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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.2012-03-16
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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.2012-03-16
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    @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. :)2012-03-16
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    The category of fields doesn't have coproducts, much less arbitrary direct limits.2012-03-16
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    Thank you for your comments.2012-03-17

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