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**1 - 2**of**2**### Pro-algebraic homotopy types

, 2008

"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic compl ..."

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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic completion of Kan’s loop group of X. There is also a notion of unpointed pro-algebraic homotopy type, replacing groups by groupoids. If X is simply connected, its pro-algebraic homotopy type is equivalent to Sullivan’s rational homotopy type. Toën’s schematic homotopy type is equivalent to the pro-algebraic homotopy type, with the pro-algebraic homotopy groups isomorphic to the schematic homotopy groups. There are spectral sequences relating pro-algebraic homotopy groups to the cohomology ring of the universal semisimple local system. For compact Kähler manifolds, the pro-algebraic homotopy groups can be described explicitly in terms of this cohomology ring. There is a notion of minimal models for pro-algebraic homotopy types, which allows us to study their automorphisms.

### Pro-algebraic homotopy types

, 2008

"... The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic compl ..."

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The purpose of this paper is to generalise Sullivan’s rational homotopy theory to non-nilpotent spaces, in such a way as to be amenable to Hodge theory. The pro-algebraic homotopy type (over a field k of characteristic zero) of a pointed space (X, x) is constructed as a certain pro-k-algebraic completion of Kan’s loop group of X. This has the property that the pro-algebraic fundamental group is the pro-algebraic completion of π1(X, x), while the nth pro-algebraic homotopy group is the completion of πn(X, x) ⊗Z k with respect to its π1(X, x)-subrepresentations of finite codimension. There is also a notion of unpointed pro-algebraic homotopy type, replacing groups by groupoids. If X is simply connected, its pro-algebraic homotopy type is equivalent to Sullivan’s rational homotopy type. Toën’s schematic homotopy type can be recovered from the pro-algebraic homotopy type, as the pro-algebraic homotopy type of X is equivalent to the homotopy type of the cochain algebra with coefficients in the universal semisimple local system on X. As an application, we show that the proalgebraic homotopy groups of a compact Kähler manifold have a canonical weight decomposition, which can be recovered explicitly from the cohomology ring of the universal semisimple local system.