nLab (n,r)-category



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



An (n,r)(n,r)-category is a higher category such that, essentially:

Put another way: given a sequence of (higher) categories C 0,C 1,...,C nC_0, C_1, ..., C_n in which each C i+1C_{i+1} is of the form Hom(A,B)Hom(A, B) for some 00-cells AA and BB from C iC_i, let us say that C nC_n is a depth-nn Hom-category of C 0C_0. (We can also cleanly extend this notion to depth-\infty Hom-categories, by taking the position that there are none). An (n,r)(n, r)-category, then, is one in which every depth-rr Hom-category is an \infty-groupoid, and, furthermore, every depth-(n+2)(n+2) Hom-category is a point. (The appearance of n+2n+2 here rather than nn allows us to make sense of this definition even when nn is as low as 2-2, and suggests that perhaps, had history gone differently, the conventions would be to number these differently.)

So (n,r)(n,r)-categories are a generalisation of both nn-categories and nn-groupoids, covering some of the ground in between (and a bit beyond). As nn increases, there are many more possibilities, until there are infinitely many kinds of (,r)(\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 kk always uniquely exists, it need not follow that a morphism at dimension k+1k+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 kk are invertible, it need not follow that all morphisms at dimension k+1k+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 nn and rr, but two subsets of \mathbb{N}.


Given a notion of \infty -category (as weak or strict as you like), then an (n,r)(n,r)-category can be defined to be an \infty-category such that

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

As explained below, we may assume that n2n \geq -2 and 0rmax(0,n+1)0 \leq r \leq \max(0, n + 1).

For finite rr, we can also define this inductively in terms of (∞,r)-categories as follows:


For 2n-2 \leq n \leq \infty, an (n,0)-category is an ∞-groupoid that is n-truncated: an n-groupoid.

For 0<r<0 \lt r \lt \infty, an (n,r)-category is an (∞,r)-category CC such that for all objects X,YCX,Y \in C the (,r1)(\infty,r-1)-categorical hom-object C(X,Y)C(X,Y) is an (n1,r1)(n-1,r-1)-category.

(Even for r=r = \infty, this definition makes sense, taking 1\infty - 1 to be \infty, as long as we know that an (1,)(-1,\infty)-category is the same thing as a (1,0)(-1,0)-category. But this may be overkill.)

You can also start with a notion of nn-poset, then define an (n,r)(n,r)-category to be an (n+1)(n+1)-poset such that any jj-morphism is an equivalence for j>rj \gt r. Or, for rnr \leq n, you can start with a notion of nn-category, then define an (n,r)(n,r)-category to be an nn-category such that any jj-morphism in an equivalence for j>rj \gt r.

To interpret the first definition above correctly for low values of jj, we must assume that all objects (00-morphisms) in a given \infty-category are parallel, which leads us to speak of the two (1)(-1)-morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any jj-morphism is an equivalence for j<1j \lt 1, so if r=0r = 0, then the condition is satisfied for any smaller value of rr. Thus, we assume that r0r \geq 0.

To say that parallel (1)(-1)-morphisms must be equivalent is meaningful; it requires that there be an object. One can continue to (2)(-2)-morphisms and so on, but there is nothing to vary about these; so we assume that n2n \geq -2. In other words, a (2)(-2)-category will automatically be an nn-category for any smaller value of nn.

If any two parallel jj-morphisms are equivalent, then any jj-morphism between equivalent (j1)(j-1)-morphisms is an equivalence (being parallel to an identity for j>0j \gt 0 and automatically for j<1j \lt 1). Accordingly, any (n,r)(n,r)-category for r>n+1r \gt n + 1 is also an (n,n+1)(n,n+1)-category. Thus, we assume that rn+1r \leq n + 1 (except when n=2n = -2, where it would conflict with the convention r0r \geq 0 and so we simply take r=0r = 0.)

Homotopy-theoretic relation

From the point of view of homotopy theory, the notion of (n,r)(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)(n,r)-category is an rr-directed homotopy nn-type.

Here we read

  • 00-directed as undirected


  • 11-directed as directed .

Then, indeed, we have for instance that

Special cases

An (n,n)-category is simply an nn-category. An (n,n+1)(n,n+1)-category is an (n+1)(n+1)-poset. Note that an \infty-category and an \infty-poset are the same thing. An (n,0)(n,0)-category is an nn-groupoid. Even though they have no special name, (n,1)(n,1)-categories are widely studied.

For low values of nn, many of these notions coincide. For instance, a 00-groupoid is the same as a 00-category, namely a set. And (1)(-1)-groupoid, (1)(-1)-category, and 00-poset all mean the same thing (namely, a truth value) while (2)(-2)-groupoid, (2)(-2)-category, and (1)(-1)-poset likewise all mean the same thing (namely, the point).

Of particular importance is the case where n=n = \infty. See

Topos cases

An analogous systematics exists for (n,r)(n,r)-categories that in additions have the property of being a topos or higher topos.

The periodic table

There is a periodic table of (n,r)(n,r)-categories:

n n
r r
2 -2 1 -1 0 0 1 1 2 2 \infty
0 0 point truth value set groupoid 2-groupoid ... ∞-groupoid
1 1 " " poset category (2,1)-category ... (∞,1)-category
2 2 " " " 2-poset 2-category ... (∞,2)-category
3 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)(n,r)-categories.


Last revised on April 30, 2024 at 01:57:29. See the history of this page for a list of all contributions to it.