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).


  • 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}.

  • 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 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 (cf. 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.


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