Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Please check if the following statement is true:

Statement Let $\varphi : X \rightarrow Y$ be a morphism of the category $\mathbf{Set}$. Let $\sim$ be an equivalence relation on $X$. Then $\varphi$ can be factored (in a unique way) $\varphi = u \circ \pi$ where $\pi$ is the projection for this equivalence and $u : (X / \sim) \rightarrow Y$.

share|improve this question
$\varphi$ needs to be constant on equivalence classes for this to hold. –  Tobias Kildetoft Oct 26 '13 at 19:02
@TobiasKildetoft: So I suppose my statement is an erroneous formulation of the UMP for coequalizers in Set. What is the correct formulation? –  porton Oct 26 '13 at 19:13
I am not sure how to formulate it in category theory terms. If you simply add the assumption that $\varphi$ is constant on equivalence classes, then the result is correct. –  Tobias Kildetoft Oct 26 '13 at 19:15

1 Answer 1

up vote 1 down vote accepted

Your statement is correct , provided that you add that $\varphi$ respects the equivalence,that is: that it is constant on all members of an equivalence class.

You can read about the UMP of coequalizers in Wikipedia

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.