nLab (n,r)-category

Idea

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.

Remark

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 $\mathbb{N}$.

Definition

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

• any $j$-morphism is an equivalence, for $j \gt r$;
• any two parallel $j$-morphisms are equivalent, for $j \gt n$.

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:

Definition

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 $r = 0$.)

Homotopy-theoretic relation

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

• $0$-directed as undirected

and

• $1$-directed as directed .

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.

Special cases

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

• (∞,n)-category .

Topos cases

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

  • in particular, a $(0,1)$-Grothendieck topos is a locale

• a $(1,1)$-topos is a topos

• an (∞,1)-topos is what Higher Topos Theory calls an $\infty$-topos

The periodic table

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?

Models for weak (n,r)-categories

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.

Related concepts

• directed (n,r)-graph

• (n,r)-site

• 0-category, (0,1)-category

• category

• 2-category

• 3-category

• n-category

• (∞,0)-category

• (n,1)-category

• (∞,1)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category

• (n × k)-category, n-fold category

References

• John Baez, Michael Shulman, Lectures on n-Categories and Cohomology, (arXiv:math/0608420), in particular pp. 34-36.