nLab cubical T-complex



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



Paths and cylinders

Homotopy groups

Basic facts


Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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.


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.