Simplicial sets and their equivalence with topological spaces

2008 ◽  
pp. 27-41
Author(s):  
David Benson ◽  
Stephen Smith
Keyword(s):  
Topological Spaces ◽  
Simplicial Sets

Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem

1993 ◽  
Vol 114 (1) ◽  
pp. 163-189 ◽  
Author(s):  
M. Bullejos ◽  
P. Carrasco ◽  
A. M. Cegarra
Keyword(s):  
Abelian Group ◽  
Homotopy Type ◽  
Abelian Groups ◽  
Group Structure ◽  
Cohomology Theory ◽  
Classification Theorem ◽  
Topological Spaces ◽  
Simplicial Sets

AbstractIn this paper we use Takeuchy–Ulbrich's cohomology of complexes of categories with abelian group structure to introduce a cohomology theory for simplicial sets, or topological spaces, with coefficients in symmetric cat-groups . This cohomology is the usual one when abelian groups are taken as coefficients, and the main topological significance of this cohomology is the fact that it is equivalent to the reduced cohomology theory defined by a Ω-spectrum, {}, canonically associated to . We use the spaces to prove that symmetric cat-groups model all homotopy type of spaces X with Πi(X) = 0 for all i ╪ n, n + 1 and n ≥ 3, and then we extend Eilenberg–MacLane's classification theorem to those spaces: .

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.

Vogt's theorem on categories of homotopy coherent diagrams

1986 ◽  
Vol 100 (1) ◽  
pp. 65-90 ◽  
Author(s):  
Jean-Marc Cordier ◽  
Timothy Porter
Keyword(s):  
Small Category ◽  
Obstruction Theory ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Enriched Category ◽  
Simplicial Sets ◽  
Continuous Maps ◽  
Compactly Generated

Let Top be the category of compactly generated topological spaces and continuous maps. The category, Top, can be given the structure of a simplicially enriched category (or S-category, S being the category of simplicial sets). For A a small category, Vogt (in [22]) constructed a category, Coh (A, Top), of homotopy coherent A-indexed diagrams in Top and homotopy classes of homotopy coherent maps, and proved a theorem identifying this as being equivalent to Ho (TopA), the category obtained from the category of commutative A-indexed diagrams by localizing with respect to the level homotopy equivalences. Thus one of the important consequences of Vogt's result is that it provides concrete coherent models for the formal composites of maps and formal inverses of level homotopy equivalences which are the maps in Ho (TopA). The usefulness of such models and in general of Vogt's results is shown in the series of notes [14–17] by the second author in which those results are applied to give an obstruction theory applicable in prohomotopy theory.

On the formula of Cohen–Vogt relatively pointed topological semi-simplicial sets

2019 ◽  
Vol 26 (1) ◽  
pp. 97-103
Author(s):  
Leonard Mdzinarishvili
Keyword(s):  
Exact Sequence ◽  
Topological Spaces ◽  
Simplicial Sets ◽  
Base Points ◽  
Analogous Formula

Abstract In the papers [1] and [6], for an inverse sequence of pointed topological spaces and fibrations preserving the base points E=E_{1}\xleftarrow{p_{1}}E_{2}\xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}E_{m+1}, there exists an exact sequence *\rightarrow{\varprojlim}^{(1)}[X,\Omega E_{m}]\rightarrow[X,\varprojlim E]% \rightarrow{\varprojlim}^{(1)}[X,E_{m}]\rightarrow*. In the present paper, for an inverse sequence of pointed topological semi-simplicial sets and fibrations preserving base points \underline{E}=\underline{E}{}_{1}\xleftarrow{p_{1}}\underline{E}{}_{2}% \xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}\underline{E}{}_{m+1}\xleftarrow{% \hphantom{p_{1}}}\cdots, an analogous formula is proved.

scholarly journals Another approach to the Kan–Quillen model structure

2019 ◽  
Vol 15 (1) ◽  
pp. 143-165
Author(s):  
Sean Moss
Keyword(s):  
Careful Analysis ◽  
Combinatorial Proof ◽  
Topological Spaces ◽  
Model Structure ◽  
Simplicial Sets ◽  
Simplicial Set ◽  
New Construction ◽  
Weak Homotopy ◽  
Basic Facts ◽  
Quillen Model

Abstract By careful analysis of the embedding of a simplicial set into its image under Kan’s $$\mathop {\mathop {\mathsf {Ex}}^\infty }$$Ex∞ functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about $$\mathop {\mathop {\mathsf {Ex}}^\infty }$$Ex∞ and hence provide a new construction of the Kan–Quillen model structure on simplicial sets, one which avoids the use of topological spaces or minimal fibrations.

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Main Tool ◽  
Homotopy Equivalence ◽  
Classifying Space ◽  
Deformation Retract ◽  
Simplicial Sets ◽  
Full Proof

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

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