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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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    Thank you for your comments.2012-03-17

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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.