nLab
(infinity,n)-category

Idea

In higher category theory an $(\mathrm{\infty },n)$-category may be thought of as

• an n-category up to coherent homotopy;

• (r,n)-category for $r=\mathrm{\infty }$;

• a weak omega-category for which all k-morphisms with $k>n$ are equivalences.

Definitions

There is an axiomatic characterization of the (∞,1)-category of $(\mathrm{\infty },n)$-categories (hence of these structures with all morphisms but only invertible transfors between them).

• Axiomatic definition

Among the concrete constructions one can roughly distinguish two flavors, those that build $(\mathrm{\infty },n)$-categories by enrichment over $(\mathrm{\infty },n-1)$-categories

• Definitions via enrichment

and those that build them by internalization in the collection of $(\mathrm{\infty },n-1)$-categories

• Definitions via internalization.

Axiomatic definition

We discuss the axiomatic characterization of the (∞,1)-category of $(\mathrm{\infty },n)$-categories due to (Barwick, Schommer-Pries).

Preliminaries

The main definition is def. 5 below, which roughly says that the collection of $(\mathrm{\infty },n)$-categories is generated from strict n-categories in a certain sense. Therefore we first need to fix some terminology and notions about strict $n$-categories and about the relevant notion of generation.

Definition

Write $\mathrm{Str}n\mathrm{Cat}$ for the 1-category of strict n-categories. Write $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}\hookrightarrow \mathrm{Str}n\mathrm{Cat}$ for the full subcategory on the gaunt $n$-categories, those $n$-categories whose only invertible k-morphisms are the identities.

This subcategory was considered in (Rezk). The term “gaunt” is due to (Barwick, Schommer-Pries).

Example

For $k\le n$ the $k$-globe is gaunt, $G_k\in \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}\hookrightarrow \in \mathrm{Str}n\mathrm{Cat}$.

Write

$${𝔾}_{\le n}\hookrightarrow \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}$$\mathbb{G}_{\leq n} \hookrightarrow Str n Cat_{gaunt}

for the full subcategory of the globe category on the $k$-globes for $k\le n$.

Being a subobject of a gaunt $n$-category, also the boundary of a globe $\partial G_k\hookrightarrow G_k$ is gaunt, i.e. the $(k-1)$-skeleton of $G_k$.

Definition

Write

$$\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}\hookrightarrow \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}$$Str n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}

for the smallest full subcategory that

1. contains the globe category ${𝔾}_{\le n}$, example 1;
2. is closed under retracts in $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gaunt}}$;
3. has all fiber products over globes.

(B-SP, def. 5.6)

Definition

The following pushout identities in $\mathrm{Str}n\mathrm{Cat}$ we call the fundamental pushouts in the following

1. Gluing two $k$-globes along their boundary gives the boundary of the $(k+1)$-globle

   $G_k{\coprod }_{\partial C_{k-1}}{G}_k\simeq \partial G_{k+1}$

2. Gluing two $k$-globes along an $i$-face gives a pasting composition of the two globles

   $G_k{\coprod }_{G_i}{G}_k$

3. The fiber product of globes along non-degenerate morphisms $G_{i+j}\to G_i$ and $G_{i+k}\to G_i$ is built from gluing of globes by

   $$G_{i+j}{\times }_{G_i}{G}_{i+k}\simeq (G_{i+j}\coprod _{G_i}{G}_{i+k})\coprod _{{\sigma }^{i+1}(G_{j-1}\times G_{k-1})}(G_{i+k}\coprod _{G_i}{G}_{i+j})$$G_{i+j} \times_{G_i} G_{i+k} \simeq (G_{i+j} \coprod_{G_i} G_{i+k}) \coprod_{\sigma^{i+1}(G_{j-1} \times G_{k-1})} (G_{i+k} \coprod_{G_i} G_{i+j})

Def. 5 considers an $(\mathrm{\infty },1)$-category generated from $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$ in the following sense

