nLab subtopos

Context

Topos Theory

Could not include topos theory - contents

Contents

Idea

A subtopos of a topos is a generalization of the concept of a subspace of a topological space.

Definition

For $\mathcal{E}$ a topos, a subtopos is another topos $\mathcal{F}$ equipped with a geometric embedding $\mathcal{F} \hookrightarrow \mathcal{E}$.

If this is an open geometric morphism (or an essential geometric morphism) one speaks of an open subtopos (an essential subtopos, respectively, also called a level of a topos).

Properties

Sheaves, localization, closure and reflection

If $\mathcal{E}$ is a Grothendieck topos then subtoposes correspond to Lawvere-Tierney topologies $j$ on $\mathcal{E}$, to localizations of $\mathcal{E}$ as well as to universal closure operators on $\mathcal{E}$.

For classifying toposes

$\mathcal{E}$ (as every Grothendieck topos over $Set$) is the classifying topos of some geometric theory $T$ and it can be shown that subtoposes of $\mathcal{E}$ correspond precisely to deductively closed quotient theories of $T$ (Caramello (2009), thm. 3.6 p.15) i.e. passage to a subtopos corresponds to adding further geometric axioms to $T$ - localizing geometrically amounts to theory refinement logically.

The lattice of subtoposes

The inclusions induce an ordering on the subtoposes of $\mathcal{E}$ that makes them a lattice with a co-Heyting algebra structure i.e. the join operator has a left adjoint subtraction operator. This corresponds to a dual Heyting algebra structure on the Lawvere-Tierney topologies.

Proposition

The atoms in the lattice of subtoposes of $\mathcal{E}$ are precisely the 2-valued Boolean subtoposes of $\mathcal{E}$.

References

Revised on March 11, 2015 22:54:40 by Mike Shulman (192.195.154.58)