Object preserving congruences of categories

Let $C$ be a category then a wide congruence $(P,Q)$ is a congruence in which no two objects are equated with one another. In that case, a wide congruence of $C$ can simply be considered to be determined by the partition $Q$ which is called an arrow congruence in toposes, triples, and theories [1].

Definition. let $C$ be a category then an arrow congruence $E$ is a partition of $Arrows(C)$ such that $E$ is:

• Quiver congruence: $f E f'$ implies that $s(f) \, E \, s(f')$ and $t(f) \, E \, t(f'))$
• Compositional congruence: $f E f'$ and $g E g'$ such that $fg$ and $f'g'$ are defined then $(fg) E (f'g')$.

These coincide with the definition of a congruence of a category, with the condition that the object partition is the trivial partition that equates no objects.

We saw in that general context of congruences of categories, that given a congruence $(P,Q)$ of the category $C$ then its quotient need not be a category. Instead, it is quite often a partial magmoid. This unusual behavior of $Cat$ is a consequence of the fact that it is not a topos. We can solve this by embedding it in something with the nicer structure of a topos like the topos $CQ$ of compositional quivers.

The issue of partiality means that for some congruences $(P,Q)$ in the quotient structure for some morphisms $f: A \to B$ and $g: B \to C$ a composite $gf$ need not exist. We would like to study the special case of object preserving congruences, to see if they have category forming quotients. In that case, that tells us something about the behavior of congruences of categories and their quotients.

Theorem. let $C$ be a category and let $E$ be an arrow congruence of $C$. Form the equivalence minimal object congruence $M$. Then the quotient $\frac{C}{(M,E)}$ is a category.

Proof. in order for something to be a category it must have a number of properties:

1. this is a quiver congruence by definition so $\frac{C}{(M,E)}$ is a quiver.
2. it is also a unital quiver congruence because $(M,E)$ is a congruence of the identity function. No two objects are equated by $M$ so no two identity morphisms need to be equated by $E$. It follows that no matter what $(M,E)$ is a unital quiver congruence.
3. $\frac{C}{(M,E)}$ is certainly a quotient unital partial magmoid because it is a compositional congruence.
4. the only thing that remains therefore is to check for the totality of $\circ$

So to prove condition four, we will let $C_1: X \to Y$ and $C_2 : Y \to Z$. The only thing that remains is for us to prove that $C_2 \circ C_1$ exists. As objects are preserved, then for each $m : X \to Y$ and $n: Y \to Z$ with $m \in C_1$ and $n \in C_2$ then their composition $n \circ m$ exists and it is in a class $C_3$. Therefore, the composition of any two classes in $E$ exists. It follows that $\frac{C}{(M,E)}$ is not simply a partial magmoid, it is also total and therefore a category. $\square$

We see that if we have a category with four objects and two non-trivial morphisms $m: A \to B$ and $n: C \to D$ then if we equate $B$ and $C$ we get a quotient partial magmoid in which the composition of arrows need not be defined. The problem here is that we are equating two objects and not two morphisms. If we don't equate any two objects then the quotient is always a category.

Equating two objects in a congruence is something you should never do in a category theory, because these partial magmoid things come out and they can no longer be considered categories. However, in the object preserving case the setting of categories is enough. This theorem can naturally be generalized to magmoids and semigroupoids.

Corollary. let $M$ be a magmoid and $E$ an arrow congruence of $M$ then $\frac{M}{E}$ is a magmoid.

As in the case of categories, it is necessary that the magmoid congruence should preserve all objects so that the quotient is not a partial magmoid. If you equate two objects then in the quotient structure the composition may not exist. So equating objects can eliminate totality of a categorical structure. These arrow congruences form a lattice.

Definition. let $C$ be a category then define its lattice of arrow congruences $AC(C)$ as the set of arrow congruences of $C$ with the intersection of equivalence relations as its meet and the arrow congruence closure of the join of partitions as its join. The lower bound of $AC(C)$ is the congruence with $C$ as its quotient and the upper bound is the congruence with the underlying preorder of $C$ as its quotient.

The thin congruence of $C$ which equates all arrows in equal hom classes is the maximal arrow congruence in the lattice $AC(C)$. It produces the underlying preorder of $C$ as a quotient. Then if we consider the full congruence lattice of a category $Con(C)$ there is an embedding functor $F: AC(C) \to Con(C)$ that turns any arrow congruence into a categorical congruence by producing the partition which preserves all objects and that equates all morphisms accordingly.

Definition. let $F: C \to D$ be a functor then $F$ is an object preserving functor if for all objects $a,b \in Ob(C)$ then $F(a) = F(b)$ implies that $a = b$.

Definition. $Cat_*$ is the category of categories with only object preserving functors between them

Then $Cat_*$ has all epi-mono factorisations and there exists a homomorphism theorem for $Cat_*$: every single object preserving functor has an arrow congruence and an image category such that the quotient category is isomorphic to the image. This is a small part of the fuller theory of subobjects, quotients, and epi-mono factorisations in a topos but it is an interesting part of the bigger picture.

See also:
Object preserving congruences of quivers

References:
[1] Toposes, triples, and theories