Notation for a representable functor

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"?

• I dunno, but I think the more interesting case is $h^X:=Hom(X,-)$, where it's a superscript because it is contravariant in $X$. Commented Feb 2, 2011 at 9:49
• @Harry: Offtopic. Commented Feb 2, 2011 at 10:54
• Hey, don't be rude about it! If my comment is off topic, then your entire question is most certainly so. Commented Feb 2, 2011 at 17:22

Let me answer the questions in order.

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.

Further evidence on a) 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 an not aware of anyone else using these ideas at that time. Cartan's seminar is available at numdam:

http://www.numdam.org/article/SHC_1960-1961__13_1_A7_0.pdf

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.
• Of course the notation $h_X$ is used extensively in EGA, but where can you find evidence that that it is Grothendieck's invention? Commented Feb 2, 2011 at 10:53