# 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?

• homomorphism (of rings, groups) I suppose
– user99914
Apr 1, 2015 at 9:54
• It is an apocope of homomorphism, that seems pretty clear… Apr 1, 2015 at 9:56
• Alternatively, in the fundamental groupoid of a topological space, $\mathsf{hom}(x,y)$ are the "homotopy classes of paths from $x$ to $y$". Jan 26, 2016 at 12:04

"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 $C$.
• the objects can be anything at all. For instance, the objects of a category could be the letters $a,b,c$. No structure, just symbols. Apr 7, 2015 at 23:10