Higher category theory
higher category theory
Extra properties and structure
A strict -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
all composition operations are strictly associative;
all composition operations strictly commute with all others (strict exchange laws);
all identity -morphisms are strict identities under all compositions.
An -category internal to is
a globular set
together with the structure of a category on all for all ;
such that for all ; is a strict 2-category.
Similarly for an -category internal to another ambient category .
The category of -categories internal to has -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.
-Categories can also be understood as the directed limit of the sequence of iterated enrichements
The category of strict -categories admits a biclosed monoidal structure called the Crans-Gray tensor product.
The category of strict -categories also admits a canonical model structure.
Terminology on -categories varies. We here follow section 2.2 of Sjoerd Crans: Pasting presentations for -categories, where it says
- Street allowed -categories to have infinite dimensional cells. Steiner has the extra condition that every cell has to be finite dimensional, and called them -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 -categories, and I agree with Verity’s suggestion to call the other ones -categories.
Simpson's conjecture, a statement about semi-strictness, states that every weak -category should be equivalent to an -category in which strictness conditions 1. and 2. hold, but not 3.
As simplicial sets
Under the ω-nerve
strict -categories yield simplicial sets that are called complicial sets.
The categories of -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.
Strict -categories have probably been independently invented by several people. Possibly the earliest definition was in
- Ronnie Brown and P.J. Higgins, The equivalence of -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 ”-categories” while ”-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 -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 -categories are called ”-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 -categories was further developed by Sjoerd Crans in parts 2 and 3 of his thesis
Sjoerd Crans, Pasting presentations for -categories (link)
Sjoerd Crans, Pasting schemes for the monoidal biclosed structure on -Cat (link)
See also the
to his thesis, in particular section I.3 ”-categories”.
The relationship between strict -categories and cubical -categories was considered in
- F.A. Al-Agl, R. Brown, 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 -categories are equivalent to -fold categories (aka “cubical -categories”) equipped with connections. This paper also develops the monoidal closed structures.
- R. Steiner, Omega-categories and chain complexes, Homology, Homotopy and Applications 6(1), 2004, pp.175–200, pdf