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Stream: practice: terminology & notation

Topic: ✔ (n, k)-categories

Nathanael Arkor (Jun 13 2025 at 14:33):

In the notion of $(n, k)$-category in higher category theory, what are $n$ and $k$ called? I am inclined to call $n$ the dimension, but I'm not so sure about $k$.

Mike Shulman (Jun 13 2025 at 17:02):

"directed dimension"?

Madeleine Birchfield (Jun 13 2025 at 17:20):

What do $(\infty,k)$-category theorists call the $k$?

$(n, k)$-categories are $n$-truncated $(\infty,k)$-categories, and I think in this case $n$ is called the "truncation level", see e.g. section 11.3 of Cisinki et al's Formalization of Higher Categories.

Amar Hadzihasanovic (Jun 14 2025 at 06:37):

I think they only deserve a name insofar as you treat them as properties of an $(\infty, \infty)$-category, i.e. "the least $n$ such that $X$ is $n$-truncated" and "the least $k$ such that $X$ has no non-invertible $(k+1)$-cells"

Amar Hadzihasanovic (Jun 14 2025 at 06:41):

But when just saying "$X$ is an $(n,k)$-category" it does not seem appropriate to call them something like "dimensions" because every $(n,k)$-category is also an $(m,j)$-category for all $m \geq n$ and $m \geq j \geq k$.

Amar Hadzihasanovic (Jun 14 2025 at 06:42):

Which is not how "dimension" behaves—n-dimensional Euclidean space is not also (n+1)-dimensional Euclidean space

Amar Hadzihasanovic (Jun 14 2025 at 06:43):

Maybe you can say that $n$ is a truncation upper bound and $k$ is a direction upper bound

Amar Hadzihasanovic (Jun 14 2025 at 06:44):

And then the least such upper bounds may be called truncation level and direction level?

Amar Hadzihasanovic (Jun 14 2025 at 06:46):

I would leave the word "dimension" for geometric, not homotopy-theoretic contexts

Morgan Rogers (he/him) (Jun 14 2025 at 06:53):

@Amar Hadzihasanovic but one doesn't have to take the terminology to mean an infinity-infinity category with extra properties :thinking:

Amar Hadzihasanovic (Jun 14 2025 at 07:00):

I am struggling to parse your sentence

Morgan Rogers (he/him) (Jun 14 2025 at 07:05):

I am not in the habit of thinking of a 1-category = (1,1)-category as a special case of an $(\infty,\infty)$-category.

Morgan Rogers (he/him) (Jun 14 2025 at 07:08):

In particular this claim doesn't type-check for the use of the term that doesn't have cells above dimension n:

every $(n,k)$-category is also an $(m,j)$-category for all $m \geq n$ and $m \geq j \geq k$.

Amar Hadzihasanovic (Jun 14 2025 at 07:50):

The term $(\infty, n)$-category or $(n, k)$-category is used in higher category theory to refer to the model-independent notion

Amar Hadzihasanovic (Jun 14 2025 at 07:51):

So I would say that what does not typecheck is speaking of "not having cells above dimension $n$" in the same sentence as $(n, k)$-category, as that can only happen within a particular model

Amar Hadzihasanovic (Jun 14 2025 at 07:52):

"Categories" are a particular model of $(1, 1)$-categories that does not have any cells in dimension > 1

Amar Hadzihasanovic (Jun 14 2025 at 07:52):

They are not the same concept as $(1, 1)$-categories

Amar Hadzihasanovic (Jun 14 2025 at 07:55):

For example, simplicial sets satisfying the Segal condition are also a model of $(1, 1)$-categories but they have $n$-cells in every $n$

Nathanael Arkor (Jun 14 2025 at 08:14):

Certainly the term $n$-category is used (for $n \leq 4$) to refer to a concrete concept. I find it unintuitive to imagine that $(n, k)$-categories should not be special cases of $n$-categories.

Nathanael Arkor (Jun 14 2025 at 08:15):

And I would use "dimension" to refer to the $n$ in "$n$-category".

James Deikun (Jun 14 2025 at 08:16):

"Directed dimension" sounds like a reasonably clear name for $k$ ...

James Deikun (Jun 14 2025 at 08:18):

(Although, $(0,1)$-categories are not really a special case of $0$-categories.)

Nathanael Arkor (Jun 14 2025 at 08:23):

Nathanael Arkor said:

I find it unintuitive to imagine that $(n, k)$-categories should not be special cases of $n$-categories.

