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Stream: deprecated: show and tell

Topic: summing series

sarahzrf (Oct 30 2020 at 15:28):

i just realized last night: if you treat the monoid ([0, ∞), +) as a 1-object category, then transfinite composition in it recovers summation of series

sarahzrf (Oct 30 2020 at 15:28):

(im sure lawvere beat me to this one but i wouldnt know where...)

sarahzrf (Oct 30 2020 at 15:29):

note that you don't need special extra equipment of any kind, literally just the monoid

sarahzrf (Oct 30 2020 at 15:30):

which kinda screws with my head because i always think of things like infinite series as requiring extra data like a topology

sarahzrf (Oct 30 2020 at 15:31):

but i guess the thing is that the topology on [0, ∞) is induced by the monoid structure, in the sense that it coincides with the order topology for the divisibility order

John Baez (Oct 30 2020 at 17:35):

sarahzrf said:

i just realized last night: if you treat the monoid ([0, ∞), +) as a 1-object category, then transfinite composition in it recovers summation of series

How does that work? I don't know what transfinite composition is. It seems you need at least the order structure on $[0,\infty)$ to do infinite sums of nonnegative numbers. Btw, $[0,\infty]$ is a nice monoid in which all infinite sums are well-defined.

sarahzrf (Oct 30 2020 at 17:39):

transfinite composition for ω-indexed sequences at least is like

sarahzrf (Oct 30 2020 at 17:43):

• treat the poset ω as a category, so it's the shape for diagrams like X → Y → Z → ...
• a ω-indexed sequence of morphisms you'd want to compose is the same as a diagram $F : \omega \to C$
• if F has a colimit, then there's injections $\iota_n : F_n \to \lim_\to F$ for each n
• you call $\iota_0 : F_0 → \lim_\to F$ the "transfinite composition" of the chain

sarahzrf (Oct 30 2020 at 17:44):

i guess this is a term used by homotopy theorists or something

sarahzrf (Oct 30 2020 at 17:51):

like the idea is "if you factor a morphism through all of the finite partial compositions, you factor it uniquely through the transfinite composition"

John Baez (Oct 30 2020 at 17:52):

I'm confused by how this meshes with your claim that you're treating $[0,\infty)$ as a 1-object category, with numbers as morphisms. A categorical limit or colimit is an object, not a morphism.

sarahzrf (Oct 30 2020 at 17:52):

read what i wrote more carefully!

sarahzrf (Oct 30 2020 at 17:53):

the "transfinite composition" is one of the legs of the universal cocone, rather than the object which is equipped with the universal cocone

John Baez (Oct 30 2020 at 17:54):

So is the information about the sum in $\iota_0$?

sarahzrf (Oct 30 2020 at 17:54):

right

sarahzrf (Oct 30 2020 at 17:56):

sarahzrf said:

like the idea is "if you factor a morphism through all of the finite partial compositions, you factor it uniquely through the transfinite composition"

"if you can write $y$ as $(\sum_i^n x_i) + \varepsilon_n$ for all n, then you can write it as $(\sum_i x_i) + \varepsilon$"

John Baez (Oct 30 2020 at 17:56):

Okay, nice. It'd also work to take $[0,\infty]$ as a monoidal poset using + and define an infinite sum as a colimit of finite sums.

sarahzrf (Oct 30 2020 at 17:57):

of course! but what i find really interesting about this is that you don't need to—a standard construction can already recover the order structure from the monoid structure

sarahzrf (Oct 30 2020 at 17:59):

there's some stuff in Taking Categories Seriously about how slicing over the object of a cancellative monoid gives you its divisibility order and i guess about how certain kinds of constructions naturally recognize order structure in monoids from the divisibility order

sarahzrf (Oct 30 2020 at 17:59):

and in particular, about how the divisibility order on [0, ∞) is the standard one

sarahzrf (Oct 30 2020 at 18:05):

also, like, i find it pretty charming to think about the fact that a functor C → B[0, ∞] that preserves ω-colimits, is some kind of metric or something

sarahzrf (Oct 30 2020 at 18:11):

hmmm... for a riemannian manifold M, could you form a category w/ objects points of M, morphisms x → y intervals [0, a] along with smooth paths [0, a] → M from x to y, concatenation for composition, no quotienting by homotopy. not a groupoid, but shd be strictly associative and unital right?
EDIT: actually okay nvm composition isnt well-defined there—youll have to either use piecewise smooth paths instead, or make the objects something like germs

sarahzrf (Oct 30 2020 at 18:11):

then arc length defines a functor to B[0, ∞] i bet—maybe one that preserves ω-colimits?

sarahzrf (Oct 30 2020 at 18:13):

conversely: if M is not a priori equipped w/ a metric, how much data would such a functor give you?

Jens Hemelaer (Oct 30 2020 at 18:24):

Interesting... what is the category $C$ in your notation?

