Meaning of (n,r)-Category

Table of Contents

A useful notation with unstable boundaries

The notation (n,r)-category is meant to classify higher categories by two parameters, where n denotes the cutoff dimension for non-trivial higher morphisms (often called cells) and r denotes the cutoff dimension for non-invertible higher morphisms.

The notation is useful because it compresses several familiar cases into one scheme: sets, preorders, groupoids, ordinary categories, bicategories with invertible 2-cells, and so on. But it is also treacherous, because the words “nontrivial,” “equivalent,” and “invertible” do not have model-independent meanings in higher category theory. They are interpreted differently in strict globular categories, weak n-categories, quasicategories, complete Segal spaces, and enrichment-based definitions, etc.

The result is that (n,r) is better understood not as a single definition, but as a notational regime: a compact way of saying what kind of higher categorical truncation and invertibility one has in mind, provided the ambient model has already been specified.

The naïve reading

The first reading one usually learns is¹:

an n-category has k-morphisms for $0\leq k\leq n$ , and no higher morphisms.

Thus 0-category is a set, 1-category is an ordinary category and 2-category is an 1-category with additional 2-morphisms along with the up to 2-dimensional coherence compatibilities. With this strict (essentially harmless) dimensional reading, a strict (n+1)-category may be defined inductively as a category enriched in strict n-categories.

But the notation (n,r) was not designed merely to say how many layers of cells exist, it also tracks invertibility as the examples display: (1,1) corresponds to an ordinary categories, (1,0) to a groupoids and $(\infty,1)$ to $\infty$ categories with all higher morphisms above 1 invertible.

Already here, the phrase “k-morphisms are trivial” is ambiguous, and indeed it is model dependent.

“Trivial” does not always mean “absent”

The main source of confusion is that “trivial above dimension n” has several possible meanings.

In a strict model, it may mean all k-morphisms for $k>n$ are identities, while in a simplicial or homotopical model, it may mean something closer to parallel higher cells are equivalent in a prescribed sense, or the relevant mapping spaces are suitably truncated.

For example, Lurie’s Higher Topos Theory gives a definition of an n-category inside simplicial sets by imposing uniqueness conditions on simplices above dimension n, and a relative homotopy condition in dimension n. This is not the same as merely saying that no higher formal symbols exist.

Similarly, Bergner–Rezk compare models for $(\infty,n)$ -categories. Separately, a common quasicategorical characterization of (n,1)-categories is that they are quasicategories with (n-1)-truncated mapping spaces, equivalently satisfying the right lifting property with respect to $\partial\Delta[m]\to\Delta[m]$ for $m\geq n+2$ [Bergner-Rezk].

One of the most revealing example is (0,1). One might think $n=0$ should mean that there are no meaningful 1-morphisms, which force us to see a (0,1)-category just as a set.

But within the flavor of modern (higher) category theory, which may be seen as in nLab/MathOverflow convention, (0,1)-categories are identified with posets or preorders, depending on whether one works strictly or up to equivalence.² The explanation is that an (n,r)-category satisfies the condition that parallel j-arrows are equivalent for $j>n$ ; hence, in a (0,1)-category, any two parallel 1-arrows are equivalent (hence coincides with classical order theoretic formulation).

So 1-morphisms are not always absent, they are either absent or exists uniquely up to equivalence for each pair of objects (we call such category thin), which leads to the interpretation of the parameter $n=0$ in that instead of erasing the arrows, it suppresses multiplicity of parallel arrows. The parameter $r=1$ allows those arrows to be non-invertible.

The preorder/poset distinction is then a separate strictness issue, whose distinction is often blurred when one works “up to equivalence,” as is common in higher-categorical exposition.

Invertibility is also relative

The other unstable word is “invertible” (i.e., to be subsumed to the notion of equivalence).

In ordinary category theory, a morphism $f:a\to b$ is (strictly) invertible if there exists $g:b\to a$ such that $gf=\operatorname{id}_a,\,fg=\operatorname{id}_b$ , hence there is no need to employ higher cell to witness the strict equivalence, or just isomorphism.

But in higher category theory with the weak notion of equivalences of cells, the equations are replaced by $gf\simeq \operatorname{id}_a,\, fg\simeq \operatorname{id}_b$ .

This difference matters, since it changes what counts as invertibile.

Recent work explicitly distinguishes different ways of defining equivalences in fully higher settings. Garner–Harpaz, for example, discuss how a saturation condition can determine equivalences for finite r, but in purely infinite-dimensional settings one obtains different notions such as inductive equivalences and reversible morphisms.

These examples confirm the point that r depends on a convention for equivalence.

Why the notation developed this way

The historical reason is coherence.

In strict low-dimensional category theory, one can say “there are no higher morphisms”. But weak higher categories require coherence cells. In sufficiently weak theories, one cannot simply delete higher cells without damaging the structure being defined, which explains the shift from no cells above dimension n to cells above dimension n are trivial, degenerate, or uniquely determined in the chosen regime.

The literature reflects this. There is no single universally accepted model of weak n-categories. MathOverflow discussions of definitions of $n$ -categories emphasize the proliferation of models after Baez–Dolan, and note that no single weak model emerged as the universal standard.

