nLab
split equalizer

A fork

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

is split if it can be embedded into a diagram

Ae Layer 1 sBg Layer 1 tfC A\xrightarrow[\quad e \quad]{\overset{\scriptsize{s}}{\begin{svg} <svg width="31" height="11" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="56258"> <g> <title>Layer 1</title> <path fill="none" stroke="#000000" d="m2.436699,8.563251c9.440479,-9.454929 17.148819,-9.609211 27.625038,0.192581" marker-start="url(#se_marker_start_svg_56258_3)" id="svg_56258_3"/> </g> <defs> <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_start_svg_56258_3"> <path stroke-width="10" stroke="#000000" fill="#000000" d="m0,50l100,40l-30,-40l30,-40l-100,40z" id="svg_56258_7"/> </marker> </defs> </svg> \end{svg}\includegraphics[width=23]{curvedArrow1} }}B\underoverset{\quad g \quad}{\mathclap{\overset{\scriptsize{t}}{\begin{svg} <svg width="33" height="18" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="69468"> <g> <title>Layer 1</title> <path marker-start="url(#se_marker_start_svg_69468_2)" id="svg_69468_2" d="m2.12632,15.467968c13.519966,-19.398649 19.108238,-19.52702 30.125008,0.027042" stroke="#000000" fill="none"/> </g> <defs> <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_start_svg_69468_2"> <path stroke-width="10" stroke="#000000" fill="#000000" d="m0,50l100,40l-30,-40l30,-40l-100,40z" id="svg_69468_6"/> </marker> </defs> </svg> \end{svg}\includegraphics[width=24]{curvedArrow2} }}f}{\rightrightarrows}C

in which se=id As e = id_A, tg=id Bt g = id_B and tf=est f = e s (we used here Leibniz order for composition of morphisms).

Surely, ff and gg 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)