nLab
reflexive coequalizer

Definitions

A reflexive pair is a parallel pair $f,g:A\rightrightarrows B$ having a common section, i.e. a map $s:B\to A$ such that $f\circ s=g\circ s=1_B$. A reflexive coequalizer is a coequalizer of a reflexive pair. A category has reflexive coequalizers if it has coequalizers of all reflexive pairs.

Dually, a reflexive coequalizer in the opposite category $C^{\mathrm{op}}$ is called a coreflexive equalizer in $C$.

Remarks

• Reflexive coequalizers should not be confused with split coequalizer?s, a distinct concept.

• Any congruence is a reflexive pair, so in particular any quotient of a congruence is a reflexive coequalizer.

Applications

• Reflexive coequalizers figure in the crude monadicity theorem.

• A theorem of Fred Linton? states that if $T$ is a monad on a cocomplete category $C$, then the category $C^T$ of Eilenberg–Moore algebras is cocomplete if and only if it has reflexive coequalizers. This is the case particularly if $T$ preserves reflexive coequalizers.

• If $F:C\times D\to E$ is a functor of two variables which preserves reflexive coequalizers in each variable separately (that is, $F(c,-)$ and $F(-,d)$ preserve reflexive coequalizers for all $c\in C$ and $d\in D$), then $F$ preserves reflexive coequalizers in both variables together. (This is emphatically not the case for arbitrary coequalizers.)

  This result is particularly interesting when $F$ is the tensor product of a cocomplete closed monoidal category $C$. In this case it implies that the free monoid monad on such a category preserves reflexive coequalizers, and thus (by Linton’s theorem) the category of monoid objects in $C$ is cocomplete.

References

• F.E.J. Linton, “Coequalizers in categories of algebras”