scholarly journals On fuzzification of topological categories

10.29007/68tw ◽  
2018 ◽  
Author(s):  
Sergejs Solovjovs
Keyword(s):  
Topological Category ◽  
Coreflective Subcategory

This talk provides a fuzzification procedure for topological categories, i.e., given a topological category A, there exists a topological category B, which contains A as a full concretely coreflective subcategory, and which can be considered as a fuzzification of A.

scholarly journals The categorical relationships between neighborhood spaces, ┬-neighborhood spaces and stratified L-neighborhood spaces

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1267-1287
Author(s):  
Lingqiang Li ◽  
Qiu Jin ◽  
Chunxin Bo ◽  
Zhenyu Xiu
Keyword(s):  
Residuated Lattice ◽  
Topological Category ◽  
Reflective Subcategory ◽  
Categorical Properties ◽  
Coreflective Subcategory ◽  
Complete Residuated Lattice

In this paper, for a complete residuated lattice L, we present the categorical properties of ?-neighborhood spaces and their categorical relationships to neighborhood spaces and stratified L-neighborhood spaces. The main results are: (1) the category of ?-neighborhood spaces is a topological category; (2) neighborhood spaces can be embedded in ?-neighborhood spaces as a reflective subcategory, and when L is a meet-continuous complete residuated lattice, ?-neighborhood spaces can be embedded in stratified L-neighborhood spaces as a reflective subcategory; (3) when L is a continuous complete residuated lattice, neighborhood spaces (resp., ?-neighborhood spaces) can be embedded in ?-neighborhood spaces (resp., stratified L-neighborhood spaces) as a simultaneously reflective and coreflective subcategory.

A cartesian closed topological category of sequential spaces

1990 ◽  
Vol 21 (2) ◽  
pp. 109-112 ◽  
Author(s):  
J. Šlapal
Keyword(s):  
Topological Category ◽  
Cartesian Closed ◽  
Sequential Spaces

The Disc Embedding Theorem

2021 ◽  
Keyword(s):  
Graduate Students ◽  
Euclidean Space ◽  
Iterative Process ◽  
Embedding Theorem ◽  
Space Theory ◽  
Topological Category ◽  
Surgery Theory ◽  
Smooth Structures ◽  
Decomposition Space ◽  
Exotic Smooth Structures

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

1991 ◽  
Vol 43 (4) ◽  
pp. 814-824 ◽  
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.

scholarly journals On projective lift and orbit spaces

1994 ◽  
Vol 50 (3) ◽  
pp. 445-449 ◽  
Author(s):  
T.K. Das
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Covariant Functor ◽  
Hausdorff Spaces ◽  
Orbit Spaces ◽  
Locally Compact ◽  
Discontinuous Group ◽  
Coreflective Subcategory ◽  
Group Of Homeomorphisms ◽  
Locally Compact Hausdorff Space

By constructing the projective lift of a dp-epimorphism, we find a covariant functor E from the category Cd of regular Hausdorff spaces and continuous dp-epimorphisms to its coreflective subcategory εd consisting of projective objects of Cd We use E to show that E(X/G) is homeomorphic to EX/G whenever G is a properly discontinuous group of homeomorphisms of a locally compact Hausdorff space X and X/G is an object of Cd.

Every topological category is convenient for Gelfand duality

1978 ◽  
Vol 25 (2) ◽  
pp. 169-204 ◽  
Author(s):  
Hans -E. Porst ◽  
Manfred B. Wischnewsky
Keyword(s):  
Topological Category ◽  
Gelfand Duality

scholarly journals Comparing Transition Systems with Independence and Asynchronous Transition Systems

1996 ◽  
Vol 3 (18) ◽  
Author(s):  
Thomas Troels Hildebrandt ◽  
Vladimiro Sassone
Keyword(s):  
Full Subcategory ◽  
Transition Systems ◽  
Simple Idea ◽  
Coreflective Subcategory ◽  
Primitive Notion

Transition systems with independence and asynchronous transition systems are non-interleaving models for concurrency arising from the same simple idea of decorating transitions with events. They differ for the choice of a derived versus a primitive notion of event which induces considerable differences and makes the two models suitable for different purposes. This opens the problem of investigating their mutual relationships,<br />to which this paper gives a fully comprehensive answer.<br />In details, we characterise the category of extensional asynchronous transitions systems as the largest full subcategory of the category of (labelled) asynchronous transition systems which admits TSI, the category of transition systems with independence, as a coreflective subcategory. In addition, we introduce event-maximal asynchronous transitions systems and we show that their category is equivalent to TSI, so providing an exhaustive characterisation of transition systems with independence in terms of asynchronous transition systems.

scholarly journals (E,M)−estructuras inducidas en categorías topológicas

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.

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

Filomat ◽  
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.

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