infinitesimal morphisms · theory: applied category theory

If you do categorical systems theory,

\circ

and

\otimes

both represent putting subsystems next to each other so in that sense "space is discrete". Similarly if you do categorical process theory

\circ

represents sequencing so in that sense "time is discrete". I'd like to be able to get "continuous time" and "continuous space" category theory. The obvious thing to try is something sheaf-y. For example if you take the interval domain

[0,1]

(ie. the poset whose elements are closed intervals in

[0,1]

ordered by reverse inclusion). Suppose you send each real number to an object of your favourite category, and each interval

[a,b]

to a morphism

F(a) \to F(b)

. Then you need a compatibility condition, so that for example if you have

[a,c]

covered by

[a,b]

and

[b,c]

then the resulting morphisms commute. Now for a discrete "path"

0 < a_1 < a_2 < \cdots < 1

you get a finite composable sequence of morphisms. Then the interesting thing is to take the "limit" where you have a continuum of degenerate intervals

[a,a]

for every real number, and "compose them together" to get the morphism for

[0,1]

Jules Hedges (May 03 2020 at 10:45):

Jules Hedges (May 03 2020 at 10:46):

(It might be more interesting if you take the domain of open intervals, so that the degenerate intervals

[a,a]

are no longer elements, but if I remember right they instead correspond to maximal ideals of the open interval domain)

Jules Hedges (May 03 2020 at 10:50):

My motivation is that I can get the (discrete time) Bellman equation to fall out from composing a bunch of optics together, and then taking a colimit to compose together an

\omega

-chain of optics, ie. "discrete time infinite horizon". What I'd like to do is wave a sheaf theory magic wand and have the (continuous time) Hamilton-Jacobi-Bellman equation fall out

Jules Hedges (May 03 2020 at 10:58):

Can also think about much more general things if you extend from categories to operad algebras. For example if you have an operad describing the combinatorics of triangulations of

[0,1]^3

, you could have an algebra of it describing using finite element methods to solve your favourite PDE on that triangulation. Then you want to be able to take a "limit" and say that as the triangulation gets arbitrarily fine your system converges to the original PDE

Jules Hedges (May 03 2020 at 10:59):

This sounds like something that geometers might have done already but not using that language...

Reid Barton (May 03 2020 at 11:31):

Jules Hedges (May 03 2020 at 11:44):

Amar Hadzihasanovic (May 03 2020 at 13:36):

I do not have a formal answer, but I've long thought that a lesson one can take from Freyd's characterisation of the real interval is that the algebraic correlate of “continuous space” is “subdivision”.
So one could work with higher-category-like gadgets which besides composition have a subdivision operation taking a morphism

f: a \to b

to a pair

f_1: a \to m(f), f_2: m(f) \to b

, where

m(f)

is the “midpoint” of

f

, such that

f = f_1;f_2

. Example: paths in a topological space taking

\gamma(t), t \in [0,1]

\gamma_1(t) := \gamma(t/2)

and

\gamma_2(t) := \gamma((1+t)/2)

. Or the same thing with cubes in higher dimensions.
But trying to impose axioms and get interesting examples, I think you would end up with something with a quite different flavour to CT -- you could think subdivision should be “weakly associative” in the path space example, but say you wanted to strictify, it would not make sense to take “paths up to homotopy” as in the fundamental groupoid, or you would quickly run into contradictions (in 'paths-up-to-homotopy' you can always pick the first or the second half to be trivial).
I imagine someone would have tried to do this as it seems like a simple idea, but I haven't looked very hard into it, and as always there's a chance it's a dead end.

Jules Hedges (May 03 2020 at 14:42):

This reminds me of convex sets (aka algebras of the f.s. probability monad), where instead of working up to homotopy you instead have a whole continuum of "weighted midpoint" operators and a slightly awkward associativity law between them

Fabrizio Genovese (May 03 2020 at 14:43):

