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What is a locally cosmall category relative to a universe?

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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 springerlink.com/content/9027521144483628) 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

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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".

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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

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