In his landmark 1969 Annals paper, Quillen showed that the rational homotopy type of a simply connected

space could be detected at the level of its singular rational chains, and furthermore, that rational chains fit

into a derived equivalence with cocommutative dg-coalgebras over the rationals, after restricting to

1-connected objects. In 1977 Sullivan subsequently proved the analogous result in the case of rational

cochains and commutative dg-algebras over the rationals. Since then topologists have worked on attempting

to establish analogous results for finite fields (Kriz, Goerss, Mandell), and more recently some partial

results have been established in the integral chains case (Mandell, Karoubi). Nevertheless, establishing

that integral chains fit into a derived equivalence has proved resistant to all attacks. In this talk I will

outline how we recently resolved, in the affirmative, the integral chains problem. Our approach exploits

a mixture of (co)simplicial techniques together with some ideas related to the framework developed in

Ching-Harper's recent resolution of the 0-connected Francis-Gaitsgory conjecture. If time permits, I

will also describe how we recently resolved that iterated suspension satisfies homotopical descent on

objects and morphisms. This is joint work with J.E. Harper.

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