Model category structures and spectral sequences

Cirici, Joana; Santander, Daniela Egas; Livernet, Muriel; Whitehouse, Sarah

doi:10.1017/prm.2019.45

Cited by 9 publications

(17 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We write n-mC R for the category of n-multicomplexes. The case n = 2 gives the category of bicomplexes and the results here recover those of [2]. Multicomplexes can be thought of as the case n = ∞ and we make frequent use of this notational device.…”

Section: Introductionsupporting

confidence: 61%

Model category structures on multicomplexes

Fu¹,

Guan²,

Livernet³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

We present a family of model structures on the category of multicomplexes.There is a cofibrantly generated model structure in which the weak equivalences are the morphisms inducing an isomorphism at a fixed stage of an associated spectral sequence. Corresponding model structures are given for truncated versions of multicomplexes, interpolating between bicomplexes and multicomplexes. For a fixed stage of the spectral sequence, the model structures on all these categories are shown to be Quillen equivalent.

show abstract

Section: Introductionsupporting

confidence: 61%

Model category structures on multicomplexes

Fu¹,

Guan²,

Livernet³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This motivates the following refined version of quasi-isomorphism (cf. [15,29] for different but related notions). Definition 10.…”

Section: And Equality Holds If and Only Ifmentioning

confidence: 99%

“…With this background at hand, let us survey the main applications: Firstly, the following refined notion of quasi-isomorphism is particularly well behaved (see also [15,29] for studies of different notions of quasi-isomorphism from a rational homotopy-theoretic point of view).…”

Section: Introductionmentioning

confidence: 99%

On the structure of double complexes

Stelzig

2021

J. London Math. Soc.

View full text Add to dashboard Cite

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of 'universal' quasi-isomorphism and the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants).Applying the theory to the double complexes of smooth complex valued forms on compact complex manifolds, we obtain a Poincaré duality for higher pages of the Frölicher spectral sequence, construct a functorial three-space decomposition of the middle cohomology, give an example of a map between compact complex manifolds which does not respect the Hodge filtration strictly, compute the Bott-Chern and Aeppli cohomology for Calabi-Eckmann manifolds, introduce new numerical bimeromorphic invariants, show that the non-Kählerness degrees are not bimeromorphic invariants in dimensions higher than three and that the ∂∂lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds.

show abstract

“…Remark 2.9. In the language of witnesses adopted in [CELW18b], the difference between the Z r (D)-cycles and the Z r -cycles is essentially the difference between specifying witnesses and just requiring the existence of them. More precisely, Z p, * r (D)/F p−r (D) corresponds to the witness r-cycles for split filtered complexes.…”

Section: The Spectral Sequence Associated To a Multicomplexmentioning

confidence: 99%

“…This corresponds to the bicomplex ZW r of [CELW18b], a representing object for the witness r-cycles.…”

Section: Examplesmentioning

confidence: 99%