regular subobject classifier.yaml

id: regular subobject classifier
relation: has a
description: >-
  A category $\C$ has a <i>regular subobject classifier</i> if it has finite limits and a regular monomorphism* $\top : 1 \hookrightarrow \Omega$ such that for every regular monomorphism $m : A \hookrightarrow B$ there is a unique morphism $\chi_m : B \to \Omega$ such that
  $$\begin{CD} A @>{m}>> B \\ @V{!}VV @VV{\chi_m}V \\ 1 @>>{\top}> \Omega \end{CD}$$
  is a pullback diagram. Equivalently, the functor $\Sub_{\reg} : \C^{\op} \to \Set^+$ is representable.
  *Every morphism $1 \to \Omega$ is a split monomorphism and hence regular anyway.
nlab_link: https://ncatlab.org/nlab/show/subobject+classifier
dual_property: regular quotient object classifier
invariant_under_equivalences: true

related_properties:
  - finitely complete
  - subobject classifier
  - quasitopos