nLab
(infinity,n)-category

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

In higher category theory an (,n)-category may be thought of as

Definitions

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

Among the concrete constructions one can roughly distinguish two flavors, those that build (,n)-categories by enrichment over (,n1)-categories

and those that build them by internalization in the collection of (,n1)-categories

Axiomatic definition

We discuss the axiomatic characterization of the (∞,1)-category of (,n)-categories due to (Barwick, Schommer-Pries).

Preliminaries

The main definition is def. 5 below, which roughly says that the collection of (,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 StrnCat for the 1-category of strict n-categories. Write StrnCat gauntStrnCat 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 kn the k-globe is gaunt, G kStrnCat gauntStrnCat.

Write

𝔾 nStrnCat gaunt\mathbb{G}_{\leq n} \hookrightarrow Str n Cat_{gaunt}

for the full subcategory of the globe category on the k-globes for kn.

Being a subobject of a gaunt n-category, also the boundary of a globe G kG k is gaunt, i.e. the (k1)-skeleton of G k.

Definition

Write

StrnCat genStrnCat gauntStr n Cat_{gen} \hookrightarrow Str n Cat_{gaunt}

for the smallest full subcategory that

  1. contains the globe category 𝔾 n, example 1;
  2. is closed under retracts in StrnCat gaunt;
  3. has all fiber products over globes.

(B-SP, def. 5.6)

Definition

The following pushout identities in StrnCat 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 C k1G kG k+1

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

    G k G iG k

  3. The fiber product of globes along non-degenerate morphisms G i+jG i and G i+kG i is built from gluing of globes by

    G i+j× G iG i+k(G i+j G iG i+k) σ i+1(G j1×G k1)(G i+k G iG 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 (,1)-category generated from StrnCat gen in the following sense

Definition

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

f:𝒞𝒟f : \mathcal{C} \to \mathcal{D}

strongly generates 𝒟 if its (,1)-Yoneda extension on the (∞,1)-category of (∞,1)-presheaves

f:𝒞yPSh (𝒞)Lan y𝒟f : \mathcal{C} \stackrel{y}{\hookrightarrow} PSh_\infty(\mathcal{C}) \stackrel{Lan_y}{\to} \mathcal{D}

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

𝒟Lan yPSh (𝒞).\mathcal{D} \stackrel{\overset{Lan_y}{\leftarrow}}{\hookrightarrow} PSh_\infty(\mathcal{C}) \,.
Remark

By definition, a strongly generated (,1)-category is in particular a presentable (∞,1)-category.

Characterization

Definition

An (,1)-category of (,n)-categories Cat (,n) is an (∞,1)-category equipped with a full and faithful functor

i:StrnCat genτ 0Cat (,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 nτ 0Cat (,n)Cat (,n) strongly generates 𝒞;

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

  3. the base change adjoint triple in Cat (,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:StrnCat gen𝒞 satisfying these three conditions it factors through i

j:StrnCatiCat (,n)L𝒞j : 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 𝒞Cat (,n).

(B-SP, def. 6.8)

Remark

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

Presentations

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

Universal presentation

Definition

Let S 0Mor(PSh (StrnCat gen)) be the class of morphism generated under fiber product X× G k() with objects XStrnCat 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 StrnCat gen, def. 2 at the class of morphism S from def. 6 is a presentation of Cat (,n), def. 5:

Cat (,n)PSh (StrnCat 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 Cat (,n) are obtained by localizations over subcategories of
i:StrnCat restrStrnCat gen at a set of morphisms TMor(PSh (R)). Write

PSh (StrnCat restr)i *i *i !PSh (StrnCat 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

Cat (,n)PSh (StrnCat gen)[S 1]i *PSh (StrnCat 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)T

  2. i !(T 0)S

  3. the counit idi *i ! has components in T;

  4. the k-globe G k is in the essential image of i, for each 0kn.

(B-SP, theorem 9.2)

Theta-spaces

Proposition

The nth Theta category is a full subcategory Θ nStrnCat gen and the localization that defines the (,1)-category Θ nSpace of (,n)-Theta-spaces satisfies the conditions of prop. 2.

Hence Θ n-spaces are a model for (,n)-categories, in the sense of def. 5:

Θ nSpaceCat (,n).\Theta_n Space \simeq Cat_{(\infty,n)} \,.

(B-SP, theorem 11.15)

Via -enrichment

General

There should be a general notion of enriched (∞,1)-category (see there) over a monoidal (∞,1)-category 𝒱. Write 𝒱Cat for the (∞,1)-category of 𝒱-enriched (,1)-categories.

Definition

For n write

Cat (,n)(((GrpdCat)Cat))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 -internalization

General

There is a general notion of internal category in an (∞,1)-category. For 𝒞 an (∞,1)-category write Cat(𝒞) for the (,1)-category of internal categories in 𝒞. Write Cat (,1) ∞Grpd for the (,1)-category of ∞-groupoids.

Definition

For n𝒩 the (,1)-category of (,n)-categories is

Cat (,n)Cat n(Grpd)Cat(Cat(Cat (,0))).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 Cat n(Grpd) is given by n-fold complete Segal spaces.

Proposition

The n-fold product of the simplex category is naturally a full subcategory Δ×nStrnCat gen and the localization that gives the (,1)-category CSS(Δ ×n) of n-fold complete Segal space satisfies the conditions of prop. 2.

Hence n-fold complete Segal space are a model for (,n)-categories, in the sense of def. 5:

Θ nSpaceCat (,n).\Theta_n Space \simeq Cat_{(\infty,n)} \,.

(B-SP, theorem 12.6)

Properties

Uniqueness and equivalences

Proposition

Let Models (,n)Cat^ (,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

Models (,n)B( 2) n,Models_{(\infty,n)} \simeq B (\mathbb{Z}_2)^n,

the delooping groupoid of the group ( 2) n, the n-fold product of the group of order 2 with itself.

The nontrivial element σ 2 in the kth slot acts by passing to the k-opposite (,n)-category.

(B-SP, theorem 8.13)

Remark

This means that

  1. the (,1)-category Cat (,n) from def. 5 is uniquely defined, up to equivalence of (∞,1)-categories;

  2. the automorphism ∞-group of Cat (,n) in Cat^ (,1) is ( 2) n, hence the only auto-equivalences are given by reversal of k-morphisms.

Examples

Special cases

In addition,

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

Specific examples

References

The definition in terms of Theta spaces is due to

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

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

An axiomatic characterization is in

Comparison of models is in