Closed model categories for the <i>n</i>-type of spaces and simplicial sets

Elvira-Donazar, Carmen; Hernandez-Paricio, L.J.

doi:10.1017/s0305004100073485

Cited by 9 publications

(4 citation statements)

References 11 publications

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“…Approximately at the same time n-homotopies (and subsequently even n-shapes [7], [6]) begun to play a substantial role in the revitalized theory of Menger manifolds [2], [5] (see [8] for a discussion of categorical connections between n-homotopies and homotopies via the theories of manifolds modeled on Menger and Hilbert cubes respectively). The following theorem has been proved in [16].…”

Section: Theorem Bmentioning

confidence: 99%

Topological Model Categories Generated by Finite Complexes

Chigogidze

Karasev

2003

Monatshefte f�r Mathematik

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Abstract. Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups -is obtained by letting L = {point}. The other -describing n-homotopy equivalences between at most (n + 1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ≤ n -by letting L = S n+1 , n ≥ 0.

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Section: Theorem Bmentioning

confidence: 99%

Topological Model Categories Generated by Finite Complexes

Chigogidze

Karasev

2003

Monatshefte f�r Mathematik

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“…Group crossed modules were firstly introduced by Whitehead in [21], [22]. They are algebraic models for homotopy 2-types, in the sense that [5], [15] the homotopy category of the model category [6], [9] of group crossed modules is equivalent to the homotopy category of the model category [11] of pointed 2-types: pointed connected spaces whose homotopy groups π i vanish, if i ≥ 3. The homotopy relation between crossed module maps A −→ A was given by Whitehead in [22], in the contex of "homotopy systems" called free crossed complexes.…”

Section: Introductionmentioning

confidence: 99%

Homotopies of 2-Algebra Morphisms

Akça

Arslan

2022

Communications in Advanced Mathematical Sciences

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“…Group crossed modules were firstly introduced by Whitehead in [33,34]. They are algebraic models for homotopy 2-types, in the sense that [5,27] the homotopy category of the model category [7,12] of group crossed modules is equivalent to the homotopy category of the model category [18] of pointed 2-types: pointed connected spaces whose homotopy groups π i vanish, if i ≥ 3. Crossed modules of groups also naturally appear in the context of simplicial homotopy theory, namely they are equivalent to simplicial groups with Moore complex of length one [13] and analogously for crossed modules of groupoids [29].…”

Section: Introductionmentioning

confidence: 99%

Pointed homotopy of maps between 2-crossed modules of commutative algebras

Akça

Emir

Martins

2016

Homology, Homotopy and Applications

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We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy relation, and prove that it yields an equivalence relation in very unrestricted cases (freeness up to order one of the domain 2-crossed module). This latter condition strictly includes the case when the domain is cofibrant. Furthermore, we prove that this notion of homotopy yields a groupoid with objects being the 2-crossed module maps between two fixed 2-crossed modules (with free up to order one domain), the morphisms being the homotopies between 2-crossed module maps.Keywords: Simplicial commutative algebra, crossed module of commutative algebras, 2-crossed module of commutative algebras, quadratic derivation.

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

Cited by 9 publications

References 11 publications

Topological Model Categories Generated by Finite Complexes

Topological Model Categories Generated by Finite Complexes

Homotopies of 2-Algebra Morphisms

Pointed homotopy of maps between 2-crossed modules of commutative algebras

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