This seems very convoluted to me. I guess this comes from the philosophical position of distinguishing between "discrete" and "continuous", which I find to be highly debatable. One may say that "continuous is when particles are so small that they blurr into shapes." Trying to formalize this intuition in mathematics has led to very different models for continuous and discrete things. Historically, this is because we focused on a potential model of infinity (that is, we modelled the "dynamics" of this blurring, the "zooming in/out" by means of limits), with the result that, in the continuous case, we are able to talk about particles and shapes but not about both at the same time (infinitesimal increments are a "dynamic thing" and not a real part of our model). The other perspective is that "blurring is just a figment of your mind", that is, "it's you that really want to distinguish between small and non-small stuff, not the models themselves". This suggests another approach, where infinitesimal and infinitary operations are not part of your dynamics (that is, modelled using a potential, limit-oriented view of infinity), but a part of your space. They are nor first-class citizens. From this "actual" point of view the idea is "just enrich the space with enough stuff so that you are able to talk about continuous stuff like you do for discrete stuff. So, in what @sarahzrf was suggesting I'd just swap

\mathbb{R}

with

\mathbb{R}^*

and see what happens.

Fabrizio Genovese (May 03 2020 at 14:45):

This said, I'm using sheaves on a daily basis right now, but I don't know if they would actually be the right tool here...

sarahzrf (May 03 2020 at 14:56):

oh i have a lot of half-formed hot takes myself on the topic of this shit, dont get me started fabrizio :smiling_face_with_horns:

sarahzrf (May 03 2020 at 14:56):

i remember one time i paced around in circles for like an hour or two thinking about internal set theory and ended up deciding that the 19th/20th century upheavals in foundations had ended up settling on the wrong way of handling infinities but then i lost my train of thought

- n-sarahzrf where n ≤ (1, 1) (@sarah_zrf)

sarahzrf (May 03 2020 at 14:58):

sarahzrf (May 03 2020 at 14:59):

Morgan Rogers (he/him) (May 03 2020 at 15:00):

sarahzrf (May 03 2020 at 15:14):

if i take a more down to earth mindset: i might want to say that it seems like theres an existing, incredibly well-studied answer to this kind of problem, which is just good old fashioned notions of integration and flows and diff geo

sarahzrf (May 03 2020 at 15:15):

sarahzrf (May 03 2020 at 15:17):

hmmmmmmm, very vague free-association thought: i'm also also reminded of having fiddled around with the fundamental group of the hawaiian earring—there's a similar flavor of "infinite composition" there

Jules Hedges (May 03 2020 at 15:17):

Oo, never heard of Lie algebroids... if only there was a directed version, "Lie categories"

Amar Hadzihasanovic (May 03 2020 at 15:17):

@Jules Hedges seemed to be taking the philosophical approach that “space and time are created by the algebra of category theory” in systems theory and process theory, and it's a view to which I'm very partial

sarahzrf (May 03 2020 at 15:18):

i mean im looking at the definition of a lie groupoid and im not sure what would stop you from making it a category instead

sarahzrf (May 03 2020 at 15:18):

Amar Hadzihasanovic (May 03 2020 at 15:18):

So it would be disappointing to just leave the continuous aspect to some other pre-existing notion of space, like a manifold

sarahzrf (May 03 2020 at 15:18):

yeah im sympathetic to that :> which is why i framed that viewpoint as "if i take a more down to earth mindset"

Jules Hedges (May 03 2020 at 15:18):

Yep, the "discreteness" comes from the fact that

X \overset{f}\longrightarrow Y \overset{g}\longrightarrow Z

is a discrete diagram

sarahzrf (May 03 2020 at 15:19):

sarahzrf (May 03 2020 at 15:20):

oh shit i think lawvere even called it that in "taking categories seriously"... which i need to fucking finish reading sometime

Amar Hadzihasanovic (May 03 2020 at 15:21):

But my point is, a diagram is different from the space in which it is interpreted (the map is not the territory). That “discrete” diagram could be interpreted as a pair of very continuous paths.

Jules Hedges (May 03 2020 at 15:21):

With

[0,1]

as a poset? Does that work? Are there any interesting functors

[0,1] \to \mathbf{Set}

for example?

sarahzrf (May 03 2020 at 15:21):

Amar Hadzihasanovic (May 03 2020 at 15:21):

sarahzrf (May 03 2020 at 15:22):

actually maybe you want R, and as a monoid, if it's a dynamical sys—fuck i need to read lawvere instead of spouting off

Amar Hadzihasanovic (May 03 2020 at 15:22):

And my point of view would be that the “meaning of discreteness” is that if we keep refining our maps, at some point we reach a scale at which we do not gain any new information.

sarahzrf (May 03 2020 at 15:23):

sarahzrf (May 03 2020 at 15:24):

Morgan Rogers (he/him) (May 03 2020 at 15:29):

Consider a solid cone of unit height and radius, take

F(r)

to be the set of points at height

1-r

and take

F(r < s)

to be the downward projection function, for example. You can think of this as a subdivision of the inclusion of the origin into the unit disk if you like.

