Coequaliser

A coequaliser of a pair of morphisms $f,g:{{X}\to{Y}}$ in a Category $\mathcal{C}$ is an object $\mathrm{coeq}(f,g):\mathcal{C}$ equipped with a morphism $q:{{Y}\to{\mathrm{coeq}(f,g)}}$ such that

• $q\circ f=q\circ g$
• For any other $h:Y\to A$ with $h\circ f=h\circ g$, there exists a unique universal map $[h]$ with $[h]\circ q=h$.

Examples

• The coequaliser of two functions $f,g:A\to B$ in the Category of Sets is the quotient $B/\sim$ by the Equivalence Relation generated by $f(x)\sim g(x)$.

Properties

• Coequalisers are Colimits over the Walking Parallel Pair.

• A diagram $X\xrightrightarrows[g]{f}Y\xrightarrow{q}Z$ is a coequaliser diagram if and only the square

  is a Pushout Square.

• Conversely, a square

  is a Pushout Square if and only if the cofork
  is a coequaliser diagram.