nLab
cotopos
Contents
Context
Category theory
category theory

Concepts
Universal constructions
Theorems
Extensions
Applications
Topos Theory
topos theory

Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
Contents
Idea
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 ).

Definition
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.

Properties
Every cotopos is a protomodular category .

Examples
A binary relation $R$ between sets $A$ and $B$ is injective if for all $a \in A$ , $b \in A$ , and $c \in B$ , if $R(a, c)$ and $R(b, c)$ then $a = b$ . A binary relation $R$ between sets $A$ and $B$ is onto if for all $b \in B$ there exists an element $a \in A$ such that $R(a, b)$ . The category of sets and injective onto binary relations is a cotopos.

See also
References
Mamuka Jibladze, “Cosheaves, coframes, cotoposes: some new facts, some old questions” (web archive )
Last revised on February 21, 2023 at 13:09:09.
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