Morgan Rogers (he/him) (May 03 2020 at 15:31):

Perhaps you want to think of your objects as metric spaces rather than sets in that setting, though, so that the ensemble of information in the functor properly recovers the cone :shrug:

T Murrills (May 03 2020 at 15:49):

one issue is that specifying “functor” alone doesn’t give you any sense of continuity or whatnot (even if you’re working in Top instead of Set or something); for example, we could have every point less than 1/2 map to set A and every point greater than or equal to 1/2 map to set B, and let any arrow between these two regions be some particular function f (while every other arrow gets mapped to the identity).

It seems like we could maybe ‘categorify’ (to maybe overuse a term) whatever structure we wanted on our archetypal category (in this case [0,1]; like, express its topology in terms of categorical relationships or something), and then demand that that was preserved somehow. (Though we might need to do something similar to our codomain category for this to make sense.)

Amar Hadzihasanovic (May 03 2020 at 15:52):

@T Murrills yep that was exactly my point. And Freyd's characterisation of

[0,1]

as the terminal coalgebra of subdivision, to me, suggests that the structure needed is subdivision.

T Murrills (May 03 2020 at 15:52):

Amar Hadzihasanovic (May 03 2020 at 15:53):

I'm sure @dusko would have something more interesting to say on this given that my intuition is largely based on his and Martin Escardó's “calculus in coinductive form”.

T Murrills (May 03 2020 at 16:08):

Also, just as a tangent, you know what would be cool? Being able to construct a “tangent space” at an object in a category! Obv there are two main routes to construction, and afaik they might well be inequivalent when categorifying without the vector space structure (the “derivatives along paths” formulation vs. the “maximal ideals in a function space” deal for the cotangent space).

My intuition says that @Amar Hadzihasanovic is right about subdivisions somehow being the key to expressing the continuous structure, for whatever that’s worth.

Jules Hedges (May 03 2020 at 16:10):

Fairly sure that's either what differential categories are for, or it's what tangent categories are for

T Murrills (May 03 2020 at 16:11):

Jules Hedges (May 03 2020 at 16:18):

It's possible, I don't know anything about them... in any case they're axiomatic, without looking I don't know any examples of them

Daniel Geisler (May 03 2020 at 17:06):

My work in extending the hyperoperators from

\mathbb{N}

\mathbb{C}

is based on taking

t \in \mathbb{N}

f^t(x)

and extending

t

\mathbb{C}

. Anyhow, I think my work can be placed completely on a CT foundation without too much difficulty.

sarahzrf (May 03 2020 at 17:50):

Daniel Geisler (May 03 2020 at 18:28):

I have a write up at Hyperoperators, pg. 8 for details.
Once I take a Taylor series of

f^t(x)

, time is still discrete. But add a symmetry condition to

f(x)

and then discrete time has a dual continuous time. The most important symmetry conditions are associated with Schroeder's functional equation (hyperbolic dynamics) and Abel's functional equation.

Schroeder's functional equation with

\lambda=Df(0)

where

|\lambda|\neq1

f^t(z)= \lambda ^t z +\frac{\lambda ^{-1+t} \left(-1+\lambda ^t\right) f''}{2 (-1+\lambda )} z^2 + \ldots

Abel's functional equation

\lambda=Df(0)=1

f^t(z)=z+\frac{1}{2}t f''(0) z^2 +\frac{1}{12}(3(t^2-t)f''(0)^2+2tf'''(0))z^3+\ldots

Morgan Rogers (he/him) (May 03 2020 at 18:43):

Moerdijk did a lot of work on topological categories, so those are also an option

sarahzrf (May 03 2020 at 19:51):

more vague spouting off: i think i remember that in non-standard analysis you can express an integral as a ordinary sum over non-standardly many terms hmm

Pastel Raschke (May 03 2020 at 19:54):

wouldn't subdivision/the coalgebra of the real interval lead to a category that is also a cocategory? i'm not sure of the ways cocategories can be, but i would think cocomposition is exactly (un)gluing intervals at a (choice/space of?) midpoint(s). so i guess in a compatible way and developed analogously to freyd's interval...