Though this does seem to depend on the definition: e.g. John and Mike introduce the term in Lectures on n-Categories and Cohomology informally as a special case of $n$-categories, but then define it to be a special case of $\infty$-categories. As James points out, these don't coincide for all $n$ and $k$.

Amar Hadzihasanovic (Jun 14 2025 at 08:34):

I think the situation for $n$-category is not at all so clear even for low $n$ (for $n=3$, should it refer to strict 3-categories, which are not even models of all (3,3)-categories?)

Amar Hadzihasanovic (Jun 14 2025 at 08:36):

The current standard in higher category theory/homotopy theory with $(\infty, n)$ being "model-independent" is a huge improvement in comparison, IMHO

Amar Hadzihasanovic (Jun 14 2025 at 08:38):

Then as usual in these matters, if one could redo the whole terminology from scratch lots of problems could be avoided, but I believe that this is the state of the matter

Amar Hadzihasanovic (Jun 14 2025 at 08:39):

I.e. $n$-category is simply not a synonym for $(n,n)$-category in practice...

Amar Hadzihasanovic (Jun 14 2025 at 08:39):

2-category almost universally refers to strict 2-categories

Amar Hadzihasanovic (Jun 14 2025 at 08:40):

But bicategories are also a model of (2,2)-categories

Amar Hadzihasanovic (Jun 14 2025 at 08:42):

Not to mention the well-known problem of $\infty$-category having been hijacked by Lurie and now being almost universally shorthand for the model-independent notion of $(\infty, 1)$-category

Amar Hadzihasanovic (Jun 14 2025 at 08:46):

I believe the whole shift to the $(n, k)$-terminology has been caused precisely by the mess of the use of $n$-category to refer to particular incomplete models

Nathanael Arkor (Jun 14 2025 at 08:47):

What do you mean by "incomplete"?

Amar Hadzihasanovic (Jun 14 2025 at 08:51):

I guess it's the wrong word, "non-universal" is better (I mean "not surjective on all intended models")

Amar Hadzihasanovic (Jun 14 2025 at 08:53):

I am thinking specifically about 3-categories being used for strict 3-categories vs tricategories, in analogy with 2-category vs bicategory

Amar Hadzihasanovic (Jun 14 2025 at 08:53):

Even though strict 3-categories are non-universal in this sense

Nathanael Arkor (Jun 14 2025 at 08:55):

I suppose I don't find it much of a pain to have to prefix "weak".

Amar Hadzihasanovic (Jun 14 2025 at 08:58):

Weak is also about models.

Amar Hadzihasanovic (Jun 14 2025 at 08:59):

Doesn't make sense to say that an $(n, k)$-category is weak or strict, a model is weak or strict

Amar Hadzihasanovic (Jun 14 2025 at 08:59):

You can say that an $(n, k)$-category has the type of a strict $n$-category, sure

Nathanael Arkor (Jun 14 2025 at 09:01):

I mean with regards to the use of "$n$-category" to refer to a "strict $n$-category".

Madeleine Birchfield (Jun 14 2025 at 09:02):

2-categories are already a pain when it comes to "weak" or "strict". You can decide whether 1. the object 2-groupoid is a set or 1-truncated or untruncated, and 2. the hom-groupoids are each sets or not. That's when people can start talking about "globally strict" and "locally strict" 2-categories, but nobody really does so. The combinatorics of weak and strict increases exponentially with increasing direction level $n$.

Amar Hadzihasanovic (Jun 14 2025 at 09:04):

I am just saying that there is a kind of standard which has formed, and it is “$(\infty, n)$-category (or $(n, k)$-category) is model-independent, so it is a type error to attribute to it model-dependent properties”; I think this is not a very hard hygiene to follow, so I do not understand the opposition

Amar Hadzihasanovic (Jun 14 2025 at 09:05):

According to this standard saying “every $(n, k)$-category is also an $(m, j)$-category for $m \geq n$ and $m \geq j \geq k$” is simply correct

Amar Hadzihasanovic (Jun 14 2025 at 09:06):

And it is completely irrelevant that there are models of $(n, k)$-categories that do not have cells in dimension higher than $n$

Madeleine Birchfield (Jun 14 2025 at 09:07):

Morgan Rogers (he/him) said:

In particular this claim doesn't type-check for the use of the term that doesn't have cells above dimension n:

every $(n,k)$-category is also an $(m,j)$-category for all $m \geq n$ and $m \geq j \geq k$.

Is a poset (by the usual definition using binary logical predicates) a category?

