Pro-algebraic homotopy types

Abstract

The purpose of this paper is to generalise Sullivan's rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toen's schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Maurer-Cartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the Baues-Lemaire conjecture. For compact Kaehler manifolds, the schematic homotopy groups can be described explicitly in terms of this cohomology ring, giving canonical weight decompositions. There are also notions of minimal models, unpointed homotopy types and algebraic automorphism groups. For a space with algebraically good fundamental group and higher homotopy groups of finite rank, the schematic homotopy groups are shown to be π_n(X)\otimes k.

72 pages; v6 Ref added to Baues-Lemaire conjecture (Rk 4.42); v7 minor changes, Sect. 5.1 corrected; v8 final version, to appear in Proc. LMS

References in corpus (2)

• Algebras and modules in monoidal model categories
• Deformation theory via differential graded Lie algebras

Cited by in corpus (16)

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• Tannaka duality and stable infinity-categories
• Algebraic and topological aspects of the schematization functor
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• Moduli problems for operadic algebras
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• Proalgebraic crossed modules of quasirational presentations
• Vanishing theorems for representation homology and the derived cotangent complex
• The de Rham homotopy theory and differential graded category
• Constructing derived moduli stacks
• The homotopy theory of strong homotopy algebras and bialgebras
• Non-abelian reciprocity laws and higher Brauer-Manin obstructions
• Hodge structures on analytic moduli of real pluriharmonic bundles
• Rational homotopy theory and differential graded category
• Analytic non-abelian Hodge theory
• All known realizations of complete Lie algebras coincide