Notation for a representable functor - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T14:35:01Z http://mathoverflow.net/feeds/question/54076 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54076/notation-for-a-representable-functor Notation for a representable functor Martin Brandenburg 2011-02-02T08:40:39Z 2011-02-03T00:13:07Z <p>For an object $X$ of a category, $h_X$ is the contravariant functor represented by $X$, i.e. $h_X = Hom(-,X)$.</p> <p><strong>Question</strong> a) Who invented this notation? (My guess: Grothendieck)</p> <p>b) Is there a special reason why the letter $h$ was chosen? Is it in an abbreviation for "homomorphism"?</p> http://mathoverflow.net/questions/54076/notation-for-a-representable-functor/54085#54085 Answer by Leo Alonso for Notation for a representable functor Leo Alonso 2011-02-02T10:47:34Z 2011-02-02T10:47:34Z <p>a) It was invented by Grothendieck, see EGA I, Springer edition, especially chapter 0, discussion of representable functors.</p> <p>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.</p> <p>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.</p>