Homotopy Categories for Simply Connected Torsion Spaces (original) (raw)

Models for torsion homotopy types

Israel Journal of Mathematics, 1998

Given an integer n > 1 and any set P of positive integers, one can assign to each topological space X a homotopy universal map X (P,n) → X where X (P,n) is an (n − 1)-connected CW-complex whose homotopy groups are P-torsion. We analyze this construction and its properties by means of a suitable closed model category structure on the pointed category of topological spaces.

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

Proceedings of the American Mathematical Society

For each integer n>0, we give a distinct closed model category structure to the category of pointed spaces, Top * such that the corresponding localized category Ho(Top * n ) is equivalent to the standard homotopy category of (n-1)-connected CW-complexes. The structure of closed model category given by D. Quillen to Top * [Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.53702)] is based on maps which induce isomorphisms on all homotopy group functors π q and for any choice of base point. For each n>0, the closed model category structure given here takes as weak equivalences those maps that for the given base point induce isomorphisms on π q for q≥n.

Closed model categories for uniquely S-divisible spaces

Journal of Pure and Applied Algebra, 2003

For each integer n ¿ 1 and a multiplicative system S of non-zero integers, we give a distinct closed model category structure to the category of pointed spaces Top ? and we prove that the corresponding localized category Ho(Top (S; n) ?), obtained by inverting the weak equivalences, is equivalent to the standard homotopy category of uniquely (S; n)-divisible, (n − 1)-connected spaces. A space X is said to be uniquely (S; n)-divisible if for k ¿ n the homotopy group k X is uniquely S-divisible. This equivalence of categories is given by an (S; n)-colocalization functor that carries a pointed space X to a space X (S; n). There is also a natural map X (S; n) → X which is (ÿnally) universal among all the maps Z → X with Z a uniquely (S; n)-divisible, (n − 1)-connected space. The structure of closed model category given by Quillen to Top ? is based on maps which induce isomorphisms on all homotopy group functors k and for any choice of base point. For each pair (S; n), the closed model category structure given here take as weak equivalences those maps that for the given base point induce isomorphisms on the homotopy groups functors k (Z[S −1 ]; −) with coe cients in Z[S −1 ]for k ¿ n. We note that the category Ho(Top (Z−{0}; 2) ?) is the homotopy category of rational 1-connected spaces.

On catn-groups and homotopy types

Journal of Pure and Applied Algebra, 1993

Bullejos, M., A.M. Cegarra and J. Duskin, On cat"-groups and homotopy types, Journal of Pure and Applied Algebra 86 (1993) 135-154. We give an algebraic proof of Loday's 'Classification theorem' for truncated homotopy types. In particular we give a precise construction of the homotopy cat"-group associated to a pointed topological space which is based on the use of the internal fundamental groupoid functor together with Illusie's 'total Dee'. modules.

CLOSED MODEL CATEGORIES FOR n; m]-TYPES

Theory and Applications of …, 1997

For m > n > 0, a map f between pointed spaces is said to be a weak n m]-equivalence if f induces isomorphisms of the homotopy groups k for n 6 k 6 m. Associated with this notion we g i v e t wo di erent closed model category structures to the category of pointed spaces. Both structures have the same class of weak equivalences but di erent classes of brations and therefore of co brations. Using one of these structures, one obtains that the localized category is equivalent to the category of n-reduced CWcomplexes with dimension less than or equal to m + 1 and m-homotopy classes of cellular pointed maps. Using the other structure we see that the localized category is also equivalent to the homotopy c a t e g o r y o f (n ; 1)-connected (m + 1)-coconnected CW-complexes. Introduction. D. Quillen 19] introduced the notion of closed model category and proved that the categories of spaces and of simplicial sets have the structure of a closed model category. This structure gives you some advantages. For instance, you can use sequences of homotopy bres or homotopy co bres associated to a map. In many cases, you can also compare two closed model categories by using a pair of adjoint functors. For example, you can prove that the localized categories of spaces and of simplicial sets are equivalent. In other cases, the co brant (or brant) approximation of an object gives objects and canonical maps with certain universal properties or can be used to construct derived functors. In this paper, for m > n > 0 , w e take a s w eak equivalences those maps of Top ? which induce isomorphisms on the homotopy group functors k for m > k > n. A m a p f with this property is said to be a weak n m]-equivalence. We complete this class of weak equivalences with brations and co brations in two di erent w ays: The authors acknowledge the nancial aid given by the U.R., project 96PYB44MRR and by D G I-CYT, project PB93-0581-C02-01.