my other thought is of quantale-enriched cats but ~metrics seem like too much for just continuity

Fabrizio Genovese (May 03 2020 at 19:57):

Fabrizio Genovese (May 03 2020 at 19:58):

sarahzrf (May 03 2020 at 19:58):

Todd Trimble (May 03 2020 at 20:30):

The unit interval is an H-cogroupoid in Top; that's how you get the fundamental groupoid. More precisely, it's an

A_\infty

-cogroupoid. This can be put to interesting uses.

dusko (May 04 2020 at 05:26):

thanks for the ping, amar. i am not going to try to post any math, but i think it might be useful for the discussion to figure out what is the goal. (provided that you still want any comments from me after i confess that i am unable to find the trailhead of the discussion.) there are a several completely different ways how infinitesimals enter categories. the first one was the synthetic way: lawvere's idea, anders kock's book, and then lots of people. or if you are willing to read XIX century stuff, it was what grassman wanted to do: forget the bureaucracy of writing limits and sequences, and compute with the continuum "as such". lawvere said that categorical axioms can do that. to me it seemed that the idea that we can capture all that they do in calculus by some essentially algebraic axioms was a little optimistic. (e.g. in his thesis, cantor wanted to characterize the domains of convergence of fourier series. they are the accummulation points of some set. by the time you reduce it to its density points you have lost some. so you repeat infinitely long, and at the infinity, you do NOT find a fixed point. so cantor discovered that there is more than one infinity. and he discovered one more thing there: the process of coinduction. looking for the greatest fixpoing of an operator...) i just wanted to reconstruct the coalgebra and coinduction which people have already been doing in math, without giving them a name. besides that paper about analytic functions, bertfried fauser and i reconstructed tangent and cotangent bundles, and he wrote down some general relativity, categorically. and then there is this whole movement of differential categories, around robin cockett and seely, and i think rick blute. i still need to read that. so the question is what you guys what to achieve.

sarahzrf (May 04 2020 at 05:35):

Pastel Raschke (May 04 2020 at 08:43):

Amar Hadzihasanovic (May 04 2020 at 11:48):

(Long sequence of posts following)
@dusko Yeah, sorry, I think the thread went in lots of different directions, so it was not very clear what exactly was “the goal” nor why I called you :) So I'm going to try to give a summary of the issue from my point of view.

Amar Hadzihasanovic (May 04 2020 at 11:48):

@Jules Hedges started from the following idea, that I find very powerful: the algebra (or combinatorics) of composition in higher categories are in themselves a “notion of space”. The power of this is that, whein in ACT present/axiomatise stuff, we are not only creating objects or processes but we are also creating the space that they inhabit (which comes with limits on how things can interact, etc).

Amar Hadzihasanovic (May 04 2020 at 11:48):

So when we give a presentation of a systems theory as a monoidal category, we are not only describing the systems but also creating a 2D background for them; when we give a presentation of a process theory as a monoidal category, we are giving the processes 1 spatial dimension and 1 time dimension, etc.
And these dimensions of space/time are intrinsic to the thing that we are presenting, not external like a Newtonian background -- say, in a process diagram, the only thing that makes time tick are the processes themselves with their “causal” connections...

Amar Hadzihasanovic (May 04 2020 at 11:48):

However, if composition of diagrams is the only way to “make time tick” or “create space”, then we end up with a discrete (or discrete-feeling) time and space -- we can “push it to infinity” by doing stuff like transfinite composition, but the fact remains that we can only get more time/space by putting together some discrete units thereof. I guess that's then reflected in the situation that Jules is trying to model, that at most one can derive a “discrete time infinite horizon” but not “continuous time” equations from a categorical presentation.

Amar Hadzihasanovic (May 04 2020 at 11:49):

So then I think the main question in this thread is: how can one “present continuous time or space”, in the spirit of presentations of (higher) categories/process theories?

Amar Hadzihasanovic (May 04 2020 at 11:49):

Which is a cognate of a much older question, “how can one algebraicise the continuum”?
As Dusko says, there are two approaches -- I'll try to summarise them in my own words.

Amar Hadzihasanovic (May 04 2020 at 11:49):

One is the “infinitesimals” approach which, like @Fabrizio Genovese explained, is based on the idea that the continuum is not unlike discrete space, in that it can still be handled in terms of “discrete units”, but now you have different types of discrete units (say, different “scales”) which have a qualitatively different behaviour. So like in SDG you can have bits of space that are non-zero but square to zero, or bits of space at one scale whose every finite concatenation is smaller than a bit of space at a different scale...

