# nLab split equalizer

A fork

$A\xrightarrow[\quad e \quad]{}B\underoverset{\quad g \quad}{f}{\rightrightarrows}C$

is split if it can be embedded into a diagram

$A\xrightarrow[\quad e \quad]{\overset{\scriptsize{s}}{\begin{svg} \end{svg}\includegraphics[width=23]{curvedArrow1} }}B\underoverset{\quad g \quad}{\mathclap{\overset{\scriptsize{t}}{\begin{svg} \end{svg}\includegraphics[width=24]{curvedArrow2} }}f}{\rightrightarrows}C$

in which $s e = id_A$, $t g = id_B$ and $t f = e s$ (we used here Leibniz order for composition of morphisms).

Surely, $f$ and $g$ can be interchanged in the definition (a matter of unimportant convention).

Every split fork is an absolute equalizer, but not conversely. See also split coequalizer.

Revised on October 7, 2010 15:16:07 by Jacques Distler (72.179.54.121)