Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.
-
(c)
The relation $R$ is of the form $f^{-1}$ (as in
) for some function $f$.
9.2.1 Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
Proof of Proposition 9.2.1.1.1.
Item 1: Non-Existence of All Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
Omitted, but will eventually follow 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 Extensions 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 extension
\[ \operatorname {\mathrm{Lan}}_{S}\webleft (R\webright )\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X \]
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 9.2.1.1.1, the left Kan extension
\[ \operatorname {\mathrm{Lan}}_{R}\colon \mathbf{Rel}\webleft (A,X\webright )\to \mathbf{Rel}\webleft (B,X\webright ) \]
along a relation of the form $R=f^{-1}$ exists. Is there a explicit description of it, similarly to the explicit description of right Kan extensions given in Proposition 9.2.3.1.1?
This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
