A Dold-Kan-type correspondence for superalgebras of differentiable functions and a "differential graded" approach to derived differential geometry
Seminar Room 1, Newton Institute
AbstractThe commutative algebra appropriate for differential geometry is provided by the algebraic theory of $C^\infty$-algebras -- an enhancement of the theory of commutative algebras which admits all $C^\infty$ functions of $n$ variables (rather than just the polynomials) as its $n$-ary operations. Derived differential geometry requires a homotopy version of these algebras for its underlying commutative algebra. We present a model for the latter based on the notion of a "differential graded structure" on a superalgebra of differentiable functions, understood -- following Severa -- as a (co)action of the monoid of endomorphisms of the odd line. This view of a differential graded structure enables us to construct, in a conceptually transparent way, a Dold-Kan-type correspondence relating our approach with models based on simplicial $C^\infty$-algebras, generalizing a classical result of Quillen for commutative and Lie algebras. It may also shed new light on Dold-Kan-type co rrespondences in other contexts (e.g. operads and algebras over them). A similar differential graded approach exists for every geometry whose ground ring contains the rationals, such as real analytic or holomorphic.
This talk is partly based on joint work with David Carchedi (arXiv:1211.6134 and arXiv:1212.3745).