What does "hom" stand for in hom-sets and hom-functors?

With given category $\mathcal{C}$ and its objects $A$ and $B$, a hom-set $\hom_\mathcal{C}(A, B)$ is the collection of all morphisms from $A$ to $B$. There is also a related notion of hom-functor from $\mathcal{C}$ to $\mathcal{Set}$, which map objects into related hom-sets.

But why the name "hom"? Who was the first to use this terminology and in what context?

Solution 1:

"Hom" stands for homomorphism, the usual name for structure preserving functions in algebra. I believe the terminology goes back to Eilenberg and Mac Lane's original article on category theory. Of course, in an arbitrary category the objects need not have structure, and the morphisms need not even be functions. Thus the terminology "morphisms" for the arrows in a category. Some texts use $Mor(x,y)$ instead of $Hom(x,y)$. It is also common to be more agnostic and simply write $C(a,b)$ for the morphisms from $a$ to $b$ in the category $C$.