-
1.
The definition of relations (Section 8.1.1).
-
2.
How relations may be viewed as decategorification of profunctors (Section 8.1.2).
-
3.
The various kinds of categories that relations form, namely:
-
(a)
A category (Section 8.3.2).
-
(b)
A monoidal category (Section 8.3.3).
-
(c)
A 2-category (Section 8.3.4).
-
(d)
A double category (Section 8.3.5).
-
(a)
-
4.
The various categorical properties of the 2-category of relations, including:
-
(a)
The self-duality of $\mathsf{Rel}$ and $\boldsymbol {\mathsf{Rel}}$ (Proposition 8.5.1.1.1).
-
(b)
Identifications of equivalences and isomorphisms in $\boldsymbol {\mathsf{Rel}}$ with bijections (Proposition 8.5.2.1.2).
-
(c)
Identifications of adjunctions in $\boldsymbol {\mathsf{Rel}}$ with functions (Proposition 8.5.3.1.1).
-
(a)
-
(d)
Identifications of monads in $\boldsymbol {\mathsf{Rel}}$ with preorders (
).
-
(e)
Identifications of comonads in $\boldsymbol {\mathsf{Rel}}$ with subsets (
).
-
(f)
A description of the monoids and comonoids in $\boldsymbol {\mathsf{Rel}}$ with respect to the Cartesian product (Remark 8.5.9.1.1).
-
(g)
Characterisations of monomorphisms in $\mathsf{Rel}$ (
).
-
(h)
Characterisations of 2-categorical notions of monomorphisms in $\boldsymbol {\mathsf{Rel}}$ (
).
-
(i)
Characterisations of epimorphisms in $\mathsf{Rel}$ (
).
-
(j)
Characterisations of 2-categorical notions of epimorphisms in $\boldsymbol {\mathsf{Rel}}$ (
).
-
(k)
The partial co/completeness of $\mathsf{Rel}$ (Proposition 8.5.14.1.1).
-
(l)
The existence or non-existence of Kan extensions and Kan lifts in $\mathsf{Rel}$ (
).
-
(m)
The closedness of $\boldsymbol {\mathsf{Rel}}$ (Proposition 8.5.19.1.1).
-
(n)
The identification of $\boldsymbol {\mathsf{Rel}}$ with the category of free algebras of the powerset monad on $\mathsf{Sets}$ (Proposition 8.5.20.1.1).
-
5.
The adjoint pairs
\begin{align*} R_{!} \dashv R_{-1} & \colon \mathcal{P}(A) \rightleftarrows \mathcal{P}(B),\\ R^{-1} \dashv R_{*} & \colon \mathcal{P}(B) \rightleftarrows \mathcal{P}(A) \end{align*}of functors (morphisms of posets) between $\mathcal{P}(A)$ and $\mathcal{P}(B)$ 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 8.7).
Of particular note are the following points:
-
(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.
-
(b)
We have $R_{-1}=R^{-1}$ iff $R$ is total and functional (
of ).
-
(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
\[ f_{!}\dashv f^{-1}\dashv f_{*} \] -
(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)
-
6.
A description of two notions of “skew composition” on $\mathbf{Rel}(A,B)$, giving rise to left and right skew monoidal structures analogous to the left skew monoidal structure on $\mathsf{Fun}(\mathcal{C},\mathcal{D})$ appearing in the definition of a relative monad (Section 8.8 and Section 8.9).
-
1.
Replicate Section 8.5 for apartness composition
-
2.
Revise Section 8.7
-
3.
Add subsection “A Six Functor Formalism for Sets, Part 2”, now with relations, building upon Section 8.7.
-
4.
Replicate Section 8.7 for apartness composition
-
5.
Revise sections on skew monoidal structures on $\mathbf{Rel}(A,B)$
-
6.
Replicate the sections on skew monoidal structures on $\mathbf{Rel}(A,B)$ for apartness composition.
-
7.
Explore relative co/monads in $\boldsymbol {\mathsf{Rel}}$, defined to be co/monoids in $\mathbf{Rel}(A,B)$ with its left/right skew monoidal structures of Chapter 8: Relations, Section 8.8 and Section 8.9
-
8.
functional total relations defined with “satisfying the following equivalent conditions:”
8 Relations
This chapter contains some material about relations. Notably, we discuss and explore:
This chapter is under revision. TODO:
