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For an object $X$ of a category, $h_X$ is the contravariant functor represented by $X$, i.e. $h_X = Hom(-,X)$.

Question a) Who invented this notation? (My guess: Grothendieck)

b) Is there a special reason why the letter $h$ was chosen? Is it in an abbreviation for "homomorphism"?

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I dunno, but I think the more interesting case is $h^X:=Hom(X,-)$, where it's a superscript because it is contravariant in $X$. – Harry Gindi Feb 2 '11 at 9:49
@Harry: Offtopic. – Martin Brandenburg Feb 2 '11 at 10:54
Hey, don't be rude about it! If my comment is off topic, then your entire question is most certainly so. – Harry Gindi Feb 2 '11 at 17:22
up vote 9 down vote accepted

a) It was invented by Grothendieck, see EGA I, Springer edition, especially chapter 0, discussion of representable functors.

b) Quite possibly is a shortcut for $Hom$. Sometimes the letter $y$ is used (for Yoneda). The trouble is when you are considering the representable functor defined over several categories, e.g. a category and a subcategory.

Bonus: If you, instead of considering contravariant functors $\mathrm{Sch}^{o} \to \mathrm{Set}$, use covariant functors $\mathrm{Aff} \to \mathrm{Set}$ the notation used in EGA is $h_X^{o}$. Perhaps the reason is that Yoneda's map is contravariant in this case.

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Of course the notation $h_X$ is used extensively in EGA, but where can you find evidence that that it is Grothendieck's invention? – Martin Brandenburg Feb 2 '11 at 10:53
Further evidence: The notation is already on SGA 3 and 4. There are several exposés by Grothendieck in Henri Cartan's seminar from 1960/61 in which he explains his point of view of Teichmüller's space through representable functors in the analytical category and he uses the notation $h_X$. I don't think anyone else was using these ideas at that time. Cartan's seminar is available at – Leo Alonso Feb 2 '11 at 11:16

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