Jens Hemelaer (Oct 30 2020 at 18:27):

Ah that is just an arbitrary category probably. I'm surprised that you can compute this kind of colimits in $[0,\infty)$.

John Baez (Oct 30 2020 at 19:34):

a standard construction can already recover the order structure from the monoid structure

Of course people usually say this particular fact as follows: for numbers in $[0,\infty)$, we have $x \le y$ iff there exists $a$ such that $x + a = y$.

John Baez (Oct 30 2020 at 19:40):

Or am I missing the point entirely?

sarahzrf (Oct 30 2020 at 19:40):

i meant that in particular, a standard category-theoretic construction can recover the order structure, as a category, from the monoid structure, as a category

sarahzrf (Oct 30 2020 at 19:40):

as in, the machinery of category theory understands all of this natively, without you having to supply the data as inputs

John Baez (Oct 30 2020 at 19:41):

What would happen if you tried doing this to $\mathbb{R}$? The recipe I just gave would break down, and I bet your recipe would break down too.

sarahzrf (Oct 30 2020 at 19:41):

yup!

sarahzrf (Oct 30 2020 at 19:41):

i wonder if that has something to do with absolute vs conditional convergence...?

John Baez (Oct 30 2020 at 19:41):

Okay, so I feel confirmed in my suspicious that they're basically the same idea.

John Baez (Oct 30 2020 at 19:43):

I always tell my real analysis students that infinite sums are trivial if we work in $[0,\infty]$ - they all converge and they behave nicely - so all the problems arise when we allow negative numbers.

John Baez (Oct 30 2020 at 19:43):

Then they may not converge, they may converge but change value if you rearrange them, etc.

Jens Hemelaer (Oct 30 2020 at 20:13):

John Baez said:

What would happen if you tried doing this to $\mathbb{R}$? The recipe I just gave would break down, and I bet your recipe would break down too.

Yes, submonoids of a group are the same things as translation-invariant orderings on the group. So looking at the group instead of the submonoid is the same thing as forgetting the ordering.

Jens Hemelaer (Oct 30 2020 at 20:30):

sarahzrf said:

then arc length defines a functor to B[0, ∞] i bet—maybe one that preserves ω-colimits?

$[0,\infty]$ is not cancellative, and as a result the colimit corresponding to the increasing sequence $0 < 1 < 2 < \dots$ doesn't seem to exist. But I would agree with you that arc length seems to preserve the colimits that do exist.

sarahzrf (Oct 30 2020 at 20:34):

oh oops i thought something was off about [0, ∞] but i missed the non-cancellativeness

sarahzrf (Oct 30 2020 at 20:35):

hmm

sarahzrf (Oct 30 2020 at 20:36):

well, shouldn't arc length for paths of the form i wrote above never take infinite values anyway?

sarahzrf (Oct 30 2020 at 20:37):

or are you saying that you can get a chain of arcs which does have a transfinite composition, and whose sequence of partial compositions has a divergent sequence of lengths?

Jens Hemelaer (Oct 30 2020 at 20:45):

As far as I can see, if the arc lengths of the paths sum to infinity, then their $\omega$-colimit already does not exist in your category of paths.

sarahzrf (Oct 30 2020 at 20:50):

cool, that's what i figured

Fawzi Hreiki (Oct 30 2020 at 21:09):

That paper you mentioned, Taking Categories Seriously, is pretty great - albeit very strange. It took me like a week to read through just the three pages where he constructs the Dedekind reals.

Fawzi Hreiki (Oct 30 2020 at 21:11):

It's surprising actually that the construction $\text{Monoids} \rightarrow \text{Posets}$ given by slicing over the unique object isn't mentioned in any other basic category theory text (at least I haven't seen it anywhere).

sarahzrf (Oct 30 2020 at 21:14):

(i'd note that slicing over the unique object only gives a poset if the monoid is cancellative, although you can certainly quotient the resulting category down to a poset regardless)

Fawzi Hreiki (Oct 30 2020 at 21:14):

Yeah that's true - my bad

Fawzi Hreiki (Oct 30 2020 at 21:15):

I wonder if anyone has pursued the Kan extension / spectral analysis thing in more detail

sarahzrf (Oct 30 2020 at 21:21):

i need to finish that paper >_<

Morgan Rogers (he/him) (Nov 02 2020 at 09:44):

Fawzi Hreiki said:

It's surprising actually that the construction $\text{Monoids} \rightarrow \text{Posets}$ given by slicing over the unique object isn't mentioned in any other basic category theory text (at least I haven't seen it anywhere).

