Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let D have small hom sets. Does this functor category Set^D have small hom-sets? I think this is not true, I mean each functor can't possibly be coded as a small set if D itself is large. I'm working within a universe.

share|improve this question

1 Answer 1

This is indeed not true. Peter Freyd and Ross Street proved in the paper on the size of categories that both $\mathcal{D}$ and $\mathbf{Set}^{\mathcal{D}}$ have small hom-sets if and only if the category $\mathcal{D}$ is essentially small, so any category which has a large set of objects will give a counterexample.

share|improve this answer
To be nitpicky, any category with a large set of objects which is ALSO not equivalent to one with a small set of objects would provide a counterexample, e.g. the category of Sets (or more generally, any Grothendieck topos). –  David Carchedi May 24 '10 at 19:17
or, a discrete category with a large set of objects. –  Mike Shulman May 24 '10 at 19:32
Not the first time the question has been asked. See "Is the presheaf category of a locally small category locally small?" –  Todd Trimble May 24 '10 at 20:40

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.