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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
 -small
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(Definition)
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Let
 be a universe (so is, in particular, a set of sets).
A set  is said to be
 -small if it is isomorphic to an element of
 (i.e., there is a bijection between  and some element of
 ).
A category  is
 -small (or just small, if no confusion is likely to arise) if the set of objects of  is isomorphic to a set in
 , and is a
 -category if for every pair of objects  ,  in  ,
 is isomorphic to a set in
 .
These definitions amount to restrictions on the cardinality of the objects involved, and are intended to provide a condition that will allow operations such as extracting the category of functors or taking the direct limit to give results that are reasonable, that is, either isomorphic to an object of
 or made up of objects of
 .
Observe that the category of subsets of
 is a
 -category but is not
 -small.
- SGA4
- Grothendieck et al., Séminaires en Gèometrie Algèbrique 4, tomes 1, 2, and 3.
- Mur68
- Murphy, O. Some modern methods in the theory of lion hunting, American Mathematical Monthly 75 (2), Feb., 1968, 185-187.
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" -small" is owned by mathcam. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: subsets, direct limit, functors, operations, cardinality, definitions, objects, category, bijection, isomorphic, universe
There are 15 references to this entry.
This is version 8 of  -small, born on 2004-03-01, modified 2008-03-18.
Object id is 5658, canonical name is Small.
Accessed 6961 times total.
Classification:
| AMS MSC: | 03E30 (Mathematical logic and foundations :: Set theory :: Axiomatics of classical set theory and its fragments) | | | 18A15 (Category theory; homological algebra :: General theory of categories and functors :: Foundations, relations to logic and deductive systems) |
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Pending Errata and Addenda
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