Amar Hadzihasanovic (May 04 2020 at 11:50):

So to translate this to the original problem, I think both what Jules was sketching and what Fabrizio suggested is that you may want a notion of category with a “type of infinitesimal morphisms” and a “type of finite morphisms”, together with, say, some way of composing infinitesimal morphisms to get a finite morphism, which relates to ordinary composition in the way that integration relates to sum.

Amar Hadzihasanovic (May 04 2020 at 11:50):

An outcome of this approach, as Fabrizio mentioned, is to get rid of the “potential”, “in-the-limit” or “dynamic” aspect of infinity, as used in (standard) analysis. I think it's fair to say that this results in something gained and something lost. Often one gets “neater” and cleaner proofs, that appeal to the aesthetics of algebraicists and category theorists; at the same time, they tend to become more “qualitative” and less “quantitative”, precisely because we have given up the computational dynamic that's embedded in stuff like epsilon-delta formulations of limits and continuity. It's harder to see stuff like convergence rates, etc. in this approach. Errett Bishop disliked nonstandard analysis for this reason.

Amar Hadzihasanovic (May 04 2020 at 11:51):

(Of course, in good cases you could say, every nonstandard proof can be interpreted systematically as a standard proof in a way which reconstructs the bounds, etc. But that's like saying you can do constructive proofs with classical logic, because anyway you can double-negate everything...)

Amar Hadzihasanovic (May 04 2020 at 11:51):

The other approach is to embrace the dynamic/computational aspect of infinity as used by analysts. Dusko and Martin's paper, and then Freyd's “algebraic real analysis”, put it into my head very convincingly that this aspect is naturally captured by coalgebra and coinduction. I'm quite partial to this approach because the quantitative and computational aspects are what I like the most in analysis! (May have something to do with the maths department of my old uni being strongest on numerical analysis... I even considered becoming a numerical analyst for some time).

Amar Hadzihasanovic (May 04 2020 at 11:51):

In particular, as I said earlier, Freyd convinced me that subdivision (or bisection) may be the key dynamical component of the continuum. What does it mean that we have processes in continuous time? That given a process

f

which takes time

t \geq 0

, then we can look at what it's doing in the first

t/2

, and then look at what it's doing in the next

t/2

. The process may ultimately be “pointwise” so that subdivision always gives

f

on one side and the identity on the other, or it could at some point split nontrivially...

Amar Hadzihasanovic (May 04 2020 at 11:52):

So I think formalising the idea of “(higher) category with subdivision” (or “(higher) graph with composition and subdivision”) would be my first approach to processes with continuous time/space. The idea would also be that in this framework one should be able to present equationally (in the style of presentations of categories) processes whose semantics are greatest fixed points, stuff like a “Cantor process”

f

defined by the fact that it subdivides into

f; e; f

for some fixed

e

...

Amar Hadzihasanovic (May 04 2020 at 11:53):

But as I said, I'd be surprised if this hadn't been tried, so it's possible it's just a dead end.

Morgan Rogers (he/him) (May 04 2020 at 12:12):

No need to end on such a pessimistic note! Even if people have thought about a thing, its absence from the mainstream could just mean that those people didn't have enough time to spend on it to extract its potential.

Jules Hedges (May 04 2020 at 13:03):

Jules Hedges (May 04 2020 at 13:12):

It might be time to pull the ultimate switcheroo and start thinking about cocategories

Jules Hedges (May 04 2020 at 13:22):

Ps. Over in #general: meta we've been talking about the tricky question of writing up things from here elsewhere (mainly the nLab) for folklore-reduction purposes. I propose this summary as something that could be written down, although it's still vague enough that I have no actual idea where it should be written down

John Baez (May 04 2020 at 15:49):

David Spivak convinced me that categories where you can "subdivide" morphisms are nicely expressed using Conduché functors. The idea is that you have a functor

p : E \to B

and if you have any morphism in

E

whose image in

B

can be factored, you can lift that factorization up to

E

So, for example,

E

could be a category whose morphisms are the motions of an object with the passage of time, and

B

could represent time, e.g. the set

\mathbb{R}

made into a poset with its usual ordering.

