Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Cauchy in terms of convergence

Gurkenglas (Oct 25 2022 at 22:11):

https://ncatlab.org/nlab/show/Cauchy+sequence says

A Cauchy sequence is an infinite sequence which ought to converge in the sense that successive terms get arbitrarily close together, as they would if they were getting arbitrarily close to a limit.

and then gives context-dependent definitions. Could one define Cauchy as "this converges in some extension", then list the "definitions" as examples?

Oscar Cunningham (Oct 26 2022 at 07:13):

Yes, in a metric space a sequence is Cauchy if and only if it converges in the completion of the space. I don't know if this extends to the more general contexts on that page.

If you want to make a change to the nLab then a better place to discuss it would be the nForum.

Peter Arndt (Oct 26 2022 at 07:38):

To define the completion you usually use equivalence classes of Cauchy sequences. If you want to define Cauchy sequences in terms of the completion, you need to come up with a way to avoid circularity.

Gurkenglas (Oct 26 2022 at 11:13):

I don't think this is circular. For Q, "a sequence N->Q is Cauchy iff there exists a distance-preserving Q->Z such that the N->Z converges" is equivalent to the usual definition of Cauchy in a metric space, and then the Cauchy completion proceeds as usual.

Gurkenglas (Oct 26 2022 at 11:15):

(Cauchy sequences would be defined equivalent iff they converge through the same choices of Q->Z.)

Gurkenglas (Oct 26 2022 at 11:20):

(I suspect you can define the Cauchy completion as the colimit of all initial (aka each Z adds at most one point to Q) such Q->Z.)

Oscar Cunningham (Oct 26 2022 at 11:48):

Neat. Wouldn't it be neater to just define the Cauchy completion as the left adjoint to the forgetful functor from complete spaces?

Mike Shulman (Oct 26 2022 at 15:31):

I think what Peter was saying is that it's not clear how to prove that those are equivalent definitions of Cauchy-ness if you don't already know that a completion exists.

Mike Shulman (Oct 26 2022 at 15:33):

It is possible to construct completions in other ways, though, such as by a Yoneda embedding.

Gurkenglas (Oct 26 2022 at 22:00):

@Oscar Cunningham, "complete space" is defined using Cauchy. Or do you mean that we define Cauchy as "converges in some extension", then define "complete", then define the Cauchy completion as that left adjoint? Seems sensible enough. Is there some way to tell from the adjunction that the completion can be constructed as the above colimit of initial Q->Z? Some of that we get from the forgetful functor being full and faithful.

Gurkenglas (Oct 26 2022 at 22:23):

@Mike Shulman: That a converging sequence is usual-Cauchy is known. For a usual-Cauchy N->Q, construct $\N \cup \{\infty\} \rightarrow \mathbb{Q} \cup \{\bullet\}$ to still be usual-Cauchy, and the sequence converges to $\bullet$ since the distance to $\bullet$ vanishes as the upper bounds from usual-Cauchy do.

Oh huh yes, Lawvere metric spaces (categories with each homset replaced by a distance) make the Cauchy completion just be "presheaves with infimum 0". Neat. (Might need to say "groupoids" instead of "categories" to get the presheaf to respect both modes of the triangle inequality.)

David Michael Roberts (Oct 27 2022 at 02:46):

Gurkenglas said:

I don't think this is circular. For Q, "a sequence N->Q is Cauchy iff there exists a distance-preserving Q->Z such that the N->Z converges" is equivalent to the usual definition of Cauchy in a metric space, and then the Cauchy completion proceeds as usual.

So N is the natural numbers, Q is the rationals, but Z is not the integers?

Mike Shulman (Oct 27 2022 at 02:58):

Gurkenglas said:

Mike Shulman: That a converging sequence is usual-Cauchy is known. For a usual-Cauchy N->Q, construct $\N \cup \{\infty\} \rightarrow \mathbb{Q} \cup \{\bullet\}$ to still be usual-Cauchy, and the sequence converges to $\bullet$ since the distance to $\bullet$ vanishes as the upper bounds from usual-Cauchy do.

Yes, that's true -- you can essentially construct just enough of the completion to characterize Cauchyness of any particular sequence.

Gurkenglas (Oct 28 2022 at 11:53):

Mike Shulman said:

just enough

When a fit is this lucky, there's surely a perspective from which it's not luck. Can you tell of another situation whose lucky fit looks the same, and describe what all such situations have in common?

Mike Shulman (Oct 28 2022 at 14:42):

I didn't use the word "lucky".

Morgan Rogers (he/him) (Oct 28 2022 at 16:32):

The inclusion of $\N$ into $\N \cup \{\infty\}$ with the right metric/topology is weakly universal amongst one-point extensions. Completeness is injectivity with respect to that map

Mike Shulman (Oct 28 2022 at 17:06):

Although that's more sensitive to the sequential nature of metric spaces than most of the things we've been saying; many of them work just as well for Cauchy nets/filters and completion of uniform spaces.

Morgan Rogers (he/him) (Oct 28 2022 at 20:55):

Right, you'd probably want to expand the singleton family of morphisms to a class of "completions of nets" for a more generic setting.

Mike Shulman (Oct 28 2022 at 21:00):

And you may get some qualitatively different behavior for size reasons, as there are a proper class of such "completions of nets".