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Nathanael Arkor (Jun 15 2025 at 16:41):Yes, I have in mind models like that of Batanin/Leinster.
Mike Shulman (Jun 15 2025 at 17:00):There's no need to mention the -morphisms explicitly when they are unique, since there's only one possible thing that you could do with them.
Amar Hadzihasanovic (Jun 15 2025 at 21:11):Also when people prove that a functor of categories "is well-defined" after giving a def on objects and morphisms, the meaning of "is well-defined" is often "respects equality of morphisms" i.e. "sends 2-cells to 2-cells".
Mike Shulman (Jun 16 2025 at 00:14):Also, I wouldn't myself use "-categories" to imply that all the higher morphisms are present but unique. I think of an -category (including the case ) as just meaning "an -category in which all morphisms above level are invertible", for any meaning of "-category", which might or might not have (unique) cells above dimension .
Notification Bot (Jun 16 2025 at 00:16):Mike Shulman has marked this topic as unresolved.
Mike Shulman (Jun 16 2025 at 00:18):I don't know of a term that distinguishes models with unique higher cells from those without. It's a bit of a funny distinction to make, since it doesn't single out a specific model, but nor is it a model-independent concept; it's partitioning all the models into two big classes. I'm having trouble thinking of a useful mathematical statement one could make that would depend on such a partition; just knowing that we have or don't have unique higher cells doesn't seem to tell us much if we haven't fixed a particular model.
Nathanael Arkor (Jun 16 2025 at 05:42):Mike Shulman said:
There's no need to mention the -morphisms explicitly when they are unique, since there's only one possible thing that you could do with them.
This may be one of the points I'm misunderstanding, then, because my interpretation of the definition on [[(n,r)-category]] indicated that there may be multiple -morphisms between any parallel pair of -morphisms in a (1, 1)-category, but they are necessarily equivalent. If this is the case, then the action of a functor between (1, 1)-categories on -morphisms (for ) is not uniquely determined, but only essentially uniquely determined, so that the action on all -morphisms must either be specified, or one must make an appeal to choice. Have I misunderstood something?
Nathanael Arkor (Jun 16 2025 at 05:46):Furthermore, in a (1,1)-category, my interpretation was that we do not necessarily have, for instance, , but rather than these two 1-morphisms are related by an essentially unique equivalence. Is this correct? If so, for any nLab page that uses the associativity equation for categories, this is an indication that the page has not been written with (1, 1)-categories in mind.
Nathanael Arkor (Jun 16 2025 at 05:48):(Obviously, even if I haven't misunderstood something, it would be easy to change these pages to use (1,1)-categories. But my point is that either I have misunderstood something, or that (1,1)-categories really do have distinctions that a low-dimensional category theorist might find objectionable for a definition of category, and I would like to understand which it is.)
Amar Hadzihasanovic (Jun 16 2025 at 05:56):I think that when working in univalent foundations or even just constructive foundations, there is literally no difference between the two (types/sets of morphisms will be equipped with an equality type/relation which must be respected but may have non-unique witnesses); and when working in classical foundations, then anyway the quotient which simply identifies all equivalent higher cells is an acyclic fibration, so it is not restrictive to always work with these objects with "strictly unique" cells up to passing to weakly equivalent objects.
Nathanael Arkor (Jun 16 2025 at 06:58):I agree these conventions could be used to address the second point.
Morgan Rogers (he/him) (Jun 16 2025 at 07:55):I have been following this with some confusion that I'm hoping some further discussion might resolve. Mike and Amar talk about "models" on n-categories or (n,k)-categories. I am aware that there are variants of n-categories with more or less strict structure, but in my experience these have specific, elementary definitions which have been studied in great detail at least up to dimension 3. Are the "models" in question models of these definitions in particular settings (in sets, types etc)? Or are you talking more broadly about models of infinity categories of which (n,k)-categories are being carved out as a subclass? Am I wrong to think of these as two different meanings of "model"?
Morgan Rogers (he/him) (Jun 16 2025 at 08:00):In the former case, while we can construct types of higher cells in various ways, the theory of n-categories doesn't talk about these, so the various things we could do should correspond to various embeddings from n-categories to (n+1)-categories. It seems strange to say that any given way is a canonical one except as far as the extensions are equivalent to one another, but I imagine this in turn may depend on the setting where models live?
Vincent R.B. Blazy (Jun 16 2025 at 08:02):Nathanael Arkor said:
Mike Shulman said:
There's no need to mention the -morphisms explicitly when they are unique, since there's only one possible thing that you could do with them.
This may be one of the points I'm misunderstanding, then, because my interpretation of the definition on [[(n,r)-category]] indicated that there may be multiple -morphisms between any parallel pair of -morphisms in a (1, 1)-category, but they are necessarily equivalent. If this is the case, then the action of a functor between (1, 1)-categories on -morphisms (for ) is not uniquely determined, but only essentially uniquely determined, so that the action on all -morphisms must either be specified, or one must make an appeal to choice. Have I misunderstood something?
