Cartesian Lift

Cartesian Lift

Let $p:\mathcal{E}\Rightarrow \mathcal{B}$ be a functor, and let $f\in \mathcal{E}[X,Y]$ be a morphism in $\mathcal{E}$. Write

$$u:=p(f)\in \mathcal{B}[pX,pY].$$

The morphism $f$ is Cartesian over $u$ if, for every object
$Z:\mathcal{E}$ and every morphism

$$v\in \mathcal{B}[pZ,pX]$$

in $\mathcal{B}$, postcomposition with $f$ induces a bijection

$$f\circ -:{\mathcal{E}}_v[Z,X]\cong {\mathcal{E}}_{u\circ v}[Z,Y].$$

Equivalently, for every morphism $g\in \mathcal{E}[Z,Y]$ in $\mathcal{E}$ and every factorisation

$$p(g)=u\circ v$$

in $\mathcal{B}$, there exists a unique morphism $h\in \mathcal{E}[Z,X]$ such that $p(h)=v$ and $f\circ h=g$.

In this case, $f$ is called a Cartesian lift of $u$.

CoCartesian Lift

The dual notion is that of a coCartesian morphism. The morphism $f$ is coCartesian over $u$ if, for every object $Z:\mathcal{E}$ and every morphism $v\in \mathcal{B}[pY,pZ]$ in $\mathcal{B}$, precomposition with $f$ induces a bijection

$$-\circ f:{\mathcal{E}}_v[Y,Z]\cong {\mathcal{E}}_{v\circ u}[X,Z].$$

Equivalently, for every morphism $g\in \mathcal{E}[X,Z]$ in $\mathcal{E}$ and every factorisation

$$p(g)=v\circ u$$

in $\mathcal{B}$, there exists a unique morphism $h\in \mathcal{E}[Y,Z]$ such that $p(h)=v$ and $h\circ f=g$.

In this case, $f$ is called a coCartesian lift of $u$.