Keith A. Hardie

BIFRAMES AND BISPACES

Abstract The concept of a biframe is introduced. Then the known dual adjunction between topological spaces and frames (i.e. local lattices) is extended to one between bispaces (i.e. bitopological spaces) and biframes. The largest duality contained in this dual adjunction defines the sober bispaces, which are also characterized in terms of the sober spaces. The topological and the frame-theoretic concepts of regularity, complete regularity and compactness are extended to bispaces and biframes respectively. For the bispaces these concepts are found to coincide with those introduced earlier by J.C. Kelly, E.P. Lane, S. Salbany and others. The Stone-Cech compactification (compact regular coreflection) of a biframe is constructed without the Axiom of Choice.

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.

TODA BRACKETS AND THE CATEGORY OF HOMOTOPY PAIRS

A new treatment is given of the cylinder-web diagram and associated diagonal sequences in homotopy pair theory. The efficiency of the diagram as a machine for computing homotopy pair groups is enhanced by a result that traces the path of a Toda bracket element through the arrows of the diagram. The diagonal factorization problem for a homotopy pair class is studied and related to the behaviour of Toda brackets. A necessary and sufficient condition for the vanishing of a Toda bracket is obtained.

The Whitehead Square of the 6-point 2-sphere

We describe a model of the Whitehead square [ι 2, ι 2] ∈ π3(S 2) in the form of an order-preserving map from a 56-point model of S 3 into the minimal model of S 2 in the category of finite posets. The simplicity of the model enables the map to be visualised.

A nontrivial pairing of finite T0 spaces

Abstract There is a four-point space S 1 weakly homotopy equivalent to the circle. The restriction to S 1 of the complex number multiplication is not continuous, nevertheless a continuous model of the multiplication with values in S 1 can be defined on an eight-point circle. Applying an analogue of Hopfs construction we obtain a finite model of Hopfs famous map S3→S2.

COMPUTING HOMOTOPY GROUPS OF A HOMOTOPY PULLBACK

Abstract Secondary structure of an exact sequence of Maya-Vietoris type associated to a simply-connected homotopy pullback is exploited to yield a technique for computing its higher homotopy groups. The information required consists of the homotopy groups of the original spaces, the homomorphisms induced by the given maps and certain matrix Toda brackets.

BIFUNLTORS AND THE DIAGONAL EXACT SEQUENCES OF A CYLINDER-WEB DIAGRAM

The doubly infinite diagram of exact sequences that an additive bifunctor T associates with a pair of short exact sequences can be regarded as a web diagram lying on the surface of a cylinder. The diagram has six diagonal sequences involving two graded derived functors that arise through the failure of T to preserve pull-backs respectively push-outs. In the case T = Hom(-,-) one of the diagonal sequences is equivalent to the bivariant Hom-Ext sequence studied by Pressmann [4].

Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.

A non-Hausdorff quaternion multiplication

We denote by (S3)? the barycentric subdivision of the minimal model S3 of the three-dimensional sphere in the category of finite posets and order-preserving functions, op(X) is the poset obtained by reversing the order relations in a poset X. We describe a finite model of a quaternion multiplication in the form of a morphism op(S3)?×(S3)??S3 that restricts to weak homotopy equivalences on the axes. For such multiplications a version of Hopfs construction can be defined that yields finite models of non-trivial homotopy classes.