The Clowder Project

Let $f\colon X\to Y$ be a function and let $R$ be a relation on $X$.

1. 1.

   As a Coequaliser. We have an isomorphism of sets

   \[ X/\mathord {\sim }^{\mathrm{eq}}_{R}\cong \operatorname {\mathrm{CoEq}}\webleft (R\hookrightarrow X\times X\stackrel{\stackrel{\operatorname {\mathrm{\mathrm{pr}}}_{1}}{\rightarrow }}{\stackrel{\rightarrow }{\scriptsize \operatorname {\mathrm{\mathrm{pr}}}_{2}}}X\webright ), \]

   where $\mathord {\sim }^{\mathrm{eq}}_{R}$ is the equivalence relation generated by $\mathord {\sim }_{R}$.

2. 2.

   As a Pushout. We have an isomorphism of sets¹

   
   where $\mathord {\sim }^{\mathrm{eq}}_{R}$ is the equivalence relation generated by $\mathord {\sim }_{R}$.

3. 3.

   The First Isomorphism Theorem for Sets. We have an isomorphism of sets^2,3

   \[ X/\mathord {\sim }_{\mathrm{Ker}\webleft (f\webright )} \cong \mathrm{Im}\webleft (f\webright ). \]
4. 4.

   Descending Functions to Quotient Sets, I. Let $R$ be an equivalence relation on $X$. The following conditions are equivalent:

   1. (a)

      There exists a map

      \[ \overline{f}\colon X/\mathord {\sim }_{R}\to Y \]

      making the diagram

      
      commute.

   2. (b)

      We have $R\subset \mathrm{Ker}\webleft (f\webright )$.

   3. (c)

      For each $x,y\in X$, if $x\sim _{R}y$, then $f\webleft (x\webright )=f\webleft (y\webright )$.

5. 5.

   Descending Functions to Quotient Sets, II. Let $R$ be an equivalence relation on $X$. If the conditions of Item 4 hold, then $\overline{f}$ is the unique map making the diagram

   
   commute.

6. 6.

   Descending Functions to Quotient Sets, III. Let $R$ be an equivalence relation on $X$. We have a bijection

   \[ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}\webleft (X/\mathord {\sim }_{R},Y\webright )\cong \operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (X,Y\webright ), \]

   natural in $X,Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, given by the assignment $f\mapsto \overline{f}$ of Item 4 and Item 5, where $\operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (X,Y\webright )$ is the set defined by

   \[ \operatorname {\mathrm{Hom}}^{R}_{\mathsf{Sets}}\webleft (X,Y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ f\in \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}\webleft (X,Y\webright )\ \middle |\ \begin{aligned} & \text{for each $x,y\in X$,}\\ & \text{if $x\sim _{R}y$, then}\\ & \text{$f\webleft (x\webright )=f\webleft (y\webright )$}\end{aligned} \right\} . \]
7. 7.

   Descending Functions to Quotient Sets, IV. Let $R$ be an equivalence relation on $X$. If the conditions of Item 4 hold, then the following conditions are equivalent:

   1. (a)

      The map $\overline{f}$ is an injection.

   2. (b)

      We have $R=\mathrm{Ker}\webleft (f\webright )$.

   3. (c)

      For each $x,y\in X$, we have $x\sim _{R}y$ iff $f\webleft (x\webright )=f\webleft (y\webright )$.

8. 8.

   Descending Functions to Quotient Sets, V. Let $R$ be an equivalence relation on $X$. If the conditions of Item 4 hold, then the following conditions are equivalent:

   1. (a)

      The map $f\colon X\to Y$ is surjective.

   2. (b)

      The map $\overline{f}\colon X/\mathord {\sim }_{R}\to Y$ is surjective.

9. 9.

   Descending Functions to Quotient Sets, VI. Let $R$ be a relation on $X$ and let $\mathord {\sim }^{\mathrm{eq}}_{R}$ be the equivalence relation associated to $R$. The following conditions are equivalent:

   1. (a)

      The map $f$ satisfies the equivalent conditions of Item 4:

      • •

        There exists a map

        \[ \overline{f}\colon X/\mathord {\sim }^{\mathrm{eq}}_{R}\to Y \]

        making the diagram

        
        commute.

      • •

        For each $x,y\in X$, if $x\sim ^{\mathrm{eq}}_{R}y$, then $f\webleft (x\webright )=f\webleft (y\webright )$.

   2. (b)

      For each $x,y\in X$, if $x\sim _{R}y$, then $f\webleft (x\webright )=f\webleft (y\webright )$.