hom(C) in category theory

Question

I know in the basic definition of a category you have the class hom(C) of morphisms between objects in the category C. What never seems to be clear from textbook definitions is this: Are the members of hom(C) the hom classes, hom(a,b), where a and b are object in C, or do they just slop all the morphisms between any of the objects in C together into a single class?

Answer 1

To my knowledge there is not the way to make precise in the language of set theory the concept of a category, but at least the following two options exist. In the first, you naturally encounter $\text{hom}({\mathcal C})$ as the class of morphism sets, while in the second, you rather get $\hom({\mathcal C})$ as the class of all morphisms.

• A category consists of a class (formula) ${\mathcal C}$ together with a class function (formula) $\text{hom}: {\mathcal C}\times{\mathcal C}\to \text{Set}$ and a $6$-ary composition predicate (formula) $\circ\subset{\mathcal C}^3\times \text{Set}^3$ satisfying the usual axioms.

In this formulation, the morphism sets need not be disjoint. The image class of the class function $\text{hom}$ has as elements the hom-sets of your category, and its union is the class of all morphisms.

Alternatively, you can use:

• A category consists of classes ${\mathcal C}_0$ and ${\mathcal C}_1$ together with class functions (formulas) $s: {\mathcal C}_1\to{\mathcal C}_0$ and $t: {\mathcal C}_1\to {\mathcal C}_0$ as well as a composition relation (formula) $\circ\subset{\mathcal C}_1\times{\mathcal C}_1\times {\mathcal C}_1$ which is a class function ${\mathcal C}_1\times_{{\mathcal C}_0} {\mathcal C}_1\to {\mathcal C}_1$, such that the fibers of $(s,t): {\mathcal C}_1\to {\mathcal C}_0\times{\mathcal C}_0$ are sets and the usual axioms are satisfied.

In this formulation the morphism sets are indeed disjoint. The class of all homomorphism is ${\mathcal C}_1$, while the class of all hom-sets is the class of fibers of the class function $(s,t)$.

The latter formulation also fits well with how you define internal category objects in cartesian closed categories.