# 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).

## Properties

• 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 $\mathcal{E}$.

• 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.

• The atoms in the lattice of subtoposes of $\mathcal{E}$ are precisely the Boolean two-valued subtoposes of $\mathcal{E}$ (cf. Caramello (2009) prop.10.1 p.58).

• $\mathcal{E}$ (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 $T$ (cf. 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.

## References

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

• O. Caramello, Lattices of theories , arXiv:0905.0299v1 (2009). (pdf)

• 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 October 22, 2014 13:46:48 by Thomas Holder? (89.15.236.120)