7.2.2 Left Kan Lifts in $\textbf{Rel}$
Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.
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Non-Existence of All Left Kan Lifts in $\textbf{Rel}$. Not all relations in $\textbf{Rel}$ admit left Kan lifts.
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Characterisation of Relations Admitting Left Kan Lifts Along Them. The following conditions are equivalent:
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The left Kan lift
\[ \text{Lift}_{R}\colon \mathbf{Rel}\webleft (X,B\webright )\to \mathbf{Rel}\webleft (X,A\webright ) \]
along $R$ exists.
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The relation $R$ admits a right adjoint in $\textbf{Rel}$.
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The relation $R$ is of the form $\text{Gr}\webleft (f\webright )$ (as in Definition 7.3.1.1.1) for some function $f$.
Item 1: Non-Existence of All Left Kan Lifts in $\textbf{Rel}$
Omitted, but will eventually follow (the dual of) Fosco Loregian’s comment on [MO 460656].
Item 2: Characterisation of Relations Admitting Left Kan Lifts Along Them
Omitted, but will eventually follow Tim Campion’s answer to to [MO 460656].
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
\[ \text{Lift}_{S}\webleft (R\webright )\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A \]
exists in terms of properties of $R$ and $S$?
This question also appears as [MO 461592].
As shown in Item 2 of Proposition 7.2.2.1.1, the left Kan lift
\[ \text{Lift}_{R}\colon \mathbf{Rel}\webleft (X,B\webright )\to \mathbf{Rel}\webleft (X,A\webright ) \]
along a relation of the form $R=\text{Gr}\webleft (f\webright )$ exists. Is there a explicit description of it, similarly to the explicit description of right Kan lifts given in Proposition 7.2.4.1.1?
This question also appears as [MO 461592].