scholarly journals A closed simplicial model category for proper homotopy and shape theories

1998 ◽  
Vol 57 (2) ◽  
pp. 221-242 ◽  
Author(s):  
J.M. García-Calcines ◽  
M. Garcia-Pinillos ◽  
L.J. Hernández-Paricio
Keyword(s):  
Exact Sequence ◽  
The Other ◽  
Model Category ◽  
Homotopy Category ◽  
Finite Covering ◽  
Covering Dimension ◽  
Homotopy Classes ◽  
Exterior Space ◽  
Proper Maps ◽  
Proper Homotopy

In this paper, we introduce the notion of exterior space and give a full embedding of the category P of spaces and proper maps into the category E of exterior spaces. We show that the category E admits the structure of a closed simplicial model category. This technique solves the problem of using homotopy constructions available in the localised category HoE and in the “homotopy category” π0E, which can not be developed in the proper homotopy category.On the other hand, for compact metrisable spaces we have formulated sets of shape morphisms, discrete shape morphisms and strong shape morphisms in terms of sets of exterior homotopy classes and for the case of finite covering dimension in terms of homomorphism sets in the localised category.As applications, we give a new version of the Whitehead Theorem for proper homotopy and an exact sequence that generalises Quigley's exact sequence and contains the shape version of Edwards-Hastings' Comparison Theorem.

scholarly journals AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

2018 ◽  
Vol 107 (02) ◽  
pp. 181-198
Author(s):  
JAMES GILLESPIE
Keyword(s):  
The Other ◽  
Homotopy Category ◽  
Gorenstein Ring ◽  
Model Structure ◽  
Projective Modules ◽  
Stable Module Category ◽  
Stable Module ◽  
Coherent Rings ◽  
Quillen Equivalent ◽  
Compactly Generated

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

scholarly journals RANK ONE CONNECTIONS ON ABELIAN VARIETIES, II

2012 ◽  
Vol 23 (12) ◽  
pp. 1250125
Author(s):  
INDRANIL BISWAS ◽  
JACQUES HURTUBISE ◽  
A. K. RAINA
Keyword(s):  
Line Bundle ◽  
Exact Sequence ◽  
Abelian Varieties ◽  
Explicit Construction ◽  
The Other ◽  
Holomorphic Line Bundle ◽  
Canonical Isomorphism ◽  
Complex Torus ◽  
Rank One

Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic ΩA-torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle [Formula: see text], where α is the addition map on A × A, and p1 is the projection of A × A to the first factor. In [I. Biswas, J. Hurtvbise and A. K. Raina, Rank one connections on abelian varieties, Internat. J. Math.22 (2011) 1529–1543], it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

scholarly journals A study of some aspects of topological groups

Filomat ◽  
2007 ◽  
Vol 21 (1) ◽  
pp. 55-65
Author(s):  
M.R. Adhikari ◽  
M. Rahaman
Keyword(s):  
Topological Group ◽  
Group Structure ◽  
Base Point ◽  
Homotopy Category ◽  
Topological Spaces ◽  
Topological Groups ◽  
Homotopy Classes ◽  
Continuous Maps

The aim of this paper is to find a generalization of topological groups. The concept arises out of the investigation to obtain a group structure on the set [X,Y], of homotopy classes of maps from a space X to a given space Y for all X which is natural with respect to X. We also study the generalized topological groups. Finally, associated with each generalized topological group we construct a contra variant functor from the homotopy category of pointed topological spaces and base point preserving continuous maps to the category of groups and homomorphism.

The Verdier Hypercovering Theorem

2012 ◽  
Vol 55 (2) ◽  
pp. 319-328 ◽  
Author(s):  
J. F. Jardine
Keyword(s):  
Broad Sense ◽  
Homotopy Category ◽  
Homotopy Classes ◽  
Theoretic Proof ◽  
Simplicial Sheaves ◽  
Simplicial Homotopy

AbstractThis note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms [X, Y] in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where Y is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object X to be locally fibrant.

Design of Balcony from the Point of View of Daylighting

2014 ◽  
Vol 899 ◽  
pp. 298-301
Author(s):  
Lenka Gábrová
Keyword(s):  
Fire Safety ◽  
Point Of View ◽  
The Other ◽  
Main Function ◽  
Visual Comfort ◽  
Static Properties ◽  
Exterior Space ◽  
Other Hand ◽  
Thermal Bridges ◽  
Building Physics

Balconies are horizontal overhanging structures whose main function is a connection between interior and exterior space. Moreover, they increase an area of a room and they also can have a presentable, architectural and aesthetic purpose. There are a lot of requirements on the balconies, for example the requirements on static properties, thermal bridges, railings or fire safety. From the point of view of building physics, we can consider the balcony as the fixed shading construction which can reduce overheating in a shaded room during the summer but on the other hand the balcony can influence visual comfort in the room in a negative way. This article presents a comparison and an evaluation of the daylight factor in the rooms which are shaded by balconies. The influence of the balconies on the daylight factor is assessed in dependence on a length of the balcony (a balcony overhang) and dimensions of the room which the balcony shades.

Compact 16-Dimensional Projective Planes with Large Collineation Groups. IV

1987 ◽  
Vol 39 (4) ◽  
pp. 908-919 ◽  
Author(s):  
Helmut Salzmann
Keyword(s):  
Transformation Group ◽  
Projective Planes ◽  
Finite Covering ◽  
Covering Dimension ◽  
Topological Projective Plane ◽  
Collineation Groups ◽  
Compact Transformation Group ◽  
Point Set ◽  
Type V ◽  
Moufang Plane

Let be a topological projective plane with compact point set P of finite (covering) dimension. In the compact-open topology (of uniform convergence), the group Σ of continuous collineations of is a locally compact transformation group of P.THEOREM. If dim Σ > 40, thenis isomorphic to the Moufang plane 6 over the real octonions (and dim Σ = 78).By [3] the translation planes with dim Σ = 40 form a one-parameter family and have Lenz type V. Presumably, there are no other planes with dim Σ = 40, cp. [17].

scholarly journals Minimal Dynamical Systems on Connected Odd Dimensional Spaces

2015 ◽  
Vol 67 (4) ◽  
pp. 870-892 ◽  
Author(s):  
Huaxin Lin
Keyword(s):  
Abelian Group ◽  
Dynamical Systems ◽  
Finite Abelian Group ◽  
Finite Covering ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Group I ◽  
Tracial Rank ◽  
Minimal Dynamical Systems ◽  
Minimal Homeomorphisms

AbstractLet be a minimal homeomorphism (n ≥1). We show that the crossed product has rational tracial rank at most one. Let Ω be a connected, compact, metric space with finite covering dimension and with . Suppose that ,where Gi is a finite abelian group, i = 0,1. Let β:Ω→Ωbe a minimal homeomorphism. We also show that has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on X✗Ω, where X is the Cantor set.

scholarly journals Stable homotopical algebra and Γ-spaces

1999 ◽  
Vol 126 (2) ◽  
pp. 329-356 ◽  
Author(s):  
STEFAN SCHWEDE
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Develop Model ◽  
Smash Product ◽  
Spectral Sequences ◽  
Homotopical Algebra ◽  
Stable Homotopy Theory ◽  
Set Up

In this paper we advertise the category of Γ-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. Γ-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (−1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for Γ-spaces. The study of ‘rings, modules and algebras’ based on Γ-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg–MacLane spectra counterparts.

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