# nLab subtopos

### Context

#### Topos Theory

Could not include topos theory - contents

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.

• $\mathcal{E}$ (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

Revised on October 13, 2014 12:33:18 by Thomas Holder? (89.15.239.244)