Cofibrant models of diagrams: mixed Hodge structures in rational homotopy. (English) Zbl 1349.18030
Summary: We study the homotopy theory of a certain type of diagram category 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.
MSC:
| 18G55 | Nonabelian homotopical algebra (MSC2010) |
| 32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |
| 55P62 | Rational homotopy theory |
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
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