Cartesian Morphism

A morphism $f:{{X}\to_{u}{Y}}$ over $u:{{A}\to{B}}$ in a Displayed Category $\mathcal{E}\rightarrowtriangle\mathcal{B}$ is cartesian if

• For all $v:{{C}\to{A}}$ and $h:{{Z}\to_{u\circ v}{Y}}$, there exists a unique $\langle v,h\rangle$ such that $f\circ\langle v,h\rangle=h$.

Properties

• Cartesian morphisms are closed under composition.

• Every invertible morphism is cartesian.

• Every cartesian morphism is Weak Monomorphism.

• If $f$ and $f\circ g$ are cartesian, then $g$ is cartesian.

• More generally, if $f\circ g$ is cartesian and $f$ is Weakly Monic, then $g$ is cartesian.

• If $f:{{}_{u}(X_{1},Y)}$ and $g:{{}_{u}(X_{2},Y)}$ are two cartesian morphisms over the same map $u$, then their domains are vertically Isomorphic.

• If $f:{{}_{u}(X,Y)}$ is cartesian and $r:{{}_{mathrm{id}{ifx..lse\par i}}(R,X)}$ is a vertical Split Epi, then $f\circ r$ is cartesian.

• Every vertical cartesian morphism is a Vertical Isomorphism.

• A morphism $f:{{}_{u}(X,Y)}$ is cartesian if and only if postcomposition with $f$ is an Equivalence.