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Leopold Schlicht (Nov 15 2021 at 20:13):To a large extent higher category theory seems to be about finding the right definition of -category (). There are many proposed definitions, and each such definition is referred to as a "model" for -categories.
Why is it the case that there are so many models? What is so hard about defining -categories? Tom Leinster explains the idea of an -category in this lecture. What goes wrong if one tries to naively define -categories along these lines? (Is there like a canonical "naive definition", which for some reason isn't satisfactory?) Is the hard part spelling out the coherence axioms or specifying the data?
Why are weak Kan complexes a natural model for -categories? I guess these were introduced as a model for spaces first. But then Joyal had the idea that they provide a good definition of the informal idea of an -category and that one can do category theory with them. Why is this a natural idea? (Here Jacob Lurie lists several reasons why simplicial sets are convenient. So apparently one can observe that this idea "works". But I want to have a more conceptual idea why one tried to use simplicial sets in the first place.)
I noticed that many models for -categories are based on geometric shapes like globes, cubes, simplices. Why is this the case? They don't seem to occur in the idea of an -category as sketched by Leinster in the linked lecture. (This might be relevant, but I cannot really say that I understand it.)
Last but not least: the definition of an -category as a weak Kan complex (a simplicial set in which each inner horn has a filler) is quite elegant. But how does this definition relate to the idea of an -category as sketched by Leinster and how does this definition capture the coherence axioms? I mean, this definition doesn't contain coherence diagrams at all, but probably somehow encodes them in an abstract way. Do you think there's a theorem in Kerodon whose proof answers this question? I thought that the theorem which states that the Duskin nerve of a (2,1)-category induces a fully faithful embedding of (2,1)-categories into -categories might answer my question, but the word "coherence" doesn't appear once in this proof.
Nathanael Arkor (Nov 15 2021 at 20:18):What goes wrong if one tries to naively define -categories along these lines?
The hard part, even for finite , is defining the coherence axioms. Todd Trimble's notes on this topic are quite illuminating.
Nathanael Arkor (Nov 15 2021 at 20:19):Though I would be interested if @Todd Trimble had any insights on whether this problem (that is, an algorithm for generating the axioms for an -category) might be more tractable now than it was when he wrote those notes.
Todd Trimble (Nov 15 2021 at 20:49):The "polyhedral shapes" of the axioms is not too horrible to describe (for the higher associativities, the shapes are essentially associahedra), and these can be nicely oriented (along the lines of Street's parity complexes or whatnot) by using the results of Christopher Nguyen's 2017 Macquarie PhD thesis. But I'm afraid I didn't solve the problem of inductively defining how pasting works in higher dimensions. :-(
John Baez has been encouraging me to post to YouTube various reflections I've had, about this (the bicategory, tricategory, tetracategory style definitions) and also about the other definition I proposed, which was written up and pursued in other papers by Tom Leinster (for example, his "10 definitions" paper) and Eugenia Cheng. Maybe some day I'll get up the gumption to do it.
Nathanael Arkor (Nov 15 2021 at 20:59):Thanks for sharing those thoughts! It's interesting to hear that there's work as recent as 2017 that would be useful. (For those who would like to read it, Nguyen's thesis can be found here: Parity structure on associahedra and other polytopes.)
John Baez has been encouraging me to post to YouTube various reflections I've had
I second that suggestion! :)
Leopold Schlicht (Nov 18 2021 at 18:58):Thanks! The answered questions are now crossed out. :smile:
Leopold Schlicht (Nov 20 2021 at 12:04):Has nobody an idea concerning the remaining questions? To me the questions seem to be quite fundamental, but as far as I can see, they aren't discussed in introductory literature such as Higher-Dimensional Categories: an illustrated guide book by Cheng and Lauda. Browsing through @John Baez 's An Introduction to -Categories, it also doesn't seem to be in there.
I'd also be interested in any other introductory book I could check out that could contain answers to my questions.
John Baez (Nov 20 2021 at 12:25):Here's one question:
To a large extent higher category theory seems to be about finding the right definition of -category ().
I don't think this is correct: I think we've found a lot of right definitions of category! There's no such thing as "the" right definition of -category unless you're using the generalized "the" here. That is, we only expect different correct definitions of -category to be equivalent, not equal or even isomorphic.
Remember, there is an -category of -categories. If you have your definition of -category you should get an -category of your -categories. If I have my definition of -category I should get an -category of my -categories. We shouldn't expect and are equal, or isomorphic. They should just be equivalent in the -category of -categories!
And then comes the question: whose -category of -categories?
