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 $\mathcal{E}$ .

(Caramello (2009), prop. 10.1 p.58 ).

Related pages
References
F. Borceux , M. Korostenski, Open Localizations , JPAA 74 (1991) pp.229-238.

O. Caramello , Lattices of theories , (2009) (arXiv:0905.0299 )

H. Forssell, Subgroupoids and quotient theories , TAC 28 no.18 (2013) pp.541-551. (pdf )

Peter Johnstone , Sketches of an Elephant I , Oxford UP 2002. pp.195-223.

G. M. Kelly , F. W. Lawvere , On the Complete Lattice of Essential Localizations , Bull.Soc.Math. de Belgique XLI (1989) pp.261-299.

Revised on March 11, 2015 22:54:40
by

Mike Shulman
(192.195.154.58)