nLab
strict omega-category

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Higher category theory

higher category theory

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Basic theorems

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Models

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Extra properties and structure

1-categorical presentations

Contents

Idea

A strict ω\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 kk-morphisms are strict identities under all compositions.

Definition

An ω\omega-category CC internal to SetsSets is

  • a globular set

    C:=(C 3C 2C 1C 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 kC l)( C_{k} \stackrel{\to}{\to} C_l ) for all k>lk \gt l;

  • such that (C kC lC m)( C_{k} \stackrel{\to}{\to} C_{l} \stackrel{\to}{\to} C_m ) for all k>l>mk \gt l \gt m; is a strict 2-category.

Similarly for an ω\omega-category internal to another ambient category AA.

The category ωCat(A)\omega Cat(A) of ω\omega-categories internal to AA has ω\omega-categories as its objects and morphism of 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.

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

    (0Cat=Set)(1Cat=SetCat)(2Cat=CatCat)(3Cat=(2Cat)Cat=(CatCat)Cat). (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 ω\omega-categories admits a biclosed monoidal structure called the Crans-Gray tensor product.

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

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

    • Street allowed ω\omega-categories to have infinite dimensional cells. Steiner has the extra condition that every cell has to be finite dimensional, and called them \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 ω\omega-categories, and I agree with Verity’s suggestion to call the other ones ω +\omega^+-categories.
  • 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.

As simplicial sets

Under the ω-nerve

N:StrωCatSSet N : Str \omega Cat \to SSet

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

Proposition

The categories of ω\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.

References

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

  • Ronnie Brown and P.J. Higgins, The equivalence of \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 “\infty-categories” while “ω\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 ω\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 ω\omega-categories are called “ω\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 ω\omega-categories was further developed by Sjoerd Crans in parts 2 and 3 of his thesis

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

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

See also the

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

The relationship between strict ω\omega-categories and cubical ω\omega-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 ω\omega-categories are equivalent to ω\omega-fold categories (aka “cubical ω\omega-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
Revised on May 29, 2012 22:10:59 by Zoran Škoda (193.51.104.33)