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This may be a repeated question, but I want to make it clear.

I am learning basic category theory by myself and came across the concepts of limits and colimits. I understand, for example, the colimit of a sequence of decreasing sets indexed by the category of natural numbers with orders as morphisms is their intersection.

But since the word "limit" originally came from analysis with the classical epsilon-delta definition, I wonder what the connection between the two kinds of limit (category-theoretical limit and limit in analysis) is? Also, can we give a new definition of the limit of a sequence or function (epsilon-delta definition) USING the category-theoretical one?

I would appreciate any help.

mrtaurho
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ScienceAge
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  • This could be useful: https://math.stackexchange.com/questions/60590/category-theoretic-limit-related-to-topological-limit , although it uses the language of filters in the more general setting of topological limits. – interregno Sep 19 '21 at 01:27
  • @whatsappcar Thank you. But I'm wondering whether there are neater explanations? – ScienceAge Sep 19 '21 at 07:16

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If you regard a partially ordered set $X$ as a category with an arrow $x \rightarrow y$ iff $x \leq y$, then the limit of a diagram in $X$ amounts to the infimum of the corresponding subset. Likewise the colimit of a diagram becomes the supremum.

This doesn’t quite answer your question. The problem is that from this perspective limit and colimit are inherently directed constructions (the smallest / largest thing satisfying some property) while an analytic limit of a sequence is in some sense undirected (a point, which is arbitrarily close to the sequence).

I believe one can remedy this by considering $[0,\infty]$-enriched categories (a categorification of metric spaces) and weighted limits, but I don’t think this will help you in understanding categorical limits.

Jonas Linssen
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