T-complexes and crossed complexes. (English) Zbl 0558.55015

Ashley, N.

School of Mathematics and Computer Science, University College of North Wales. 76 p. (1978).

Let \(\mathbf{X}: X_0\subset X_1\subset\cdots \subset X_n\subset\cdots \subset X\) be a filtered space. The homotopy (or fundamental) crossed complex \(\pi\mathbf{X}\) of \(\mathbf{X}\) consists of the fundamental groupoid \(\pi_1\underline X=\pi_1(X_1,X_0)\), the family \(\pi_n\underline X = \{\pi_n(X_n,X_{n-1},p): p\in X_0\}\) of relative homotopy groups, for \(n\geq 2\), together with the usual boundaries \(\delta: \pi_n\underline X\to \pi_{n-1} \underline X\) and the operations of \(\pi_1\underline X\) on \(\pi_n\underline X\), \(n\geq 2\). A crossed complex \(C\) consists of a similar structure, the laws being those universally satisfied by the basic example \(\pi\mathbf{X}\). Such a structure expresses many standard facts about relative homotopy theory, and was first considered in the reduced case \((X_0\) a point) by A. L. Blakers [Ann. Math. (2) 49, 428–461 (1948; Zbl 0040.25701)] under the name group system. Crossed complexes have since been used in a number of areas of mathematics, see the reviewer’s article in [Categorical topology, Proc. int. Conf. Toledo/Ohio 1983, Sigma Ser. Pure Math. 5, 108–146 (1984; Zbl 0558.55001)] reviewed above.
The main purpose of this thesis is to prove a non-abelian version of the Dold-Kan theorem which gives an equivalence between chain complexes and simplicial abelian groups. The new equivalence is between crossed complexes and simplicial \(T\)-complexes. A functor \(D: Crs \to SS\), from crossed complexes to simplicial sets, was defined by Blakers (in the reduced case). It is essentially given by \((DC)_n)=\underline{Crs}(\pi \underline {\Delta}^n,C)\), where \(\underline{\Delta}^n\) is the standard \(n\)-simplex with its skeletal filtration, and \(DC\) should be thought of as the nerve of \(C\). Indeed, if \(C_n\) is trivial for \(n\geq 2\), then \(DC\) is just the usual nerve of the groupoid \(C_1\). However, \(DC\) has extra structure, first envisaged by M. K. Dakin in his 1977 University of Wales Ph. D. Thesis.
A (simplicial) \(T\)-complex \((K,T)\) consists of a simplicial set \(K\) together with a family of sets \(T_n\subset K_n\), \(n\geq 1\). The elements of \(T_n\) are called thin. These must satisfy Keith Dakin’s axioms: T1) Degenerate elements are thin. T2) Any horn in \(K\) has a unique thin filter. T3) If all faces except possibly one of a thin element are thin, then so also is the remaining face. A \(T\)-complex is of rank \(\leq n\) if \(T_m=K_m\) for \(m>n\). Dakin proved that the functor \(D\) gives an equivalence of categories between groupoids and \(T\)-complexes of rank 2 with one vertex. He conjectured that \(D\) gives an equivalence between crossed complexes and \(T\)-complexes. It is this result that is proved by Ashley in Chapter 1, the first 45 pages of his thesis. It is quite a technical achievement.
Chapter 2 of Ashley’s thesis explores simplicial analogues of cubical results of the reviewer and P. J. Higgins [J. Pure Appl. Algebra 22, 11–41 (1981; Zbl 0475.55009)], and in particular the geometrical definition of a functor \[\rho\colon \text {(filtered spaces)}\to (T\text{-complexes)} \] such that \(D\pi\cong \rho\). Chapter 3 has some interesting results on simplicial groups. For example, a simplicial abelian group is a \(T\)-complex if thin elements are defined to be sums of degenerate elements. If \(S_n\subset K_n\) consists of products of degenerate elements in the simplicial group \(K\), then the family \((S_n)_{n\geq 1}\) need not satisfy the \(T\)-complex axioms – conditions are given for this to occur. Chapter 4 has minor results.
{It is remarkable to the reviewer how much algebra is generated by Dakin’s innocent seeming axioms. His idea of \(T\)-complex played a crucial role in its cubical version in work of Brown-Higgins [loc. cit. and Cah. Topol. Géom. Différ. 22, 349–370 (1981; Zbl 0486.55011)]. There are a number of problems still open. For example, Brown-Higgins (U.C.N.W., Pure Math. Preprint 85.1) have recently described on the category Crs a monoidal closed structure appropriate for discussing homotopies and higher homotopies. The corresponding notion for simplicial \(T\)-complexes has not been described in detail. The notion of \(T\)-complex is related to that of hypergroupoid due to Duskin and Lawvere, and discussed by P. Glenn in [J. Pure Appl. Algebra 25, 33–105 (1982; Zbl 0487.18015)]}.

Reviewer: Ronald Brown (Bangor)

MSC:

55U10	Simplicial sets and complexes in algebraic topology
55Q20	Homotopy groups of wedges, joins, and simple spaces
18G50	Nonabelian homological algebra (category-theoretic aspects)
18G55	Nonabelian homotopical algebra (MSC2010)
18G30	Simplicial sets; simplicial objects in a category (MSC2010)

Keywords:

homotopy crossed complex; filtered space; fundamental groupoid; relative homotopy groups; non-abelian version of the Dold-Kan theorem; simplicial T-complexes; simplicial groups; hypergoupoid

Citations:

Zbl 0040.25701; Zbl 0475.55009; Zbl 0486.55011; Zbl 0487.18015; Zbl 0558.55001