Fibration of Points

The fibration of points of a Category $\mathcal{C}$ is the Displayed Category $\mathrm{Pt}(\mathcal{C})\rightarrowtriangle\mathcal{C}$ where

• Objects over $I:\mathcal{C}$ are Split Epimorphisms $p:X\leftrightarrows I:s$

• Morphisms over $u:{{}(I,J)}$ are commuting squares

  with $t\circ u=f\circ s$ and $u\circ p=q\circ f$.

This is a Cartesian Fibration when $\mathcal{C}$ has Binary Pullbacks of Split Epimorphisms, and the cartesian morphisms are squares where the downward directed component is a pullback.

Intuitively, the fibration of points encodes the data of Bundles with a chosen cross-section.

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.