scholarly journals A closed model category for $(n - 1)$-connected spaces

1996 ◽  
Vol 124 (11) ◽  
pp. 3545-3553 ◽  
Author(s):  
J. Ignacio Extremiana Aldana ◽  
L. Javier Hernández Paricio ◽  
M. Teresa Rivas Rodríguez
Keyword(s):  
Model Category ◽  
Closed Model ◽  
Closed Model Category ◽  
Connected Spaces

scholarly journals Stability in pro-homotopy theory

1990 ◽  
Vol 33 (3) ◽  
pp. 419-441
Author(s):  
R. M. Seymour
Keyword(s):  
Stability Problem ◽  
Homotopy Theory ◽  
Category Structure ◽  
Local Condition ◽  
Model Category ◽  
Homotopy Category ◽  
Suitable Model ◽  
Closed Model ◽  
Closed Model Category ◽  
The Stability

If is a category, an object of pro- is stable if it is isomorphic in pro- to an object of . A local condition on such a pro-object, called strong-movability, is defined, and it is shown in various contexts that this condition is equivalent to stability. Also considered, in the case is a suitable model category, is the stability problem in the homotopy category Ho(pro-), where pro- has the induced closed model category structure defined by Edwards and Hastings [6].

Closed Model Category Structures

2010 ◽  
pp. 169-182
Author(s):  
Leonid Positselski ◽  
Sergey Arkhipov ◽  
Dmitriy Rumynin
Keyword(s):  
Model Category ◽  
Closed Model ◽  
Closed Model Category

scholarly journals Closed model category structures on the category of chain functors

2003 ◽  
Vol 131 (2) ◽  
pp. 101-128 ◽  
Author(s):  
Friedrich W. Bauer ◽  
Tamar Datuashvili
Keyword(s):  
Model Category ◽  
Closed Model ◽  
Closed Model Category

Closed model categories for the n-type of spaces and simplicial sets

1995 ◽  
Vol 118 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Carmen Elvira-Donazar ◽  
Luis-Javier Hernandez-Paricio
Keyword(s):  
Base Point ◽  
Category Structure ◽  
Model Category ◽  
Homotopy Groups ◽  
Weak Equivalence ◽  
Model Categories ◽  
Closed Model ◽  
Simplicial Sets ◽  
Closed Model Categories ◽  
Closed Model Category

AbstractFor each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.

On the simple object associated to a diagram in a closed model category

1986 ◽  
Vol 100 (3) ◽  
pp. 459-474 ◽  
Author(s):  
Pere Pascual-Gainza
Keyword(s):  
Model Category ◽  
General Context ◽  
Algebraic Varieties ◽  
Sheaf Cohomology ◽  
Graded Algebras ◽  
Closed Model ◽  
Coherent Sheaves ◽  
Noetherian Scheme ◽  
Differential Graded Algebras ◽  
Closed Model Category

In this paper we develop a descent technique for generalized (co)-homology theories defined in the category of algebraic varieties. By such a theory we mean a functor Sch→C, where C is a closed model category in the sense of Quillen satisfying certain axioms (cf. §4). We have chosen to work in such a general context so as to include two situations for which the results of SGA 4 of Deligne and Saint-Donat are not applicable: descent of multiplicative structures (i.e. of differential graded algebras) and descent for generalized sheaf cohomology (such as the algebraic K-theory of coherent sheaves over a noetherian scheme).

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.

Stable Homotopy Theory of Simplicial Presheaves

1987 ◽  
Vol 39 (3) ◽  
pp. 733-747 ◽  
Author(s):  
J. F. Jardine
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Closed Model ◽  
Stable Homotopy Theory ◽  
Simplicial Sets ◽  
Stable Homotopy Category ◽  
Closed Model Category ◽  
Simplicial Presheaves ◽  
Direct Use

Let C be an arbitrary Grothendieck site. The purpose of this note is to show that, with the closed model structure on the category S Pre(C) of simplicial presheaves in hand, it is a relatively simple matter to show that the category S Pre(C)stab of presheaves of spectra (of simplicial sets) satisfies the axioms for a closed model category, giving rise to a stable homotopy theory for simplicial presheaves. The proof is modelled on the corresponding result for simplicial sets which is given in [1], and makes direct use of their Theorem A.7.This result gives a precise description of the associated stable homotopy category Ho(S Pre(C))stab, according to well known results of Quillen [6]. One will recall, however, that it is preferable to have several different descriptions of the stable homotopy category, for the construction of smash products and the like.

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