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 some 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.
This definition does not attempt to cover all possible scenarios regarding the triviality or invertibility of morphisms at different dimensions. In particular:
In general, if a morphism at dimension $k$ always uniquely exists, it need not follow that a morphism at dimension $k+1$ always uniquely exists. A counterexample are monoid-enriched categories, where we view monoids as single-object categories, so that we can see monoid-enriched categories as 2-categories whose sets of 1-morphisms are singletons, but whose sets of 2-morphisms are not.
In general, if all morphisms at dimension $k$ are invertible, it need not follow that all morphisms at dimension $k+1$ are invertible. A counterexample is ordered groupoids, and in particular the single object case of ordered groups such as $(\mathbb{Z}, +, \leq)$. They have invertible 1-morphisms, but non-invertible 2-morphisms.
A more fine-grained classification of higher categories could thus be imagined, where we would not specify two numbers $n$ and $r$, but two subsets of $\mathbb{N}$.
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 $0 \leq r \leq \max(0, n + 1)$.
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 the first definition above 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 $r \leq n + 1$ (except when $n = -2$, where it would conflict with the convention $r \geq 0$ and so we simply take $r = 0$.)
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.
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:
$n$→ $r$↓ |
$-2$ | $-1$ | $0$ | $1$ | $2$ | … | $\infty$ |
---|---|---|---|---|---|---|---|
$0$ | point | truth value | set | groupoid | 2-groupoid | ... | ∞-groupoid |
$1$ | " | " | poset | category | (2,1)-category | ... | (∞,1)-category |
$2$ | " | " | " | 2-poset | 2-category | ... | (∞,2)-category |
$3$ | " | " | " | " | 3-poset | ... | (∞,3)-category? |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
$\infty$ | point | truth value | poset | 2-poset | 3-poset | ... | (∞,∞)-category/∞-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
Last revised on April 30, 2024 at 01:57:29. See the history of this page for a list of all contributions to it.