I hope you see why this is a bit complicated. There are various ways to try to avoid some of these problems, e.g. using model categories, but they still take work.
There are many proposed definitions, and each such definition is referred to as a "model" for -categories.
Most of these proposed definitions are likely to be "right", i.e. equivalent to most of the rest.
Why is it the case that there are so many models? ~~What is so hard about defining -categories?
One reason is that there are a lot of -categories that are all equivalent. If you thought about -categories for long enough, it's quite likely you'd make up your own definition!
John Baez (Nov 20 2021 at 12:27):Notice that there are also lots of definitions of "group". I could list a bunch. But these definitions produce equivalent (or maybe even isomorphic) categories , so we don't consider these definitions "truly different" - we rightly consider the differences to be "unimportant".
John Baez (Nov 20 2021 at 12:28):The situation is similar with -categories, but because the concept of equivalence is much more flexible, definitions of -category can look much more different yet still be equivalent.
Andrea Gentili (Nov 20 2021 at 16:05):As for the question about the naturality of the model, keep in mind that a presheaf can be seen as a formal gluing of objects.
On the other hand, a category can be seen as a gluing of objects, maps, and associative compositions of all finite sequences of composable maps; all of these, by definition, are the objects of . From this point of view, not only to see a category as a simplicial set is natural, but it is hard to tell the difference between the two.
Of course, not every simplicial set "is" a category, and Joyal lifting property singles out those that are, by imposing that if there is a sequence of successive maps there must also be all the compositions satisfying compatible associativity properties. (in the 1-categorical case, also uniqueness has to be imposed; interestingly, imposing all the associativity conditions gives uniqueness up to coherent homotopies for free, so these conditions are the only ones to be imposed in the higher case).
Finally, the Kan lifting property just adds to Joyal's one the invertibility up to coherent homotopies (or "higher isomorphisms", if one prefers a more categorical terminology) of maps.
Leopold Schlicht (Nov 22 2021 at 15:41):@John Baez Thanks! That clarifies some things. But it's still mysterious to me why quasicategories are a good model for -categories, in particular, it seems weird to me that it uses simplices instead of globes, and it's completely unclear to me how that definition encodes the coherence laws. (These are actually my main questions in this thread.)
Leopold Schlicht (Nov 22 2021 at 15:45):@Andrea Gentili What is a "formal gluing"? Why is a category a "gluing of objects"? What is the "Joyal lifting property"? (Google yields 0 occurrences.) Do you mean the condition that characterizes nerves of categories? What is an "associativity property"?
John Baez (Nov 22 2021 at 15:50):Leopold Schlicht said:
John Baez Thanks! That clarifies some things. But it's still mysterious to me why quasicategories are a good model for -categories, in particular, it seems weird to me that it uses simplices instead of globes, and it's completely unclear to me how that definition encodes the coherence laws.
The first thing to remember is that quasicategories are only good for -categories, which are -categories where all the morphisms above 1-morphisms are invertible (up to equivalence). A lot of people get sloppy and use -category to mean -category. (Jacob Lurie started it, probably because his 800-page book Higher Topos Theory would be 1000 pages if he didn't make that abbreviation.)
John Baez (Nov 22 2021 at 15:51):It's indeed somewhat mysterious how quasicategories capture all the higher coherence laws, so I urge you to dig in, do some computations, and see that it really works. As Joyal has said: simplicial sets, like the complex numbers, are magic. We don't completely understand why they are so great.
John Baez (Nov 22 2021 at 15:53):To get started, it's easier to see how Kan complexes capture all the higher coherence laws for -groupoids.
(Kan complexes are a special case of quasicategories. -groupoids are a special case of -categories: they are -categories where all morphisms are invertible up to equivalence.)
John Baez (Nov 22 2021 at 15:54):For example, starting with a Kan complex see how you get the associator for composition, and then the pentagonator, and so on.
Zhen Lin Low (Nov 22 2021 at 22:13):John Baez said:
It's indeed somewhat mysterious how quasicategories capture all the higher coherence laws, so I urge you to dig in, do some computations, and see that it really works. As Joyal has said: simplicial sets, like the complex numbers, are magic. We don't completely understand why they are so great.
I would say the mystery is that there is a model structure on simplicial _sets_ that works. But I find it completely natural that the complete Segal space model structure on simplicial _spaces_ works. I would be very interested to know general conditions under which a model structure on -spaces can be transferred to a Quillen-equivalent model structure on -sets along the functor that extracts the 0th level in the simplicial direction.
