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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"?

flag	asked Feb 2 2011 at 8:40 Martin Brandenburg 29.1k●2●45●129

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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 2011 at 9:49

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@Harry: Offtopic. – Martin Brandenburg Feb 2 2011 at 10:54

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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 2011 at 17:22

Leo Alonso · Accepted Answer · 2011-02-02 10:47:34Z UTC

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.

Notation for a representable functor

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