In this paper we develop homotopy theoretical methods for studying diagrams. In particular we explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept we introduce is that of a model approximation. A model approximation of a category C with a given class of weak equivalences is a model category M together with a pair of adjoint functors M ⇄ C which s...

The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coh...

1 Rational Homotopy Theory 4 1.1 Cohomological Eilenberg-Moore Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 k-Rational Homotopy Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Rational Homotopy Theory and E∞-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Differential Graded Lie Algebras . . . . . . . . . . ...