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: link between limits and "usual" limits for sequence ?

Vincent L (Jul 04 2021 at 13:39):

Hi,
I was wondering if there is a formal connection between limits in the categorical sense, and the "traditional" limits of function/sequence.
I get that the limit can be seen as a lower upper bound in case of ordered elements, but to me it seems like a specific case of limit.

সায়ন্তন রায় (Jul 04 2021 at 14:17):

@Vincent L See this.

Vincent L (Jul 04 2021 at 14:33):

thank you

Vincent L (Jul 04 2021 at 15:28):

diagonal functor $\Delta$ is sending $X$ to $X\times X$ ?

John Baez (Jul 04 2021 at 15:53):

Yes.

Henry Story (Jul 04 2021 at 15:54):

I drew up some pictures of an adjunction with the diagonal functor a week ago on math stack exchange

Vincent L (Jul 04 2021 at 15:59):

thank you !

Ralph Sarkis (Jul 04 2021 at 16:01):

Vincent L said:

diagonal functor $\Delta$ is sending $X$ to $X\times X$ ?

I think the diagonal functor mentioned in the accepted answer on MO is more general, it implicitly depends on the indexing category. Here is how I define it in my (unfinished) notes ($[J,C]$ is the category of functors from $J$ to $C$):
image.png

Ralph Sarkis (Jul 04 2021 at 16:02):

In their example, if you consider the functor $E:\mathcal F_{x,F}\hookrightarrow\mathcal F(X)$ and a filter $F$, then $\Delta(F)$ is the constant functor $\mathcal F_{x,F}\rightarrow \mathcal F(X)$ that sends everything to $F$.

Vincent L (Jul 04 2021 at 16:08):

thank you

Vincent L (Jul 04 2021 at 16:08):

was a bit confused because $\lambda(G)$ was from $F$ to $G$.

Vincent L (Jul 04 2021 at 16:08):

and G wasn't really a product

Vincent L (Jul 04 2021 at 16:17):

I don't really understand "$\lambda$ is a limit", as far as I understand it's a natural transformation, and a limit is a functor ?

Vincent L (Jul 04 2021 at 16:21):

or is it the natural transformation between the apex of the cone and the base ?

Fawzi Hreiki (Jul 04 2021 at 16:28):

$\Delta(F)$ is just the constant diagram (i.e. functor) at $F$. A cone is a natural transformation from a constant diagram to a diagram. The statement that $\lambda$ is a limit means that $\lambda$ is the universal cone.

Vincent L (Jul 04 2021 at 16:29):

thank for the clarification

Fawzi Hreiki (Jul 04 2021 at 16:32):

This is all pretty well explained here https://ncatlab.org/nlab/show/diagram

Fawzi Hreiki (Jul 04 2021 at 16:32):

And the relationship to the more straightforward definition of limit as an object equipped with some maps such that ...

Fawzi Hreiki (Jul 04 2021 at 16:32):

This is all pretty well explained here https://ncatlab.org/nlab/show/diagram

Fawzi Hreiki (Jul 04 2021 at 16:32):

And the relationship to the more straightforward definition of limit as an object equipped with some maps such that ...

Davi Sales Barreira (May 16 2022 at 20:33):

I'm also interested in this question. I know that a link to the MathOverflow answer was posted, but there is this another answer here in the same question:

I am not completely satisfied by the accepted answer because the functor which characterizes the convergence of a filter depends on the limit. I therefore add another quite simple answer (written for sequences but this easily generalizes to filters and nets) to this old post...

So, I was wondering if there were any books/articles going a bit further on this topic. Do you people agree with the criticism posted above?

Davi Sales Barreira (May 16 2022 at 20:40):

In the same MO post, someone suggested the following. Consider a category $I$ which is a poset on the natural numbers, i.e. $(\mathbb N, \leq)$, where there is a morphism between $x$ and $y$ if $x\leq y$. Now, given a sequence of real numbers $(x_n)$, make a functor $F:I\to\mathbb R$, such that $F(n) = x_n$. One can then show that a cone in $F$ as an upper-bound, and the limit of $F$ would be the least upperbound.

So, I suppose that the colimit of $F$ would be the least lowerbound. Hence, would it be correct to say that if the cone category has a zero object, then the sequence has a limit, which would be such zero object?

