Equalizer

The equalizer $(𝐸, e q)$ of a collection of morphisms $𝑀 \subseteq 𝖢 (𝑋, 𝑌)$ is the limit of the diagram containing these morphisms. #m/def/cat Thus $e q 𝑓 = e q 𝑔$ for any $𝑓, 𝑔 \in 𝑀$ , and given any other morphism $𝑞 : 𝑄 \to 𝑋$ with this property there exists a unique $¯ 𝑞 : 𝑄 \to 𝐸$ such that the following diagram, except for $𝑓 = 𝑔$ , commutes

Note that in case $𝑀 = \emptyset$ we take the diagram consisting of only $𝑋$ . Thus the equalizer is the “most general” Subobject for which the morphisms $𝑀$ concur.

The coëqualizer $(𝑄, 𝑞)$ of a collection of morphisms $𝑀 \subseteq 𝖢 (𝑌, 𝑋)$ is the colimit of the diagram containing these morphisms. #m/def/cat Thus $𝑓 𝑞 = 𝑔 𝑞$ , and given any other morphism $ℎ : 𝑋 \to 𝑍$ there exists a unique $¯ ℎ : 𝑄 \to 𝑍$ such that the following diagram commutes, except for $𝑓 = 𝑔$ :

Note that in case $𝑀 = \emptyset$ we take the diagram consisting of only $𝑋$ . Thus the coëqualizer is the “most general” quotient object onto which the morphisms concur.

Properties

The equalizer $e q$ is always a Regular monomorphism.
The coëqualizer $𝑞$ is always a Regular epimorphism.

Equalizer

Properties

See also