# Insights Why Higher Category Theory in Physics? - Comments

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1. Apr 4, 2018

### 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:

"Higher geometry for non-geometric T-duals"
(arXiv:1804.00677)

2. Apr 4, 2018

### 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 cant 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

3. Apr 5, 2018

### Urs Schreiber

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.

Last edited: Apr 5, 2018
4. Apr 5, 2018

### 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 wich great things!
/Fredrik

5. Apr 7, 2018

### Greg Bernhardt

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 $\neg \neg A$)