Definition

For $𝒟$ an (∞,1)-category with all small (∞,1)-colimits, say that an (∞,1)-functor

$$f:𝒞\to 𝒟$$f : \mathcal{C} \to \mathcal{D}

strongly generates $𝒟$ if its $(\mathrm{\infty },1)$-Yoneda extension on the (∞,1)-category of (∞,1)-presheaves

$$f:𝒞\stackrel{y}{\hookrightarrow }{\mathrm{PSh}}_{\mathrm{\infty }}(𝒞)\stackrel{{\mathrm{Lan}}_y}{\to }𝒟$$f : \mathcal{C} \stackrel{y}{\hookrightarrow} PSh_\infty(\mathcal{C}) \stackrel{Lan_y}{\to} \mathcal{D}

is the reflector of a reflective sub-(∞,1)-category

$$𝒟\stackrel{\stackrel{{\mathrm{Lan}}_y}{\leftarrow }}{\hookrightarrow }{\mathrm{PSh}}_{\mathrm{\infty }}(𝒞).$$\mathcal{D} \stackrel{\overset{Lan_y}{\leftarrow}}{\hookrightarrow} PSh_\infty(\mathcal{C}) \,.

Remark

By definition, a strongly generated $(\mathrm{\infty },1)$-category is in particular a presentable (∞,1)-category.

Characterization

Definition

An $(\mathrm{\infty },1)$-category of $(\mathrm{\infty },n)$-categories ${\mathrm{Cat}}_{(\mathrm{\infty },n)}$ is an (∞,1)-category equipped with a full and faithful functor

$$i:\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}\hookrightarrow {\tau }_{\le 0}{\mathrm{Cat}}_{(\mathrm{\infty },n)}$$i : Str n Cat_{gen} \hookrightarrow \tau_{\leq 0}Cat_{(\infty,n)}

from the generating strict $n$-categories, def. 2 into its category of 0-truncated objects, such that

1. ${𝒴}_n\to {\tau }_{\le 0}{\mathrm{Cat}}_{(\mathrm{\infty },n)}\hookrightarrow {\mathrm{Cat}}_{(\mathrm{\infty },n)}$ strongly generates $𝒞$;

2. $i$ preserves the fundamental pushouts; and sends (…) to an equivalence.

3. the base change adjoint triple in ${\mathrm{Cat}}_{(\mathrm{\infty },n)}$ exists along morphisms with codomain in the image of $i$;

and such that $𝒞$ is universal with respect to these properties in that for any other $j:\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}\hookrightarrow 𝒞$ satisfying these three conditions it factors through $i$

$$j:\mathrm{Str}n\mathrm{Cat}\stackrel{i}{\to }{\mathrm{Cat}}_{(\mathrm{\infty },n)}\stackrel{L}{\to }𝒞$$j : Str n Cat \stackrel{i}{\to} Cat_{(\infty,n)} \stackrel{L}{\to} \mathcal{C}

by an (∞,1)-functor $L$ which is the reflector of a reflective inclusion $𝒞\hookrightarrow {\mathrm{Cat}}_{(\mathrm{\infty },n)}$.

(B-SP, def. 6.8)

Remark

By the first axiom, the localization demanded in the universal property is essentially unique. In particular, therefore, ${\mathrm{Cat}}_{(\mathrm{\infty },n)}$ is defined uniquely, up to equivalence of (∞,1)-categories. For more on this see prop. 5 below.

Presentations

By def. 5 ${\mathrm{Cat}}_{(\mathrm{\infty },n)}$ is equivalence of (∞,1)-categories to a localization of the (∞,1)-category of (∞,1)-presheaves on $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$. In fact, various subcategories of $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$ are already sufficient. Here we discuss these presentations.

Universal presentation

Definition

Let $S_0\subset \mathrm{Mor}({\mathrm{PSh}}_{\mathrm{\infty }}(\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}))$ be the class of morphism generated under fiber product $X{\times }_{G_k}(-)$ with objects $X\in \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$ over globes by the following finite sets of morphisms

1. (…)

2. (…)

3. (…)

4. (…)

Write $S$ for the strongly saturated class of morphisms (see reflective sub-(∞,1)-category) generated by $S_0$.

Proposition

The localization of the (∞,1)-category of (∞,1)-presheaves over $\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$, def. 2 at the class of morphism $S$ from def. 6 is a presentation of ${\mathrm{Cat}}_{(\mathrm{\infty },n)}$, def. 5:

$${\mathrm{Cat}}_{(\mathrm{\infty },n)}\simeq {\mathrm{PSh}}_{\mathrm{\infty }}(\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}})[S^{-1}].$$Cat_{(\infty,n)} \simeq PSh_\infty(Str n Cat_{gen})[S^{-1}] \,.

(B-SP, theorem 7.6).

Proof

The three axioms of def. 5 are satisfied effectively by construction of $S$ (…). Conversely, every localization satisfying the second and third axiom must invert the morphisms in $S$, hence must be a sub-localization.

