MathOverflow is a question and answer site for professional mathematicians. 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

A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I.

The question is then, does Fields have a weakly initial set? This is equivalent to the collection of prime fields being a set.

The converse is, is there a (fairly natural) example of a category without a weakly initial set? Aside from obvious things like the discrete category on the objects of a large category.

share|cite|improve this question
Actually, this is a bit of a folly. My guess is that the fields Q, F_p are enough, but I wonder if there are crazy model theory type things that go on at large cardinals. Or am I worrying about nothing? – David Roberts Oct 6 '10 at 4:33
Sorry, David, I'm not sure what you're worried about. Any field contains a smallest field (the smallest field containing 1) where 1 is either torsion (making the smallest field a finite field F_p) or not (making it Q). – Todd Trimble Oct 6 '10 at 5:13
Well, there's the opposite of Fields, if that's natural enough? – George Lowther Oct 6 '10 at 6:58
Ah, that's nice. So Fields has a weakly initial set, and Fields^op doesn't. Thanks. For the amount of space I want to talk about these two examples, that should be better than Todd's answer. – David Roberts Oct 6 '10 at 8:00
And I realised how stupid my question regarding Fields was, but the second part of the question was less stupid, so I left it up. – David Roberts Oct 6 '10 at 8:02
up vote 1 down vote accepted

Regarding the second question: I'm not sure what should count as "natural", but couldn't you just work with examples where the solution set condition in an adjoint functor theorem fails? The solution set is a weakly initial set in a comma category.

For example, there is no left adjoint to the underlying-set functor $U$ from complete Boolean algebras to sets, and in particular no free complete Boolean algebra on a countably infinite set. But the category of complete Boolean algebras is small-complete and $U$ preserves all small limits. So it's the solution set condition that fails, and therefore the comma category

$$\mathbb{N} \downarrow U$$

has no weakly initial set.

Edit: After reading David's request for really simple, I offer instead $Ord^{op}$, where $Ord$ is the class of ordinals ordered by inclusion. I acknowledge the influence of Laurent's answer.

share|cite|improve this answer
HI Todd - took away the 'accepted answer' tick, as more answers are rolling in, and I really need simple examples (really simple!) – David Roberts Oct 6 '10 at 8:53

How about the category of sets with injective maps as morphisms? As an ad hoc example, this may not count as "natural", but it's simple enough.

[EDIT] following Martin's comment: take the dual, or replace "injective" by "surjective".

share|cite|improve this answer
I think you mean "surjective" (with "injective", the empty set is initial). Then this is almost the same example as the dual category of Fields. – Martin Brandenburg Oct 6 '10 at 8:50
Sorry, "surjective" indeed. Actually I had in mind the dual category, in analogy with the dual of Fields. I agree about your other comment; my point is only that this example is in a sense more straightforwrd. – Laurent Moret-Bailly Oct 6 '10 at 10:54

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.