R. W. Kieboom

A Homotopy 2-Groupoid of a Hausdorff Space

If X is a Hausdorff space we construct a 2-groupoid G 2 X with the following properties. The underlying category of G 2 X is the ‘path groupoid’ of X whose objects are the points of X and whose morphisms are equivalence classes , of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G 2 X is a quotient groupoid Π X/NX,where ΠX is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. NX is a normal subgroupoid of fIX determined by the thin relative homotopies. There is an isomorphism G 2 X ( , ) ≈ π2(X, f(0)) between the 2-endomorphism group of and the second homotopy group of X based at the initial point of the path f. The 2groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.

A Homotopy Bigroupoid of a Topological Space

In this paper we give an explicit description of a homotopy bigroupoid of a topological space as a 2-dimensional structure in homotopy theory which allows one to derive some basic properties in 2-dimensional homotopical algebra using purely algebraic arguments. The main results are valid in the general setting of a bigroupoid.

Fibrations of bigroupoids

Abstract Using Browns construction (J. Algebra 15 (1970) 103) of an exact 6-term sequence for a fibration of groupoids we show how an exact 9-term sequence can be associated to a fibration of bigroupoids. Applications to topology and algebra are given.

A pullback theorem for cofibrations

In this note we prove a pullback theorem for cofibrations, which extends a well known theorem of Strøm [5]. It also implies the pullback theorem of Heath [4] for locally equiconnected spaces. In addition, we comment on the dual problem of attaching fibrations.

Dold Type Theorems in Cubical Homotopy Theory

It is shown how a generalised comparison theorem for topological spaces under a fixed space A and over a fixed space B due to the third author can be transferred to cubical homotopy theory.

Stable Homotopy Monomorphisms

It is well known that the concept of monomorphism in a category can be defined using an appropriate pullback diagram. In the homotopy category of TOP pullbacks do not generally exist. This motivated Michael Mather to introduce another notion of homotopy pullback which does exist. The aim of this paper is to investigate the modified notion of homotopy monomorphism obtained by applying the pullback characterization using Mathers homotopy pullback.The main result of Section 1 shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology “stable” homotopy monomorphism. We also link these stable homotopy monomorphisms to monomorphisms and products in the track homotopy category over a fixed space. In Section 2 we answer the question: when is a (weak) fibration also a stable homotopy monomorphism? In the final section it is shown that the class of (weak) fibrations with this additional property coincides with the class of “double” (weak) fibrations. The double (weak) covering homotopy property being introduced here is a stronger version of the (W) CHP in which the final maps of the homotopies involved play the same role as the initial maps.