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What is the relation between inverse limit (and direct limit) with limits in calculus? Are there some special cases that an inverse limit (or direct limit) is a limit in calculus (for example, the limit of a sequence $a_0, a_1, \ldots$). Thank you very much.

LJR
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    My intuition is to say that limits in calculus gave limits in category theory their name. (If you consider $\mathbb{R}$ as a poset, then a diagram of shape $\mathbb{N}$ in $\mathbb{R}$ is an increasing sequence $a_0, a_1, \dots$ in $\mathbb{R}$, and a direct limit of the diagram is a limit in the classical sense. This probably isn't deep, though; can we (easily) talk about $\mathbb{C}$, or other topologies, or non-monotone sequences? Still, somehow they both talk about increasingly accurate approximations.) Look up the p-adic integers (and their topology), though. – Billy Jun 15 '13 at 03:09
  • A very similar question: http://math.stackexchange.com/questions/60590/category-theoretic-limit-related-to-topological-limit – Adam Saltz Jun 15 '13 at 03:10

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