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

What is a locally cosmall category relative to a universe?

share|cite|improve this question
I've never heard of the phrase 'cosmall'. What do you mean by this? Where did you hear it? – David Roberts May 18 '12 at 10:03
PS I edited the tags. – David Roberts May 18 '12 at 10:04
I have seen this concept in a paper of Enochs(see section 7, of in module theory where the author extends a module version of a theorem in general in a locally small and cosmall category relative to a Grothendieck universe. So the category of right modules over a unitary ring may be an example of a locally cosmall category! – Jamal May 18 '12 at 12:16
up vote 3 down vote accepted

Unfortunately, there is a clash of terminology: Some people call a category locally small if the hom-classes are sets. These people call a category well-powered if the subobjects of any object form a set; dually, co-well-powered refers to quotients. For these people, there is no notion of locally cosmall. Other people (and this seems to be the context of the question) call a category locally small if the subobjects of any object form a set (i.e. what the former call well-powered), and the dual notion concerning quotients locally cosmall (i.e. what the former call co-well-powered). For a reference, see Pareigis' "Categories and functors", Section 1.6. If you work with universes, replace "set" by "element of U" and "class" with "subset of U".

share|cite|improve this answer
Dear Martin, thank you very much for your answer and the reference. I have seen this concept in a paper in module theory (cited in comment, above). So the second definition of yours may be abut the subject. – Jamal May 18 '12 at 12:33
Yes, Enochs also uses this terminology (locally cosmall = co-well-powered). – Martin Brandenburg May 18 '12 at 14:42
Good knowledge! I hadn't heard this other usage of "locally small". For what it's worth, some people prefer "well-copowered" to "co-well-powered": it's more logical, if you think about it... – Tom Leinster May 18 '12 at 22:21

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.