Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

2004 ◽  
Vol 12 (3) ◽  
pp. 225-243 ◽  
Author(s):  
J. M. Garcia-Calcines ◽  
M. Garcia-Pinillos ◽  
L. J. Hernandez-Paricio
Keyword(s):  
Homotopy Theory ◽  
Proper Homotopy ◽  
Model Structures

A theoretical framework for proper homotopy theory

1990 ◽  
Vol 107 (3) ◽  
pp. 475-482 ◽  
Author(s):  
R. Ayala ◽  
A. Quintero ◽  
E. Dominguez
Keyword(s):  
Homotopy Theory ◽  
Theoretical Framework ◽  
Homotopy Groups ◽  
Theoretical Treatment ◽  
Proper Homotopy

AbstractFollowing the techniques of ordinary homotopy theory, a theoretical treatment of proper homotopy theory, including the known proper homotopy groups, is provided within Baues's theory of cofibration categories.

scholarly journals Homotopy theory of cyclic presheaves

2010 ◽  
Vol 107 (1) ◽  
pp. 30 ◽  
Author(s):  
Sigurd Seteklev ◽  
Paul Arne Østvær
Keyword(s):  
Homotopy Theory ◽  
Local Model ◽  
Simplicial Presheaves ◽  
Model Structures ◽  
Cyclic Sets

We generalize the homotopy theory of cyclic sets to cyclic presheaves on small Grothendieck sites. This is achieved by constructing pointwise and local model structures reminiscent of the homotopy theory of simplicial presheaves.

scholarly journals Quillen model structures for relative homological algebra

2002 ◽  
Vol 133 (2) ◽  
pp. 261-293 ◽  
Author(s):  
J. DANIEL CHRISTENSEN ◽  
MARK HOVEY
Keyword(s):  
Homotopy Theory ◽  
Homological Algebra ◽  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Category Structure ◽  
Model Category ◽  
Projective Class ◽  
Chain Complexes ◽  
Model Structures

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.

Proper homotopy theory in simplicial complexes

1974 ◽  
pp. 41-46 ◽  
Author(s):  
Edward M. Brown
Keyword(s):  
Homotopy Theory ◽  
Simplicial Complexes ◽  
Proper Homotopy

scholarly journals Homotopy theory of Moore flows (II)

2021 ◽  
Vol 36 (2) ◽  
pp. 157-239
Author(s):  
Philippe Gaucher
Keyword(s):  
Internal Structure ◽  
Homotopy Theory ◽  
Left Adjoint ◽  
Main Step ◽  
Derived Functor ◽  
Category Equivalence ◽  
Q Model ◽  
Quillen Equivalent ◽  
Model Structures

This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.

scholarly journals Lusternik-Schnirelmann invariants in proper homotopy theory

1992 ◽  
Vol 153 (2) ◽  
pp. 201-215 ◽  
Author(s):  
R. Ayala ◽  
Eladio Domínguez Murillo ◽  
Alberto Márquez Pérez ◽  
A. Quintero
Keyword(s):  
Homotopy Theory ◽  
Proper Homotopy ◽  
Lusternik Schnirelmann

An Elementary Approach to the Projective Dimension in Proper Homotopy Theory

2003 ◽  
Vol 31 (12) ◽  
pp. 5995-6017 ◽  
Author(s):  
R. Ayala ◽  
M. Cárdenas ◽  
F. Muro ◽  
A. Quintero
Keyword(s):  
Homotopy Theory ◽  
Projective Dimension ◽  
Elementary Approach ◽  
Proper Homotopy

scholarly journals Whitehead theorems in proper homotopy theory

1976 ◽  
Vol 82 (1) ◽  
pp. 59-61
Author(s):  
David A. Edwards ◽  
Harold M. Hastings
Keyword(s):  
Homotopy Theory ◽  
Proper Homotopy

scholarly journals On Proper Homotopy Theory for Noncompact 3-Manifolds

1974 ◽  
Vol 188 ◽  
pp. 105 ◽  
Author(s):  
E. M. Brown ◽  
T. W. Tucker
Keyword(s):  
Homotopy Theory ◽  
Proper Homotopy
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