topos theory

# 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 an elementary 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

Every Grothendieck topos $\mathcal{E}$ over $Set$ is (equivalent to) 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) 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.

The lattice structure is further analyzed in (Johnstone (2002), Caramello (2009)). We contend us to report

###### Proposition

Let $Sh_j(\mathcal{E})$, $Sh_k(\mathcal{E})$ be two subtoposes of a topos $\mathcal{E}$ and $j,k$ the corresponding topologies with $j\vee k$ their join in the lattice of topologies. Then the following holds: $Sh_j(\mathcal{E})\wedge Sh_k(\mathcal{E})=Sh_{j\vee k}(\mathcal{E})=Sh_j(\mathcal{E})\cap Sh_k(\mathcal{E})$.

(cf. Johnstone (2002), p.217). In other words, the meet of two subtoposes is just their intersection (and is equivalently given by the subtopos corresponding to the join of their topologies).

###### Proposition

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

(Caramello (2009); prop. 10.1). This follows from the fact that two-valued and Boolean toposes are opposite extremes when it comes to dense subtoposes: in a two-valued topos every non-trivial subtopos is dense, whereas a Boolean topos has no non-trivial dense subtopos (cf. at dense subtopos for further details).

## References

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (exposé IV, section 9, pp.431ff)

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

• O. Caramello, Lattices of theories , arXiv:0905.0299 (2009). (abstract, pp.15,58)

• 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)

• Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014, section 3.5)

• 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 September 4, 2015 18:20:26 by Urs Schreiber (195.82.63.197)