Why Higher Category Theory in Physics? - Comments

[Urs Schreiber]

Urs Schreiber submitted a new PF Insights post

Why Higher Category Theory in Physics?

Continue reading the Original PF Insights Post.

 

[mitchell porter]

Just a few days ago, for the first time, a paper by Patricia Ritter gave me an explanation that I could understand, for the relevance of n-categories to objects like branes - the categorical identities express the equivalence of different ways of doing certain integrals over a volume, e.g. where in effect you might first integrate in the x direction, then along the xy plane, then throughout the xyz volume; but you might have done all that for a different order of x,y,z... the result needs to be the same for all orderings, and that leads to the categorical formulation of higher gauge theory.

I want to emphasize, that's not exactly what she says, that's me trying to dumb it down to the simplest way of saying it. But am I even approximately correct in this interpretation?

 

[Urs Schreiber]

Just a few days ago, for the first time, a paper by Patricia Ritter gave me an explanation that I could understand, for the relevance of n-categories to objects like branes - the categorical identities express the equivalence of different ways of doing certain integrals over a volume, e.g. where in effect you might first integrate in the x direction, then along the xy plane, then throughout the xyz volume; but you might have done all that for a different order of x,y,z... the result needs to be the same for all orderings, and that leads to the categorical formulation of higher gauge theory.

I want to emphasize, that's not exactly what she says, that's me trying to dumb it down to the simplest way of saying it. But am I even approximately correct in this interpretation?

Yes, that's one good way of thinking about it. This is the motivation from "higher parallel transport".

Like so: the structure of a group (an ordinary group) is exactly what one needs in order to label the edges in a lattice gauge theory: the group product and associativity give that edge labels may be composed, and inverses gives that going back and forth along the same edge picks up no curvature. Of course this is not restricted to the lattice. In general, group-valued gauge fields are exactly the right data to have consistent Wilson line observables

Now a 2-group (categorical group) is, similarly, exactly the data needed to consistencly label edges AND plaquettes in a consistent way (with possibly different labels for each). For instance associativity now includes a 2-dimensional codition which says that with four plaquettes arranged in a square, then first composing horizontally and then vertically is the same (in fact: is gauge equivalent to) first composing vertically and then horizontally.

Again this is not restricted to the lattice. Generally, 2-group valued gauge fields are exactly what one needs for consistent Wilson surfaces.

 

[Greg Bernhardt]

Fascinating topic Urs!

 

[Crass_Oscillator]

This is not even unconvincing, to coin a phrase. If you cannot connect this mathematics to a physical problem other than gravity, it's almost certainly useless, since you are unable to make tangible statements about experimental reality.

There's lots of physics out there in need of new mathematical models. Non-equilibrium quantum/classical physics of few or many bodies, soft matter, fluids, strongly correlated materials etc. Why, a great example would be in 2D materials; we already know that some exhibit Dirac excitations, and speculation has been ongoing for years on condensed phase simulations of gravity. Hawking radiation in dumb holes has already been verified or at least strongly suggested in Bose gases if I am not mistaken.

Without that, there is absolutely no reason for any physicist to take this seriously.

 

[MathematicalPhysicist]

It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?

BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.

Will it work?

Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.

BTW, @[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] do you work at the maths or physics department? :-D

Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".

 

[Crass_Oscillator]

It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?

BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.

Will it work?

Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.

BTW, @[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] do you work at the maths or physics department? :-D

Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".

Well, Feynman is also reputed to have said that "Math is to physics as masturbation is to sex." ;)

However that's not really the point of my gripe. Urs is trying to sell higher category theory to physicists judging by the title alone, but he doesn't really seem to grasp how to make a compelling case to physicists. I could have been polite, but instead I decided to give him a hard prod. I'm very mathematically conservative and am no fan of math driven physics research, but I do recognize the value of more sophisticated mathematics. He needs to make a much better case.

