Topos Theory

Could not include topos theory - contents



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


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).


Sheaves, localization, closure and reflection

If \mathcal{E} is a Grothendieck topos then subtoposes correspond to Lawvere-Tierney topologies jj 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 SetSet) is the classifying topos of some geometric theory TT and it can be shown that subtoposes of \mathcal{E} correspond precisely to deductively closed quotient theories of TT (Caramello (2009), thm. 3.6 p.15) i.e. passage to a subtopos corresponds to adding further geometric axioms to TT - 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 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).


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