Andrea Gentili (Nov 22 2021 at 22:16):Leopold Schlicht said:
Andrea Gentili What is a "formal gluing"? Why is a category a "gluing of objects"? What is the "Joyal lifting property"? (Google yields 0 occurrences.) Do you mean the condition that characterizes nerves of categories? What is an "associativity property"?
By "Joyal lifting property" I meant the lifting property with respect to inner horns that characterizes the fibrant objects with respect to the Joyal model structure, that is, quasi-categories (in order to obtain a characterization of nerves of 1-categories you also have to require uniqueness of the lifting).
"Assiociativity property" was a misspelling for "associative property".
When I wrote "gluing of objects, maps" etc. I did not mean that you glue one object to another, but that you put together objects, maps, etc.
"Formal gluing" is not technical terminology. Here is what I meant: you know what Yoneda embedding is. In a sense (I'm very far from being formal here), it says that an object can be safely identified with the presheaf it represents. Now, see this presheaf as the "description" the objects of the category make of . For example, suppose we are in the category Top of topological spaces; a map from the one point space to "is" simply a point of ; so we can say that what the point "sees", and thus "describes" of is its underlying set. As another example, the unit interval "describes" the paths in . These two "descriptions" are not enough to recover , but Yoneda lemma says, roughly speaking, that putting together all the "descriptions" that all the objects make of (that is, the overall functor ) is enough.
When you pass to the category of presheaves the colimits are no more colimits of objects in the original category, but colimits of "descriptions". This is why I loosely called a presheaf a "formal gluing" of objects (a classical result says indeed that not just some presheaf, but any presheaf can be obtained as a colimit of representables).
For example, the succession of two arrows does not belong to , but can nonetheless be obtained by gluing two copies of the object (which is the category ) of , and is thus a simplicial set. You see now why this allows you to construct a category (say, as when you "draw it" on a piece of paper).
Since you can also obtain something that is not a quasi-category (as in the example above, in which a composition of the two arrows is missing), you need to impose some conditions in order to characterize those simplicial sets that are quasi-categories, and the lifting properties above give these conditions; for example, using Yoneda lemma you can see that the lifting property with respect to says that any two successive maps have a composite.
Leopold Schlicht (Nov 25 2021 at 13:19):Thanks. I don't see why a category should be a formal gluing in this sense, though.
Morgan Rogers (he/him) (Nov 25 2021 at 13:46):Consider it as a directed graph, constructed by gluing together its vertices/objects and arrows.
Leopold Schlicht (Nov 26 2021 at 13:48):John Baez said:
The situation is similar with -categories, but because the concept of equivalence is much more flexible, definitions of -category can look much more different yet still be equivalent.
Would you expect that two models for -categories are equivalent even if they use different geometric shapes? I think a "globular" approach is what comes closest to the naive idea of an -category. On the other hand, given a quasicategory, I guess one can define a globular structure (globular -morphisms and globular compositions, as we are discussing in another thread). In this sense I have the feeling quasicategories might contain more information than just a "globular -category". (Maybe it would be more appropriate to call quasicategories weak Kan complexes and say that each weak Kan complex has an underlying -category.) Or, conversely, do you think given a globular -category one can derive a simplicial -category, i.e., quasicategory?
John Baez (Nov 26 2021 at 14:41):Leopold Schlicht said:
John Baez said:
The situation is similar with -categories, but because the concept of equivalence is much more flexible, definitions of -category can look much more different yet still be equivalent.
Would you expect that two models for -categories are equivalent even if they use different geometric shapes?
Oh, of course! If people thought different shapes gave inequivalent definitions, we wouldn't be calling all these different definitions definitions of -categories, or -categories.
The shapes are ultimately just "machinery". When Grothendieck started working on -categories back in 1983 his first step was to try to develop a general theory of shapes (which he called "modelizers"), aiming at a way of showing that many different shapes could give equivalent.
People have recently been trying to understand and continue his work.
Morgan Rogers (he/him) (Nov 27 2021 at 11:51):You have to keep in mind that the category of shapes is not itself the infinity category; rather, we equip such a category with some extra structure, of "weak equivalences", and the infinity category is obtained by inverting those. Somehow it's less surprising that quotients of different structures can produce equivalent results.
Leopold Schlicht (Dec 02 2021 at 17:55):John Baez said:
Oh, of course! If people thought different shapes gave inequivalent definitions, we wouldn't be calling all these different definitions definitions of -categories, or -categories.
Thanks! When are two definitions of "-category" equivalent? (I want to get a precise, nonhandwavy answer to that question.)