Mathematicians often get lost in how, for instance, the machinery they've constructed shortens previously elaborate proofs, or paves the way to proving that certain mathematical structures have exciting, exotic properties, but I use these mathematical structures to construct models. I don't care about proofs, save quick and dirty ones, and much of physics still doesn't get far past super-charged 19th century calculus, with topology and advanced algebra sprinkled in here and there.

Perhaps starting with Lawvere's book would be the place to begin. And it had better simplify calculations, not make them more complicated. Or, allow me to calculate something that previously was difficult. If Lawvere's book suggests algorithms for solving continuum mechanics problems that nobody had thought of which are competitive with simpler 20th or 19th century algorithms or allows you to build models that obtain experimental observables that are hard to describe otherwise, it's interesting to a physicist. Otherwise, it's a waste of time.

 

[Urs Schreiber]

I understand that not everybody is inclined to follow and follow through the arguments and pointers that I gave, many of them related to string theory. Were it not for the fact that this particular article originates in a personal reply to a question in an interview that Greg was (trying to) do with me, as briefly explained at the beginning, I should have given a more broadly targeted exposition which would have pre-empted some of the misunderstandings that are surfacing above. I have to apologize for this neglect. I believe though that I had included pointers to more general expositions which I have produced elsewhere, a good point to start may be my Oberwolfach talk: Higher Structures in Mathematics and Physics A conspiring phenomenon which I don't feel responsible for is that not everyone cares about the fundamental issues at stake in the first place, and ignorance of a problem may cause underestimation of its solution.

But even string theory with its explicit higher gauge fields aside, there is no room left for the standpoint that higher homotopy structure may be ignored in the formulation of accurate physical theory, certainly not in fundamental physics, but increasingly also in physics relevant for desktop experiments.

Regarding the former I now use the occasion of this addendum to highlight what in a more pedagogical and less personal account would have been center stage right in the introduction, namely the developments propelled by A. Schenkel and M. Benini in the last years, regarding the foundations of quantum field theory. Curiously, it had been a well kept secret for more than half a century that the mathematical formulation of Lorentzian QFT in terms of the Haag-Kastler axioms (AQFT) is incompatible with local gauge theory. At the QFT meeting in Trento 2014 I had pointed out (here) that this may be seen irrespective of details of formulation from basic principles of gauge fields, which is what in mathematics is the principle of "stacks" (higher sheaves). By a curious coincident, at the same meeting Alexander Schenkel presented (here) a detailed analysis of the AQFT construction of free QED (without matter) showing explicitly how it fails the locality axioms. As I had explained (here, see also this BA thesis for a still simple but more technical introduction ) the solution to this problem is higher homotopy/category theory, namely the local net of quantum observables has to be promoted to its homotopy version, sometimes called a co-stack or similar. Since then Beninin, Schenkel at al. have be been demonstrating this in increasing detail, I recommend to try to look at least at the introductions of these articles:

Marco Benini, Alexander Schenkel, Richard J. Szabo
"Homotopy colimits and global observables in Abelian gauge theory"
Lett. Math. Phys. 105, 1193-1222 (2015)
https://arxiv.org/abs/1503.08839

Marco Benini, Alexander Schenkel
"Quantum field theories on categories fibered in groupoids"
https://arxiv.org/abs/1610.06071

Next, regarding the second point of higher mathematical structures required in solid state physics, I'd just draw your attention to a little mini revolution in the field that has been going on the last years, and which is reflected in the last round of Physics Nobel Prizes: the understanding of topological phases in solid state physics. The influential publication here is

Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen,
"Symmetry protected topological orders and the group cohomology of their symmetry group",
Phys. Rev. B 87, 155114 (2013) arXiv:1106.4772; Shortt version in: Science 338, 1604-1606 (2012)

which spurred much activity in the use of higher mathematical structures for the description of topological phenomena in solid states, such as notably certain configurations of Graphene. It turns out that the stable homotopy theory of twisted generalized cohomology theory is required to understand the special topological behaviour of these gapped physical systems

