# Exact Squares and Relation Lifting

(under construction) ... ([index](https://hackmd.io/@alexhkurz/Hy1oUrS4u))

## History

From the start with Puppe, given a category $C$, a relation is regarded as an arrow in a symmetrisation $s:C\to K$ of $C$. As part of the definition, any symmetrisation $K$ comes equipped with an involution $(-)^\ast$ fixing the objects of $C$ and mapping isos of $C$ to their inverses.[^symm-weakening] Symmetrisations are generalised relational extensions. It seems worth studying relational extensions inside this larger picture.

At first, only the abelian enriched situation was of interest. Later this method was extended to include ordinary categories (Klein, Grandis) and category-enriched categories (Guitart). [^symm-guitart]

Before Puppe, there was a paper by Mac Lane, but he credits unpublished work by Puppe, who also seems to have been the first to have a theorem characterising symmetrisations.

- [Puppe (1962)](https://link.springer.com/article/10.1007/BF01438388) defines relational morphisms for abelian categories and shows that every abelian (or Puppe exact) category $C$ can be embedded into a category $Rel(\mathcal C)$. Importantly, he has a universal characterisation of $Rel(\mathcal C)$ that involves an involution and is called the symmetrisation of $\mathcal C$. So he is the first to study **symmetrisations**, that is, relational extensions, of categories and to characterise them by universal properties.

- [Hilton (1966)](https://link.springer.com/chapter/10.1007/978-3-642-99902-4_11) introduces, still[^abelian] in the abelian setting, **exact squares**. Is this also the first place where relations as equivalence classes of spans, or cospans, appear? His interpretation of cospans $(i,j)$ as "right fractions" $i/j$ and of spans $(p,q)$ as "left fractions" $p\backslash q$ is very beautiful.

- [Brinkmann (1969)]: Constructs the symmetrisation of a category as the category of all zig-zags quotiented by taking pullbacks.

- [Brinkmann and Puppe (1969)]: This textbook is mainly concerned with the abelian situation, but starts out with ordinary categories.

- [Klein (1970)](https://projecteuclid.org/euclid.ijm/1256052950) proves results analogous to those of Puppe (and Mac Lane and Hilton), replacing abelian categories by **general categories**.[^bicategory] Klein does require $\mathcal C$ to have a factorisation system and he composes relations via pullbacks and image factorisation, but he does not insist on $\mathcal C$ being regular. Accordingly, he calls a *near-category* a category in which composition is not associative. Thm 2.5 characterises those categories for which relations are associative.[^regular]

- [Klein (1971)](https://www.jstor.org/stable/2038255)

- [Grandis (1975)](http://www.numdam.org/item/RSMUP_1975__54__271_0/) contains a useful overview of what has been done on symmetrisation (=relational extensions) of categories until this point. Was Grandis the first to see that $f^\ast$ is right-adjoint in $K$ to $f$?
- Chapter 1:
- (2.10) Maximal symmetrisation $C^M$ of a category $C$ as the category of "zig-zags" quotiented by composition and identities. Symmetrisations form a complete lattice and the minimal symmetrisation $C^m$ is given by the equivalence relation of connected components of $C$.
- (2.18) Regular symmmetrisations are those in which all arrows (relations) satisfies $R\cdot R^{-1}\cdot R=R$
- (2.22) Exact squares are defined wrt to a symmetrisation $s:C\to K$ as being commutative and translated under $s$ to "bicommutative" squares (ie the span and the cospan represent the same relation).
- (2.24) A symmetrisation $K$ has (co)binary factorisations if every arrow (relation) can be (co)tabulated. A sufficient condition can be formulated in terms of the existence of enough exact squares.
- Chapter 4: (4.1) A square type $q$ is a set of commutative squares. Every symmetrisation induces a square type and each square type induces a symmetrisation, forming a Galois connection. (4.10) Not every symmetrisation is induced by a square type. (4.13) Those induced by square types are characterised by the symmetrisation preserving exact squares.

- [Conte (1981)](http://www.numdam.org/item/CTGDC_1981__22_4_429_0/) Again, the introduction gives a good survey. Given a category $C$ there is a maximum symmetrisation $C^M$, see Grandis (1975). Equivalence relations on $C^M$ compatible with composition and involution are called congruences. Every involution category extending $C$ is a quotient of $C^M$ by a congruence. Such quotients can be specified by subcategories of $C$. In fact, any subcategory $E$ of $C$ spans a congruence ${R}_E$ on homs of the symmetrisation via
- in any pullback the span and the cospan define the same relation;
- if $e\in E$ is an arrow between two spans, then they define the same relation;

...

- [Guitart (1981)]() ... First to study the $Cat$-enriched situation ...

## Basic Facts

**Proposition:** In posets and preorders, exact squares are weak comma squares.

*Proof:* Straightforward from spelling out the definitions in the category of posets.

...

## Further References

When we wrote [Relation Lifting: A Survey](https://alexhkurz.github.io/papers/kv-relation-lifting.pdf), I was not aware of the history of the subject before Guitart's paper. These rudimentary notes show that I should spend some more time on this topic ...

...

[^symm-weakening]: This is the part that does not generalise to weakening the relations: The converse of a relation $A\to B$ is a relation $B^{op}\to A^{op}$.

[^symm-guitart]: As far as I remember, but I need to re-read Guitart, he drops symmetrisations because they are not compatible with the enriched viewpoint.

[^abelian]: One should remark here that the abelian setting is particularly rich and interesting. Composition of relations generalises multiplication of fractions and in abelian categories on can also define a corresponding addition on relations. Hilton continues, "moreover we make heavy use of the presence, within every equivalence class, of a minimal representative whose role is much like that of the expression of a fraction in its lowest terms". A law that fails for these "generalised fractions" in general is the distributive law, but there are conditions under which it can be obtained. Interestingly, Hilton credits Lambek and Beberman's New Mathematics for helping him discover exact squares.

[^bicategory]: When Klein says "bicategory", he means a category with a factorisation system.

[^regular]: If I understand correctly, it is enough to have a category with pullbacks and a factorisation system in which the "epis" are stable under pullbacks. This is weaker than a regular category.