Joint Cartesian Family

Let $\mathcal{E}\rightarrowtriangle\mathcal{B}$ be a Displayed Category, and $u_{i}:{{A}\to{B_{i}}}$ be a Source in $\mathcal{B}$. A family of morphisms $f_{i}:{{X}\to_{u_{i}}{Y_{i}}}$ is jointly cartesian if for every other $v:{{C}\to{B}}$ and $h_{i}:{{Z}\to_{u_{i}\circ v}{Y_{i}}}$, there exists a unique $\langle v,h_{i}\rangle:{{Z}\to_{v}{X}}$ such that $f_{i}\circ\langle v,h_{i}\rangle=h_{i}$ for every $i:I$.

Examples

• In the Displayed Category of Topologies, the joint cartesian morphisms classify Codiscrete Topologies.

• What about the Fibration of Signatures? Let's consider a Span $I\leftarrow J\to K$ in $\mathrm{Sets}{.}$, and a pair of signatures $\mathbb{S}\rightarrowtriangle I$, $\mathbb{T}\rightarrowtriangle K$. If there is a joint cartesian family $\mathbb{S}\leftarrow\mathbb{A}\to\mathbb{T}$

Properties

• A morphism $f$ is Cartesian if and only if $f$ is a singleton jointly cartesian family.

• If $f_{i,j}:{{Y_{i}}\to_{u_{i,j}}{Z_{i,j}}}$ is an $I$-indexed family of $J_{i}$ jointly cartesian families and $g_{i}:{{Y_{i}}\to_{X}{missing}}$ is an $I$-indexed jointly cartesian family, then the composite

  $\par\lambda(i,j).\;f_{i,j}\circ g_{i}\par$

  is a $\Sigma(i:I).\;J_{i}$ indexed jointly cartesian family.

  This theorem has some very nice corollaries:

  • If $f_{i}:{{Y_{i}}\to_{u_{i}}{Z_{i}}}$ is an $I$-indexed family of cartesian morphisms and $g_{i}:{{X}\to_{v_{i}}{Y_{i}}}$ is jointly cartesian, then $\lambda i.\;f_{i}\circ g_{i}$ is jointly cartesian.
  • If $f_{i}:{{Y}\to_{u_{i}}{Z_{i}}}$ is jointly cartesian and $g:{{Y}\to_{X}{missing}}$ is cartesian, then $f_{i}\circ g$ is jointly cartesian.

• Conversely, if $f_{i,j}$ is an $I$-indexed family of $J_{i}$-indexed Jointly Weak Monic Families and $\lambda(i,j).\;f_{i,j}\circ g_{i}$ is jointly cartesian, then $g_{i}$ is jointly cartesian.

• If we view a category $\mathcal{C}$ as a displayed category $\mathcal{C}\rightarrowtriangle 1$, then the joint cartesian families are precisely the Indexed Cartesian Products.

• A joint cartesian lift of a family $\lambda(i:I).\;X_{i}\to A$ along the constant family $\lambda(i:I).\;mathrm{id}{ifx..lse\par i}$ in the Codomain Fibration is a Wide Pullback in $\mathcal{C}$.

References

• Abstract and Concrete Categories: The Joy of Cats, Definition 10.57.
• Topological Functors as Familiarly-Fibrations, though they refer to joint cartesian families as "initial families".