Daniel S. Freed, Gregory W. Moore,
"Twisted equivariant matter",
Annales Henri Poincaré December 2013, Volume 14, Issue 8, pp 1927–2023
arxiv/1208.5055

You see these solid state physicists indulge in higher category theory for their solid needs, such as

Liang Kong, Xiao-Gang Wen
"Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions"
arXiv: 1405.5858

To dispel the idea that this is maths too far ahead of its physics development, it may be instructive to see the evidence that instead the maths is lagging behind, see Edward Witten's question to the maths community: Group cohomology and condensed matter

So much for tonight. If you have further questions or remarks, I'll offer to react, but please take a moment to make sure that before you go the easy route and conveniently declare as irrelevant what is unfamiliar and potentially scary, to take the chance to read up and learn first of all about open problems in physics that you may have been unaware of, and second about their mathematical answers.

Beware the instructive events in history where this attitude of ignorance backfired. There was a time when people rejected complex numbers as overly fancy mathematics. Interestingly, it was largely the observation of complex numbers in physics experiment, namely in the guise of quantum mechanical phases, which revealed this attitude as born out of ignorance and laziness. What complex numbers were for the physics of the beginning 20th century, so higher homotopy/category theory is for the physics of the beginning 21st century. Don't be left behind.

 

[RockyMarciano]

Regarding the former I now use the occasion of this addendum to highlight what in a more pedagogical and less personal account would have been center stage right in the introduction, namely the developments propelled by A. Schenkel and M. Benini in the last years, regarding the foundations of quantum field theory. Curiously, it had been a well kept secret for more than half a century that the mathematical formulation of Lorentzian QFT in terms of the Haag-Kastler axioms (AQFT) is incompatible with local gauge theory. At the QFT meeting in Trento 2014 I had pointed out (here) that this may be seen irrespective of details of formulation from basic principles of gauge fields, which is what in mathematics is the principle of "stacks" (higher sheaves). By a curious coincident, at the same meeting Alexander Schenkel presented (here) a detailed analysis of the AQFT construction of free QED (without matter) showing explicitly how it fails the locality axioms. As I had explained (here, see also this BA thesis for a still simple but more technical introduction ) the solution to this problem is higher homotopy/category theory, namely the local net of quantum observables has to be promoted to its homotopy version, sometimes called a co-stack or similar. Since then Beninin, Schenkel at al. have be been demonstrating this in increasing detail, I recommend to try to look at least at the introductions of these articles:

Diagnosing that there is a problem(I'd say a serious one), like the mentioned about gauge fields and locality in the way they are usually displayed, and finding a good path towards its solution are two processes that not always come together.
How would you convince a physicist that at least gets to glimpse the issue (the well kept secret) that the solution lies in higher homotopy/category. I mean I guess going to higher homotopy allows you to obtain finer distinctions and gives you more flexibility to accommodate locality in a way that the more rigid lower categories can't, but how is this actually done and connected to the actual physics? what is the physical correlate of the 2-bundle?, how do the different physical interactions fit in all this?
On the other hand it seems this is a movement in the direction of greater complexity, while traditionally generalizing theories have worked in physics in the direction of simplifying the local apparent disparately unrelated observations(well admittedly this trend is not so clear in the case of QFT). It would seem that going to higher categories adds in complexity, so it would be great if examples were given about how it could be applied to specific physical problems and the unification of observations or maybe give more details about the above mentioned applications to solid state physics.

 

[Urs Schreiber]

How would you convince a physicist that at least gets to glimpse the issue (the well kept secret) that the solution lies in higher homotopy/category.

By explaining it, as I did. Do have a look at the slides . They are expository. They are aimed at a QFT audience. They were invited at a QFT meeting. Do have a look. It's not black magic.

I mean I guess going to higher homotopy allows you to obtain finer distinctions and gives you more flexibility to accommodate locality in a way that the more rigid lower categories can't, but how is this actually done and connected to the actual physics?