Various other presentations of ${\mathrm{Cat}}_{(\mathrm{\infty },n)}$ are obtained by localizations over subcategories of
$i:\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{restr}}\hookrightarrow \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$ at a set of morphisms $T\subset \mathrm{Mor}({\mathrm{PSh}}_{\mathrm{\infty }}(R))$. Write

$${\mathrm{PSh}}_{\mathrm{\infty }}(\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{restr}})\stackrel{\stackrel{i_{!}}{\to }}{\stackrel{\stackrel{i^{*}}{\leftarrow }}{\underset{i_{*}}{\to }}}{\mathrm{PSh}}_{\mathrm{\infty }}(\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}})$$PSh_\infty(Str n Cat_{restr}) \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}} PSh_\infty(Str n Cat_{gen})

for the induced essential geometric morphism.

Proposition

The following conditions are sufficient in order that

$${\mathrm{Cat}}_{(\mathrm{\infty },n)}\simeq {\mathrm{PSh}}_{\mathrm{\infty }}(\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}})[S^{-1}]\stackrel{i^{*}}{\to }{\mathrm{PSh}}_{\mathrm{\infty }}(\mathrm{Str}n{\mathrm{Cat}}_{\mathrm{restr}})[T^{-1}]$$Cat_{(\infty,n)} \simeq PSh_\infty(Str n Cat_{gen})[S^{-1}] \stackrel{i^*}{\to} PSh_\infty(Str n Cat_{restr})[T^{-1}]

is an equivalence of (∞,1)-categories:

1. $i^{*}(S_0)\subset T$

2. $i_{!}(T_0)\subset S$

3. the counit $\mathrm{id}\to i^{*}{i}_{!}$ has components in $T$;

4. the $k$-globe $G_k$ is in the essential image of $i$, for each $0\le k\le n$.

(B-SP, theorem 9.2)

Theta-spaces

Proposition

The $n$th Theta category is a full subcategory ${\Theta }_n\hookrightarrow \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$ and the localization that defines the $(\mathrm{\infty },1)$-category ${\Theta }_n\mathrm{Space}$ of $(\mathrm{\infty },n)$-Theta-spaces satisfies the conditions of prop. 2.

Hence ${\Theta }_n$-spaces are a model for $(\mathrm{\infty },n)$-categories, in the sense of def. 5:

$${\Theta }_n\mathrm{Space}\simeq {\mathrm{Cat}}_{(\mathrm{\infty },n)}.$$\Theta_n Space \simeq Cat_{(\infty,n)} \,.

(B-SP, theorem 11.15)

Via $\mathrm{\infty }$-enrichment

General

There should be a general notion of enriched (∞,1)-category (see there) over a monoidal (∞,1)-category $𝒱$. Write $𝒱\mathrm{Cat}$ for the (∞,1)-category of $𝒱$-enriched $(\mathrm{\infty },1)$-categories.

Definition

For $n\in \mathbb{N}$ write

$${\mathrm{Cat}}_{(\mathrm{\infty },n)}≔(((\mathrm{\infty }\mathrm{Grpd}\mathrm{Cat})\mathrm{Cat})\cdots )\mathrm{Cat}.$$Cat_{(\infty,n)} \coloneqq (((\infty Grpd Cat) Cat) \cdots) Cat \,.

Presentation by Segal $n$-categories

A presentation of the notion of $n$-fold enrichement of def. 7 is given by Segal n-categories.

(…)

Via $\mathrm{\infty }$-internalization

General

There is a general notion of internal category in an (∞,1)-category. For $𝒞$ an (∞,1)-category write $\mathrm{Cat}(𝒞)$ for the $(\mathrm{\infty },1)$-category of internal categories in $𝒞$. Write ${\mathrm{Cat}}_{(\mathrm{\infty },1)}≔$ ∞Grpd for the $(\mathrm{\infty },1)$-category of ∞-groupoids.

Definition

For $n\in 𝒩$ the $(\mathrm{\infty },1)$-category of $(\mathrm{\infty },n)$-categories is

$${\mathrm{Cat}}_{(\mathrm{\infty },n)}≔{\mathrm{Cat}}^n(\mathrm{\infty }\mathrm{Grpd})≔\mathrm{Cat}(\cdots \mathrm{Cat}({\mathrm{Cat}}_{(\mathrm{\infty },0)})\cdots ).$$Cat_{(\infty,n)} \coloneqq Cat^n(\infty Grpd) \coloneqq Cat(\cdots Cat(Cat_{(\infty,0)}) \cdots) \,.

