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?

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    $\begingroup$ homomorphism (of rings, groups) I suppose $\endgroup$ – user99914 Apr 1 '15 at 9:54
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    $\begingroup$ It is an apocope of homomorphism, that seems pretty clear… $\endgroup$ – Bernard Apr 1 '15 at 9:56
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    $\begingroup$ Alternatively, in the fundamental groupoid of a topological space, $\mathsf{hom}(x,y)$ are the "homotopy classes of paths from $x$ to $y$". $\endgroup$ – JustAskin Jan 26 '16 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 $C$.

  • $\begingroup$ What is meant by "Of course, in an arbitrary category the objects need not have structure"? $\endgroup$ – Martin Brandenburg Apr 7 '15 at 13:06
  • $\begingroup$ 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. $\endgroup$ – Ittay Weiss Apr 7 '15 at 23:10
  • $\begingroup$ I would say that the hom-sets are the structure on the objects. $\endgroup$ – Martin Brandenburg Apr 7 '15 at 23:11
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    $\begingroup$ that may be a bit misleading. The hom sets are the categorical structure on the objects induced by that particular category. Different categories may have the same objects, so the objects themselves do not have structure. $\endgroup$ – Ittay Weiss Apr 7 '15 at 23:14
  • $\begingroup$ Yes, sure, I fix a category ... $\endgroup$ – Martin Brandenburg Apr 7 '15 at 23:17

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