@Jens Hemelaer and I have an article on the back-burner examining what Connes and Consani call the Root of a monoid $M$, which is the topos obtained by slicing the topos of right $M$-sets over $M$. It can be equivalently expressed as the topos of presheaves over the slice of $M$, viewed as a category, over its unique object.
We mention in the conclusion of a previous paper that $M$ is (left) cancellative if and only if some slice of its topos of actions is localic, but this turns out to be equivalent to that particular slice being localic, which occurs if and only if the category mentioned above is a poset. We believe that the root of a monoid could be a very convenient construction for characterising other properties of monoids, too.

sarahzrf (Nov 02 2020 at 17:05):

oh, lawvere discusses that exact construction right afterward!

sarahzrf (Nov 02 2020 at 17:06):

not in extreme generality or for the purposes / from the pov you're describing, but certainly it seems extremely relevant

sarahzrf (Nov 02 2020 at 17:07):

image.png

sarahzrf (Nov 02 2020 at 17:08):

"Cayley-Dedekind-Grothendieck-Yoneda lemma"

sarahzrf (Nov 02 2020 at 17:08):

typical lawvere

Morgan Rogers (he/him) (Nov 02 2020 at 17:09):

It's as diplomatic as it is pretentious :rolling_on_the_floor_laughing:

sarahzrf (Nov 02 2020 at 17:10):

the famously diplomatic lawvere :upside_down:

Todd Trimble (Nov 02 2020 at 17:48):

Are we publicly mocking famous category theorists now?

Morgan Rogers (he/him) (Nov 02 2020 at 18:33):

I was criticising a literary decision that one made :shrug: Being famous doesn't make one immune to that!

Nikolaj Kuntner (Nov 02 2020 at 18:47):

There's a nice 500 page book on the Baker–Campbell–Hausdorff formula and it spends 2 pages with surveys and a discussion on all the options and permutations of the name.

Todd Trimble (Nov 02 2020 at 18:51):

@Morgan Yes, you're right, as long as the criticism is aimed at the writing and not at the person. And yours was.

Amar Hadzihasanovic (Nov 03 2020 at 09:19):

I read @sarahzrf's comment as a gentle mockery of the suggestion that Lawvere would use the multiple attribution for reasons of “diplomacy”, not as a mockery of Lawvere...

sarahzrf (Nov 03 2020 at 09:20):

reasonable interpretation, but i was actually poking fun at the fact that lawvere is known for, say, getting into a fistfight over issues of maoism

Amar Hadzihasanovic (Nov 03 2020 at 09:20):

Lawvere subscribes to a polemical tradition of philosophy, so I think saying “the famously diplomatic Lawvere” ironically is as fair as saying “the famously right-wing Lawvere” :)

sarahzrf (Nov 03 2020 at 09:21):

if you think both are fair to say facetiously, then it sounds like we're on the same page

Amar Hadzihasanovic (Nov 03 2020 at 09:21):

Yep

Todd Trimble (Nov 03 2020 at 10:09):

I think what I'm saying, folks, is to keep in mind the possibility that whoever you talk about here might be reading it some day. Put differently, would you ever have said this over at the categories list? And that's the last I'll say on the matter.

Robert Seely (Nov 03 2020 at 18:05):

Todd Trimble said:

I think what I'm saying, folks, is to keep in mind the possibility that whoever you talk about here might be reading it some day. ...

Without aiming my comment at anyone in particular (esp sarahzrf), in a general way I'd like to second Todd's suggestion - maybe even go further and suggest it's a good idea online to say about someone only what you'd happily say to their face, and even then, maybe be a bit more cautious than you'd be in person (where facial gestures, tone of voice, etc, can moderate the effect). Over the years, I've seen folks forget this, and sometimes unintentionally cause serious psychic damage (although unlikely in this case - I think Bill is pretty resilient). It's good to be on one's "best behaviour" when online, especially in a friendly, technical forum such as this.

John Baez (Nov 03 2020 at 19:15):

Without aiming my comment at anyone in particular (esp sarahzrf)

:upside_down:

Robert Seely (Nov 03 2020 at 20:38):

John Baez said:

Without aiming my comment at anyone in particular (esp sarahzrf)

:upside_down:

Oh dear - this is getting out of hand ... I'm sorry you took offense at my comment, John - I know it seems unlikely, but I really did mean that, no "frowny" needed. Of course hers was the post that started this discussion, but it wasn't to her that my comment was aimed, but more generally - and I hoped to make that clear. I thought originally that no response to her post was needed, but once others did so, it was only the content of Todd's post that I wanted to address and support. Not the target, if there was one. sarahzrf is a valued and frequent participant to this forum (a lot more so that I am, for example), and her participation is indeed valuable. My intention in mentioning her was to emphasize that point, however badly I did so.

BTW - I was once personally involved - though neither the target nor the source of the relevant text - in an incident where an unintentionally thoughtless remark did considerable harm, and that incident has coloured my view of "casual talk" online ever since. This isn't just an "academic" issue. Having said that, I doubt this incident is in that category at all - sarahzf's remark was innocuous and marginally amusing, and that's how I personally took it originally - and still do. As I said, my reply addressed only Todd's comment, which I think was a wise one, whether relevant to the context or not.

