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Stream: theory: category theory

Topic: Categories of Abstract Relations

Chad Nester (Apr 16 2020 at 13:19):

Hello all. This is something I'm pretty interested in recently, and I'm slowly learning about the ways you can have a "category of relations". Maybe we can have a topic about them?

My more immediate motivation is that I'm reading about relations in regular categories (i.e., subobjects given by jointly monic spans), and I can't figure out how to prove that the relation given by $(f,g)$ is simple if and only if $f$ is monic. I feel like I am missing something obvious, and I wonder if anyone knows how this works? By simple I mean that $R^\circ ; R \leq 1$.

Morgan Rogers (he/him) (Apr 16 2020 at 13:22):

You need double $ to make TeX formatting work :+1:
Try drawing the pullback diagram + image factorisation that defines that composite

Chad Nester (Apr 16 2020 at 13:46):

Thanks for the formatting tip!

I understand that if $f$ is monic, then immediately we have that $R^\circ ; R$ is simple, but am still struggling with the other direction

Morgan Rogers (he/him) (Apr 16 2020 at 13:52):

"1" is the diagonal relation, right? So that if $R^\circ R$ factors through it, both legs of this composite must be equal?