nLab cotopos



Category theory

Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




The notion of cotopos is dual to that of a topos.

A co-elementary cotopos should be the 1-categorical version of a co-Heyting algebra (which model dual intuitionistic logic), and a co-Grothendieck cotopos should be the 1-categorical version of a coframe (which model closed sublocales in topology).


Co-elementary cotopos

A co-elementary cotopos is a finitely cocomplete cocartesian coclosed category with a quotient object coclassifier.

Co-Grothendieck cotopos

A co-Grothendieck cotopos is a locally small finitely cocomplete category with a small cogenerator, with all small products which are codisjoint and pushout-stable, and where all internal bisimulations have effective subobjects, which are also pushout-stable.


Every cotopos is a protomodular category.


A binary relation RR between sets AA and BB is injective if for all aAa \in A, bAb \in A, and cBc \in B, if R(a,c)R(a, c) and R(b,c)R(b, c) then a=ba = b. A binary relation RR between sets AA and BB is onto if for all bBb \in B there exists an element aAa \in A such that R(a,b)R(a, b). The category of sets and injective onto binary relations is a cotopos.

See also


  • Mamuka Jibladze, “Cosheaves, coframes, cotoposes: some new facts, some old questions” (web archive)

Last revised on February 21, 2023 at 13:09:09. See the history of this page for a list of all contributions to it.