nLab simplicial T-complex

Idea

A simplicial T-complex is a Kan complex equipped with a choice of horn fillers, which satisfy certain ‘equations’. It is thus a special case of an algebraic Kan complex.

There is quite a difference between the Kan complex structure

1. of the nerve $N(\mathcal{G})$ of a groupoid, $\mathcal{G}$,

2. and that of, say, a singular complex $Sing X$ of some topological space $X$.

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; however the fact that this simplicial set is Kan is a property of retractions on standard simplices, and is not specifically a property of the space $X$ - that is the basic intuition.

Abstracting, in part, from this idea, Brown and Higgins developed the idea of a cubical T-complex. This was a cubical 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 initiated in a simplicial context in the 1977 Bangor thesis of Keith Dakin listed below, and used by Brown and Higgins who showed that cubical $T$-complexes were equivalent to crossed complexes. The corresponding simplicial $T$-complex theory was further developed in the 1978 Bangor thesis of Nick Ashley, (see below for publication).

Simplicial $T$-complex: the definition

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

Examples

1. A closely related idea is that of group T-complex. Group $T$-complexes form a category equivalent to reduced crossed complexes. Any group $T$-complex has an underlying simplicial set, which is a simplicial $T$-complex.

2. The nerve of a crossed complex has a natural T-complex structure. In a bit more detail, if $\mathsf{C}$ is a crossed complex, its nerve is given by $Ner(\mathsf{C})_n = Crs(\pi(n),\mathsf{C})$, where $\pi(n)$ is the free crossed complex on the $n$-simplex, $\Delta[n]$. This singular complex description shows that if we have an $n$-simplex $f : \pi(n) \to \mathsf{C}$, and declare it to be \emph{thin} if the image $f(\iota_n)$ of the top dimensional generator in $\pi(n)$ is trivial, then the resulting collection of thin simplices determines a $T$- complex structure on the nerve.

Results

• Simplical $T$-complexes together with maps between them which preserve ‘thinness’ form a category that is equivalent to that of crossed complexes and thus to strict $\infty$-groupoids. The uniqueness of the thin filler is exactly what gives a definite composition in the model.

Related concepts

• T-complex

  • simplicial $T$-complex

  • cubical T-complex

  • group T-complex

• algebraic Kan complex

Together with very similar ideas of John Roberts, adapted by Ross Street, the notion of $T$-complex is one of the precursors of Dominic Verity‘s notion of complicial set.

They are also related to Jack Duskin‘s notion of hypergroupoid. (The connection is explored in the papers by Nan Tie listed below.)

References

Relevant references for simplicial T-complexes include:

• Ronnie Brown, An Introduction to simplicial T-complexes, Esquisses Math. 32 (1983) Part 1

• M.K. Dakin, Kan complexes and multiple groupoid structures, Ph.D Thesis, University of Wales, Bangor, 1977. Esquisses Math. (1983) 32 Part 2

• N. Ashley, Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan, Ph.D Thesis University of Wales, Bangor, 1978; Dissertationes Math., 165, (1989), 11 – 58. Esquisses Math. (1983) 32 Part 3

• G. Nan Tie, A Dold-Kan theorem for crossed complexes, J. Pure Appl. Alg., 56, (1989.), 177 -– 194.

• G. Nan Tie, Iterated W and T-groupoids, J. Pure Appl. Alg., 56, (1989), 195 -– 209.

• Ronnie Brown, and P.J. Higgins, On the algebra of cubes

  J. Pure Appl. Algebra 21 (1981) 233–260.

• R. Brown, and P.J. Higgins, The equivalence of $\omega$-groupoids and cubical $T$-complexes Cahiers Topologie G'eom. Diff'erentielle 22 (1981) 349–370.