nLab simplicial T-complex (Rev #5, changes)

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Idea

There is quite a difference between the Kan complex structure of the nerve of a groupoid, $G$, and that of, say, a singular complex. In the first, if we are given a $(n,i)$-horn, then there is exactly one $n$-simplex in $Ner(G)$, since the $(n,i)$-horn has a chain of $n$-composable arrows of $G$ in it (at least unless $(n,i) = (2,0)$ or $(2,2)$, which cases are slightly different) and that chain gives the required $n$-simplex. In other words, there is a ‘canonical’ filler for any horn. In $Sing(X)$, there will usually be many fillers.

Abstracting from this idea, Brown and Higgins developed the idea of a cubical $T$T-complex -complex. . This was acubical set with in each dimension $n$, a subset of the $n$-cubes being declared ‘thin’. The term was adopted to indicate that they, somehow, were of lower dimension than they looked to be. The theory was initially developed in the thesis of Keith Dakin, and used by Brown and Higgins who showed that cubical $T$-complexes were equivalent to crossed complexes. The corresponding simplicial theory was developed in another thesis, this time by Nick Ashley.

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 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.

Definition

A simplicial $T$-complex is a pair $(K,T)$, where $K$ is a simplicial set and $T = (T_n)_{n\geq 1}$ is a graded subset of $K$ with $T_n\subseteq K_n$. Elements of $T$ are called thin. The thin structure satisfies the following axioms:

• Every degenerate element is thin.
• Every horn in $K$ has a unique thin filler.
• A thin filler of a thin box also has its last face thin.

Remark

1. Together with very similar ideas of John Roberts, adapted by Ross Street, the notion of $T$-complex is one of the precursors of Verity’s notion of complicial set.

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

• R. 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).

• R. 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.

and generally

• Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian algebraic topology (web).

A closely related idea is that of group T - complex.

See also:

• 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.