Closed model structures for algebraic models of n-types

Cabello, Julia García; Garzón, Antonio R.

doi:10.1016/0022-4049(94)00105-r

Cited by 17 publications

(18 citation statements)

References 14 publications

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“…This should be compared with what would be guaranteed from the model category [16] point of view, where we would expect homotopy of maps G → G ′ to be an equivalence relation only when G = (∂ : E → G, ◮) is cofibrant (given that any object is fibrant). In the well-known model category structure in the category of crossed modules [12], obtained by transporting the usual model category structure of the category of simplicial sets, G = (∂ : E → G, ◮) is cofibrant if, and only if, G is a free group ( [30]).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…, of groups, is called free up to order one if G is a free group, A being called free up to order two if, furthermore, (∂ : E → G, ◮) is a free pre-crossed module; [19,23]. In the model category of 2-crossed modules of groups, [11,12,23,26],…”

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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“…In fact, we prove that the category Tn] (Gpd) is a closed Quillen model category with structure defined via that of Simp(Gpd), (cf. [5]) and this closed model structure in Tn](Gpd) coincides, for n = 1, with that given to the category of 2-groupoids in [22] (cf. [3]).…”

Section: Introductionmentioning

confidence: 99%

Models for homotopy n-types in diagram categories

Garzón

Miranda

1996

Appl Categor Struct

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The classical Mac Lane-Whitehead equivalence showing that crossed modules of groups are algebraic models of connected homotopy 2-types has found a corresponding equivariant version by Moerdijk and Svensson ([22]). In this paper we show that this equivariant result has a higherdimensional version which gives an equivalence between the homotopy category of diagrams of certain objects indexed by the orbit category of a group H and H-equivariant homotopy n-types forn~ 1.Mathematics Subject Classifications (1991). 55P91, 55U35, 55P15.

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“…However we have a map {, } : E × E → L, called the Peiffer lifting, measuring how far Peiffer 2 is from being satisfied, namely:The category of 2-crossed modules is equivalent to a reflexive subcategory of the category of simplicial groups [11,23,24]. Thus [10,19] an adjunction:exists. Such is constructed in the obvious way from the Dwyer-Kan loop-group G ⊣ W adjunction between the category of simplicial sets and the category of simplicial groups; [14,16,26].…”

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confidence: 99%

Two-fold homotopy of 2-crossed module maps of commutative algebras

Akça

Emir

Martins

2018

Communications in Algebra

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We address the homotopy theory of 2-crossed modules of commutative algebras. In particular, we define the concept of a 2-fold homotopy between a pair of 1-fold homotopies connecting 2-crossed module maps A → A ′ . We also prove that if the domain 2-crossed module A is free up to order one (i.e. if the bottom algebra is a polynomial algebra) then we have a 2-groupoid of 2-crossed module maps A → A ′ and their homotopies and 2-fold homotopies. is what is called a pre-crossed module. The second Peiffer relation does not hold in general. However we have a map {, } : E × E → L, called the Peiffer lifting, measuring how far Peiffer 2 is from being satisfied, namely:The category of 2-crossed modules is equivalent to a reflexive subcategory of the category of simplicial groups [11,23,24]. Thus [10,19] an adjunction:exists. Such is constructed in the obvious way from the Dwyer-Kan loop-group G ⊣ W adjunction between the category of simplicial sets and the category of simplicial groups; [14,16,26]. B is the simplicial classifying space functor [7] and Π 3 was explicitly constructed in [15] from triad homotopy groups of the geometric realisation. We thus [9,16] have a model category structure in the category of 2-crossed modules of groups in which all objects are fibrant and which renders a 2-crossed module A = (L → E → G, ◮, {, }) cofibrant if, and only if, it is a retract of a totally free 2-crossed module, the latter meaning that G is a free group and that (∂ : E → G, ◮) is a free pre-crossed module [15,19,23,24].Following previous constructions in the context of quadratic modules (a particular case of 2crossed modules) [7] and Gray categories [12], a homotopy relation between group 2-crossed module maps was addressed in [18,19], via path-objects, and proven in [15] to faithfully model homotopy classes of maps between 3-types. Given 2-crossed modules A and B, of groups, if A is totally free (therefore cofibrant), if follows that homotopy between maps A → B is an equivalence relation. A surprising result of [18,19] was that, in order for homotopy between maps A → B to be an equivalence relation, we solely need to impose that A is free up to order one, meaning that G is a free group. Moreover in the latter case we can define, if we are given a free basis of G, a 2-groupoid of maps A → B, homotopies between maps, and 2-fold homotopies between homotopies.This paper, which is a follow-up of [1], contains a proof that the latter property (1-freeness suffices to compose homotopies and 2-fold homotopies) also holds for 2-crossed modules of commutative algebras.All algebras in this paper will be commutative and over a fixed ring κ, not necessarily with 1. Crossed modules and 2-crossed modules of algebras [4,5,6,13,25] are defined in the same way as in the group case, essentially switching actions by automorphisms to actions by multipliers, where a multiplier in an algebra R is a linear map f : R → R such that f (ab) = f (a)b, for each a, b ∈ R. A 2-crossed module of algebras A = (L ∂2 −→ E ∂1 −→ R, ◮, {, }) has an underlying comp...

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Closed model structures for algebraic models of n-types

Cited by 17 publications

References 14 publications

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

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

Models for homotopy n-types in diagram categories

Two-fold homotopy of 2-crossed module maps of commutative algebras

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