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 ∞-category. Even today, most authors who use that term still mean this notion.
This means that
all composition operations are strictly associative;
all composition operations strictly commute with all others (strict exchange laws);
all identity $k$-morphisms are strict identities under all compositions.
An $\omega$-category $C$ internal to $Sets$ is
together with the structure of a category on all $( C_{k} \stackrel{\to}{\to} C_l )$ for all $k \gt l$;
such that $( C_{k} \stackrel{\to}{\to} C_{l} \stackrel{\to}{\to} C_m )$ for all $k \gt l \gt m$; is a strict 2-category.
Similarly for an $\omega$-category internal to another ambient category $A$.
The category $\omega Cat(A)$ of $\omega$-categories internal to $A$ has $\omega$-categories as its objects and morphism of the underlying globular objects respecting all the above extra structure as morphisms.
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 enrichments
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
Simpson's conjecture, a statement about semi-strictness, states that every weak $\infty$-category should be equivalent to an $\infty$-category in which strictness conditions 1. and 2. hold, but not 3.
Under the ∞-nerve
strict $\omega$-categories yield simplicial sets that are called complicial sets.
The categories of $\omega$-categories and complicial sets are equivalent.
This is sometimes called the Street-Roberts conjecture. It was completely proven in
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 ∞-category.
Strict $\omega$-categories have probably been independently invented by several people.
According to Street 09, p. 10 the concept was first brought up in
in an attempt to define non-abelian cohomology (of local nets of observables in algebraic quantum field theory).
Possibly the earliest published definition is due to
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 “$\infty$-categories” while “$\omega$-groupoid” is used to mean a cubical set with connections and compositions, each a groupoid, as in
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:
Another 1980s reference is
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
and also in
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
to his thesis, in particular section I.3 “$\omega$-categories”.
The relationship between strict $\omega$-categories and cubical $\omega$-categories was considered in
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.
A direct proof that the category of strict $\omega$-categories is monadic over the category of polygraphs is given in:
Last revised on June 6, 2023 at 11:48:39. See the history of this page for a list of all contributions to it.