File:Equalizer as Limit.gif

English: Diagram D contains objects X and Y and parallel morphisms ${\displaystyle f,g:X\rightarrow Y}$. A cone with apex E has arrows ${\displaystyle eq:E\rightarrow X}$ and ${\displaystyle h:E\rightarrow Y}$ such that ${\displaystyle f\circ eq=h}$ and ${\displaystyle g\circ eq=h}$, so that ${\displaystyle f\circ eq=g\circ eq}$. The cone with apex E is limiting in the sense that for any other cone, say with apex O and arrows ${\displaystyle m:O\rightarrow X}$ and ${\displaystyle j:O\rightarrow Y}$ such that ${\displaystyle f\circ m=j}$ and ${\displaystyle g\circ m=j}$, so that ${\displaystyle f\circ m=g\circ m}$, the apex of such a cone must factor through the apex of the limiting cone; i.e., there is an arrow ${\displaystyle u:O\rightarrow E}$ such that ${\displaystyle m=eq\circ u}$.

Commutations of commutative diagrams are depicted as triangular cobordisms where the composition of two edges equals the third edge. The result is a pair of digonal cones (but without their bases). Cobordisms ${\displaystyle f\circ eq=h}$, ${\displaystyle f\circ m=j}$, ${\displaystyle eq\circ u=m}$, and ${\displaystyle h\circ u=j}$ are shown mesh-like and thereby see-through. The image is animated (GIF) so that cobordisms are peeled off like layers of an onion so that the overall geometric structure may be more easily seen.

Made with Grapher and Paint Pad Lite.