(Lurie).

Presentation by $n$-fold complete Segal spaces

A presentation of ${\mathrm{Cat}}^n(\mathrm{\infty }\mathrm{Grpd})$ is given by n-fold complete Segal spaces.

Proposition

The $n$-fold product of the simplex category is naturally a full subcategory $\Delta \times n\hookrightarrow \mathrm{Str}n{\mathrm{Cat}}_{\mathrm{gen}}$ and the localization that gives the $(\mathrm{\infty },1)$-category $\mathrm{CSS}({\Delta }^{\times n})$ of n-fold complete Segal space satisfies the conditions of prop. 2.

Hence $n$-fold complete Segal space are a model for $(\mathrm{\infty },n)$-categories, in the sense of def. 5:

$${\Theta }_n\mathrm{Space}\simeq {\mathrm{Cat}}_{(\mathrm{\infty },n)}.$$\Theta_n Space \simeq Cat_{(\infty,n)} \,.

(B-SP, theorem 12.6)

Properties

Uniqueness and equivalences

Proposition

Let ${\mathrm{Models}}_{(\mathrm{\infty },n)}\hookrightarrow {\widehat{\mathrm{Cat}}}_{(\mathrm{\infty },1)}$ be the core (maximal ∞-groupoid inside) the full sub-(∞,1)-category of (∞,1)Cat on those that satisfy the definition 5.

This is equivalent to

$${\mathrm{Models}}_{(\mathrm{\infty },n)}\simeq B({\mathbb{Z}}_2{)}^n,$$Models_{(\infty,n)} \simeq B (\mathbb{Z}_2)^n,

the delooping groupoid of the group $({\mathbb{Z}}_2{)}^n$, the $n$-fold product of the group of order 2 with itself.

The nontrivial element $\sigma \in {\mathbb{Z}}_2$ in the $k$th slot acts by passing to the $k$-opposite $(\mathrm{\infty },n)$-category.

(B-SP, theorem 8.13)

Remark

This means that

1. the $(\mathrm{\infty },1)$-category ${\mathrm{Cat}}_{(\mathrm{\infty },n)}$ from def. 5 is uniquely defined, up to equivalence of (∞,1)-categories;

2. the automorphism ∞-group of ${\mathrm{Cat}}_{(\mathrm{\infty },n)}$ in ${\widehat{\mathrm{Cat}}}_{(\mathrm{\infty },1)}$ is $({\mathbb{Z}}_2{)}^n$, hence the only auto-equivalences are given by reversal of k-morphisms.

Examples

Special cases

• ∞-groupoid = $(\mathrm{\infty },0)$-category
• (∞,1)-category
• (∞,2)-category
• etc …

In addition,

• (m,n)-categories can be obtained as particular $(\mathrm{\infty },n)$-categories whose $k$-cells are trivial for $k>m$.
• In particular, n-categories = $(n,n)$-categories can be so obtained.

Specific examples

• One motivating example for $(\mathrm{\infty },n)$-categories is the (∞,n)-category of cobordisms which plays a central role in the formalization of the cobordism hypothesis.

• Another class of examples are (∞,n)-categories of spans.

Related concepts

• 0-category, (0,1)-category

• category

• 2-category

• 3-category

• n-category

• (∞,0)-category

• (∞,1)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category

References

The definition in terms of Theta spaces is due to

• Charles Rezk, A cartesian presentation of weak n-categories (arXiv:0901.3602)

An iterartive definition in terms of n-fold complete Segal spaces is given in

• Jacob Lurie, $(\mathrm{\infty },2)$-Categories and the Goodwillie Calculus I (arXiv:0905.0462)

A summary of definitions and some known comparison results can be found at

• Julie Bergner, Models for $(\mathrm{\infty },n)$-categories and the cobordism hypothesis, arXiv:1011.0110

• Julie Bergner, Models for $(\mathrm{\infty },n)$-Categories and the Cobordism Hypothesis , in Mathematical Foundations of Quantum Field and Perturbative String Theory

An axiomatic characterization is in

• Clark Barwick, Chris Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories (arXiv:1112.0040, unusual slides)

Comparison of models is in

• Julie Bergner, Charles Rezk, Comparison of models for $(\mathrm{\infty },n)$-categories (arXiv:1204.2013)