nLab quotient object coclassifier

Idea

The notion of a quotient object coclassifier in a finitely cocomplete category is dual to that of a subobject classifier in a finitely complete category.

 Definition

In a category $C$ with finite colimits, a quotient object coclassifier is an object $Q$ with an epimorphism $\epsilon:Q \to \mathbb{0}$ into the initial object $\mathbb{0}$, such that for every epimorphism $f:X \to U$ there is a unique morphism $\digamma_U:Q \to X$ such that there is a pushout diagram of the form

$$\array{ Q &\overset{\epsilon}{\longrightarrow}& \mathbb{0} \\ \big\downarrow {}^{\mathrlap{\digamma_U}} && \big\downarrow {}^{\mathrlap{\exists !}} \\ X &\underset{f}{\longrightarrow}& U } \,.$$

See also

• subobject classifier

• finitely cocomplete category

• cotopos