How to prove that for all parallel pairs of arrows in category of monoids there exists a coequalizer? And how to define this coequalizer?
2026-03-25 10:55:32.1774436132
Coequalizer in the category of monoids
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The coequalizer of $f,g:A\rightrightarrows B$ is the quotient of $B$ by the relation $R=\{(f(a),g(a))\}$ on $B$. To construct this quotient, we fatten $R$ up into a congruence $R'$, which is an equivalence relation that respects the multiplication on $B$ in the sense that $x\sim y$ and $z\sim w$ ipmlies $xz\sim yw$. So the coequalizer is the set of equivalence classes $B/R'$, where $R'$ is the least congruence containing $R$, with multiplication induced from $B$. Any monoid homomorphism $h:B\to C$ which identifies pairs of elements related by $R$ factors uniquely through $B/R'$ via the formula $[b]\mapsto h(b)$.
There are a few more details, in a more general situation, in 2.4 and 2.5 here.