The point is that homotopy in mathematics is exactly the formalization of gauge transformation in physics. If you want to be serious about describing a system with gauge symmetries, the relevant mathematics is, by necessity, homotopy theory. See the slides.

what is the physical correlate of the 2-bundle?,

It's the correct field bundle such that the field content does contain the global instanton sectors, and not just the patchwise gauge field information. See towards the end of the slides where this is explained.

how do the different physical interactions fit in all this?

They are encoded by the Lagrangian density, as usual. The difference is this: textbooks say that the Lagrangian density is a horizontal differential form on the jet bundle of a field bundle. But this is just not true in general. For gauge theories there is no field bundle that captures the full global field content. Instead the correct field bundle is a 2-bunde. So is it's jet bundle. And the Lagrangian density which encodes the gauge QFT (interactions and all) is a differential form on that jet 2-bundle.

On the other hand it seems this is a movement in the direction of greater complexity,

This is an illusion, coming from confusing unfamiliarity with complexity. Similarly, originally people said that complex numbers are overly complex, whence the name. Later they realized that, on the contrary, many a thing in real analysis becomes simpler when passing to the complex domain.

Homotopy theory is conceptually most simple. But rich in phenomena. It is just mathematics with the gauge principle natively built in.

 

[Debaa]

It's maths, @Crass_Oscillator ,if one day some physicists will find applications for it, then why not?

BTW there's the book by Lawvere on Categories in continuum mechanics, so it seems mathematicians are working hard of finding applications of this work in the physical sciences.

Will it work?

Who knows, but for the maths sake I still will like to learn this stuff, eventually something beneficial will come from it, even if not in physics.

BTW, @[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] do you work at the maths or physics department? :-D

Anyway, how did Feynman once said about sex and physics:" sure, it's practical, but that's not why we do it".

[emoji23] [emoji23]

 

[RockyMarciano]

By explaining it, as I did. Do have a look at the slides . They are expository. They are aimed at a QFT audience. They were invited at a QFT meeting. Do have a look. It's not black magic.
The point is that homotopy in mathematics is exactly the formalization of gauge transformation in physics. If you want to be serious about describing a system with gauge symmetries, the relevant mathematics is, by necessity, homotopy theory. See the slides.
It's the correct field bundle such that the field content does contain the global instanton sectors, and not just the patchwise gauge field information. See towards the end of the slides where this is explained.
They are encoded by the Lagrangian density, as usual. The difference is this: textbooks say that the Lagrangian density is a horizontal differential form on the jet bundle of a field bundle. But this is just not true in general. For gauge theories there is no field bundle that captures the full global field content. Instead the correct field bundle is a 2-bunde. So is it's jet bundle. And the Lagrangian density which encodes the gauge QFT (interactions and all) is a differential form on that jet 2-bundle.

This is an illusion, coming from confusing unfamiliarity with complexity. Similarly, originally people said that complex numbers are overly complex, whence the name. Later they realized that, on the contrary, many a thing in real analysis becomes simpler when passing to the complex domain.

Homotopy theory is conceptually most simple. But rich in phenomena. It is just mathematics with the gauge principle natively built in.

Actually I had read the slides. The problem with slides separated from their oral presentation is that they are not nearly as explanatory as the actual talk. I understand you might be rationing your efforts to spread higher category among physicists though, it is a long distance race. Hope it gets somewhere, I think it is a step in the right direction.
Thanks for bothering to answer, had more questions but I guess I'll just see the slides again.

 

[RockyMarciano]

It's the correct field bundle such that the field content does contain the global instanton sectors, and not just the patchwise gauge field information.

Some would argue that it is the demand of locality that prevents from having a field that contains global instanton sectors.

They are encoded by the Lagrangian density, as usual. The difference is this: textbooks say that the Lagrangian density is a horizontal differential form on the jet bundle of a field bundle. But this is just not true in general. For gauge theories there is no field bundle that captures the full global field content. Instead the correct field bundle is a 2-bunde. So is it's jet bundle. And the Lagrangian density which encodes the gauge QFT (interactions and all) is a differential form on that jet 2-bundle.