dusko (May 05 2020 at 04:26):

or in other words, you can formalize nonstandard proofs in the internal language of a suitable topos (which i think tierney spelled out in the 70s). forcing is "just" standard reasoning in the internal language of a topos in a similar way. that was all spelled out in the 70s, but noone took it to the heart: analysts continued speaking their language, standard or nonstandard, set theorists continued constructing their universes their way. people are fluent in their languages, and they like their stories in their languages, even if it takes more words to tell them. grothendieck spelled out in his thesis all that was to be known about TVS (as he told laurent schwartz, i think) and moved on, and the functional analysts moved on as well, and they still write tensors as matrices of matrices of matrices.

the other way around holds as well. you begin from presenting spaces by categories. i suppose you have grothendieck's "homotopy hypothesis" in mind. commenting about "pursuing stacks", eilenberg pointed out that that idea was going in the oposite direction of his homotopy theory. simplicial complexes were supposed to make computing homotopies simpler, and not more complicated. but on the other hand, people built a beautiful theory of highere categories. eilenberg was probably right that it was an overkill for computing homotopy groups or groupoids. but everyone is having good ideas, and pursuing them. there can't be anything wrong with that.

the ideas about "infinitesimal morphisms" sound very attractive, but it would be good to set some tasks that they should achieve (some problems to solve, some numbers to compute) --- so that the effort can fail. it is not good to pursue a project that always succeeds, no matter what...(except for people who want to be presidents.)

dusko (May 05 2020 at 04:51):

it is interesting that there seems to be a loop in the effort: we capture spaces as categories, because we can capture categories by diagrams. from geometry to algebra to geometry. if we go beyond monoidal categories to 2-categories, the faces in string diagrams become 0-cells. soon we encounter string/face diagrams that unfold in complicated ways. they could be simplified if we extract some algebraic invariants. but to simplify computing them, we categorify, which is nice because we can use diagrams.

Jules Hedges (May 05 2020 at 10:18):

I remember saying something similar to this once, roughly "most people want to use categories to study geometry, but I'm more interested in using geometry to study categories"

Amar Hadzihasanovic (May 05 2020 at 10:29):

Jules did give us a "task" for infinitesimal morphisms or whatever other setup we adopt to succeed or fail at: get the (continuous time) Hamilton-Jacobi-Bellmann equation in a setup that generalises the one in which he obtains the discrete time version.
Of course we would need more details to proceed with that ;)

Amar Hadzihasanovic (May 05 2020 at 10:42):

Re: John's comment on Conduché functors, I don't think there's a lack of category-theoretic means of "describing" processes in continuous time, or in some specific space, etc.

But here I'm looking for intrinsic ways -- in a sense which may be a bit vague. A functor to

\mathbb{R}

as a poset seems to be exactly an "external clock": we are forcing every morphism to "take a certain time" in reference to the clock. So now there's two process worlds, my process world and the clock world. That's very different from the "intrinsic" discrete time given by causal connections of processes in a diagram.

Amar Hadzihasanovic (May 05 2020 at 10:50):

To me this is similar to how "categories enriched in topological spaces" are a disappointing model of

\infty

-categories compared to, say, quasicategories -- and not just because they are harder to work with. It's because they are a chimera put together from two ways of "creating space": an algebro-combinatorial one in dimensions 0 and 1 and a point-set one in higher dimensions...

John Baez (May 05 2020 at 15:34):

Yes, the distinction between intrinsic and extrinsic "clocks" is interesting and also important in physics.

John Baez (May 05 2020 at 15:36):

In general relativity if you say a rock was at one position at t = 0 and another at t = 1 it doesn't really make sense to ask where it was at t = 1/2 without providing any additional information, because there's no god-given best time coordinate.

John Baez (May 05 2020 at 15:36):

John Baez (May 05 2020 at 15:38):

David and I were thinking about Markov processes, where there's an external clock; then given the state of the system at t = 0 and t = 1 you can ask for the probability that it has any particular state at time t where 0

\le

\le

John Baez (May 05 2020 at 15:38):

John Baez (May 05 2020 at 15:40):

So these could probably be studied using Conduché functors (mixed with probability theory in some way).

