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K.A. Hardie

Other affiliations: University of the Western Cape
Bio: K.A. Hardie is an academic researcher from University of Cape Town. The author has contributed to research in topics: Homotopy & Homotopy category. The author has an hindex of 10, co-authored 45 publications receiving 347 citations. Previous affiliations of K.A. Hardie include University of the Western Cape.

Papers
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Journal ArticleDOI
TL;DR: In this paper, the concept of a biframe is introduced and the known dual adjunction between topological spaces and frames is extended to one between bispaces (i.e. bitopological spaces) and biframes.
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.

70 citations

Journal ArticleDOI
TL;DR: This paper gives 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.
Abstract: 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.

43 citations

Book ChapterDOI
TL;DR: In this article, a 2-groupoid G 2 X with the following properties is constructed: if X is a Hausdorff space, the underlying category 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.
Abstract: 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.

42 citations

Journal Article
TL;DR: In this article, the authors associate a Hausdorff space with a double groupoid, ρ 2 (X), based on the notion of thin square, and show how to construct a homotopy 2-groupoid with connection given in (BM).
Abstract: We associate to a Hausdorff space, X, a double groupoid, ρ 2 (X), the homotopy double groupoid of X. The construction is based on the geometric notion of thin square. Under the equivalence of categories between small 2-categories and double categories with connection given in (BM) the homotopy double groupoid corresponds to the homotopy 2-groupoid, G2(X), constructed in (HKK). The cubical nature of ρ 2 (X )a s opposed to the globular nature of G2(X) should provide a convenient tool when handling 'local-to-global' problems as encountered in a generalised van Kampen theorem and dealing with tensor products and enrichments of the category of compactly generated Hausdorff spaces.

38 citations

Journal ArticleDOI

19 citations


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Book
01 Jan 1962
TL;DR: In this paper, Compositional Methods in Homotopy Groups of Spheres (AM-49) have been described, and a detailed description of the methods can be found in the Appendix.
Abstract: The description for this book, Compositional Methods in Homotopy Groups of Spheres. (AM-49), will be forthcoming.

309 citations

Posted Content
TL;DR: A detailed introduction to weak and coherent 2-groups can be found in this paper, where weak 2-group is defined as a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x tensor y and y tensor x are isomorphic to 1.
Abstract: A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x tensor y and y tensor x are isomorphic to 1. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse x* and isomorphisms i_x: 1 -> x tensor x* and e_x: x* tensor x -> 1 forming an adjunction. We describe 2-categories of weak and coherent 2-groups and an "improvement" 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories. We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups. We give a tour of examples, including the "fundamental 2-group" of a space and various Lie 2-groups. We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G_hbar (for integral values of hbar) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2-groups are built using Chern-Simons theory, and are closely related to the Lie 2-algebras g_hbar (for real hbar) described in a companion paper.

285 citations

Book
27 Aug 2011
TL;DR: The Andrews-Curtis conjecture and the Lefschetz number of finite spaces were studied in this article, where they were shown to be a conjecture of Quillen.
Abstract: 1 Preliminaries.- 2 Basic topological properties of finite spaces.- 3 Minimal finite models.- 4 Simple homotopy types and finite spaces.- 5 Strong homotopy types.- 6 Methods of reduction.- 7 h-regular complexes and quotients.- 8 Group actions and a conjecture of Quillen.- 9 Reduced lattices.- 10 Fixed points and the Lefschetz number.- 11 The Andrews-Curtis conjecture.

182 citations

Book ChapterDOI
01 Jan 2001
TL;DR: The historic development of what is now often called “Nonsymmetric or Asymmetric Topology” is summarized in Section 2 and in the following, more specific sections the authors discuss thehistoric development of some of the main ideas of the area in greater detail.
Abstract: We begin with some remarks explaining the structure of this article. After some introductory statements in the following paragraphs, we summarize the historic development of what is now often called “Nonsymmetric or Asymmetric Topology” in Section 2. In the following, more specific sections we discuss the historic development of some of the main ideas of the area in greater detail. The list of sections and keywords given above should help the specialist to find his way through the various sections.

178 citations

01 Jan 2002
TL;DR: In this paper, the authors give a complete characterization of simplicial sets which are the nerves of bicategories as certain 2-dimensional Postnikov complexes which satisfy certain restricted horn-lifting conditions whose satisfaction is controlled by subsets of (abstractly) invertible 2 and 1-simplices.
Abstract: To a bicategory B (in the sense of Benabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2-Coskeleton and itself isomorphic to its 3-Coskeleton, what we call a 2-dimensional Postnikov complex. We then give, somewhat more delicately, a complete characterization of those simplicial sets which are the nerves of bicategories as certain 2-dimensional Postnikov complexes which satisfy certain restricted “exact horn-lifting” conditions whose satisfaction is controlled by (and here defines) subsets of (abstractly) invertible 2 and 1-simplices. Those complexes which have, at minimum, their degenerate 2-simplices always invertible and have an invertible 2-simplex χ2(x12, x01) present for each “composable pair” (x12, , x01) ∈ ∧1 2 are exactly the nerves of bicategories. At the other extreme, where all 2 and 1-simplices are invertible, are those Kan complexes in which the Kan conditions are satisfied exactly in all dimensions > 2. These are exactly the nerves of bigroupoids–all 2-cells are isomorphisms and all 1-cells are equivalences.

102 citations