nLab
closed subtopos

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

The concept of a closed subtopos generalizes the concept of a closed subspace from topology to toposes.

Definition

Let UU be a subterminal object of a topos \mathcal{E}. Then c U(V)UVc_U(V) \coloneqq U\cup V defines a Lawvere-Tierney topology on \mathcal{E}, whose corresponding subtopos is called the closed subtopos associated to UU.

The reflector into the sheaves can be constructed explicitly as the join with UU, i.e. C U(X)C_U(X) is the pushout of XX and UU under X×UX\times U.

A topology jj that is of this form for some subterminal object UU is called closed.

Example

In case =Sh(X)\mathcal{E}=Sh(X) is the topos of sheaves on a topological space XX, a subterminal object is just an open subset UU of XX and the closed subtopos corresponding to it is equivalent to Sh(XU)Sh(X\setminus U).

Remark

The subterminal object UU in \mathcal{E} is associated with an open subtopos Sh o(U)()Sh_{o(U)}(\mathcal{E}) as well e.g. in the case of =Sh(X)\mathcal{E}=Sh(X) on a space XX this yields Sh(U)Sh(U).

Moreover, given a Lawvere-Tierney topology jj on a topos \mathcal{E} with corresponding subtopos Sh j()Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}, we a get a canonical subterminal object ext(j)ext(j) associated to jj by taking the jj-closure of O1O\rightarrowtail 1. The corresponding closed and open subtoposes associated to ext(j)ext(j) provide a ‘closureSh j()Sh c(ext(j))()Sh_j(\mathcal{E})\hookrightarrow Sh_{c(ext(j))}(\mathcal{E}), called l’adhérence in (SGA4, p.461), respectively, an ‘exteriorSh o(ext(j))()Sh_{o(ext(j))}(\mathcal{E}) for Sh j()Sh_j(\mathcal{E}).

Properties

Proposition

Let UU a subterminal object and Sh c(U)()Sh_{c(U)}(\mathcal{E}) and Sh o(U)()Sh_{o(U)}(\mathcal{E}) the corresponding closed, resp. open subtoposes. Then Sh c(U)()Sh_{c(U)}(\mathcal{E}) and Sh o(U)()Sh_{o(U)}(\mathcal{E}) are complements for each other in the lattice of subtoposes.

For the proof see (Johnstone (2002), pp.212,215).

Closed subtoposes that are Boolean are parametrized by automorphisms of the subobject classifier as the next proposition documents:

Proposition

Let \mathcal{E} be a topos. Then automorphisms of Ω\Omega correspond bijectively to closed Boolean subtoposes. The group operation on Aut(Ω)Aut(\Omega) corresponds to symmetric difference of subtoposes.

This result appears in Johnstone (1979).

The following result is a part of the so called (dense,closed)-factorization.

Proposition

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

This says that the only closed subtopos of a topos \mathcal{E} that is dense, is \mathcal{E} itself.

It follows e.g. that Boolean toposes have no non-trivial dense subtoposes since all their subtoposes are closed (and open, as well). Of course, this follows also from the fact that a Boolean topos coincides with its double negation subtopos and the latter is the smallest dense subtopos!

References

Last revised on May 11, 2016 at 08:44:47. See the history of this page for a list of all contributions to it.