nLab quotient object coclassifier




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 CC with finite colimits, a quotient object coclassifier is an object QQ with an epimorphism ϵ:Q𝟘\epsilon:Q \to \mathbb{0} into the initial object 𝟘\mathbb{0}, such that for every epimorphism f:XUf:X \to U there is a unique morphism ϝ U:QX\digamma_U:Q \to X such that there is a pushout diagram of the form

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

See also

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