For $(\infty,n)$ -categories, the situation is better. There are comparison and unicity results for major proposed models: Bergner–Rezk establish Quillen equivalences among several model structures for $(\infty,n)$ -categories, while Barwick–Schommer-Pries give an axiomatic unicity theorem for the homotopy theory of $(\infty,n)$ -categories. But this does not automatically settle all questions about finite weak n-categories, nor about the expressive completeness of the two-parameter notation (n,r) [Bergner-Rezk][Bergner-Rezk2][Barwick-Pries].

For convenience, I internally read the dimension n as truncation, thinness or nontriviality parameter, while the dimension r is read as the invertibility parameter, both depending on the prescribed equivalence or ambient structure.

The current landscape

The current landscape seems to have three layers.

First, in informal exposition, (n,r) is used as a mnemonic. It tells the reader approximately where nontriviality and non-invertibility stop. This is common in web/blog style explanation and casual higher-categorical discussion.

Second, in serious technical work, authors usually specify a model: quasicategories, complete Segal spaces, marked or scaled simplicial sets, enriched $\infty$ -categories, as I have so far detected. Once the model is fixed, “trivial,” “equivalent,” and “invertible” acquire precise meanings.

Third, in recent foundational work, there is renewed interest in organizing the entire family $\mathrm{Cat}(n,r)$ systematically. Goldthorpe gives a formal enrichment-recursive construction of categories $\mathrm{Cat}(n,r)$ , using n-groupoids as the $r=0$ base and enrichment as the recursive step; this is a serious structured use of (n,r)-notation, but it is not primarily a meta-analysis of the notation itself [Goldthorpe].

There is also work on the tower $\cdots\mathrm{Cat}(\infty,n)\subseteq \mathrm{Cat}(\infty,n+1)\cdots$ with adjoints that either invert or discard certain higher morphisms. One recent formulation distinguishes $\pi_{\le r}$ , which formally inverts (r+1)-morphisms, from $\kappa_{\le r}$ , which keeps the maximal sub- $(\infty,r)$ -category by discarding noninvertible $(r+1)$ -morphisms [Goldthorpe2].

The problem of representativeness

Does (n,r) represent all meaningful higher-categorical distinctions?

The answer is No.

It captures two important dimensions:

where higher structure becomes trivial or truncated;
where higher morphisms become invertible.

But it does not record:

whether equality is strict or weak;
whether equivalent objects are identified;
whether higher equivalences are inductive, coinductive, saturated, or reversible;
whether the model is globular, simplicial, opetopic, cubical, Segal-type, or type-theoretic;
whether one works internally to spaces, sets, simplicial sets, model categories, or $\infty$ -topoi;
whether truncation means absence, identity-only, propositionality, contractibility, or a lifting property.

Thus (n,r) is expressive but incomplete. It is a coordinate label, not a full specification.

A convention for a better use

The (0,1)-case is especially pedagogically dangerous as I described in earlier section when it is naively expressed as

(0,1)-category = partially ordered set,

or something similar.

The same issue recurs at higher dimensions. A weakly truncated object is not necessarily strictly truncated (more to be truncated). An equivalence class of structures is not the same as a chosen skeletal representative. The (n,r)-notation alone does not say which level of identification is intended.

The notation remains valuable. But its value is heuristic and classificatory before it is definitional. A mathematically precise use of (n,r) should always say what model, what equivalence, and what truncation convention is being used.

Some authors ask the existence of morphisms in older literatures.
In any interpretation of a $(0,1)$ -category, every $k$ -morphism with $k \geq 1$ is regarded as trivial in the sense that any two parallel morphisms are equivalent under a prescribed convention; write $f \sim g : a \to b$ for this relation. Each hom-set is then, up to $\sim$ , either empty or a singleton.

The conventional narrative attributes the antisymmetry of the poset interpretation to $r$ , the cutoff dimension of invertibility. This attribution misallocates the work. Given composable morphisms $f : a \to b$ and $g : b \to a$ , the relation $\mathrm{id}_a \sim gf$ is forced already by the condition $n = 0$ , since $\mathrm{id}_a$ and $gf$ are parallel endomorphisms of $a$ ; symmetrically, $\mathrm{id}_b \sim fg$ . The pair $(f,g)$ thus exhibits $a$ as isomorphic to $b$ within the $(0,1)$ -category, and this isomorphism is a consequence of $n = 0$ alone, independent of any condition on $r$ .

Whether the isomorphism $a \cong b$ is read as the identification $a = b$ is a matter of interpretive convention external to the $(n,r)$ -data. The preorder interpretation treats $\sim$ as a genuine equivalence under which $a \cong b$ does not entail $a = b$ , so antisymmetry fails. The poset interpretation imposes the additional convention that parallel morphisms equivalent under $\sim$ are identified outright — equivalently, that isomorphic objects coincide — and antisymmetry is recovered.

The passage from preorder to poset is therefore effected by an explicit skeletal replacement — a quotient by isomorphism on objects — and this datum lies outside what $(n,r)$ encodes. In particular, the skeletal property of the poset interpretation is irrelevant to $r$ .