuous map if the preimage of every open set in YY is open in XX. 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 XX, YY, the projection map π:X×YY\pi \colon X \times Y \to Y is open.

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

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

For morphisms between locales

A continuous map f:XYf\colon X \to Y of topological spaces defines a homomorphism f *:Op(Y)Op(X)f^*\colon Op(Y) \to Op(X) between the frames of open sets of XX and YY. If ff is open, then this frame homomorphism is also a complete Heyting algebra homomorphism; the converse holds for sober spaces (maybe as long as YY is T 0T_0?). Accordingly, we define a map f:XYf\colon X \to Y of locales to be open if it is, as a frame homomorphism f *:Op(Y)Op(X)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 *:Op(Y)Op(X)f^*\colon Op(Y) \to Op(X) has a left adjoint f !f_! (by the adjoint functor theorem for posets) which satisfies the Frobenius reciprocity condition that f !(Uf *V)=f !(U)Vf_!(U \cap f^* V) = f_!(U) \cap V.

For geometric morphisms of toposes

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

For morphisms in a topos

A class RMor()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 RR;

  2. The pullback of a morphism in RR belongs to RR.

  3. If the pullback of a morphism ff along an epimorphism lands in RR, then ff is also in RR.

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

  5. For {X if iY i} iIR\{X_i \stackrel{f_i}{\to} Y_i\}_{i \in I} \subset R then also the coproduct iX i iY i\coprod_i X_i \to \coprod_i Y_i is in RR.

  6. If in a diagram of the form

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

    we have that pp is an epimorphism and gg is in RR, then ff is in RR.

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

  1. for f:XYf : X \to Y in RR also the diagonal YY× XYY \to Y \times_X Y is in RR.

  2. If in

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

    we have that pp is an epimorphism, and p,gRp, g \in R, then fRf\in R.

For instance (JoyalMoerdijk, section 1).

References

An application:

Revised on December 9, 2013 11:23:45 by Urs Schreiber (89.204.138.139)