Contents

category theory

topos theory

# 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.