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.

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 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].

scholarly journals Exact Sequences and Closed Model Categories

2009 ◽  
Vol 18 (4) ◽  
pp. 343-375 ◽  
Author(s):  
Mónica García Pinillos ◽  
Luis Javier Hernández Paricio ◽  
María Teresa Rivas Rodríguez
Keyword(s):  
Model Categories ◽  
Closed Model ◽  
Closed Model Categories ◽  
Exact Sequences

scholarly journals Props in model categories and homotopy invariance of structures

2010 ◽  
Vol 17 (1) ◽  
pp. 79-160
Author(s):  
Benoit Fresse
Keyword(s):  
General Setting ◽  
Model Category ◽  
Topological Spaces ◽  
Algebra Structure ◽  
Model Structure ◽  
Homotopy Invariance ◽  
Model Categories ◽  
Arbitrary Ring ◽  
General Argument ◽  
Simplicial Sets

Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴∞-algebras.

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
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