Is there a way to see limits of nets as categorical limits?

Asked Sep 05 '15 at 18:09

Active Sep 05 '15 at 18:19

Viewed 100 times

This is just for fun(ctors).

I was wondering if limits in category theory are a generalization of limits (of nets) in topology. I found here a complete and satisfactory answer for limits of filters, and I know that there is a correspondence between filters and nets that preserves its limits.

However, I have friends who dislike filters (sad, but true), and in order to prove this equivalence to them I wish to know if there is a direct way to prove this without explicitly using filters. More precisely, if $\mathscr{I}$ is a directed set and $f:\mathscr{I}\rightarrow X$ is a net, there is some functor $F:\mathscr{I}\rightarrow\mathscr{C}$ (for any nice category $\mathscr{C}$) such that $f$ converges to $x$ iff "something" related to $x$ in $\mathscr{C}$ is the limit of $F$?

Thank you!

edited Apr 13 '17 at 12:21

Community

asked Sep 05 '15 at 18:09

Renan Mezabarba

1,949

1

There is some kind of correspondence between nets and filters. But the correspondence is not one-to-one. It just preserves converging definition. – Minimus Heximus Sep 05 '15 at 18:16
1

Oh, thanks! I'll edit my question. – Renan Mezabarba Sep 05 '15 at 18:17
see http://mathoverflow.net/questions/9951/limits-in-category-theory-and-analysis – Matematleta Sep 05 '15 at 18:56
@Chilango The answer there is the same answer I linked. Anyway, thanks. – Renan Mezabarba Sep 05 '15 at 19:00
1

I don't think my question is a duplicate. I specifically ask for an answer that doesn't use the concept of filters. Anyway... – Renan Mezabarba Sep 12 '15 at 14:23

Is there a way to see limits of nets as categorical limits?

0 Answers0