this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:

Abstract. 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 coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.

Spaces of maps into classifying spaces for equivariant crossed complexes, II:

The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, [26], or any equivalent text if one can be found! What do the notes set out to do? “Aims and Objectives! ” or should it be “Learning Outcomes”? • To revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today’s resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; • To introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the older ideas; • To introduce Joyal’s quasicategories, (previously called weak Kan complexes but I agree with André that his nomenclature is better so will adopt it) and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and myself; • To ask lots of questions of myself and of the reader. The notes include some material from the ‘Cubo ’ article, [35], which was itself based on notes for a course at the Corso estivo Categorie e Topologia in 1991, but the overlap has been kept as small as is feasible as the purpose and the audience of the two sets of notes are different and the abstract homotopy theory has ‘moved on’, in part, to try the new methods out on those same ‘old ’ problems and to attack new ones as well. As usual when you try to specify ‘learning outcomes ’ you end up asking who has done the learning, the audience? Perhaps. The lecturer, most certainly! 1