and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
The fundamental theorem of rational homotopy theory modeled by dgc-algebras.
(nilpotent and finite rational homotopy types)
Write
for the full subcategory of the classical homotopy category (homotopy category of the classical model structure on simplicial sets) on those homotopy types which are
connected: ;
nilpotent: the fundamental group is a nilpotent group and the higher homotopy groups are nilpotent modules over it;
rationally of finite type: the rational cohomology-groups are finite-dimensional, , for all .
and
for the further full subcategory on those homotopy types that are already rational.
Similarly, write
for the full subcategory of the homotopy category of the projective model structure on connective dgc-algebras on those dgc-algebras over the rational numbers which are
connected:
finite type: the cochain cohomology groups are finite dimensional, , for all .
(fundamental theorem of dg-algebraic rational homotopy theory)
of the Quillen adjunction between simplicial sets and connective dgc-algebras (whose left adjoint is the PL de Rham complex-functor) has the following properties:
on connected, nilpotent rationally finite homotopy types (1) the derived adjunction unit is rationalization
on the full subcategories of nilpotent and finite rational homotopy types from Def. it restricts to an equivalence of categories:
(Bousfield-Gugenheim 76, Theorems 9.4 & 11.2)
Often it is desireable to work with dgc-algebras not over the rational numbers but over the real numbers, because these relate to de Rham theory (e.g.: the PL de Rham complex of a smooth manifold is equivalent to the de Rham complex). While a PL de Rham complex-Quillen adjunction (“piecewise -linear”) exists over all ground fields of characteristic zero, with induced derived adjunction
this does not model -localization of spaces unless . However, it does still relate to rationalization under extension of scalars, given by the derived adjunction (via this Prop.)
in that the following holds:
For be a field of characteristic zero, the following diagram of derived functors commutes up to natural isomorphism:
This is effectivley the statement of Bousfield&Gugenheim 1976, Lem. 11.7.
This state of affairs may be recast as follows (FSS 2020):
For any field of characteristic zero, abbreviate
keeping in mind that this is a localization of spaces only if .
Then for a pair of connected nilpotent ℚ-finite homotopy types, define the -Chern-Dold character on the non-abelian -cohomology of to be the cohomology operation induced by the derived adjunction unit of the PL de Rham adjunction (this Prop.):
Moreover, define extension of scalars on non-abelian rational cohomology to be the comoposite
(where denotes hom-sets in the respective homotopy category)
of:
the hom-isomorphisms of the derived PL de Rham adjunction;
the corresponding hom-component of the derived functor of extension of scalars (this Prop.).
(This is essentially the construction of “tensoring a homotopy type with ” that is mentioned in DGMS 1975, Footnote 5.)
Then:
The -Chern-Dold character (4) factors through the rational Chern-Dold character via the extension-of-scalars-transformation (5).
Consider the following diagram of hom-sets (shown for , just for definiteness):
Here:
the commutativity of the bottom part is Prop. ;
the triangle on the left as well as as the outer diagram commute by basic properties of adjoint functors (this Lemma);
the square on the right commutes by definition (5);
Together this implies that the top rectangle commutes, which is the claim to be shown.
The full-blown equivalence first appears in
Concise review (without model category-theory, but discussion as an sSet-enriched adjunction nonetheless) and generalization both to Borel-equivariant rational homotopy theory (of covering spaces of non-nilpotent spaces), as well as to real homotopy theory:
Further related discussion over the real numbers:
Inventiones mathematicae volume 29, pages 245–274 (1975) (doi:10.1007/BF01389853)
Re-derivation in a context of derived algebraic geometry:
Review and interpretation in terms of non-abelian Chern-Dold character-theory:
Last revised on August 17, 2023 at 12:01:59. See the history of this page for a list of all contributions to it.