6.2.1 Left Kan Extensions in $\textbf{Rel}$

Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.

  1. Non-Existence of All Left Kan Extensions in $\textbf{Rel}$. Not all relations in $\textbf{Rel}$ admit left Kan extensions.
  2. Characterisation of Relations Admitting Left Kan Extensions Along Them. The following conditions are equivalent:
    1. The left Kan extension
      \[ \text{Lan}_{R}\colon \mathbf{Rel}\webleft (A,X\webright )\to \mathbf{Rel}\webleft (B,X\webright ) \]

      along $R$ exists.

    2. The relation $R$ admits a left adjoint in $\textbf{Rel}$.
    3. The relation $R$ is of the form $f^{-1}$ (as in Definition 6.3.2.1.1) for some function $f$.

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 6.2.1.1.1, 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 6.2.3.1.1?

This question also appears as [MO 461592].


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