John Baez (May 05 2020 at 15:40):

The continuous-time Hamilton-Jacobi-Bellmann equation also has an extrinsic clock, so we shouldn't be shy of using it.

dusko (May 05 2020 at 18:25):

the main task seems to be: formalize such intuitions in a productive way. E.g., the idea of mappings from a manifold to

\mathbb{R}

as a space is productively contrasted with mappings from

\mathbb{R}

to mainifolds: one leads to cotangent bundles, the other to tangent bundles. (Incindentally, both turn out to be comonads, and if we spell them out over some gros topos, albeit of all spaces, we can at least reason about it all internally. I think for Grothendieck, a productive meaning of the word intrinsic was internal in a topos.) Another duality of the mappings from and the mappints into leads to homology vs cohomology. So there are many directions in which such intuitions can be tested on a formal language.

Without specifying a particular language, there is a chance that the informal conversation may be some sort of "intuition telepathy": the blind men sharing with excietment their different views of the elephant, until it turns out that one of them is actually talking about the tiger, another one about the skunk, and two of them about each other. E.g., my interpretation of your idea that I offered above stands a good chance of be an "understanding" of that kind. Such is the breadth of English as a carrier of mathematical thoughs.

So the biggest challenge of it all is perhaps to reconcile the two aspects of science: science as a conversation with nature, and science as a conversation with people. How much time should I spend writing symbols on a piece of paper, and how much time should I go out and try to translate that into something people would get interested in?

E.g., you and Jules and Fabrizio have been going about the continuum. Cantor literally had a voice (of Nature?) telling him about the continuum, and he was writing it all down; and then he would go out and try to tell people about it, and there he would hear the voices, such as e.g. of Kronecker, who was trying to presevent such sick ideas from spoling the youth. (There is a book of Cantor's correspondence with Dedekind, edited by Emmy Noether.) Although most of us think less deep than Cantor, we are all in the business of trying to reconcile those two kinds of voices.

Amar Hadzihasanovic (May 05 2020 at 18:35):

On this note, @Jules Hedges, would you be willing to share your notes on optics and the Bellman equation to see if we can make any progress?

Paolo Perrone (May 07 2020 at 15:28):

I thought I'd mention this: there are notions of infinite monoidal products. This leads to the idea of complete monoids (https://en.wikipedia.org/wiki/Monoid#Complete_monoids), and one level up, to the idea of Kolmogorov products that Tobias and Eigil developed to talk about probability (https://arxiv.org/abs/1912.02769).
Maybe, just maybe, what you people are after is the many-object version of these.

sarahzrf (May 08 2020 at 14:11):

sarahzrf (May 08 2020 at 14:12):

John Baez (May 08 2020 at 17:07):

sarahzrf (May 08 2020 at 17:27):

sarahzrf (May 08 2020 at 17:28):

John Baez (May 08 2020 at 17:47):

sarahzrf (May 08 2020 at 17:47):

sarahzrf (May 08 2020 at 17:48):

natural morphisms are the "scott-continuous" functions, i.e., directed-sup-preserving ones

sarahzrf (May 08 2020 at 17:48):

sarahzrf (May 08 2020 at 17:51):

when you have a programming language that allows arbitrary recursion, and you want to interpret its terms as directly denoting mathematical objects rather than ongoing processes that must be gradually resolved (and you're not a brouwerian so you don't think mathematical objects are already like that), you have a problem

John Baez (May 08 2020 at 17:52):

(All I know is that my student Mike Stay suggested a Star Wars remake featuring a character called DCPO.)

sarahzrf (May 08 2020 at 17:52):

Dan Doel (May 08 2020 at 17:52):

sarahzrf (May 09 2020 at 01:32):

sarahzrf (May 09 2020 at 01:33):

it reminds me of the whole "any set is the filtered colimit of its finite subsets" thing

sarahzrf (May 09 2020 at 01:33):

sarahzrf (May 09 2020 at 01:35):

it looks like that equation is basically saying that ∑ turns filtered colimits—or at least this particular filtered colimit—into directed suprema (which are just depleted filtered colimits)

sarahzrf (May 09 2020 at 01:35):

sarahzrf (May 09 2020 at 01:36):

Daniel Geisler (May 09 2020 at 10:26):

R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps, - experimentally consistent with my research
http://arxiv.org/abs/physics/9712026
J. Math. Phys. 39, 5324 (1998)

Stream: theory: applied category theory

Topic: infinitesimal morphisms

Jules Hedges (May 03 2020 at 10:44):

Jules Hedges (May 03 2020 at 10:45):

Jules Hedges (May 03 2020 at 10:46):

Jules Hedges (May 03 2020 at 10:50):