SCNR. Extraordinary claims require extraordinary proofs. The above are very strong statements, the physical validity of which can only be demonstrated by proving how the 2-bundle contains the global instanton sector and captures the global field content. First you have to prove that the usual textbook procedure can't do it(in other words that the Yang-Mills existence and mass gap problem cannot be solved within the current framework), this by itself is a highly prized enterprise, and then you have to prove that the 2-bundles of higher category do indeed give rigorous path integrals(that need to capture that global content) and solve the Yang-Mills problem.

Without all this backing the above statements, certain scepticism from physicists is granted, not towards the milder claim that this COULD be a good research program to eventually be able to make them , but about doing it as of now.

 

[Urs Schreiber]

Some would argue that it is the demand of locality that prevents from having a field that contains global instanton sectors.

Sorry to say this once more, but this is just the point explained in the slides.

If you tell me to which point you follow the argument, and where you first feel you're thrown, I'll help you out with further comments at that point.

Extraordinary claims require extraordinary proofs. The above are very strong statements, the physical validity of which can only be demonstrated by proving how the 2-bundle contains the global instanton sector and captures the global field content.

That part is easy. For more details than in those slides, you could see the review

A higher stacky perspective on Chern-Simons theory
Domenico Fiorenza, Hisham Sati, [URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL]
https://arxiv.org/abs/1301.2580

First you have to prove that the usual textbook procedure can't do it(in other words that the Yang-Mills existence and mass gap problem cannot be solved within the current framework), this by itself is a highly prized enterprise, and then you have to prove that the 2-bundles of higher category do indeed give rigorous path integrals(that need to capture that global content) and solve the Yang-Mills problem.

This paragraph is mixing up a few things. What is your background? Do you know the concept of a fiber bundle? Maybe at the level of Fiber bundles in physics?

Without all this backing the above statements, certain scepticism from physicists is granted,

I don't think it's fair towards the physics community to continuously refer to them as a crowd unable to understand some basics. None of the authors or references that I listed here in this thread are in pure mathematics, all of them are physicists who understand this business.

 

[RockyMarciano]

If you tell me to which point you follow the argument, and where you first feel you're thrown, I'll help you out with further comments at that point.

Thanks for the offer. I've found a video with the presentation of the slides, I haven't finished it yet.

This paragraph is mixing up a few things.

My background includes fiber bundles. I might be mixing up things but if the claim is that the usual QFT textbooks have a wrong mathematical framework(Yang-Mills gauge fields and gauge bundles) to handle locality, that should have some consequence in the form of rigorous proof, I would honestly like to know how is this reasoning confused?(P.S.:Maybe it's just my perception but your tone in part of your replies to my comments comes across as defensive/dismissive, not sure why. If I ask all these questions is because the approach interests me, anyway I'll keep leaving out the condescending parts).

 

[RockyMarciano]

So after watching the talk I have a very basic question. When going from the usual gauge fields with their gauge equivalence relations and with the gauge principle defined informally as the fact that these gauge equivalence relations that allow the concept of gauge transformations are supposed to leave the physics unaffected,
to the gauge fields as groupoids in which the choice of equivalence(g,h,...) is "remembered" and matters physically(in the talk it is stressed that instantons are physically important in baryogenesis and the observed manifestations of the QCD vacuum), the added step about the choice of gauge actually being physically important and observable, even if it looks as a good step locality-wise as explained in the talk, appears to me go counter the very gauge principle it is trying to extend.

I mean if the gauge principle is about redundancy and gauge equivalence as "physical undistinguishability" of the gauge fields under gauge transformations, I'm not sure how introducing formally(the move from pre-stack to stack and from bundle to 2-bundle) this "memory of the choice of equivalence" that actually is inserting physical distinguishability, can keep the original notion o gauge equivalence. Is this modification perhaps changing the meaning of gauge symmetry, so it doesn't require for the gauge fields under gauge transformations to be physicall indistinguishable anymore? But this seemed to be the essence of the gauge principle.

