A Cartan–Eilenberg approach to homotopical algebra

Guillén, F.; Navarro, Vı́ctor; Pascual, P.; Roig, Agustí

doi:10.1016/j.jpaa.2009.04.009

Cited by 15 publications

(22 citation statements)

References 24 publications

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“…Proof. By Lemma 2.8 the map ι A : A → P (A) is a homotopy equivalence for all A ∈ C. The result follows from Proposition 1.3.3 of [GNPR10].…”

Section: 4mentioning

confidence: 69%

Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy

Cirici

2014

Trans. Amer. Math. Soc.

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Abstract. We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is applied to the category of mixed Hodge diagrams of differential graded algebras. Using Sullivan's minimal models, we prove a multiplicative version of Beilinson's Theorem on mixed Hodge complexes. As a consequence, we obtain functoriality for the mixed Hodge structures on the rational homotopy type of complex algebraic varieties. In this context, the mixed Hodge structures on homotopy groups obtained by Morgan's theory follow from the derived functor of the indecomposables of mixed Hodge diagrams.

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“…Proof. By Lemma 2.8 the map ι A : A → P (A) is a homotopy equivalence for all A ∈ C. The result follows from Proposition 1.3.3 of [GNPR10].…”

Section: 4mentioning

confidence: 69%

Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy

Cirici

2014

Trans. Amer. Math. Soc.

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“…Note that Propodition 6.8 together with Lemma 6.11 make Sullivan minimal algebras in Alg, minimal in an abstract categorical sense (c.f. [Roi94b], [Roi93], [GNPR10]). Theorem 6.13.…”

Section: Algebras Over Variable Operadsmentioning

confidence: 99%

Sullivan minimal models of operad algebras

Cirici¹,

Roig²

2019

Publicacions Matemàtiques

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We prove the existence of Sullivan minimal models of operad algebras, for a quite wide family of operads in the category of complexes of vector spaces over a field of characteristic zero. Our construction is an adaptation of Sullivan's original step by step construction to the setting of operad algebras. The family of operads that we consider includes all operads concentrated in degree 0 as well as their minimal models. In particular, this gives Sullivan minimal models for algebras over Com, Ass and Lie, as well as over their minimal models Com∞, Ass∞ and Lie∞. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study.2010 Mathematics Subject Classification. 18D50, 55P62.

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“…Generalizing the notion of filtered injective complex of Illusie, we introduce r-injective complexes and show that these are fibrant objects in the sense of [GNPR10], Definition 2.2.1, with respect to the classes of r-homotopy equivalences and E r -quasi-isomorphisms. We then prove the existence of r-injective models for bounded below filtered complexes (a similar result is proved by Paranjape in [Par96]), giving rise to a Cartan-Eilenberg structure.…”

Section: R-injective Modelsmentioning

confidence: 99%

“…Hence the triple (C + (FA), S r , E r ) is a Cartan-Eilenberg category with fibrant models in C + r (FInjA). The equivalence of categories follows from Theorem 2.3.4 of [GNPR10].…”

Section: R-injective Modelsmentioning

confidence: 99%

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Homotopy theory of mixed Hodge complexes

Cirici¹,

Guillén²

2016

Tohoku Math. J. (2)

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Abstract. We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg structure, a notion introduced in [GNPR10] leading to a good calculation of the homotopy category in terms of (co)fibrant objects. This result provides a conceptual framework from which Beilinson's [Bei86] and Carlson's [Car80] results on mixed Hodge complexes and extensions of mixed Hodge structures follow easily.

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A Cartan–Eilenberg approach to homotopical algebra

Cited by 15 publications

References 24 publications

Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy

Cofibrant models of diagrams: Mixed Hodge structures in rational homotopy

Sullivan minimal models of operad algebras

Homotopy theory of mixed Hodge complexes

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