Spaces and cochains -- yet another approach

 Rationally, the homotopy type of any reasonable space is completely determined by (a minimal model of) the Sullivan cochain algebra of the space. If you want to be non-rational, then Mandell's result says that the $E_\infty$-algebra structure of the cochains determines the homotopy type. In joint work with Steffen Sagave we construct a strictly commutative model of the cochains of a space using the diagram category of finite sets and injections in order to free things up. We show that this cochain algebra determines the homotopy type of (finite type, nilpotent) spaces