An $(n,r)$-category is a higher category such that, essentially:
all k-morphisms for $k \gt n$ are trivial.
all k-morphisms for $k \gt r$ are reversible.
Put another way: given a sequence of (higher) categories $C_0, C_1, ..., C_n$ in which each $C_{i+1}$ is of the form $Hom(A, B)$ for some $0$-cells $A$ and $B$ from $C_i$, let us say that $C_n$ is a depth-$n$ Hom-category of $C_0$. (We can also cleanly extend this notion to depth-$\infty$ Hom-categories, by taking the position that there are none). An $(n, r)$-category, then, is one in which every depth-$r$ Hom-category is an $\infty$-groupoid, and, furthermore, every depth-$(n+2)$ Hom-category is a point. (The appearance of $n+2$ here rather than $n$ allows us to make sense of this definition even when $n$ is as low as $-2$, and suggests that perhaps, had history gone differently, the conventions would be to number these differently.)
So $(n,r)$-categories are a generalisation of both $n$-categories and $n$-groupoids, covering all of the ground in between (and a bit beyond). As $n$ increases, there are many more possibilities, until there are infinitely many kinds of $(\infty,r)$-categories.
Eric: What is the category of all (small) $(n,r)$-categories? An $(n+1,r+1)$-category?
Urs Schreiber: yes, that should be right. Roughly the argument is that a $(k+1)$-morphism of $(n,r)Cat$ is a (n,k)-transformation:
a 1-morphism in $(n,r)Cat$ is an $n$-functor $C \stackrel{F}{\to} D$ , hence an “$(n,0)$-transformation”
a 2-morphism is a transformation between $n$-functors, hence a “(n,1)-transformation”.
and so on
finally an $(n+1)$-morphism is an $(n,n)$-transformation.
So $(n,r)Cat$ is an $(n+1)$-category.
The invertibiliy of the $(n,k)$-transformations is that of their components which are $(\ell \geq k)$-morphisms in the target $n$-category $D$. So if all $(\ell \gt r)$-morphisms in $D$ are invertible, then so are all $(n,\ell \gt r)$-transformations between $C$ and $D$ hence all $(\ell \gt r+1)$-morphisms in $(n,r)Cat$. So $(n,r)Cat$ is an $(n+1,r+1)$-category.
Given a notion of $\infty$-category (as weak or strict as you like), then an $(n,r)$-category can be defined to be an $\infty$-category such that
As explained below, we may assume that $n \geq -2$ and $0 \leq r \leq n + 1$.
For finite $r$, we can also define this inductively in terms of (∞,r)-categories as follows:
For $-2 \leq n \leq \infty$, an (n,0)-category is an ∞-groupoid that is n-truncated: an n-groupoid.
For $0 \lt r \lt \infty$, an (n,r)-category is an (∞,r)-category $C$ such that for all objects $X,Y \in C$ the $(\infty,r-1)$-categorical hom-object $C(X,Y)$ is an $(n-1,r-1)$-category.
(Even for $r = \infty$, this definition makes sense, taking $\infty - 1$ to be $\infty$, as long as we know that an $(-1,\infty)$-category is the same thing as a $(-1,0)$-category. But this may be overkill.)
You can also start with a notion of $n$-poset, then define an $(n,r)$-category to be an $(n+1)$-poset such that any $j$-morphism is an equivalence for $j \gt r$. Or, for $r \leq n$, you can start with a notion of $n$-category, then define an $(n,r)$-category to be an $n$-category such that any $j$-morphism in an equivalence for $j \gt r$.
To interpret this correctly for low values of $j$, we must assume that all objects ($0$-morphisms) in a given $\infty$-category are parallel, which leads us to speak of the two $(-1)$-morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any $j$-morphism is an equivalence for $j \lt 1$, so if $r = 0$, then the condition is satisfied for any smaller value of $r$. Thus, we assume that $r \geq 0$.
To say that parallel $(-1)$-morphisms must be equivalent is meaningful; it requires that there be an object. One can continue to $(-2)$-morphisms and so on, but there is nothing to vary about these; so we assume that $n \geq -2$. In other words, a $(-2)$-category will automatically be an $n$-category for any smaller value of $n$.
If any two parallel $j$-morphisms are equivalent, then any $j$-morphism between equivalent $(j-1)$-morphisms is an equivalence (being parallel to an identity for $j \gt 0$ and automatically for $j \lt 1$). Accordingly, any $(n,r)$-category for $r \gt n + 1$ is also an $(n,n+1)$-category. Thus, we assume that $r \leq n + 1$.
From the point of view of homotopy theory, the notion of $(n,r)$-categories may be understood as a combination of the notion of homotopy n-type and that of directed space.
Recall that an (∞,0)-category is an ∞-groupoid. In light of the homotopy hypothesis – that identifies $\infty$-groupoids with (nice) topological spaces and n-groupoids with homotopy n-types – and in view of the notion of directed space, the following terminology is suggestive:
An $(n,r)$-category is an $r$-directed homotopy $n$-type.
Here we read
and
Then, indeed, we have for instance that
a (1,0)-category is an undirected 1-type: a 1-groupoid,
a (2,0)-category is an undirected 2-type: a 2-groupoid,
etc.
a (1,1)-category is directed 1-type : a category,
an (n,n)-category is an $n$-directed $n$-type: an n-category,
etc.
an (∞,0)-category is an undirected space: an ∞-groupoid,
an (∞,1)-category is a directed space: a quasi-category,
an (∞,n)-category is an $n$-directed space
etc.
