7.2.1 Left Kan Extensions 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 Extensions in $\textbf{Rel}$. Not all relations in $\textbf{Rel}$ admit left Kan extensions.
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Characterisation of Relations Admitting Left Kan Extensions Along Them. The following conditions are equivalent:
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The left Kan extension
\[ \text{Lan}_{R}\colon \mathbf{Rel}\webleft (A,X\webright )\to \mathbf{Rel}\webleft (B,X\webright ) \]
along $R$ exists.
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The relation $R$ admits a left adjoint in $\textbf{Rel}$.
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The relation $R$ is of the form $f^{-1}$ (as in Definition 7.3.2.1.1) for some function $f$.
Item 1: Non-Existence of All Left Kan Extensions in $\textbf{Rel}$
Omitted, but will eventually follow Fosco Loregian’s comment on [MO 460656].
Item 2: Characterisation of Relations Admitting Left Kan Extensions 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 extension
\[ \text{Lan}_{S}\webleft (R\webright )\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X \]
exists in terms of properties of $R$ and $S$?
This question also appears as [MO 461592].
As shown in Item 2 of Proposition 7.2.1.1.1, the left Kan extension
\[ \text{Lan}_{R}\colon \mathbf{Rel}\webleft (A,X\webright )\to \mathbf{Rel}\webleft (B,X\webright ) \]
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 7.2.3.1.1?
This question also appears as [MO 461592].