Suppose we decided on a formal definition of "globular -category". Probably you are saying that globular -categories are "equivalent" to quasicategories. I guess in particular that means that there is a bijection between the collection of all equivalence classes of quasicategories and the collection of all equivalence classes of "globular -categories". Given a globular -category, how do you define the corresponding quasicategory? I just want to know the rough idea (without details): what are the -simplices when we are given for each a collection of -cells?
The shapes are ultimately just "machinery". When Grothendieck started working on -categories back in 1983 his first step was to try to develop a general theory of shapes (which he called "modelizers"), aiming at a way of showing that many different shapes could give equivalent.
People have recently been trying to understand and continue his work.
I can't find a comparison of different definitions of -categories based on different geometric shapes in that book.
Leopold Schlicht (Dec 02 2021 at 18:02):Morgan Rogers (he/him) said:
You have to keep in mind that the category of shapes is not itself the infinity category;
I know. Why do you feel like remarking that I should keep that in mind? (I think I never claimed the opposite.)
Also I don't know what you mean by "the infinity category". There are several infinity categories.
rather, we equip such a category with some extra structure, of "weak equivalences", and the infinity category is obtained by inverting those. Somehow it's less surprising that quotients of different structures can produce equivalent results.
I don't understand that. The definition of -category that I know doesn't talk about weak equivalences. Also it's unclear to me what you mean by "inverting those" and "quotients".
Reid Barton (Dec 02 2021 at 18:07):Leopold Schlicht said:
John Baez said:
Oh, of course! If people thought different shapes gave inequivalent definitions, we wouldn't be calling all these different definitions definitions of -categories, or -categories.
Thanks! When are two definitions of "-category" equivalent? (I want to get a precise, nonhandwavy answer to that question.)
I guess this will become fairly obvious (maybe not totally obvious) once you have a precise definition of "definition of "-category"".
Leopold Schlicht (Dec 02 2021 at 18:10):That doesn't help me. When people say that quasicategories are equivalent to, say, complete Segal spaces (whatever that is), what do they mean?
Reid Barton (Dec 02 2021 at 18:14):Complete Segal spaces and quasicategories both form model categories, and those model categories are Quillen equivalent (more precisely, some specific adjunctions between them are Quillen equivalences).
Leopold Schlicht (Dec 02 2021 at 18:15):Does that induce a bijection between equivalence classes of quasicategories and equivalence classes of complete Segal spaces (or something like that)?
Leopold Schlicht (Dec 02 2021 at 18:17):I don't know anything about model categories, except that a model category is something like a "place to do homotopy theory in". But then your above statement just shows that quasicategories and complete Segal spaces yield the same homotopy theory, not that they yield the same category theory...
Reid Barton (Dec 02 2021 at 18:28):Model categories are one way to present -categories (of a special kind) in terms of a 1-category with some additional structure. In other words, there is a way (in fact, several equivalent ways) to extract from a model category an -category, which could be regarded as the model category's "homotopy theory". Quillen equivalent model categories present equivalent -categories.
Leopold Schlicht (Dec 02 2021 at 18:51):What does "present" mean? Which question are you answering here? I asked whether a "Quillen equivalence" between complete Segal spaces and quasicategories induces a bijection between equivalence classes of quasicategories and equivalence classes of complete Segal spaces (or something like that).
Reid Barton (Dec 02 2021 at 18:59):Sure, something like that.
Reid Barton (Dec 02 2021 at 19:16):But that is only a rather weak form of equivalence.
John Baez (Dec 02 2021 at 19:21):Leopold Schlicht said:
John Baez said:
Oh, of course! If people thought different shapes gave inequivalent definitions, we wouldn't be calling all these different definitions definitions of -categories, or -categories.
Thanks! When are two definitions of "-category" equivalent? (I want to get a precise, nonhandwavy answer to that question.)
I've already explained this but with replacing . If you have a definition of -categories, it's your job to use your definition to construct the -category of small -categories, say . Similarly, if I have a definition, I should construct .
Then your definition is equivalent to mine iff
I've already discussed some of the difficulties here: for example, the above equivalence only makes sense if both and have been constructed as objects in the same -category of -categories.
This is one reason progress has been slow.
John Baez (Dec 02 2021 at 19:28):People often introduce model categories to sidestep these difficulties. A model category is a way to get an -category. "Quillen equivalence" of model categories is a way to formalize equivalence of -categories.
See this paper to see how this plays out in a simpler example:
She explains four definitions of -category. For each one she defines a model category of -categories. At the end, she outlines proofs that all four model categories are Quillen equivalent.
