Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In the definition of a natural transformation from

it is said that a natural transformation is $\alpha : F \Rightarrow G$ (and the definition continues)

But what does that $\Rightarrow$ mean, how is it called?

share|cite|improve this question
I read that as "maps to"... – J. M. Aug 21 '11 at 11:57
Do you mean $\mapsto$ or $\to$? – Yrogirg Aug 21 '11 at 12:00
It's called a arrow/morphism in the functor category if you insist on giving it a name. @J.M. I read that as "$\alpha$ is a natural transformation from $F$ to $G$." or simply "$\alpha$ from $F$ to $G$" – t.b. Aug 21 '11 at 12:09
I think the use of $\Rightarrow$ for natural transformations is more common among higher categorists, who think of them as $2$-morphisms in the $2$-category $\textbf{Cat}$. – Zhen Lin Aug 21 '11 at 14:39
@Zhen: that's a very good point. In fact, I don't know where the notation comes from but it is pretty standard in homological algebra nowadays even without mention of $2$-categorical ideas. I checked the usual suspects and the first reference I've found up to now using the $\Rightarrow$ notation is Bénabou's Introduction to bicategories. It doesn't appear that it was used earlier by any of the founding fathers Eilenberg-Mac Lane, Ehresmann, Freyd, Gabriel, Grothendieck, Kan, etc. – t.b. Aug 21 '11 at 22:27
up vote 0 down vote accepted

Just look in the comments, especially the one by t.b., $\Rightarrow$ is just the notion of natural transformation, nothing more general.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.