Punctual Span

A Span $A\xleftarrow{l}R\xrightarrow{r}B$ in a Pointed Category $\mathcal{C}$ is left punctual if there is a morphism $s:A\to R$ such that

• $l\circ s=mathrm{id}{ifx..lse\par i}$
• $r\circ s=0$.

Dually, a span $A\xleftarrow{l}R\xrightarrow{r}B$ is right punctual if there is a morphism $t:B\to R$ such that

• $l\circ t=0$
• $r\circ t=mathrm{id}{ifx..lse\par i}$

A span is punctual if it is left and right punctual.

Intuitively, a punctual span is a span equipped a pair of orthogonal cross-sections.

Examples

• If $\mathcal{C}$ has Binary Cartesian Products, the span $X\leftarrow X\times Y\to Y$ is punctual, where the left cross section is given by $\langle mathrm{id}{ifx..lse\par i},0\rangle$ and the right by $\langle 0,mathrm{id}{ifx..lse\par i}\rangle$.

References

• Borceux, Francis, and Dominique Bourn. Mal’cev, Protomodular, Homological and Semi-Abelian Categories. Dordrecht: Springer Netherlands, 2004. https://doi.org/10.1007/978-1-4020-1962-3.