Group-theoretic algebraic models for homotopy types

Journal of Pure and Applied Algebra, 1991

In this paper a nonabelian version of the Dold-Kan-Puppe theorem is provided. showing hou the Moore-complex functor defines a full equivalence between the category of simplicial groups and the category of what is called 'hypcrcrcssed complexes of groups'. i.e. chain complexes of nonabelian groups (G,,. S,,) with an additional structure in the form of binary operations G, x G,* G,. We associate to a pointed topological space X a hypercrossed complex A(X): and the functor 2 induces an equivalence between the homotopy category of connected CW-complexes and a localization of the category of hypercrossed complexes. The relationshjp between s(X) and Whitehead's crossed complex II(X) is established by a cano:!ical surjection p : ii(X)-* II(X). which is a quasi-isomorphism if and only if X is a J-complex. Algebraic rnodels consisting of truncated chain-complexes with binary operations I;re deduced for rl-types. and as an application we deduce a group-theoretic interpretation of the cohomology groups H"(G. A).

Generalized Homotopy theory in Categories with a Natural Cone

In proper homotopy theory, the original concept of point used in the classical homotopy theory of topological spaces is generalized in order to obtain homotopy groups that study the infinite of the spaces. This idea: "Using any arbitrary object as base point" and even "any morphism as zero morphism" can be developed in most of the algebraic homotopy theories. In particular, categories with a natural cone have a generalized homotopy theory obtained through the relative homotopy relation. Generalized homotopy groups and exact sequences of them are built so that respective classical pointed ones are a particular case of these.

Colimit theorems for relative homotopy groups

Journal of Pure and Applied Algebra, 1981

to take into account later views of both authors, and to make minor clarifications. The main change is to avoid the J0 condition on filtered space which was used in the published version -this is done by defining higher homotopy groupoids using homotopy classes rel vertices of I n . This makes the theory nearer to standard homotopy theory, and is also essential for later work in defining the homotopy crossed complexes for filtered function spaces, when the J0 condition is unlikely to be fulfilled. It is hoped that this version will be useful to readers.

On the homotopy types

2022

This dissertation is concerned with the foundations of homotopy theory following the ideas of the manuscripts Les Dérivateurs and Pursuing Stacks of Grothendieck. In particular, we discuss how the formalism of derivators allows us to think about homotopy types intrinsically, or, even as a primitive concept for mathematics, for which sets are a particular case. We show how category theory is naturally extended to homotopical algebra, understood here as the formalism of derivators. Then, we proof in details a theorem of Heller and Cisinski, characterizing the category of homotopy types with a suitable universal property in the language of derivators, which extends the Yoneda universal property of the category of sets with respect to the cocomplete categories. From this result, we propose a synthetic re-definition of the category of homotopy types. This establishes a mathematical conceptual explanation for the the links between homotopy type theory, ∞-categories and homotopical algebra, and also for the recent program of re-foundations of mathematics via homotopy type theory envisioned by Voevodsky. In this sense, the research on foundations of homotopy theory reflects in a discussion about the re-foundations of mathematics. We also expose the theory of Grothendieck-Maltsiniotis ∞-groupoids and the famous Homotopy Hypothesis conjectured by Grothendieck, which affirms the (homotopical) equivalence between spaces and ∞-groupoids. This conjectured, if proved, provides a strictly algebraic picture of spaces.