This is a very basic question that might come from some deep confusion of mine but I think is worth clarifying.

 

[RockyMarciano]

To be more specific, there is a subtle mathematical nontriviality in applying the cocycle condition to fix the above that seems to be overlooked in the slide presentation.

 

[Urs Schreiber]

I mean if the gauge principle is about redundancy

That's the thing, it is not. Only globally, when all the dust has settled, at the very end of a computation, we are intersted in passing to gauge equivalene classes. But if you insist on doing this throughout, then either locality goes out of the window or else all topological effects such as instantons go away.

Consider the simple case of an SU(2)-instanton; its instanton number is all in the gauge transformation, it's the winding number of the gauge transformation. If you forget which specific gauge transformations you use, then you destroy all nontrivial instanton sectors.

 

[Urs Schreiber]

To be more specific,

That would be good

there is a subtle mathematical nontriviality in applying the cocycle condition to fix the above that seems to be overlooked in the slide presentation.

That's not specific at all, it is impossible to tell what you mean. Try to ask a concrete precise question regarding the first point in the presentation where you are not following.

 

[RockyMarciano]

Thanks for your replies.

That's the thing, it is not. Only globally, when all the dust has settled, at the very end of a computation, we are intersted in passing to gauge equivalene classes. But if you insist on doing this throughout, then either locality goes out of the window or else all topological effects such as instantons go away.

I can see this. By redundancy I meant that global passage to gauge equivalence at the end that you are also assuming when the dust settles. Now the intermediate steps where you are very reasonably demanding locality have to be compatible with the end global result. I was stuck trying to justify the compatibility of these two realms(local or intermediate and global or final) in a fundamental level. But I guess in doing this you are simply relying(as everyone) on the usual concept of actual infinity and the axiom of choice, right? And for this the whole apparatus of higher categories and homotopies ad infinitum fits really well so it makes sense to promote it.

Try to ask a concrete precise question regarding the first point in the presentation where you are not following.

Hope I managed above.

 

[Urs Schreiber]

For readers interested in the topic of the above PF Insights:

The British Physics Research Council and the London Mathematical Socienty are funding a Symposium on
Higher Structures in M-Theory
this August at Durham.

Unfortunately there are no funds left to support travel or accomodation, but if you are interested and have means to get there, you should be welcome.

 

[Fra]

For someone not actively into this, from that angle, the insight article is dense containing references to plenty of more dense papers, but even so i really enjoy seeing Urs passion and red line of argument! I am sure this is not effortlessly conveyed to physicists that does not have the strong mathematical inclination that Urs has.

While i can't claim to have digested the insight article in any detail, i just make associations from my perspective when trying to understand Urs suggestion that the mathematical tools for understanding the theory of theories we end up with in physics. I don't know what Urs things about these associations, but they arent mathematical statements, they are just supposedly a conceptual bridge to the higher category abstractions and the abstractions of spontaneous information processing that is use.

So Urs, does my association make any sense from your formal technical perspective? In order not to derail your thread here, i will keep the comments minimal.

In my view, there are something we can call "objects" that corresponds to physical structures and encode information about their own environment. we can call these also information processing agents. We can also associate them to matter.

Then the set of these objects can morph into other objects, in the same we we recode information. In these transformations, anything can change, even topology.

Without going into details, loosely speaking this is the basis for a category, right? Then if we consider that the set of morphisms (which can be understood as a set of possible computation programs) are acutally evolving, and are restricted by the strucutre of the objects, we here get a higher category as the morphisms themselves are resulting from another process. (I see thigs starting from permutations of discrete states), and the set of morphisms get richer and richer the more complex the objects get (getting bigger information capacity or mass generation).

What i envision here is that the "order" of the categories will essentiall be a dynamical process, except one without external description, so its better described as an evolutionary process.

