8:23 AM
I think at this point I should be clear that my previous comment was not a criticism of Lurie's book(s). I have not really read them, nor have I had cause to, really. But, as an outsider to higher category theory, they are kind of intimidating in scale. Pointing people to the whole 950 page thing (like "give HTT another try") doesn't help. I guess my point is that it's great that reference texts exist, but they also aren't really an introduction.
2 hours later…
10:29 AM
Following the comments about the foundational issues in higher category theory, I feel like choosing that synthetic model for groupoids (Segal stuff or Quillen fibra the simplicial sets) is essentially cheating. On the other side, Grothendieck and Batanin models, the analytics models in the theoretic and operadic sense respectively, simulate the usual intuition of how infinity groupoids should be defined.
Also Riehl and Verity approach that was cited by many people, relies on the usual choice of the model for an infinity groupoid, so it’s not really model independent. So maybe things should be performed in a axiomatic setting where no model for homotopy theory must be chosen. Maybe test categories or something like this would do the work. Of course, there’s another approach. One could start with HoTT as a metatheory and then build higher category theory, but this is another way of cheating.
10:56 AM
There are lots of introductory texts on higher categories, some of which go quite far into things. Check out Rezk's course notes: faculty.math.illinois.edu/~rezk/595-fal16/quasicats.pdf ... They're brilliant, and there one of dozens of great bits of writing on this subject.
I disagree with the attitude that the technical details bear little resemblance to the 'true' way of thinking about these objects. I claim that there is no unicity in the space of legitimate ways of thinking about these things. Additionally, I know from experience that taking the time to 'get inside' the way of thinking represented by this perspective offers more insight, not less.
11:17 AM
A friend of mine suggested that instead of a "car" a better analogy is a programming language. Usual category theory and set theory are like Assembly while higher categories are more high level languages (Fortran, C, Python, etc...). These languages were written on top of Assembly, they didn't replace it. Moreover knowing some deep stuff about a programming language (as opposed to just knowing the syntax) gives you more possibilities and power when you're trying to do stuff with it.
@ClarkBarwick : thanks for your answer. I would like to address each of the point you raise but at the moment I can only stub an item list:
1. The familiarity with Joyal/Lurie's approach is a consequence of your having "tried and failed". You distinctly feel you need something, you already know where the language fails to be expressive or lacks foundation. To a category theorist, HTT seems to be a cluttered amount of ways to complicate your life, hiding fundamental questions in favour of applications
1. The familiarity with Joyal/Lurie's approach is a consequence of your having "tried and failed". You distinctly feel you need something, you already know where the language fails to be expressive or lacks foundation. To a category theorist, HTT seems to be a cluttered amount of ways to complicate your life, hiding fundamental questions in favour of applications
In a few words you don't have problems with HTT because you already know which issues it tried to solve and you already attempted to draw a map of the land you were roaming when someone handed you the orange book
A category theorist's faith is somewhere else, they need different tools because they have different questions, and the computational complexity of the arguments in HTT is not only daunting, but disqualifies them as being "categorical".
Sure it's a titanic effort to bring clarity in the dark, but it's leaving outside the outsiders, and it's not category theory.
I'd say that from the outside it seems an elitary attempt to let a crew of algebraic topologists speak their private dialect, and solving their problems, not ours. What is sad, is that many people tend to forget the phenomenon known as "everybody knows what a derived category is". (a statement which is true if "everybody" is a subset of Paris' athematicians); it already happened once, why are we letting it happen again, in 201x when mathematics is globalize and happens on facebook?
11:40 AM
11:53 AM
Are you familiar with the etiquette of software developing? HTT is an outstanding application with zillions of uses, but it was deployed too early. And you shouldn't give people bad code, because unfortunately they tend to use it first, instead than making it bulletproof (improving 100 lines of code: few days; improving 10K lines: ~years).
@user40276 This "intuition" for how infinity-groupoids "should be defined" is presumably based on the standard definitions of things like 2-categories. But what is the reason for assuming that the standard definition is the best one (as opposed to just the historically original one)? You could equally well start with a "homotopical" definition of 2-categories, for example as functors from $\Theta_2^{op}$ to groupoids that satisfy Segal conditions (in terms of equivalences of groupoids).
I would guess that such an approach would have some advantages over the traditional one (for example, if you kept going in this style you'd probably avoid writing down multiple pages of commutative diagrams whenever you define something). Are there (and I mean this as an actual question, not a rhetorical one) reasons to prefer the traditional approach (which to me seems often quite cumbersome) to a homotopical version, and to think it gives better intuitions about higher categories?
@DisappointedCategoricien I'm sorry to say that I disagree with virtually everything you've said here. Let me agree with the bits I can agree with: Yes, higher category theory, in the grand scheme of things, is still in its infancy. That seems unproblematic to me.
No, HTT doesn't trivialise the coherence diagrams that define a tetracategory. HTT only deals with (∞,1)-categories. Tetracategories are (4,4)-categories. Trivialising those axioms was done by Charles Rezk (gosh, his name keeps turning up!): faculty.math.illinois.edu/~rezk/cs-objects-revised-arxiv.pdf
12:56 PM
@RuneHaugseng The advantage of an analytic theory is that every coherence is explicitly written down. You re not choosing an n-ary composition, which hides the coherences. So the benefit is the same one that any analytic theory offers:everything is explicit and constructive. Of course, having both the synthetic and analytic theories as tools would be the better option. And as I said an even better option would be an axiomatic characterization of the category of infinity groupoids. You may
2 hours later…
3:06 PM
@DisappointedCategoricien I take issue with you speaking for all category theorists and putting opinions in their mouths. I consider myself a category theorist and certainly consider HTT to be a great piece of category theory. Even more, all of the important theorems in the book with the essence of their proofs can be quickly intuitively understood by anyone sufficiently familiar with categorical thinking.
