nLab
strict omega-category

Idea

A strict $\omega$-category is a globular ∞-category in which all operations obey their respective laws strictly.

This was the original notion of ∞-category, and the original meaning of the term ω-category. Even today, most authors who use that term still mean this notion.

This means that

1. all composition operations are strictly associative;

2. all composition operations strictly commute with all others (strict exchange laws);

3. all identity $k$-morphisms are strict identities under all compositions.

Definition

An $\omega$-category $C$ internal to $\mathrm{Sets}$ is

• a globular set

  $$C:=(\cdots C_3\overrightarrow{\to }{C}_2\overrightarrow{\to }{C}_1\overrightarrow{\to }{C}_0)$$C := (\cdots C_3 \stackrel{\to}{\to} C_2 \stackrel{\to}{\to} C_1 \stackrel{\to}{\to} C_0 )

• together with the structure of a category on all $(C_k\overrightarrow{\to }{C}_l)$ for all $k>l$;

• such that $(C_k\overrightarrow{\to }{C}_l\overrightarrow{\to }{C}_m)$ for all $k>l>m$; is a strict 2-category.

Similarly for an $\omega$-category internal to another ambient category $A$.

The category $\omega \mathrm{Cat}(A)$ of $\omega$-categories internal to $A$ has $\omega$-categories as its objects and morphism og the underlying globular objects respecting all the above extra structure as morphisms.

Remarks

• The last condition in the above definition says that all pairs of composition operations satisfy the exchange law.

• $\omega$-Categories can also be understood as the directed limit of the sequence of iterated enrichements

  $$(0\mathrm{Cat}=\mathrm{Set})\hookrightarrow (1\mathrm{Cat}=\mathrm{Set}-\mathrm{Cat})\hookrightarrow (2\mathrm{Cat}=\mathrm{Cat}-\mathrm{Cat})\hookrightarrow (3\mathrm{Cat}=(2\mathrm{Cat})-\mathrm{Cat}=(\mathrm{Cat}-\mathrm{Cat})-\mathrm{Cat})\hookrightarrow \cdots .$$(0 Cat = Set) \hookrightarrow (1 Cat = Set-Cat) \hookrightarrow (2 Cat = Cat-Cat) \hookrightarrow \left(3 Cat = (2Cat)-Cat = (Cat-Cat)-Cat\right) \hookrightarrow \cdots \,.

• The category of strict $\omega$-categories admits a biclosed monoidal structure called the Crans-Gray tensor product.

• The category of strict $\omega$-categories also admits a canonical model structure.

• Terminology on $\omega$-categories varies. We here follow section 2.2 of Sjoerd Crans: Pasting presentations for $\omega$-categories, where it says

  • Street allowed $\omega$-categories to have infinite dimensional cells. Steiner has the extra condition that every cell has to be finite dimensional, and called them $\mathrm{\infty }$-categories, following Brown and Higgins. I will use Steiner’s approach here since that’s the one that reflects the notion of higher dimensional homotopies closest, but I will nonethless call them $\omega$-categories, and I agree with Verity’s suggestion to call the other ones ${\omega }^{+}$-categories.

• Simpson's conjecture, a statement about semi-strictness, states that every weak $\mathrm{\infty }$-category should be equivalent to an $\mathrm{\infty }$-category in which strictness conditions 1. and 2. hold, but not 3.

As simplicial sets

Under the ω-nerve

$$N:\mathrm{Str}\omega \mathrm{Cat}\to \mathrm{SSet}$$N : Str \omega Cat \to SSet

strict $\omega$-categories yield simplicial sets that are called complicial sets.

Proposition

The categories of $\omega$-categories and complicial sets are equivalent.

This is sometimes called the Street-Roberts conjecture. It was completely proven in

• Dominic Verity, Complicial sets (arXiv)

which also presents the history of the conjecture.

Based on this fact, there are attempts to weaken the condition on a simplicial set to be a complicial set so as to obtain a notion of simplicial weak ω-category.

Literature

Strict $\omega$-categories have probably been independently invented by several people. Possibly the earliest definition was in

• R. Brown and P.J. Higgins, The equivalence of $\mathrm{\infty }$-groupoids and crossed complexes, Cah. Top. Géom. Diff. 22 (1981) no. 4, 371-386 web.

which also contains the definitions of n-fold category and of what was later called globular set. There these strict, globular higher categories are called ”$\mathrm{\infty }$-categories” while ”$\omega$-groupoid” is used to mean a cubical set with connections and compositions, each a groupoid, as in

• R. Brown and P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981) 233-260.

Applications to homotopy theory were given in

• R. Brown and P.J. Higgins, Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981) 11-41.

• R. Brown, Non-abelian cohomology and the homotopy classification of maps, in Homotopie algébrique et algebre locale, Conf. Marseille-Luminy 1982, ed. J.-M. Lemaire et J.-C. Thomas, Astérisques 113-114 (1984), 167-172.

Related monoidal closed structures were developed in:

• R. Brown and P.J. Higgins, Tensor products and homotopies for $\omega$-groupoids and crossed complexes, J. Pure Appl. Alg. 47 (1987) 1-33.

Another 1980s reference is

• Ross Street, The algebra of oriented simplices, J. Pure Appl. Algebra 49 (1987) 283-335; MR89a:18019 (pdf),

in which strict $\omega$-categories are called ”$\omega$-categories.” This paper was also the first to define orientals.

A review of some of the theory in the context of some of the history is given in

• Ross Street, An Australian conspectus of higher categories (pdf)

and also in

• Ross Street, Categorical and combinatorial aspects of descent theory (arXiv)

The theory of $\omega$-categories was further developed by Sjoerd Crans in parts 2 and 3 of his thesis

• Sjoerd Crans, Pasting presentations for $\omega$-categories (link)

• Sjoerd Crans, Pasting schemes for the monoidal biclosed structure on $\omega$-Cat (link)

See also the

• Introduction

to his thesis, in particular section I.3 ”$\omega$-categories”.

The relationship between strict $\omega$-categories and cubical $\omega$-categories was considered in

• F.A. Al-Agl, R. Brown, and R. Steiner Multiple categories: the equivalence of a globular and a cubical approach, Adv. Math. 170 (2002), no. 1, 71–118

where they prove that strict globular $\omega$-categories are equivalent to $\omega$-fold categories (aka “cubical $\omega$-categories”) equipped with connections. This paper also develops the monoidal closed structures.