Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

If I have functors from C to Set for a small category C and a natural transformation between them, how can I show that this natural transformation is monomorphic iff each of its components, indexed by the objects of C, is an injection of Set?

share|cite|improve this question
If a natural transformation is componentwise monic, then it is monic as a natural transformation – this direction is easy. The converse can be proven in several different ways, but you have to use something about the category $\textbf{Set}$. The first one that comes to my mind is to use the fact that $\textbf{Set}$ has equalisers, but this can be done by completely elementary means as well. – Zhen Lin Oct 14 '12 at 14:03

Hint: Read the definitions and try to work out what they mean. The first direction, to prove that if it is monic in each of its components is easy. For the other direction, try the following: Try to see if you can find any properties characterising a monomorphism in the category of functors from C to Set. Set has pullbacks, what relates pullbacks and monomorphisms? Construct the pullback of your natural transformation with itself, and check what this implies levelwise.

share|cite|improve this answer
Which natural monomorphisms (in general) aren't collections of monomorphism? I'm having trouble thinking of an example. – PyRulez May 18 '15 at 10:44

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.