Is there a meaningful difference betwen formulating categories via hom-sets vs. via assigning domains and codomains to morphisms?

Question

I usually see categories defined as having a class of objects, a class of morphisms, and operations that send each morphism to its domain and its codomain. I tend to find it more natural to think of a category as having a class of objects, and an assignment, to each pair of objects, of a hom-class. In material set theoretic foundations, the former kind of category always uniquely defines one of the latter kind of category, but the latter kind of category need not have disjoint homs and hence need not determine a category by the former definition. I usually handwave this, but I do wonder: Can this ever be the source of an actual problem?

Answer 1

For any category $\mathbf{A}$ in the second sense you always have a category $\mathbf{B}$ which is isomorphic to $\mathbf{A}$ and is such that its hom-sets are disjoint. The construction of $\mathbf{B}$ is as follows,

• its objects are the same as that of $A$

• for each $\mathbf{A}$-morphism $f\in \operatorname{hom}_{\mathbf{A}}(A,B)$ for some $\mathbf{A}$-objects $A,B$ the corresponding $\mathbf{B}$-morphism is defined by $(A,f,B)$. In other words, $f\in \operatorname{hom}_{\mathbf{A}}(A,B)$ iff $(A,f,B)\in \operatorname{hom}_{\mathbf{B}}(A,B)$.

Since the categories are isomorphic. You don't really lose anything important so long as you are focusing only on the categorical properties. But having disjoint hom-sets sometimes gives you an advantage by letting you prove things more easily. Whereas when you are trying to show that some mathematical structure is a category, it is easier to show that by appealing to the later definition.

As a final remark, note that I kind of cheated when I said that, "You don't really lose anything important so long as you are focusing only on the categorical properties." because 'categorical properties' are precisely those properties of a category that are preserved by categories isomorphic to it. So, a more precise answer to your question would depend on the precise sense of the term "important".

Hope this helps.