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wellpowered category
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(Definition)
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Wellpoweredness is a kind of smallness condition on a category.
Let  be a class of monomorphisms. A category is said to be  -wellpowered if for any object any class of parwise non-isomorphism  -subobjects is a set. (By a  -subobject of an object  we understand a pair  , where  is a morphism belonging to  .) In other words, if we consider isomorphic objects as the same object, the class of all  -subobjects is a set.
More precisely, for any  there exists a set of  -subobjects  ,  such that for any extremal subobject  of the object  there exists  and
an isomorphism
 such that
 .
If  is the class of all regular monomorphisms, extremal monomorphisms, monomorphisms, we speak about regular wellpowered, extremally wellpowered, wellpowered category.
Dual notions: regular cowellpowered, extremally cowellpowered, cowellpowered category.
- 1
- J. Adámek, H. Herrlich, and G. Strecker.
Abstract and Concrete Categories.
Wiley, New York, 1990.
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"wellpowered category" is owned by kompik.
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See Also: subobject
| Other names: |
wellpowered, well-powered, locally small |
| Also defines: |
extremally wellpowered, regular wellpowered, cowellpowered, extremally cowellpowered, regulary cowellpowered |
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Cross-references: regular, extremal monomorphisms, regular monomorphisms, isomorphism, subobject, isomorphic, morphism, object, monomorphisms, class, category
There are 5 references to this entry.
This is version 6 of wellpowered category, born on 2006-06-30, modified 2008-11-07.
Object id is 8113, canonical name is WellpoweredCategory.
Accessed 2719 times total.
Classification:
| AMS MSC: | 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations) |
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Pending Errata and Addenda
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