This all illustrates the main point - text alone, without the usual social graces that accompanies speech, is ambiguous, and can often be misunderstood, especially the "implications", intended or not, one adds to it. And we should try to keep that in mind. Even if (as I did) we often fail ...

(And I'll bow out of this conversation now, having said too much maybe!)

John Baez (Nov 03 2020 at 20:39):

That's not a "frowny", that's the "smiley" I usually use in online forums.

I thought your remark was funny: "not anyone in particular... especially sarazf" is such an oxymoron I thought you were joking.

"I'm not talking about anyone in particular.... especially you."

Robert Seely (Nov 03 2020 at 20:41):

John Baez said:

That's not a "frowny", that's a "smiley". I thought your remark was funny: "not anyone in particular... especially sarazf" is such an oxymoron I thought you were joking.

Oh dear!! I thought it was a frowny, i.e. an upside down smiley!! :-( vs :-)
This just proves how difficult text can be !!! :-) (and that is a smiley!!)

However, my rather laboured response should clarify things, I trust!

John Baez (Nov 03 2020 at 20:56):

Your misunderstanding of my smiley is indeed further proof that one needs to be careful.

:upside_down: $\ne$ :frown:

John Baez (Nov 03 2020 at 20:57):

But anyway, I'm not feeling at all upset by anything in this conversation.

Jules Hedges (Nov 03 2020 at 21:10):

:eyes: $\longleftarrow$ looking at this thread to see if it needs moderator involvement.... seems to be all under control

Robert Seely (Nov 03 2020 at 21:26):

Jules Hedges said:

:eyes: $\longleftarrow$ looking at this thread to see if it needs moderator involvement.... seems to be all under control

Yup! "I just said what I said and it was wrong. Or it was taken wrong. And now it's all this."

David Michael Roberts (Nov 03 2020 at 23:00):

Coming back to mathematics, I would be interested to know where in Dedekind's and Grothendieck's writing Lawvere sees the Yoneda lemma (Cayley is less mysterious, it's one of my go-to examples to demystify the Yoneda Lemma). I don't have a hard time imagining the Yoneda lemma/embedding is in SGA4 somewhere, but one can really point to these things; it's not like trying to track down the first statement of Cayley's theorem: https://en.wikipedia.org/wiki/Cayley%27s_theorem#History

Although Burnside8 attributes the theorem to Jordan,9 Eric Nummela[10] nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,[11] showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an embedding). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.

The theorem was later published by Walther Dyck in 1882[12] and is attributed to Dyck in the first edition of Burnside's book.[13] (1897)

As far as Dedekind goes, "Es steht alles schon bei Dedekind." and all, but where?

Fawzi Hreiki (Nov 03 2020 at 23:57):

Well in that same paper Lawvere constructs the real numbers as Dedekind cuts using the Yoneda embedding.

More classically, the embedding of a poset $P$ into the poset $2^{P^{op}}$ by sending an element to its downset is just the Yoneda lemma but enriched in 2.

Fawzi Hreiki (Nov 03 2020 at 23:58):

You can have a look at his Perugia notes where he elaborates on the Cayley and Dedekind connections more.

Fawzi Hreiki (Nov 04 2020 at 00:12):

With regards to Grothendieck, I think it’s more a case of paying respect to the fact that he recognised its fundamental importance and used it all over the place in his work.

I think when Lawvere writes things like ‘Cayley-Dedekind-Grothendieck-Yoneda lemma’ he isn’t actually attributing the lemma to all of those people but just using it as a way to teach some history.

In another of his papers he speaks of the ‘Pythagoras-Steiner-Burnside abstraction process’ for example by which he just means passing to isomorphism classes.

David Michael Roberts (Nov 04 2020 at 01:44):

Hmm. ok :-) I'd not seen the '‘Pythagoras-Steiner-Burnside abstraction process' before, that's a good one.

Todd Trimble (Nov 04 2020 at 01:46):

Yes, Dedekind is Yoneda for posets.

Todd Trimble (Nov 04 2020 at 01:47):

I'd not heard of Pythagoras-Steiner-Burnside. Which paper is that?

Fawzi Hreiki (Nov 04 2020 at 07:34):

Categories of Space and Quantity

David Michael Roberts (Nov 04 2020 at 22:15):

Link: https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1992-categories-of-space-and-quantity.pdf

Todd Trimble (Nov 04 2020 at 23:29):

Thanks Fawzi and David. He does explain a little of of the thinking behind that conjunction Pythagoras-Steiner-Burnside, so it's not absolutely gratuitous (or pretentious if you prefer). Whether one agrees with his history is of course another matter.