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categorical direct product is an inverse limit
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(Theorem)
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Proof. Suppose we have a direct product of
 for some (
 ) set  . Consider  as a category whose arrows are only the identity arrows. Then we can define a functor  by  . It is then clear that the universal property of an inverse limit is equivalent to the universal property defining a categorical direct product. 
Reversing the arrows, it is also clear that the categorical direct sum is an example of a direct limit.
These results are of interest when one is looking to prove exactness of sums and products in a category: often it is easier to address exactness of direct and inverse limits, and the result then applies to many other constructions as well.
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"categorical direct product is an inverse limit" is owned by archibal.
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Cross-references: products, sums, categorical direct sum, equivalent, universal property, clear, functor, identity, category, inverse limit, categorical direct product
There is 1 reference to this entry.
This is version 1 of categorical direct product is an inverse limit, born on 2004-02-25.
Object id is 5620, canonical name is CategoricalDirectProductIsAnInverseLimit.
Accessed 1409 times total.
Classification:
| AMS MSC: | 18A30 (Category theory; homological algebra :: General theory of categories and functors :: Limits and colimits ) |
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Pending Errata and Addenda
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