Proposition 9.2.1.1.1 (00NJ): Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$

Table of Contents
Part 3: Relations
Chapter 9: Constructions With Relations
Section 9.2: Kan Extensions and Kan Lifts in the $2$-Category of Relations
Subsection 9.2.1: Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
Proposition 9.2.1.1.1: Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$ (cite)

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9.2.1 Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$

Proposition 9.2.1.1.1 ▶ Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$

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 $\boldsymbol {\mathsf{Rel}}$. Not all relations in $\boldsymbol {\mathsf{Rel}}$ admit left Kan extensions.
2.
Characterisation of Relations Admitting Left Kan Extensions Along Them. The following conditions are equivalent:
1. (a)
  The left Kan extension
  
  \[ \operatorname {\mathrm{Lan}}_{R}\colon \mathbf{Rel}\webleft (A,X\webright )\to \mathbf{Rel}\webleft (B,X\webright ) \]
  
  along $R$ exists.
2. (b)
  The relation $R$ admits a left adjoint in $\boldsymbol {\mathsf{Rel}}$.
3. (c)
  The relation $R$ is of the form $f^{-1}$ (as in ) for some function $f$.

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].

Question 9.2.1.1.2 ▶ Existence of Specific Left Kan Extensions of Relations

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].

Question 9.2.1.1.3 ▶ Explicit Description of Left Kan Extensions Along Functions

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].

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