On realizing diagrams of Π–algebras

Blanc, David; Johnson, Mark; Turner, James

doi:10.2140/agt.2006.6.763

Cited by 15 publications

(37 citation statements)

References 38 publications

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“…In addition, all the resolution model categories of Section 1.4 are semi-spherical (see [17,Section 3]). We note that for the ''algebraic'' categories C = sΘ-Alg of simplicial universal algebras (Section 1.4(b)),π 1 X • is just π 0 X • ,and Coef(C) is Θ-Alg itself.…”

Section: Examples Of Resolution Model Categoriesmentioning

confidence: 99%

“…The approach of [35,35,16] to realizing Π-algebras can be generalized somewhat (see [17]), but it still does not apply to arbitrary resolution model categories (for example, it does not even apply to topological spaces if A consists of mod-p Moore spaces -see [13,Section 4.6]). We therefore restrict to the following setting:…”

Section: Realizing π a -Algebrasmentioning

confidence: 99%

“…In [17,, it was shown that all the examples of Section 1.4 are E 2 -model categories, which satisfy a somewhat weaker set of axioms (see [17,Definition 3.12]). However, there are a number of additional examples satisfying these stricter conditions -T * can be replaced by S red or G, or various categories of spectra, or DG-categories; or we can take diagrams in these categories.…”

Section: Example 73 the Main Example That We Have In Mind Ismentioning

confidence: 99%

“…in sC/BΛ such that π A X ⟨n⟩ • ≃ E Λ (Ω n+1 Λ, n + 2) (as for the usual Postnikov system of a realization of BΛ in sC -see [17,Section 5.8]). The object X ⟨n⟩ • ∈ sC will be called an n-th quasi-Postnikov section for Λ.…”

Section: Example 73 the Main Example That We Have In Mind Ismentioning

confidence: 99%

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Comparing cohomology obstructions

Baues

Blanc

2011

Journal of Pure and Applied Algebra

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Communicated by C.A. Weibel MSC: Primary: 55S35 Secondary: 55Q05; 18G55; 55T25 a b s t r a c t We show that three different kinds of cohomologies -Baues-Wirsching cohomology, the (S * , O)-cohomology of Dwyer and Kan, and the André-Quillen cohomology of a Π-algebra -are isomorphic, under certain assumptions. This is then used to identify the cohomological obstructions in three general approaches to realizability problems: the track category version of Baues and Wirsching, the diagram rectifications of Dwyer, Kan,and Smith, and the Π-algebra realization of Dwyer, Kan, and Stover. Our main tool in this identification is the notion of a mapping algebra: a simplicially enriched version of an algebra over a theory.

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Section: Examples Of Resolution Model Categoriesmentioning

confidence: 99%

Section: Realizing π a -Algebrasmentioning

confidence: 99%

Section: Example 73 the Main Example That We Have In Mind Ismentioning

confidence: 99%

Section: Example 73 the Main Example That We Have In Mind Ismentioning

confidence: 99%

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Comparing cohomology obstructions

Baues

Blanc

2011

Journal of Pure and Applied Algebra

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“…We do not need the modified Postnikov system in this case: the obstructions to realizing Λ (or G) are just the classes χ n ∈ H n+3 (Λ; Ω n+1 Λ), and the difference obstructions distinguishing between the different realizations are δ n ∈ H n+2 (Λ; Ω n+1 Λ) (n ≥ 1). See [BDG] and [BJT,§5] for two descriptions of this case. 7.5.…”

Section: Applying the Theorymentioning

confidence: 99%

Comparing homotopy categories

Blanc

2007

J K-Theor

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Abstract. Given a suitable functor T : C → D between model categories, we define a long exact sequence relating the homotopy groups of any X ∈ C with those of T X, and use this to describe an obstruction theory for lifting an object G ∈ D to C. Examples include finding spaces with given homology or homotopy groups. IntroductionA number of fundamental problems in algebraic topology can be described as measuring the extent to which a given functor T : C → D between model categories induces an equivalence of homotopy categories: more specifically, which objects (or maps) from D are in the image of T , and in how many different ways. For example: a) How does one distinguish between different topological spaces with the same homology groups, or with chain-homotopy equivalent chain complexes? How can one realize a given map of chain complexes up to homotopy? b) When do two simply-connected topological spaces have the same rational homotopy type? c) When is a given topological space a suspension, up to homotopy? Dually, how many distinct loop space structures, if any, can a given topological space carry? d) Is a given Π-algebra (that is, a graded group with an action of the primary homotopy operations) realizable as the homotopy groups of a topological space, and if so, in how many ways?Our goal is to describe a unified approach to such problems that works for functors between spherical model categories, for which several familiar concepts and constructions are available. These include a set A of models (to play the role of spheres, in particular determining the corresponding homotopy groups π

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The realizability of operations on homotopy groups concentrated in two degrees

Baues

Frankland

2014

J. Homotopy Relat. Struct.

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The homotopy groups of a space are endowed with homotopy operations which define the Π-algebra of the space. An Eilenberg-MacLane space is the realization of a Π-algebra concentrated in one degree. In this paper, we provide necessary and sufficient conditions for the realizability of a Π-algebra concentrated in two degrees. We then specialize to the stable case, and list infinite families of such Π-algebras that are not realizable. Keywords Realization • Homotopy operation • Homotopy group • 2-Stage • Π-algebra • Whitehead product Mathematics Subject Classification (2000) 55Q35 • 55Q40 • 55Q45 • 55Q15 • 55P20 1 Realization problem for homotopy operations The homotopy groups π * X of a pointed space X are not merely a list of groups, but carry the additional structure of an action of the (primary) homotopy operations, which are natural transformations Communicated by Mark Behrens.

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On realizing diagrams of Π–algebras

Cited by 15 publications

References 38 publications

Comparing cohomology obstructions

Comparing cohomology obstructions

Comparing homotopy categories

The realizability of operations on homotopy groups concentrated in two degrees

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