Mike Shulman: I am not convinced that the homotopy hypothesis applies to anything directed. I’ll believe that maybe an $r$-directed $n$-type (whatever that means) should have a fundamental $(n,r)$-category, and that this operation has a left adjoint that geometrically realizes an $(n,r)$-category as an $r$-directed $n$-type. But I don’t see why to expect this adjunction to be an equivalence in the directed world, unless all of your $r$-directed $n$-types come equipped with a chosen CW-complex-like $n$-skeleton which you restrict your fundamental categories to.
More concretely: take the interval category. Realize it as a directed space; presumably you get a directed topological interval $[0,1]$. Now take the fundamental category of this space: you get the ordered set $[0,1]$ considered as a category—quite different from the interval category! In order to get back the interval category, you need to do something like remember the endpoints of the directed topological interval, and only use these chosen points as the objects of your fundamental category. Perhaps everyone talking about identifying directed homotopy types with higher categories has some fix like this in mind, but if so I think it should be stressed. (Alternately, maybe someone can tell me why I’m completely wrong.)
David Roberts: Perhaps one could take a leaf out of Ronnie Brown’s book and consider filtered/stratified directed spaces. The relative fundamental category is, as you point out, the 'correct' answer.
Urs Schreiber: right. I didn’t mean to imply that there is an established theory of directed spaces that yields a directed homotopy hypothesis-theorem yet. Instead the idea was that “in view of the homotopy hypothesis” we should be entitled to think of an $(n,r)$-category as an $r$-directed $n$-type. Over at directed space I say more explicitly that one option is to defined what a (nice) $r$-directed $n$-type is this way. I have very little online time today, otherwise I would now add a paragraph along these lines to the above. Maybe one of you feels like doing it. I still think that th slogan “An $(n,r)$-category is an $r$-directed $n$-type.” is a very useful guiding principle, and be it for the right definition of directed space. My impression is that the theory of directed spaces is at the time still tentative and not set in stonee. But if that’s wrong, then I’d still keep the above slogan but put an explicit caveat that this uses the notion “diected space” differently to that established in the literature.
David Corfield: During a discussion on fundamental categories with duals of statified spaces, we had this description of a project to provide a geometric picture of directed homtopy. Speaking of categories with duals, couldn’t nLab do with some more pages on them?
Mike Shulman: Your definition at directed space (“a directed space is a topological space in which not every cell is traversable in all directions”) doesn’t say anything about a stratification, so I think it’s misleading to then say that they could be defined as $(n,r)$-categories without making a point that this would change the notion. My impression from the very little I’ve read about directed spaces is that they don’t necessarily come with any sort of stratification. Do we have any reason to want to define “$r$-directed $n$-type” to mean “$(n,r)$-category”, other than that it would be cute if the homotopy hypothesis could be generalized? We like $(n,r)$-categories for lots of reasons—but would calling them $r$-directed $n$-types really be useful to us or anyone else?
Toby: I don't think that it helps our understanding of $(n,r)$-categories, at least not yet, which is why I moved this section down here. But I think that it may help us to understand directed spaces, particularly to suggest the idea that spaces might be $r$-directed.
Urs Schreiber: I agree with Mike that the statements may currently be too misleading, and with Toby about what they should still achieve for us. Will try to improve on the state of the two entries a bit tomorrow – unless someone beats me to it.
David Roberts: Going back to Mike’s original comment, having read a little about fundamental categories (fingers automatically started typing ‘groupoid’ there :), the concept of equivalence of categories has to be expanded so as to capture ‘directed homotopy equivalence’. In particular, there is the notions of past retract? and future retract? - these should be considered as equivalences, but are not equivalences of categories in the usual sense. From memory they are more like (co)relexive subcategories.
An (n,n)-category is simply an $n$-category. An $(n,n+1)$-category is an $(n+1)$-poset. Note that an $\infty$-category and an $\infty$-poset are the same thing. An $(n,0)$-category is an $n$-groupoid. Even though they have no special name, $(n,1)$-categories are widely studied.
For low values of $n$, many of these notions coincide. For instance, a $0$-groupoid is the same as a $0$-category, namely a set. And $(-1)$-groupoid, $(-1)$-category, and $0$-poset all mean the same thing (namely, a truth value) while $(-2)$-groupoid, $(-2)$-category, and $(-1)$-poset likewise all mean the same thing (namely, the point).
Of particular importance is the case where $n = \infty$. See
An analogous systematics exists for $(n,r)$-categories that in additions have the property of being a topos or higher topos.
a (0,1)-topos is a Heyting algebra
a $(1,1)$-topos is a topos
an (∞,1)-topos is what Higher Topos Theory calls an $\infty$-topos
There is a periodic table of $(n,r)$-categories:
$r$↓\$n$→ | $-2$ | $-1$ | $0$ | $1$ | $2$ | … | $\infty$ |
---|---|---|---|---|---|---|---|
$0$ | trivial | truth value | set | groupoid | 2-groupoid | ... | infinity groupoid |
$1$ | " | " | poset | category | (2,1)-category | ... | (infinity,1)-category |
$2$ | " | " | " | 2-poset | 2-category | ... | (infinity,2)-category |
$3$ | " | " | " | " | 3-poset | ... | (infinity,3)-category? |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
$\infty$ | trivial | truth value | poset | 2-poset | 3-poset | ... | (infinity,infinity)-category/infinity poset? |
There are various model category models for collections of $(n,r)$-categories.
The standard model structure on simplicial sets models (∞,0)-categories.
The Joyal-model structure on simplicial sets models (∞,1)-categories.
The Charles Rezk-model structure for Theta spaces models general $(n,r)$-categories.
(n,r)-category