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I came across the notion of inductive limits in C*-algebras, where they exist. Except for the category of finite sets, what are natural examples of categories which fail to have inductive limits?

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I guess examples like "a non-complete boolean algebra considered as a category" don't count for you either? Obviously, categories with all (co)limits are nice, and one tries to avoid studying categories which are not nice. –  Zhen Lin Mar 16 '12 at 20:54
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Since your question is tagged functional analysis: The categories of Banach/Fréchet/Hilbert spaces with bounded linear maps or separable $C^\ast$-algebras or general Banach algebras would be natural examples. –  t.b. Mar 16 '12 at 20:58
    
@Zhen: from an analytic perspective completeness is often a bit too much to ask for and I insist that the categories I mentioned are nice and natural. :) –  t.b. Mar 16 '12 at 21:00
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The category of fields doesn't have coproducts, much less arbitrary direct limits. –  SL2 Mar 16 '12 at 21:01
    
Thank you for your comments. –  user26770 Mar 17 '12 at 16:19
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The comments contain the following examples:

  • A non-complete Boolean algebra considered as a category;
  • The category of Banach/Fréchet/Hilbert spaces with bounded linear maps as morphisms;
  • Categories of separable $C^*$-algebras and Banach algebras;
  • The category of fields.
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