Every topological category is convenient for Gelfand duality

The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman’s proof of the disc embedding theorem is given, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem’s key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures.

Homotopy Theory of Diagrams and CW-Complexes Over a Category

Canadian Journal of Mathematics ◽

10.4153/cjm-1991-046-3 ◽

1991 ◽

Vol 43 (4) ◽

pp. 814-824 ◽

Cited By ~ 8

Author(s):

Robert J. Piacenza

Keyword(s):

Classical Theory ◽

Homotopy Theory ◽

Homotopy Groups ◽

Weak Equivalence ◽

Equivariant Homotopy ◽

Topological Category ◽

Sheaf Theory ◽

Closed Model ◽

Base Category ◽

Closed Model Category

The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

Constructive Gelfand duality for C*-algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004109002515 ◽

2009 ◽

Vol 147 (2) ◽

pp. 339-344 ◽

Cited By ~ 8

Author(s):

THIERRY COQUAND ◽

BAS SPITTERS

Keyword(s):

Constructive Proof ◽

C Algebras ◽

Gelfand Duality

AbstractWe present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras.

A Gelfand duality for compact pospaces

Algebra Universalis ◽

10.1007/s00012-018-0519-7 ◽

2018 ◽

Vol 79 (2) ◽

Author(s):

Laurent De Rudder ◽

Georges Hansoul

Keyword(s):

Gelfand Duality

SEPARATION PROPERTIES AT p FOR THE TOPOLOGICAL CATEGORY OF CLOSURE SPACES

Journal of Pure and Applied Mathematics Advances and Applications ◽

10.18642/jpamaa_7100121656 ◽

2016 ◽

Vol 15 (2) ◽

pp. 93-106

Keyword(s):

Topological Category ◽

Closure Spaces

(E,M)−estructuras inducidas en categorías topológicas

Revista Integración ◽

10.18273/revint.v39n2-2021006 ◽

2021 ◽

Vol 39 (2) ◽

Author(s):

Juan Angoa Amador ◽

Agustín Contreras Carreto ◽

Jesús González Sandoval

Keyword(s):

Topological Category ◽

Projective Structures ◽

Categorical Structure

In this paper, we describe a convenient categorical structure with respect to a class of monomorphisms M and epimorphisms E for any topological category. We show in particular that the structure that we introduce here, which is induced by topological functors and their initial liftings, allows the study of some M−coreflective subcategories of a topological category. We pay special attention to projective structures.

Closedness, separation and connectedness in pseudo-quasi-semi metric spaces

Filomat ◽

10.2298/fil2014757b ◽

2020 ◽

Vol 34 (14) ◽

pp. 4757-4766

Author(s):

Tesnim Baran

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Closure Operator ◽

Topological Category ◽

The Relationship

In this paper, we give the characterization of closed and strongly closed subsets of an extended pseudo-quasi-semi metric space and show that they induce closure operator. Moreover, we characterize each of Ti, i = 0, 1, 2 and connected extended pseudo-quasi-semi metric spaces and investigate the relationship among them. Finally, we introduce the notion of irreducible objects in a topological category and examine the relationship among each of irreducible, Ti,i = 1,2, and connected extended pseudo-quasi-semi metric spaces.

Pre-Hausdorff and Hausdorff proximity spaces

Filomat ◽

10.2298/fil1712837k ◽

2017 ◽

Vol 31 (12) ◽

pp. 3837-3846 ◽

Cited By ~ 2

Author(s):

Muammer Kula ◽

Samed Özkan ◽

Tuğba Maraşlı

Keyword(s):

Usual Sense ◽

Topological Category ◽

Explicit Characterization ◽

Proximity Spaces

In this paper, an explicit characterization of the separation properties for T0, T1, PreT2 (pre-Hausdorff) and T2 (Hausdorff) is given in the topological category of proximity spaces. Moreover, specific relationships that arise among the various Ti,i = 0,1,2 and PreT2 structures are examined in this category. Finally, we investigate the relationships among generalized separation properties for Ti,i = 0,1,2 and PreT2 (in our sense), separation properties at a point p and separation properties for Ti, i = 0,1,2 in the usual sense in this category.

Reflexive objects in topological categories

Mathematical Structures in Computer Science ◽

10.1017/s0960129500001079 ◽

1996 ◽

Vol 6 (4) ◽

pp. 375-386

Author(s):

Michael D. Rice

Keyword(s):

Geometric Structure ◽

Unit Interval ◽

Continuous Image ◽

Topological Category ◽

Hausdorff Spaces ◽

Tychonoff Space ◽

Countably Compact ◽

Discrete Subspace ◽

Cartesian Closed

This paper presents several basic results about the non-existence of reflexive objects in cartesian closed topological categories of Hausdorff spaces. In particular, we prove that there are no non-trivial countably compact reflexive objects in the category of Hausdorff k-spaces and, more generally, that any non-trivial reflexive Tychonoff space in this category contains a closed discrete subspace corresponding to a numeral system in the sense of Wadsworth. In addition, we establish that a reflexive Tychonoff space in a cartesian-closed topological category cannot contain a non-trivial continuous image of the unit interval. Therefore, if there exists a non-trivial reflexive Tychonoff space, it does not have a nice geometric structure.