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.

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

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

Last revised on June 4, 2022 at 05:42:51. See the history of this page for a list of all contributions to it.