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38
Homotopy Coherent Category Theory
, 1996
"... 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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Cited by 22 (6 self)
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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:
Homotopy limits and colimits and enriched homotopy theory
, 2006
"... 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 equiv ..."
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Cited by 15 (2 self)
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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
Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... 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 ..."
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Cited by 10 (7 self)
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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.
THE MORPHIC ABELJACOBI MAP
"... Abstract. The morphic AbelJacobi map is the analogue of the classical AbelJacobi 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 rcycles on a complex variety that are algebraically equivalent to zero to a ..."
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Cited by 9 (0 self)
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Abstract. The morphic AbelJacobi map is the analogue of the classical AbelJacobi 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 rcycles 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 AbelJacobi map, establish its foundational properties, and then apply these results to the study of algebraic cycles. In particular, we show the classical AbelJacobi 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 AbelJacobi map that are not in the kernel of the morphic one. We also investigate the behavior of the morphic AbelJacobi map on the torsion subgroup of the Chow group of cycles algebraically equivalent to zero modulo
On Equivariant Algebraic Suspension
"... 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 Gmodule, 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 representati ..."
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Cited by 8 (4 self)
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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 Gmodule, 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 codimensions for all s dimX \Gamma e(X) where e(X) is the maximal dimension of gfixed 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...
Algebraic model structures
"... Abstract. We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove “algebraic ” analogs of classical results. Using a modified version of Quillen’s small object argument, we show that e ..."
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Cited by 7 (5 self)
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Abstract. We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove “algebraic ” analogs of classical results. Using a modified version of Quillen’s small object argument, we show that every cofibrantly generated model structure in the usual sense underlies a cofibrantly generated algebraic model structure. We show how to pass a cofibrantly generated algebraic model structure across an adjunction, and we characterize the algebraic Quillen adjunction that results. We prove that pointwise algebraic weak factorization systems on diagram categories are cofibrantly generated if the original ones are, and we give an algebraic generalization of the projective model structure. Finally, we prove that certain fundamental comparison maps present in any cofibrantly generated model category are cofibrations when the cofibrations are monomorphisms, a conclusion that does not seem to be provable in the classical, nonalgebraic, theory. Contents