Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.
-
(c)
The relation $R$ is of the form $\operatorname {\mathrm{Gr}}\webleft (f\webright )$ (as in
) for some function $f$.
9.2.2 Left Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
Proof of Proposition 9.2.2.1.1.
Item 1: Non-Existence of All Left Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
Omitted, but will eventually follow (the dual of) Fosco Loregian’s comment on [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations I].
Item 2: Characterisation of Relations Admitting Left Kan Lifts Along Them
Omitted, but will eventually follow Tim Campion’s answer to to [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations I].
Given relations $S\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X$ and $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$, is there a characterisation of when the left Kan lift
\[ \operatorname {\mathrm{Lift}}_{S}\webleft (R\webright )\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A \]
exists in terms of properties of $R$ and $S$?
This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
As shown in Item 2 of Proposition 9.2.2.1.1, the left Kan lift
\[ \operatorname {\mathrm{Lift}}_{R}\colon \mathbf{Rel}\webleft (X,B\webright )\to \mathbf{Rel}\webleft (X,A\webright ) \]
along a relation of the form $R=\operatorname {\mathrm{Gr}}\webleft (f\webright )$ exists. Is there a explicit description of it, similarly to the explicit description of right Kan lifts given in Proposition 9.2.4.1.1?
This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