You should think of this as a way of making precise this idea: each of these four definitions of -category gives an equivalent -category of -categories.
So, as an easier alternative to what I said above, we can expect that anyone providing a definition of -category will define a model category of -categories, and that people will prove that different definitions give Quillen equivalent model categories. In some ways this is not as good as what I originally proposed - but it's pretty good, and it's a lot easier.
John Baez (Dec 02 2021 at 19:40):Leopold Schlicht said:
That doesn't help me. When people say that quasicategories are equivalent to, say, complete Segal spaces (whatever that is), what do they mean?
That is answered in Julie Bergner's paper.
Zhen Lin Low (Dec 02 2021 at 22:03):Leopold Schlicht said:
I don't know anything about model categories, except that a model category is something like a "place to do homotopy theory in". But then your above statement just shows that quasicategories and complete Segal spaces yield the same homotopy theory, not that they yield the same category theory...
The "category theory" is also the same, at least to the extent that it can be captured by the homotopy bicategory. I wrote some remarks on that topic here.
Leopold Schlicht (Dec 03 2021 at 00:30):Thanks so much for all the explanations! That helps a lot.
John Baez (Dec 03 2021 at 02:35):Good!
Morgan Rogers (he/him) (Dec 03 2021 at 10:19):Leopold Schlicht said:
Morgan Rogers (he/him) said:
You have to keep in mind that the category of shapes is not itself the infinity category;
I know. Why do you feel like remarking that I should keep that in mind? (I think I never claimed the opposite.)
I said this because you seemed surprised that infinity categories built from different shapes could produce equivalent results, but my phrasing was off; I should have said "collection of shapes" in the first instance. What I was trying to get at is that, just as we typically work with categories up to equivalence, in the sense that we study properties that are preserved by equivalence, the same is true for categories of infinity categories (as was discussed above), but moreover it's helpful to think that any given infinity category in your favourite context (a quasi-category, say) is a presentation of some essential object which can also be expressed in other ways (as a globular infinity category, say). We do this in maths all the time. For example, we are used to viewing the real numbers as a partial order, a field and a space, and thinking of these as different perspectives on the same essential object.
For the time being, many results about categories of infinity categories are about making sure that different constructions are equally expressive, and these results are necessary because, as your intuition suggested, it's not obvious a priori that this must be the case. Eventually, these results have the potential to have a big payoff, since we will be able to leverage the different presentations of infinity categories to understand them better in lots of different ways.
rather, we equip such a category with some extra structure, of "weak equivalences", and the infinity category is obtained by inverting those. Somehow it's less surprising that quotients of different structures can produce equivalent results.
I don't understand that. The definition of -category that I know doesn't talk about weak equivalences. Also it's unclear to me what you mean by "inverting those" and "quotients".
This was in reference to model structure (specifically inverting weak equivalences), and once again my comment was off, since I was describing the construction of the homotopy category of infinity categories rather than the construction of any individual infinity category.
Jonathan Weinberger (Dec 03 2021 at 12:53):A 2-dimensional notion of equivalence of theories of higher categories is provided in Riehl--Verity's -cosmos theory. E.g. quasi-categories, complete Segal spaces, and Segal categories form biequivalent -cosmoses. This means that the formal category theories of each of these models are equivalent. Indeed such a biequivalence between -cosmoses induces a biequivalence of double-categories of profunctors internal to the respective -cosmoses (which capture the respective formal category theory, in a "strict" way). Moreover, this induces a biequivalence on the level of homotopy 2-categories. The result appears as Theorem 11.1.6 in Riehl, Verity: Elements of -Category Theorry
Peter Arndt (Mar 18 2022 at 10:44):I think it hasn't been mentioned yet that Toen axiomatized the theory of -categories and showed among other things that up to equivalence there is only one model. Barwick and Schommer-Pries did the same for -categories. Even more amazingly they work out what equivalences there can be!
Peter Arndt (Mar 18 2022 at 10:44):These works are a few miles further down the bootstrapping process of defining higher categories: They already assume that higher categories can be somehow modelled by simplicial categories/model categories/or something along those lines. So they won't help with the original questions of this thread, but I thought it's worth mentioning here.
Nathanael Arkor (Mar 18 2022 at 11:45):Peter Arndt said:
I think it hasn't been mentioned yet that Toen axiomatized the theory of -categories and showed among other things that up to equivalence there is only one model. Barwick and Schommer-Pries did the same for -categories. Even more amazingly they work out what equivalences there can be!
By the way, the first link points to the wrong paper.
Leopold Schlicht (Mar 18 2022 at 12:11):Thanks!