Your statements show only that you have entrenched yourself in one specific narrow point of view and method and unwilling to familiarize yourself with other approaches - which is against the spirit of category theory. Making forced choices is bad, and so is making once and for all choices about your point of view.
Is HTT an easy read or a good introduction to the subject for the newbies? No, certainly not, nor is it intended to be. Neither are Borceux's "Handbook" or Johnstone's "Elephant", but they are both good category theory. It is an encyclopedic foundational reference in the spirit of EGA/SGA, intended foremost to convince the experts of the merits of quasicategories, and secondly, to give a comprehensive treatment of all foundational questions in a single language.
Your software development analogy is good, except that you have got it exactly the wrong way. There is no such thing as "bulletproof code", neither 10 000 lines nor 100 one. I can crash something as simple as "Hello World" in multiple bad ways, given sufficient system access. So a good piece of code is the one that solves most of users' problems in the shortest and most robust sort of way, where "robust" again means solve all critical bugs and most common non-critical bugs.
For many kinds of software the best way to develop isn't to plan and do everything in advance but to roll out the minimal working version to the users as soon as possible and then work guiding yourself by the users' feedback, by their problems and use cases. You can spend years building your "awesome app" only to learn that you solved a problem that no one cared about, and the actual use cases can be entirely different.
3:49 PM
Regarding your linked comments: 1 and 2 are just you enforcing your stereotypes, so I will ignore it. You should approach a new subject with an open mind, not try to fit it into preconceived notions modeled on a few simple cases. Regarding 3: I don't know what you mean by "sound and expressive coend calculus". Coends can certainly be defined via HTT methods and they will obey the same formal properties.
Roughly, if you have a proof which relies only on universal properties and not on calculations in specific model, then it would be translatable verbatim into higher category theory. This is true not only of coends but of absolutely most category theoretical results.
"there are many other toposes" - most of HTT can be done in any topos, since the definitions and proofs are constructive. You would at the very least require the existence of natural numbers, otherwise discussing any algebra is a mess, but other than that most proofs should work. The name of the book is "Higher topos theory", so it's obviously concerned with toposes other than Set and sSet.
One quote I am reminded strongly of is from Illusie: "That was one principle of Grothendieck: every assertion should be justified, either by a reference or by a proof. Even a “trivial” one. He hated such phrases as “It’s easy to see,” “It’s easily checked.” When he was writing EGA, you see, he was in unknown territory. Though he had a clear general picture, it was easy to go astray. That’s
partly why he wanted a justification for everything." I would compare this very much with the situation in higher categories. It is relatively easy to have correct intuition, but a…
partly why he wanted a justification for everything." I would compare this very much with the situation in higher categories. It is relatively easy to have correct intuition, but a…
It would be nice to have foundations that would abolish the use of sets entirely, but you can't do that in classical mathematics since morphisms form a set and objects form a set. In fact higher category theory is the only way to move forward: if all that exist for you are the categories, then the morphisms themselves must form a category, and the morphisms in this category, etc.
Internalization - internal (00,1)-categories are very easy to talk about in quasicategorical language, the foundations are already stated in this way. Enrichment - here we have a problem. We can talk about enriched categories, but that would be messy since already defining (00,1)-categories as enriched in spaces is messy (compare with classical case, where all categories are enriched in Set).
Indexation - the fibrational language is well-suited for that. I believe that arguments using indexed-category approach are already common in HTT.
"Trivialize the coherences" - I don't know what do you mean by the word "trivialize" since higher coherences are a very nontrivial piece of structure. You can neither drop nor collapse them. This is fundamental and it would be obvious if you were familiar with algebraic topology: it is the same as having spaces which are not products of Eilenberg-Maclane spaces, i.e. almost always.
1 hour later…
5:10 PM
@user40276 But why is it an advantage to have the coherences explicit? My impression is that even with 2-categories they are quite messy to work with. "What is the minimal amount of data you need to specify to define a symmetric monoidal 2-category?" is a reasonable question to ask, but it seems to me that it has a very different answer than "What is the cleanest definition of symmetric monoidal 2-category to write down, work with, and give examples of?"
5:28 PM
I guess the point I'm trying to make is just that the homotopical approach seems to me to be the right one for actually doing stuff with higher categories (as opposed to just studying definitions of higher categories for their own sake), arguably even for 2-categories. (Perhaps nobody here actually disagreed with that claim.)
Of course that doesn't mean that there isn't some interesting mathematics to be done concerning other approaches (and relating them to the homotopical ones).
@AntonFetisov Segal spaces (at least when regarded as simplicial objects in the infinity-category of spaces, rather than as bisimplicial sets) can reasonably be regarded as defining infinity-categories as "infinity-categories enriched in spaces". I guess I also have to object to the idea that defining enriched infinity-categories is "messy" (though perhaps the papers on the subject are...)
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