Nathanael Arkor (Jun 14 2025 at 09:07):

Madeleine Birchfield said:

2-categories are already a pain when it comes to "weak" or strict. You can decide whether 1. the object 2-groupoid is a set or 1-truncated or untruncated, and 2. the hom-groupoids are each sets or not. That's when people can start talking about "globally strict" and "locally strict" 2-categories, but nobody really does so. The combinatorics of weak and strict increases exponentially with increasing direction level $n$.

In classical category theory, the "object 2-groupoid" is not a natural object to consider; this is a particularly homotopy-theoretic point of view. So I think it would be unreasonable to say that the classical notions of weakness in low-dimensional category theory are painful, because they are motivated by different perspectives.

Amar Hadzihasanovic (Jun 14 2025 at 09:10):

Within a particular paper, of course, it is perfectly fine for someone to use $(n, k)$-category as a shorthand for $(n+1)$-coskeletal complete $k$-fold Segal space or whatever

Nathanael Arkor (Jun 14 2025 at 09:10):

Amar Hadzihasanovic said:

I am just saying that there is a kind of standard which has formed, and it is “$(\infty, n)$-category (or $(n, k)$-category) is model-independent, so it is a type error to attribute to it model-dependent properties”; I think this is not a very hard hygiene to follow, so I do not understand the opposition

I accept your point. It is annoying that the terminology for low dimensional category theory and high dimensional category theory are not entirely compatible, but it is something we must live with now.

Nathanael Arkor (Jun 14 2025 at 09:12):

In any case, my original question was answered (namely, there doesn't appear to be standard terminology, but several reasonable suggestions were put forward), so I'll mark the topic as resolved. Thanks for everyone for the discussion!

Notification Bot (Jun 14 2025 at 09:12):

Nathanael Arkor has marked this topic as resolved.

Amar Hadzihasanovic (Jun 14 2025 at 09:12):

Yes, although I think that the shift to the $(n, k)$ terminology has been exactly a way to create a new "namespace" to avoid confusion with low-dimensional category theory

Nathanael Arkor (Jun 14 2025 at 09:13):

Although that shift has proven relatively ineffective as higher category theorists start using the same notation as lower category theorists.

Amar Hadzihasanovic (Jun 14 2025 at 09:14):

I mean, I guess some people do write things as $(2, 1)$-categories for strict 2-categories with invertible 2-cells, but what can you do

Amar Hadzihasanovic (Jun 14 2025 at 09:14):

“Locally groupoidal" would work just fine

Madeleine Birchfield (Jun 14 2025 at 09:29):

I already see people in the higher category literature writing "groupoid" for $\infty$-groupoid and "category" for $(\infty,1)$-category.

Nathanael Arkor (Jun 14 2025 at 09:33):

(This is the kind of practice I was referring to in my previous message.)

Madeleine Birchfield (Jun 14 2025 at 10:28):

So then a $k$-category is likely to mean an $(\infty,k)$-category, and an $(n,k)$-category in their language is just an $n$-truncated $k$-category.

The situation remains though, regardless the natural number $k$ currently has no standard vocabulary.

Amar Hadzihasanovic (Jun 14 2025 at 10:38):

I think it's still standard practice to state something like "category means $\infty$-category" in the preamble of an article, that is, omitting the $\infty$ is just to not flood the notation with $\infty$ symbols... I don't think it's exactly the same problem

Amar Hadzihasanovic (Jun 14 2025 at 10:39):

I'd compare it more with the way people say "space" or "topological space" to mean "compactly generated Hausdorff space" or "ring" to mean "Noetherian commutative ring" or other such examples

Amar Hadzihasanovic (Jun 14 2025 at 10:43):

Like, from the point of view of a homotopy theorist it arguably serves the same purpose of "making things nicer", as from that point of view 1-categories have bad quotients etc. so even if you only care about categories you should treat them as $\infty$-categories...

Nathanael Arkor (Jun 14 2025 at 15:31):

Is it reasonable to take away from this discussion that $(n, n)$-categories really are not the same as weak $n$-categories, from the perspective of weak $n$-category theory rather than $(\infty, n)$-category theory? Presumably the appropriate notion of equivalence of $(n, n)$-categories is weaker than that of weak $n$-categories? (E.g. a $(0, 0)$-category contains much more data than a set, even if it is homotopically equivalent to a set.)

Nathanael Arkor (Jun 14 2025 at 15:34):

Perhaps a slightly more concrete question: for any model of $(2, 2)$-categories, should there be a tricategory of $(2, 2)$-categories triequivalent to the tricategory of bicategories?

