The rational homology of function spaces
The category DGCA of differential \({\mathbb{Z}}\)-graded commutative algebras over \({\mathbb{Q}}\) is a closed model category. For any object B of finite type and bounded below, the functor \(-{\hat \otimes}B\) has a left adjoint \((-:B).\) Using this functor, the authors prove: Theorem. Let S denote the category of simplicial sets. Let \(X,Y\in obj(S)\) be connected, with X finite and Y fibrant and nilpotent of finite type. Then for any map \(\psi\) : \(X\to Y\) and any weak equivalence \(\beta\) : \(M^*(X)\to B^*(X)\) with \(B^*(X)\) non-negative of finite type, there is an equivalence in ho-DGCA \((M^*(Y):B^*(X))_{\beta \psi^*}\simeq M^*(Map(X,Y)_{\psi})\). This sheds new and interesting light on Haefliger's construction.
differential \({\mathbb{Z}}\)-graded commutative algebras over \({\mathbb{Q}}\)
scientific article; zbMATH DE number 4103975