Well then what would the n mean more than the k? Usually it’s the higher arrows there are (or they all are trivial above n, if we mean a degenerated m-category for ω>=m>n), and they are all invertible above k…
Nathanael Arkor (Jun 16 2025 at 08:12):Morgan Rogers (he/him) said:
Are the "models" in question models of these definitions in particular settings (in sets, types etc)? Or are you talking more broadly about models of infinity categories of which (n,k)-categories are being carved out as a subclass?
The latter. I think one would use a word like "foundation" to speak of the former.
Nathanael Arkor (Jun 16 2025 at 08:14):Vincent R.B. Blazy said:
Well then what would the n mean more than the k? Usually it’s the higher arrows there are (or they all are trivial above n, if we mean a degenerated m-category for ω>=m>n), and they are all invertible above k…
Yes, they are all invertible, but that doesn't uniquely determine the assignment of a functor on them. Am I misunderstanding your point?
Vincent R.B. Blazy (Jun 16 2025 at 08:34):Nathanael Arkor said:
Vincent R.B. Blazy said:
Well then what would the n mean more than the k? Usually it’s the higher arrows there are (or they all are trivial above n, if we mean a degenerated m-category for ω>=m>n), and they are all invertible above k…
Yes, they are all invertible, but that doesn't uniquely determine the assignment of a functor on them. Am I misunderstanding your point?
I guess so: They are all invertible above k<=n indeed, but all trivial — or even nonexistent — above n itself! So in particular if k=n, you have no invertible nontrivial arrow above n to worry about?
Nathanael Arkor (Jun 16 2025 at 08:49):But "trivial" does not mean "equal to the identity" (indeed, this doesn't even type-check), so the action of a functor is not uniquely determined on trivial morphisms.
Nathanael Arkor (Jun 16 2025 at 08:51):(If I have not misunderstood) one could work with a notion of [[anafunctor]] instead, and this would resolve my complaint. However, it wasn't my impression that functors of (1,1)-functors are defined like this (as opposed to simply typically considering (1,1)-functors up to equivalence).
Nathanael Arkor (Jun 16 2025 at 08:58):Perhaps it's helpful for me to share some context, because it could be that my confusion is really stemming from somewhere else. In this MathOverflow thread, Cisinski seems to indicate that Thorgott's answer does not provide a reference in the setting of weak -categories. However, Thorgott does provide a reference in the setting of -categories. This led me to understand that Cisinski views the two concepts as distinct, which seems in constrast to the nLab's PoV.
It could be that Cisinski only views them as distinct because a Batanin/Leinster-style model of weak -categories has not yet been proven equivalent to one of the models of -categories (and that, assuming such an equivalence is proven, the objection would no longer hold). However, even in this case, why is this not a point of concern for the nLab?
Vincent R.B. Blazy (Jun 16 2025 at 13:10):Nathanael Arkor said:
But "trivial" does not mean "equal to the identity" (indeed, this doesn't even type-check), so the action of a functor is not uniquely determined on trivial morphisms.
Oh yeah I confused myself here sorry. Maybe @Mike Shulman had talked about unicity just because in the case of what y’all call (n,n)-categories here, (n,n)-functors also belong to higher categories hence their action is uniquely determined in their relevant sense, that is, essentially. Just as n-functors have a uniquely determined action between n-categories… :thinking:
Nathanael Arkor (Jun 16 2025 at 13:15):I'm not sure what you mean by this.
Vincent R.B. Blazy (Jun 16 2025 at 13:34):Nathanael Arkor Nevermind, I’m only an interested amateur in those matters, I might still be confused. I only tried to say it may be senseful that what hereabove you said was essentially uniquely determined in an infinite-categorical setting (even if degenerate above n), correspond to unique determination in the n-categorical one, without needing specification of the action on >n-arrows because all potential such specifications would essentially result in the same action…
Sorry for your time :slight_smile:
Mike Shulman (Jun 16 2025 at 17:07):Let me be sure to say out loud that the authors of nLab pages about 1-categories certainly do generally have one of the traditional definitions in mind where there are only objects and morphisms. My point is just that anything "category-theoretic" that one says about 1-categories in these classical senses can equivalently be interpreted as a statement about the -categories that are equivalent to them, so I don't think it's inconsistent for the nLab to say that it considers these the same.
Mike Shulman (Jun 16 2025 at 17:08):It's true that when -categories are defined in set-theoretic foundations, if the -morphisms are only essentially unique, one would in general need choice to define an action on them as an ordinary functor (rather than an anafunctor). However, classical mathematicians use choice all the time without stating it as an assumption, and in particular this is the case in homotopy theory where it is used in the small object argument, so I don't see this as an obstacle.
Mike Shulman (Jun 16 2025 at 17:10):Similarly, the process of "interpreting" a statement about classical 1-categories to refer to their homotopical incarnation translates "equals" for morphisms to the existence of a 2-cell.