Jules Hedges (May 03 2020 at 10:58):

Jules Hedges (May 03 2020 at 10:59):

Reid Barton (May 03 2020 at 11:31):

Jules Hedges (May 03 2020 at 11:44):

Amar Hadzihasanovic (May 03 2020 at 13:36):

Jules Hedges (May 03 2020 at 14:42):

Fabrizio Genovese (May 03 2020 at 14:43):

Fabrizio Genovese (May 03 2020 at 14:45):

sarahzrf (May 03 2020 at 14:56):

sarahzrf (May 03 2020 at 14:56):

sarahzrf (May 03 2020 at 14:58):

sarahzrf (May 03 2020 at 14:58):

sarahzrf (May 03 2020 at 14:59):

Morgan Rogers (he/him) (May 03 2020 at 15:00):

sarahzrf (May 03 2020 at 15:14):

sarahzrf (May 03 2020 at 15:15):

sarahzrf (May 03 2020 at 15:17):

Jules Hedges (May 03 2020 at 15:17):

Amar Hadzihasanovic (May 03 2020 at 15:17):

sarahzrf (May 03 2020 at 15:18):

sarahzrf (May 03 2020 at 15:18):

Amar Hadzihasanovic (May 03 2020 at 15:18):

sarahzrf (May 03 2020 at 15:18):

Jules Hedges (May 03 2020 at 15:18):

sarahzrf (May 03 2020 at 15:19):

sarahzrf (May 03 2020 at 15:19):

sarahzrf (May 03 2020 at 15:20):

Amar Hadzihasanovic (May 03 2020 at 15:21):

Jules Hedges (May 03 2020 at 15:21):

sarahzrf (May 03 2020 at 15:21):

Amar Hadzihasanovic (May 03 2020 at 15:21):

sarahzrf (May 03 2020 at 15:22):

Amar Hadzihasanovic (May 03 2020 at 15:22):

sarahzrf (May 03 2020 at 15:23):

sarahzrf (May 03 2020 at 15:23):

sarahzrf (May 03 2020 at 15:23):

sarahzrf (May 03 2020 at 15:24):

sarahzrf (May 03 2020 at 15:24):

Morgan Rogers (he/him) (May 03 2020 at 15:29):

Morgan Rogers (he/him) (May 03 2020 at 15:31):

T Murrills (May 03 2020 at 15:49):

Amar Hadzihasanovic (May 03 2020 at 15:52):

T Murrills (May 03 2020 at 15:52):

Amar Hadzihasanovic (May 03 2020 at 15:53):

T Murrills (May 03 2020 at 16:08):

Jules Hedges (May 03 2020 at 16:10):

T Murrills (May 03 2020 at 16:11):

Jules Hedges (May 03 2020 at 16:18):

Daniel Geisler (May 03 2020 at 17:06):

sarahzrf (May 03 2020 at 17:50):

Daniel Geisler (May 03 2020 at 18:28):

Morgan Rogers (he/him) (May 03 2020 at 18:43):

sarahzrf (May 03 2020 at 19:51):

Pastel Raschke (May 03 2020 at 19:54):

Fabrizio Genovese (May 03 2020 at 19:57):

Fabrizio Genovese (May 03 2020 at 19:58):

sarahzrf (May 03 2020 at 19:58):

Todd Trimble (May 03 2020 at 20:30):

dusko (May 04 2020 at 05:26):

sarahzrf (May 04 2020 at 05:35):

sarahzrf (May 04 2020 at 05:35):

Pastel Raschke (May 04 2020 at 08:43):

Amar Hadzihasanovic (May 04 2020 at 11:48):

Amar Hadzihasanovic (May 04 2020 at 11:48):

Amar Hadzihasanovic (May 04 2020 at 11:48):

Amar Hadzihasanovic (May 04 2020 at 11:48):

Amar Hadzihasanovic (May 04 2020 at 11:49):

Amar Hadzihasanovic (May 04 2020 at 11:49):

Amar Hadzihasanovic (May 04 2020 at 11:49):

Amar Hadzihasanovic (May 04 2020 at 11:50):

Amar Hadzihasanovic (May 04 2020 at 11:50):

Amar Hadzihasanovic (May 04 2020 at 11:51):

Amar Hadzihasanovic (May 04 2020 at 11:51):

Amar Hadzihasanovic (May 04 2020 at 11:51):