Amar Hadzihasanovic (Jun 14 2025 at 16:17):

A model of (n,k)-categories also needs to model the notion of equivalence, which will not be the same as the "external" notion of equivalence (the analogy is: when topological spaces are used as a model for homotopy types, the model of equivalence is weak equivalence and not homeomorphism)

Amar Hadzihasanovic (Jun 14 2025 at 16:17):

I think if you see it with the analogy of topological spaces vs homotopy types it's quite enlightening

Amar Hadzihasanovic (Jun 14 2025 at 16:19):

Like, your question would be analogous to: is there a topological space whose points are small homotopy 2-types and is weakly equivalent to the 3-type of small 2-types? I think the part about the points does not quite typecheck (I mean, you can ask it but it is not very meaningful)

Amar Hadzihasanovic (Jun 14 2025 at 16:23):

What I'm trying to say is, it is not entirely meaningless but seems wrong to "feed back" the model-independent things into a specific model, to speak of something like "the tricategory of (2,2)-categories" in the same way as it feels wrong to speak of a "topological space of homotopy 2-types"

Nathanael Arkor (Jun 14 2025 at 18:21):

Thanks, I think this makes sense. In that case, I feel the way the [[(n,r)-category]] nLab article is written is somewhat misleading (e.g. in saying that "an $(n, n)$-category is simply an $n$-category"), because it indicates that it is reasonable to directly compare the two notions.

Amar Hadzihasanovic (Jun 14 2025 at 18:33):

I think the nLab does choose to use "n-category" in the sense that I am attributing to "(n, n)-category", and is mostly consistent about it; but in practice this seems to lead to confusion...

Nathanael Arkor (Jun 14 2025 at 18:51):

On any page about classical category theory concepts, "category" refers to the traditional set-theoretic definition, and not to (1, 1)-categories, and "2-category" refers to the notion of bicategory, rather than (2, 2)-categories.

Nathanael Arkor (Jun 14 2025 at 18:52):

My impression is that it is primarily on the pages specifically about higher category theory that the convention of n-category = (n, n)-category may be in use.

Nathanael Arkor (Jun 14 2025 at 19:03):

Furthermore, if "n-category = (n, n)-category", then what does one call the non-homotopical variants? "Algebraic n-categories"?

Mike Shulman (Jun 14 2025 at 21:41):

Nathanael Arkor said:

On any page about classical category theory concepts, "category" refers to the traditional set-theoretic definition, and not to (1, 1)-categories, and "2-category" refers to the notion of bicategory, rather than (2, 2)-categories.

Hmm... why do you say that? Certainly pages about definitions of category refer to specific models, but I feel like a randomly chosen page about some categorical concept could be interpreted just as well in any model for 1-categories, and similarly for 2-categories.

Mike Shulman (Jun 14 2025 at 21:41):

Nathanael Arkor said:

Furthermore, if "n-category = (n, n)-category", then what does one call the non-homotopical variants? "Algebraic n-categories"?

I thought the claim was that both are "model-independent" concepts, hence neither intrinsically homotopical nor algebraic.

Nathanael Arkor (Jun 15 2025 at 07:35):

Mike Shulman said:

Nathanael Arkor said:

On any page about classical category theory concepts, "category" refers to the traditional set-theoretic definition, and not to (1, 1)-categories, and "2-category" refers to the notion of bicategory, rather than (2, 2)-categories.

Hmm... why do you say that? Certainly pages about definitions of category refer to specific models, but I feel like a randomly chosen page about some categorical concept could be interpreted just as well in any model for 1-categories, and similarly for 2-categories.

I may not be understanding the distinction appropriately, but on many pages about categories/functors, there are explicit references to collections of objects and morphisms, but I would have thought if we were talking about (1, 1)-categories, we would have to provide assignments for all $n$-morphisms, so that these definitions are not well-typed in that context? (E.g. [[full image]].)

Nathanael Arkor (Jun 15 2025 at 07:39):

Mike Shulman said:

Nathanael Arkor said:

Furthermore, if "n-category = (n, n)-category", then what does one call the non-homotopical variants? "Algebraic n-categories"?

I thought the claim was that both are "model-independent" concepts, hence neither intrinsically homotopical nor algebraic.

I may be emphasising the wrong aspect, then. What words can I use to distinguish between the notion of weak n-categories where there are only cells up to level n, and (n, n)-categories?

James Deikun (Jun 15 2025 at 10:06):

I think we only have a "classical algebraic" notion of weak $n$-categories for $n=2,3$ and arguably $n=4$ ... aside from that we would need to pick some generic globular model like the Batanin or Leinster model on the $n$-globe category to say that higher morphisms "don't exist".