Mike Shulman (Jun 16 2025 at 17:15):Morgan Rogers (he/him) said:
Are the "models" in question models of these definitions in particular settings (in sets, types etc)? Or are you talking more broadly about models of infinity categories of which (n,k)-categories are being carved out as a subclass?
Neither. There is a particular -category called , determined up to unique equivalence, and a "model" of -categories is any way of constructing some -category equipped with an equivalence to .
Mike Shulman (Jun 16 2025 at 17:16):Of course it would be even better to regard as an -category, and in practice most models give you one of those too.
Nathanael Arkor (Jun 16 2025 at 17:27):in practice most models give you one of those too
How can an -category equipped with an equivalence to give rise to an -category?
Nathanael Arkor (Jun 16 2025 at 17:30):Mike Shulman said:
However, classical mathematicians use choice all the time without stating it as an assumption, and in particular this is the case in homotopy theory where it is used in the small object argument, so I don't see this as an obstacle.
I accept that choice is used all over the place in homotopy theory, but my questions are really motivated by pure category theory, where choice tends to be used more carefully. In particular, the nLab tends to be careful about stating when choice is a necessary assumption.
Nathanael Arkor (Jun 16 2025 at 17:31):I feel this distinction is significant enough to warrant making a distinction between -categories and -categories.
Mike Shulman (Jun 16 2025 at 18:15):Nathanael Arkor said:
in practice most models give you one of those too
How can an -category equipped with an equivalence to give rise to an -category?
The model isn't the -category, it's a way of constructing some -category. I'm saying such methods generally also give you a way of constructing an -category.
Nathanael Arkor (Jun 16 2025 at 18:17):Just to clarify, is "model" being used in an informal sense here, or do you have a precise meaning in mind?
Mike Shulman (Jun 16 2025 at 18:19):Nathanael Arkor said:
I accept that choice is used all over the place in homotopy theory, but my questions are really motivated by pure category theory, where choice tends to be used more carefully. In particular, the nLab tends to be careful about stating when choice is a necessary assumption.
Nonalgebraic approaches to higher categories are basically indistinguishable from homotopy theory, and in particular can't really be done without choice either.
Mike Shulman (Jun 16 2025 at 18:20):Nathanael Arkor said:
Just to clarify, is "model" being used in an informal sense here, or do you have a precise meaning in mind?
If you want to make it more precise, you could take "way of constructing an -category" to mean "a [[relative category]]". I think that includes all models I'm aware of.
Mike Shulman (Jun 16 2025 at 18:24):Nathanael Arkor said:
In this MathOverflow thread, Cisinski seems to indicate that Thorgott's answer does not provide a reference in the setting of weak -categories.
If you read the comments, Cisinski does seem to be using "weak -category" there to refer to Batanin-Leinster algebraic definitions. That's not how I would use the word, and not I think how the nLab does, but opinions can certainly differ.
It could be that Cisinski only views them as distinct because a Batanin/Leinster-style model of weak -categories has not yet been proven equivalent to one of the models of -categories (and that, assuming such an equivalence is proven, the objection would no longer hold). However, even in this case, why is this not a point of concern for the nLab?
I can't speak for Cisinski, so I don't know why he wants to emphasize the difference in this terminological way. (Of course there is a significant mathematical difference, whatever terminology we use for it.) I would say it's not a point of concern for the nLab because the nlab doesn't use algebraic higher categories, since not only are they not yet proven equivalent to the nonalgebraic models, they also haven't proven useful for much of anything yet.
Nathanael Arkor (Jun 16 2025 at 18:33):Mike Shulman said:
Nonalgebraic approaches to higher categories are basically indistinguishable from homotopy theory, and in particular can't really be done without choice either.
That's fair, but shouldn't "n-category" reasonably refer to algebraic approaches as well as geometric approaches?
Nathanael Arkor (Jun 16 2025 at 18:34):Mike Shulman said:
If you read the comments, Cisinski does seem to be using "weak -category" there to refer to Batanin-Leinster algebraic definitions. That's not how I would use the word, and not I think how the nLab does, but opinions can certainly differ.
Okay, thanks for clarifying!
Nathanael Arkor (Jun 16 2025 at 18:35):Mike Shulman said:
I would say it's not a point of concern for the nLab because the nlab doesn't use algebraic higher categories, since not only are they not yet proven equivalent to the nonalgebraic models, they also haven't proven useful for much of anything yet.
Oh, this was not clear to me. The algebraic approaches seem much closer to intuition from a low dimensional category theory PoV, so it's counterintuitive to me that the nLab would commit to the geometric approach.
Madeleine Birchfield (Jul 08 2025 at 11:24):Mike Shulman said:
I would say it's not a point of concern for the nLab because the nlab doesn't use algebraic higher categories, since not only are they not yet proven equivalent to the nonalgebraic models, they also haven't proven useful for much of anything yet.
There is now this paper by Chavanat & Hadzihasanovic that does prove an equivalence between an algebraic model of higher categories and a nonalgebraic model: