A $2$-category (Section 8.3.4).

A double category (Section 8.3.5).

The various categorical properties of the $2$-category of relations, including:

1. (a)

   The self-duality of $\mathsf{Rel}$ and $\boldsymbol {\mathsf{Rel}}$ (Proposition 8.4.1.1.1).

2. (b)

   Identifications of equivalences and isomorphisms in $\boldsymbol {\mathsf{Rel}}$ with bijections (Proposition 8.4.2.1.1).

3. (c)

   Identifications of adjunctions in $\boldsymbol {\mathsf{Rel}}$ with functions (Proposition 8.4.3.1.1).

4. (d)

   Identifications of monads in $\boldsymbol {\mathsf{Rel}}$ with preorders (Proposition 8.4.4.1.1).

5. (e)

   Identifications of comonads in $\boldsymbol {\mathsf{Rel}}$ with subsets (Proposition 8.4.5.1.1).

6. (f)

   A description of the monoids and comonoids in $\boldsymbol {\mathsf{Rel}}$ with respect to the Cartesian product (Remark 8.4.6.1.1).

7. (g)

   Characterisations of monomorphisms in $\mathsf{Rel}$ (Proposition 8.4.7.1.1).

8. (h)

   Characterisations of $2$-categorical notions of monomorphisms in $\boldsymbol {\mathsf{Rel}}$ (Proposition 8.4.8.1.1).

9. (i)

   Characterisations of epimorphisms in $\mathsf{Rel}$ (Proposition 8.4.9.1.1).

10. (j)

    Characterisations of $2$-categorical notions of epimorphisms in $\boldsymbol {\mathsf{Rel}}$ (Proposition 8.4.10.1.1).

11. (k)

    The partial co/completeness of $\mathsf{Rel}$ (Proposition 8.4.11.1.1).

12. (l)

    The existence or non-existence of Kan extensions and Kan lifts in $\mathsf{Rel}$ (Remark 8.4.12.1.1).

13. (m)

    The closedness of $\boldsymbol {\mathsf{Rel}}$ (Proposition 8.4.13.1.1).

14. (n)

    The identification of $\boldsymbol {\mathsf{Rel}}$ with the category of free algebras of the powerset monad on $\mathsf{Sets}$ (Proposition 8.4.14.1.1).

The adjoint pairs

of functors (morphisms of posets) between $\mathcal{P}\webleft (A\webright )$ and $\mathcal{P}\webleft (B\webright )$ induced by a relation $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$, as well as the properties of $R_{!}$, $R_{-1}$, $R^{-1}$, and $R_{*}$ (Section 8.5).

Of particular note are the following points:

1. (a)

   These two pairs of adjoint functors are the counterpart for relations of the adjoint triple $f_{!}\dashv f^{-1}\dashv f_{*}$ induced by a function $f\colon A\to B$ studied in Chapter 4: Constructions With Sets, Section 4.6.

2. (b)

   We have $R_{-1}=R^{-1}$ iff $R$ is total and functional (Item 8 of Proposition 8.5.2.1.3).

3. (c)

   As a consequence of the previous item, when $R$ comes from a function $f$, the pair of adjunctions

   \[ R_{!}\dashv R_{-1}=R^{-1}\dashv R_{*} \]

   reduces to the triple adjunction

   \[ f_{!}\dashv f^{-1}\dashv f_{*} \]

   from Chapter 4: Constructions With Sets, Section 4.6.

4. (d)

   The pairs $R_{!}\dashv R_{-1}$ and $R^{-1}\dashv R_{*}$ turn out to be rather important later on, as they appear in the definition and study of continuous, open, and closed relations between topological spaces (, ).

A description of two notions of “skew composition” on $\mathbf{Rel}\webleft (A,B\webright )$, giving rise to left and right skew monoidal structures analogous to the left skew monoidal structure on $\mathsf{Fun}\webleft (\mathcal{C},\mathcal{D}\webright )$ appearing in the definition of a relative monad (Section 8.6 and Section 8.7).