nLab cubical T-complex

Definition

A cubical $T$-complex is a cubical set $K$ which is also a special kind of Kan complex, in that it has a family $T_n \subseteq K_n$ (for $n \geq 1$) 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 $K$ 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.

Related concepts

• T-complex

  • simplicial T-complex

  • cubical $T$-complex

• algebraic Kan complex

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 $\omega$-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 $\omega$-groupoids and cubical $T$-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 $\omega$-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 $\omega$-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.