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).


  • 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. 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 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.


  • 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 (