On the levels of maps and topological realization of objects in a triangulated category

Kuribayashi, Katsuhiko

doi:10.1016/j.jpaa.2011.08.009

Cited by 4 publications

(20 citation statements)

References 36 publications

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“…We here recall from [32] and [33] two numerical topological invariants defined by the level in a triangulated category D(R). Unless otherwise explicitly stated, it is assumed that a space has the homotopy type of a connected CW complex whose cohomology with coefficients in the underlying field is locally finite.…”

Section: Resultsmentioning

confidence: 99%

“…This work is a sequel to previous one [32,33] in which new topological invariants have been studied.…”

Section: Introductionmentioning

confidence: 94%

“…Jørgensen [24,25,26] developed categorical representation theory of spaces employing the singular (co)chain complexes of spaces. In the context of such work, the cochain and chain type levels of maps between topological spaces have been defined and studied in [32,33].…”

Section: Introductionmentioning

confidence: 99%

“…The cochain type level of a map α : Y → X indeed provides a lower bound on the number of spherical fibrations which describe a factorization of α in a relevant sense; see [32,Proposition 2.11]. On the other hand, the chain type level of the identity map on a space Y gives an upper bound of the L.-S. category of Y in rational case; see [33,Corollary 2.9].…”

Section: Introductionmentioning

confidence: 99%

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The ghost length and duality on the chain and cochain type levels

Kuribayashi

2016

Homology, Homotopy and Applications

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Abstract. We establish equalities between cochain and chain type levels of maps by making use of exact functors which connect appropriate derived and coderived categories. Relevant conditions for levels of maps to be finite are extracted from the equalities which we call duality on the levels. Moreover, we give a lower bound of the cochain type level of the diagonal map on the classifying space of a Lie group by considering the ghostness of a shriek map which appears in derived string topology. A variant of Koszul duality for a differential graded algebra is also discussed.

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Section: Resultsmentioning

confidence: 99%

“…This work is a sequel to previous one [32,33] in which new topological invariants have been studied.…”

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The ghost length and duality on the chain and cochain type levels

Kuribayashi

2016

Homology, Homotopy and Applications

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“…2.1] where a cluster tilting subcategory was also shown. The Hall algebra of D was computed in [15], and some relations with algebraic topology were investigated in [11] and [17].…”

Section: Theorem C the Clusters Form A Cluster Structure In Dmentioning

confidence: 99%

On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon

2010

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Mathematics Subject Classification (2010) IntroductionLet k be a field and let D be a k-linear algebraic triangulated category with split idempotents. Let be the suspension functor of D and let s be a 2-spherical object of D, that is, the morphism space D(s, i s) is k for i = 0 and i = 2 and vanishes otherwise. Assume that s classically generates D, that is, each object of D can be built from s using (de)suspensions, direct sums, direct summands, and distinguished triangles.It was proved in [15, thm. 2.1] that D is uniquely determined by these properties. As we will explain, D is a good candidate for a cluster category of Dynkin type A ∞ . For instance, we show that there is a bijection between the cluster tilting subcategories of D and certain triangulations of the ∞-gon.We use this to give an example of a subcategory A which is weak cluster tilting, that is, satisfies A = ( −1 A ) ⊥ = ⊥ ( A ), but fails to be functorially finite. Perpendicular

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Upper and Lower Bounds of the (co)chain Type Level of a Space

Kuribayashi

2011

Algebr Represent Theor

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We establish an upper bound for the cochain type level of the total space of a pull-back fibration. It explains to us why the numerical invariants for principal bundles over the sphere are less than or equal to two.Moreover computational examples of the levels of path spaces and Borel constructions, including biquotient spaces and Davis-Januszkiewicz spaces, are presented. We also show that the chain type level of the homotopy fibre of a map is greater than the E-category in the sense of Kahl, which is an algebraic approximation of the Lusternik-Schnirelmann category of the map. The inequality fits between the grade and the projective dimension of the cohomology of the homotopy fibre.

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On the levels of maps and topological realization of objects in a triangulated category

Cited by 4 publications

References 36 publications

The ghost length and duality on the chain and cochain type levels

The ghost length and duality on the chain and cochain type levels

On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon

Upper and Lower Bounds of the (co)chain Type Level of a Space

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