Given a category $\mathcal{C}$, it follows from Proposition 4.1 of Day's
Brian Day, Limit spaces and closed span categories, Lecture Notes in Mathematics, 420, 1974 (doi:10.1007/BFb0063100).
That the bicategory $\mathsf{Span}_\mathcal{C}$ of spans in $\mathcal{C}$ has right Kan extensions and right Kan lifts iff $\mathcal{C}$ is locally Cartesian closed.
- Does $\mathsf{Span}_\mathcal{C}$ also have left Kan extensions and left Kan lifts if (and perhaps only if?) $\mathcal{C}$ is locally Cartesian closed too?
- What would be explicit descriptions for $\mathrm{Ran}$ and $\mathrm{Rift}$ in $\mathsf{Span}_{\mathcal{C}}$? What about for $\mathrm{Lan}$ and $\mathrm{Lift}$ if these also exist?
Edit: I was reading
Renato Betti, Robert F. C. Walters, Closed bicategories and variable category theory, Universita degli Studi di Milano (1985), reprinted in: Reprints in Theory and Applications of Categories, 26 (2020) 1–27 [tac:tr26]
where I found the following excerpt from Example 2.2:
[...] the formulae for right extensions and right liftings [in $\mathsf{Span}_{\mathsf{Set}}$] become \begin{align*} hom_u(\alpha,\beta)_{ik} &= \Pi_j\hom(\alpha_{ij},\beta_{kj})\\ hom^u(\alpha,\beta)_{ik} &= \Pi_j\hom(\alpha_{ji},\beta_{kj}). \end{align*}
Here $hom^u$ and $hom_u$ denote right Kan liftings and extensions respectively, where I believe the matrix notation should be interpreted as follows: given a span $A\xleftarrow{f}S\xrightarrow{g}B$, an element $a\in A$, and an element $b\in B$, we write $S_{ab}$ for the set $$S_{ab}=\{s\in S\ |\ \text{$f(s)=a$ and $g(s)=b$}\}.$$ So given spans $A\xleftarrow{f}S\xrightarrow{g}B$ and $A\xleftarrow{\phi}K\xrightarrow{\psi}X$, the un/straightening isomorphism for fibred and indexed sets would then give $$\mathrm{Ran}_{S}(K)\cong\coprod_{(b,x)\in B\times X}\prod_{a\in A}\mathsf{Set}(S_{ab},K_{ax}),$$ with the maps $B\leftarrow\mathrm{Ran}_{S}(K)\rightarrow X$ being given by sending an element to its index in the coproduct above.
Is this formula correct, and is there a nicer one?