These are abstractions i am working with as well, but to be honest i do not yet know which mathematics in the end that will end up be the right thing. And i see at least a possibility that this can be describe as higher categories as well. But unlike you, i am not convinced that the mathematical angle itselt is the right "guide". My own internal guidance is much more intuitive, and based on a vision of interacting "computer codes". But it might well be that this converges to something that is characterised by higer cateogories.

/Fredrik

 

[Urs Schreiber]

My own internal guidance is much more intuitive, and based on a vision of interacting "computer codes". But it might well be that this converges to something that is characterised by higher categories.

The modern theory of computation is secretly essentially the same as category theory. This remarkable confluence has been called computational trinitarianism.

This has more recently been reinforced by the understanding that the foundations of computation is in fact a foundation for homotopy theory; this insight is now known as homotopy type theory.

I recommend this introduction:

• Mike Shulman,
  Homotopy Type Theory: The Logic of Space,
  in "New Spaces for Mathematics and Physics", 2018
  (arXiv:1703.03007)

 

[Fra]

The modern theory of computation is secretly essentially the same as category theory. This remarkable confluence has been called computational trinitarianism.

This has more recently been reinforced by the understanding that the foundations of computation is in fact a foundation for homotopy theory; this insight is now known as homotopy type theory.

I recommend this introduction:

• Mike Shulman,
  Homotopy Type Theory: The Logic of Space,
  in "New Spaces for Mathematics and Physics", 2018
  (arXiv:1703.03007)

Thanks, I will try to get around to check that paper.

Just to take a huge jump ahead of things - regardless of the value of a developed branch of mathematics that can bring some order and structure into some of the crazy things that is going on at the foundations of theoretical physics, i mainly wonder if once the correspondence is established, wether there is a nice wealth of theorems etc, that can immediately be "translated" into conjectures about the marriage of the standard model of particle physics which IMO "lives" in and is dependent on a rigid classical frame of reference, and the inside vides that are implies either by earh based cosmological theories, or the "inside views" implicit in understand how forces are unified at high energies?

For example, would the mathematical machinery of higher category theory, provide a physicist with any brilliant shortcuts to understand unification?
etc. I somehow would not expect things to be that nice, but perhaps you see this differentlty?

/Fredrik

 

[Urs Schreiber]

For example, would the mathematical machinery of higher category theory, provide a physicist with any brilliant shortcuts to understand unification?
etc.

There are theorems that indicate that this is the case. I must have mentioned this before:

• "Supercocyles and the Brane scan" (pdf)
  talk at SuperPhysMatics18, NYU AD, March 2018
  (based on talk at StringMath2017, see exposition at PF Insights)

 

[Urs Schreiber]

Incidentally, today Thomas Nikolaus and Konrad Waldorf came out with their long-announced discussion of how Hull's T-Folds (a subtle "non-geometric" generalization of perturbative string backgrounds) are realized in the higher differential geometry of higher principal bundles (catgorical bundles) for higher structure group the T-duality 2-group:

Thomas Nikolaus, Konrad Waldorf,
"Higher geometry for non-geometric T-duals"
(arXiv:1804.00677)

 

[Fra]

As I am not a fan of getting lost in details, before you know youre in the right forest, may i ask you another "way ahead of things" question. (Maybe the answer is in your references though)

There is no question that i see the abstraction here, where one can describe theories, relations between theories, and theories about theories in a more abstract way of higher categories. But in my view, and an terms of the computational picture of interacting "information processing agents", the computational capacity and memory capacity must put constraints on how complex the "theory of theory" can be, and still be computable. After all, from the point of view of an information processing agent, simply trying to survive in the black box environment, an algorithm that is too complex to run (and can't be scaled down) to the limited hardware in question is useless, it will get the agent "killed".

What i am trying to say is, what prevents this n-category from just inflating into a turtle tower of infinitity-category? And how do you attach such complexiy to experimental contact? After all, this is what i see as the problem so far. You can ALWAYS inflat and theory, and create a bigger theory. But my hunch is that there is a physical cutoff (relating to observers mass) that must fix the maximum complexity here, and thus for any specific case, we should find some kind of maximal n?

