In this paper we elaborate a general homotopy-theoretic framework in which to
study problems of descent and completion and of their duals, codescent and
cocompletion. Our approach to homotopic (co)descent and to derived
(co)completion can be viewed as $\infty$-category-theoretic, as our framework
is constructed in the universe of simplicially enriched categories, which are a
model for $(\infty, 1)$-categories.

We provide general criteria, reminiscent of Mandell's theorem on
$E_{\infty}$-algebra models of $p$-complete spaces, under which homotopic
(co)descent is satisfied. Furthermore, we construct general descent and
codescent spectral sequences, which we interpret in terms of derived
(co)completion and homotopic (co)descent.

We prove that Baum-Connes and Farrell-Jones-type isomorphism conjectures for
assembly can be expressed in the language of derived cocompletion and show that
a number of very well-known spectral sequences, such as the unstable and stable
Adams spectral sequences, the Adams-Novikov spectral sequence and the descent
spectral sequence of a map, are examples of general (co)descent spectral
sequences. There is also a close relationship between the Lichtenbaum-Quillen
conjecture and homotopic descent along the Dwyer-Friedlander map from algebraic
K-theory to etale K-theory.