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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:

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Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense, thus partially bridging the gap between classical homotopy theory and modern abstract homotopy theory. To do this we introduce a notion of “enriched homotopical categories”, which are more general than enriched model categories, but are still a good place to do enriched homotopy theory. This demonstrates that the presence of enrichment often simplifies rather than complicates matters, and goes some way toward achieving a better understanding of “the role of homotopy in homotopy theory.” Contents

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.

Abstract. The morphic Abel-Jacobi map is the analogue of the classical Abel-Jacobi map one obtains by using Lawson and morphic (co)homology in place of the usual singular (co)homology. It thus gives a map from the group of r-cycles on a complex variety that are algebraically equivalent to zero to a certain “Jacobian ” built from the Lawson homology groups viewed as inductive limits of mixed Hodge structures. In this paper, we define the morphic Abel-Jacobi map, establish its foundational properties, and then apply these results to the study of algebraic cycles. In particular, we show the classical Abel-Jacobi map (when restricted to cycles algebraically equivalent to zero) factors through the morphic version, and show that the morphic version detects cycles that cannot be detected by its classical counterpart — that is, we give examples of cycles in the kernel of the classical Abel-Jacobi map that are not in the kernel of the morphic one. We also investigate the behavior of the morphic Abel-Jacobi map on the torsion subgroup of the Chow group of cycles algebraically equivalent to zero modulo

Equivariant versions of the Suspension Theorem [L 1 ] for algebraic cycles on projective varieties are proved. Let G be a finite group, V a projective G-module, and X ae P C (V ) an invariant subvariety. Consider the algebraic join \Sigma= V0 X = X#P C (V 0 ) of X with the regular representation V 0 = C G of G. The main result asserts that algebraic suspension induces a G- homotopy equivalence Z s (X) \Gamma! Z s (\Sigma= V0 X) of topological groups of algebraic cycles of codimension-s for all s dimX \Gamma e(X) where e(X) is the maximal dimension of g-fixed point sets in \Sigma= V0 X for g 6= 1. This leads to a Stability Theorem for equivariant algebraic suspension. The result enables the determination of coefficients in certain equivariant cohomology theories based on algebraic cycles, and it enables the definition of cohomology operations in such theories. The methods also yield a Quaternionic Suspension Theorem for cycles in P C (H n ) under the antiholomor...

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Abstract. We extend to the non-connective case a lemma of Bökstedt about the equivalence of the telescope with a more complicated homotopy colimit of symmetric spectra used in the construction of THH. 1.