nLab cubical T-complex

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Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Definition

A cubical TT-complex is a cubical set KK which is also a special kind of Kan complex, in that it has a family T nK nT_n \subseteq K_n (for n1n \geq 1) of thin elements with the properties, (due first to Keith Dakin, for the simplicial case, in his doctoral thesis):

  • T1) Degenerate elements are thin.

  • T2) Every box in KK has a unique thin filler.

  • T2) If all but one face of a thin element are thin, then so also is the remaining face.

Cubical T-complexes are equivalent to crossed complexes, and also to cubical omega-groupoid?s with connections.

A modification of axiom T2) to be more like an axiom for a quasi-category is suggested in the paper by Steiner referenced below.

References

T-complexes first appeared in:

  • M.K. Dakin, Kan complexes and multiple groupoid structures , PhD Thesis, University of Wales, Bangor, (1976).

A main application of the idea of thin elements in the cubical case was to define the notion of commutative cube. In the case of cubical ω\omega-groupoids with connection, the boundary of a cube is commutative if and only if the boundary has a thin filler. One consequence is that any well defined composition of thin elements is thin. This is a key element of the proof of the Higher Homotopy van Kampen theorem in the colimits paper, which thus nicely generalises the proof of the 1-dimensional van Kampen theorem for the fundamental groupoid on a set of base points.

Cubical T-complexes are thus a key concept in the long series of articles by Brown and Higgins on strict omega-groupoids. For instance

  • Ronnie Brown, P.J. Higgins, The equivalence of ω\omega-groupoids and cubical TT-complexes, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 22 no. 4 (1981), p. 349-370 (pdf).

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

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

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

  • R. Brown, R. Street, Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product, Cah. Top. G'eom. Diff. Cat., 52 (2011) 188-208.

and generally

  • Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol 15 (2011). (web).

For more on ‘thinness’ in a cubical context, see:

  • P.J. Higgins, Thin elements and commutative shells in cubical ω\omega-categories, Theory Appl. Categ., 14 (2005) 60–74.

  • R. Steiner, Thin fillers in the cubical nerves of omega-categories, Theory Appl. Categ. 16 (2006), 144–173.

Last revised on June 8, 2012 at 18:51:58. See the history of this page for a list of all contributions to it.