Now, does the n-cat machinery really provide any insight to THIS point?

/Fredrik

 

[Urs Schreiber]

You can ALWAYS inflat and theory, and create a bigger theory.

Remarkably, homotopy theory is a "smaller theory": it arises from classical theory by removing axioms from classical logic. This is the fantastic insight of homotopy type theory.

The story goes like so: In the 70s logician Per Martin-Löf comes up with the modern foundation of computer science, now known as intuitionistic type theory (where "type" is short for "data type", such as "Boolean" or "Integer".) This type theory is an absolute minimum of logic, as Martin-Löf lays out very enjoyably here:

Per Martin-Löf,
"On the Meanings of the Logical Constants and the Justifications of the Logical Laws",
Nordic Journal of Philosophical Logic, 1(1): 11–60, 1996,
(pdf)

Moreover, this type theory is fully constructive, meaning roughly that it regards everything as a computer program.

In particular it thus regards assertions of equality as computer programs: The assertion $x = y$ is to be understood as a computer program $\gamma_1$ which checks that indeed $x$ equals $y$. This brings with it the curious possibility that there can be another program $\gamma_2$ which also proves that $x$ equals $y$.

First, Martin-Löf distrusts his own theory, feeling that this would be weird, and imposes the ad hoc extra rule (the axiom of uniqueness of identity proofs) saying that any two such computer programs, proving $x = y$ must in fact be equal. But eventually it is realized that such an ad hoc rule is awkward and breaks various nice properties of the system. Hence it is removed again, sticking with the minimal theory.

But this minimal theory now has a maximally rich behaviour: To ask whether the two programs $\gamma_1$ and $\gamma_2$ are equal or not, one now needs to invoke yet another program $\kappa$ which checks $\gamma_1 = \gamma_2$.

And again there may be two different such programs, $\kappa_1$ and $\kappa_2$, and to tell whether they are equal, we need to invoke yet another computer program

And so ever on. (Graphics taken from Higher Structures in Mathematics and Physics.)

For decades nobody in computer science new what to make of this. Then suddenly there was a little revolution, when it was realized that this minimal theory of computation with this curious rich inner structure is secretly the formal computer language for homotopy theory and higher category theory: It automatically regards data types as homotopy types. Ever since, Per Martin-Löf's type theory is now called homotopy type theory.

This a new foundation of mathematics rooted in computer science and flourishing into higher category theory. It may be argued that it also provides foundations for modern physics, see at Modern Physics formalized in Modal Homotopy Type Theory.

 

[Fra]

It is indeed interesting to note how what you know says in itself is an abstraction that is analogous to the observer problem! It is a totally different but intuitive way to raise questions that just by replacing labels look similar to yours. This is cool indeed!

For example: observer equivalence (beeing at heart of physical gauge theory) may ask. How do we prove that observers inferences (which indeed i viee as special computations) are as consistent as we think they "must be"? This imo hits at the heart of the problems of fundamentaö physics. Like the reaction of Per some are tempting to put this as an axiom! Which translates to saying there must be observer invariant eternal physical laws.

However i think, this is a fallacy and it is remarkably analogoua to what smolin calls cosmological fallacy.

The resolution is that the only way to "compare" inferences between two observers is to let them interact AND have a third observer to judge.

Thanks for enriching the forum which great things!
/Fredrik

 

[Greg Bernhardt]

FYI,

David Spivak has a new version of the e-book out:

Seven Sketches in Compositionality: An Invitation to Applied Category Theory
http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf

 

[stevendaryl]

Remarkably, homotopy theory is a "smaller theory": it arises from classical theory by removing axioms from classical logic. This is the fantastic insight of homotopy type theory.

Well, larger versus smaller or simpler versus more complex depends on how you measure things. You can understand classical mathematics as being a simplification of constructive mathematics, in which you toss out distinctions. (such as the distinction between a proof of $A$ and a proof of $\neg \neg A$)