I went through the definition of the quotient of a category $\mathcal{C}$ by a congruence relation $\sim$ on the morphisms. I was very surprised to find that you can prove $\mathcal{C}/{\sim}$ is a category, with only the assumptions that $\sim$:
- is reflexive: $f \sim f$
- respects composition: if $f \sim f'$ and $g \sim g'$, then $fg \sim f'g'$.
In particular, nowhere did I need to assume that $\sim$ was symmetric or transitive.
This led me to thinking: what happens if you try to define the quotient by a "congruence" relation which is not necessarily symmetric or transitive? I figure that either:
- something goes wrong; or
- this turns out to be equivalent to e.g. the quotient by the symmetric transitive closure of $\sim$.