Say I have an abstract set $X$ (could be points, functions, functors or whatever). Say I have an equivalence relation $R\in X\times X$.

What would be the category-theory way to express $X/R$, that is, the set of subsets of $X$ equivalent by $R$. Or using another formalism, $\{xR^*| x\in X\}\in{\frak P}(X)$.

  • $\begingroup$ Quotients of equivalence relations are a special case of coequalisers. $\endgroup$ – Zhen Lin Feb 22 '16 at 21:49
  • $\begingroup$ @ZhenLin I feel that category theory is a very powerful tool to express abstract relations, but (shame on me) I could never gain much experience and understanding. Could you pleas explain and make a connection to layman's language? (Or, give a link on where I should look and learn about this?) $\endgroup$ – Gyro Gearloose Feb 22 '16 at 21:55
  • $\begingroup$ My (selfish) goal is to improve my question here math.stackexchange.com/q/1658608/290307 . $\endgroup$ – Gyro Gearloose Feb 22 '16 at 21:57

Equivalences relations in category theory are used by Giraud to characterize a Grothendieck topos.

Let $C$ be a category, an equivalence relation defined on the object $X$ is an object $R$ together with a morphism $R\rightarrow X\times X$ such that for every object $Y$ in $C$, $Hom_C(Y,R)\rightarrow Hom_C(Y,X)\times Hom_C(Y,X)$ defines an equivalence relation on $Hom_C(Y,X)$.

If $X$ is a set, and $R\subset X\times X$ an equivalence relation, you can define $f,g:R\rightarrow X$ such that for $(x,y)\in R$, $f(x,y)=x$ and $g(x,y)=y$. The quotient $X/R$ is the coequalizer of $f$ and $g$.


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