Davi Sales Barreira (May 16 2022 at 20:45):

The author that critiqued the given answer provided the following definition:

Let now $(x_n)$ be a sequence in some topological space $X$. Then the limit of the contravariant functor $F:\mathbb N \to \mathcal P (X)$ assigning to $n$ the set $F(n)=\overline{\{x_k : k \geq n\}}$ is the set of all limit points of the sequence.
I think that this is a strong relation between analytical and functorial limits although it does not yet characterize convergence of sequences.

Unfortunately, the author states that this definition is still lacking.

Ralph Sarkis (May 17 2022 at 06:11):

(EDIT: I was confusing cones with cocones before, it is now fixed but my comment is less clear.)

A functor between two posets is the same thing as an order preserving functions (the condition that morphisms between $a$ and $b$ are taken to morphisms between $Fa$ and $Fb$ implies $a\leq b \implies Fa \leq Fb$). So your functor $F:I \to \mathbb{R}$ is a non-decreasing function from $\N$ to $\R$, and the corresponding sequence is also non-decreasing.

A cocone under $F$ is an upper bound of the set $\{F(n)\mid n \in \N\}$, and the tip of the colimit cocone (it exists if and only that set has an upper bound --- that is usually an axiom put on $\R$ in early analysis classes) is the least upper bound and the limit of the sequence (because the sequence is non-decreasing).

A cone over $F$ is a lower bound of the set $\{F(n)\mid n \in \N\}$, and the tip of the limit cone (it always exists) is $F(0)$ which is guaranteed to be the greatest lower bound (because the sequence is non-decreasing).

(There is probably a better way to write this so that the categorical limit (and not the colimit) is the analytical limit, I leave this task to the reader.)

Morgan Rogers (he/him) (May 17 2022 at 06:24):

Right, for the inf of a non-increasing sequence you would want a colimit of a functor out of $\N^{\mathrm{op}}$, but neither limit nor colimit recovers the analytic notion of convergence of a sequence.

Ralph Sarkis (May 17 2022 at 06:28):

Davi Sales Barreira said:

Hence, would it be correct to say that if the cone category has a zero object, then the sequence has a limit, which would be such zero object?

You are technically correct, but not morally so (I assume you write 'zero object' for an object that is both initial and terminal). The cocone category of $F:I \to \R$ never has a terminal object. A cocone over $F$ is simply a real number bigger than all $F(n)$, thus if $r \geq F(n)$ is a cocone, then $r+1 \geq F(n)$ is a cocone, and there is no morphism of cocone from the latter to the former because $r+1\not\leq r$. We conclude that $r$ is not terminal.

Therefore, after dualizing what I said (sorry for this step), we find that your statement "if the cone category has a zero object, then the sequence has a limit" is vacuously true.

Ralph Sarkis (May 17 2022 at 06:31):

It is extremely atypical for a cone category to have a zero object (but I don't have an intelligent explanation for why that is). What we are usually interested in is a terminal object in a cone category (i.e. a limit) or, dually, an initial object in a cocone category (i.e. a colimit).

Davi Sales Barreira (May 17 2022 at 12:15):

Yeah, I've realized that the functor in I've talked about would only work for monotonic sequences, hence, the limit of $F$ would be the sup for the increasing sequence, and the colimit would be the inf for the decreasing sequence.

Davi Sales Barreira (May 17 2022 at 12:17):

Davi Sales Barreira said:

The author that critiqued the given answer provided the following definition:

Let now $(x_n)$ be a sequence in some topological space $X$. Then the limit of the contravariant functor $F:\mathbb N \to \mathcal P (X)$ assigning to $n$ the set $F(n)=\overline{\{x_k : k \geq n\}}$ is the set of all limit points of the sequence.
I think that this is a strong relation between analytical and functorial limits although it does not yet characterize convergence of sequences.

Unfortunately, the author states that this definition is still lacking.

Now, what about this construction here.

James Deikun (May 17 2022 at 12:35):

I don't know about that construction, but the limit of a Cauchy sequence in a Lawvere metric space is an enriched-categorical limit. (And also a colimit, since it's an absolute limit.)