Timeline for Limits in category theory and analysis
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 16, 2022 at 17:18 | comment | added | Davi Barreira | So, @Tyler, would the actual limit of a sequence be a zero object in the category of cones? I mean, if the limit of $F$ and the colimit are the same? | |
| Feb 14, 2020 at 23:15 | comment | added | Tyler | @Nathaniel Consider the category $J$ whose objects are natural numbers and there is a single morphism in $\mathop{\mathrm{Hom}}_J(x,y)$ whenever $x \le y$ and empty otherwise. Given a sequence of real numbers, $(a_n)$, there is a functor $F\colon J \to R$ assigning $F(n) = a_n$. The functor $F$ is the diagram in this example. A cone to $F$ is a real number which is a lower bound for the sequence $(a_n)$, so the limit of $F$ is the least upper bound, i.e. $\lim_{n\to\infty} a_n$ in the classical sense. | |
| Dec 5, 2017 at 3:50 | comment | added | N. Virgo | I hope you don't mind a dumb question on an 8 year old post, but how should I think of a sequcence of numbers as being a diagram in category theory? (I hope that's the right question to ask. My confusion is that in category theory we speak of the limit of a diagram, so in order for a sequence to have a limit in the categorical sense, it seems like the sequence would have to correspond to a diagram.) | |
| Feb 23, 2012 at 22:51 | history | edited | David White | CC BY-SA 3.0 |
Texified since the question was already on the front page
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| Dec 28, 2009 at 18:07 | comment | added | Mike Shulman | Much more generally, metric spaces can be identified with certain categories enriched over [0,∞] with the opposite of the usual order, as realized by Lawvere in his paper "Metric spaces, generalized logic, and closed categories." At least some limits of sequences or nets in metric spaces can then be identified with limits in the categorical sense. | |
| Dec 28, 2009 at 16:25 | history | answered | Reid Barton | CC BY-SA 2.5 |