Twisting Morphisms

Abstract

In this chapter, we introduce the bar construction and the cobar construction as follows. A twisting morphism is a linear map f:C→A, from a dga coalgebra C to a dga algebra A, which satisfies the Maurer–Cartan equation:

$$\partial (f) + f\star f =0 .$$

The set of twisting morphisms Tw(C,A) is shown to be representable both in C and in A:

$$\operatorname{Hom}_{\mathsf{dga}\ \mathsf{alg}}(\Omega C, A) \cong \mathrm{Tw}(C,A) \cong \operatorname{Hom}_{\mathsf{dga}\ \mathsf{coalg}}(C, \mathrm{B}A).$$

Then we investigate the twisting morphisms which give rise to quasi-isomorphisms under the aforementioned identifications. We call them Koszul morphisms.

…remember young fellow, Ω is left adjoint …

Dale Husemöller, MPIM (Bonn),

private communication