What is a locally cosmall category relative to a universe?
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2$\begingroup$ I've never heard of the phrase 'cosmall'. What do you mean by this? Where did you hear it? $\endgroup$– David Roberts ♦May 18, 2012 at 10:03
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$\begingroup$ PS I edited the tags. $\endgroup$– David Roberts ♦May 18, 2012 at 10:04
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$\begingroup$ 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! $\endgroup$– JamalMay 18, 2012 at 12:16
1 Answer
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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$\begingroup$ 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. $\endgroup$– JamalMay 18, 2012 at 12:33
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$\begingroup$ Yes, Enochs also uses this terminology (locally cosmall = co-well-powered). $\endgroup$ May 18, 2012 at 14:42
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$\begingroup$ 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... $\endgroup$ May 18, 2012 at 22:21