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34
A generalization of Quillen’s small object argument
 J. Pure Appl. Algebra
"... Abstract. We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen’s small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were ..."
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Abstract. We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen’s small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were proven to be noncofibrantly generated [2, 6, 8, 20]. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces [10] and diagrams of chain complexes. We also formulate a nonfunctorial version of the argument, which applies in two different model structures on the category of prospaces [11, 20]. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a classcofibrantly
Duality and ProSpectra
, 2004
"... Abstract Cofiltered diagrams of spectra, also called prospectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of prospectra and to study its relation to the usual homotopy theory of spectra ..."
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Abstract Cofiltered diagrams of spectra, also called prospectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of prospectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of prospectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of SpanierWhitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the SpanierWhitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the SpanierWhitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of indspectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories.
An étale approach to the Novikov Conjecture
 Comm. Pure Appl. Math
"... Abstract. We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the ..."
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Abstract. We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups. Ten years ago, the most popular approach to the Novikov conjecture went via compactifications. If a compact aspherical manifold, say, has a universal cover which suitably equivariantly compactifies, already Farrell and Hsiang [FH] proved that the Novikov conjecture follows. Subsequent work by many authors weakened
EpsilonDelta surgery over Z
, 2003
"... This manuscript fills in the details of the lecture I gave on “squeezing structures in Trieste in June, 2001. The goal is to develop a controlled surgery theory of the sort discussed/used in [3]. This material will appear as part of the writeup of my CBMS lectures. ..."
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This manuscript fills in the details of the lecture I gave on “squeezing structures in Trieste in June, 2001. The goal is to develop a controlled surgery theory of the sort discussed/used in [3]. This material will appear as part of the writeup of my CBMS lectures.
Flexible sheaves
, 1996
"... This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector b ..."
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This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to JardineIllusie’s theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in “Homotopy over the complex numbers and generalized de Rham cohomology”. 1 Added in August 1996: I am finally sending this to “Duke eprints ” because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopysheafification (by doing a certain operation n+2 times) seems to be useful, for example I need
Resolutions of spaces by cubes of fibrations
 Hagen Germany T. Porter School of Mathematics University of Wales Bangor Bangor
, 1986
"... J.L. Loday has used wcubes of fibrations, where n is a nonnegative integer, in his study of spaces with finitely many nontrivial homotopy groups [4]. His main result is the construction of an algebraic category equivalent to the weak homotopy category of pathconnected spaces Z with TI^Z = 0 for ..."
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J.L. Loday has used wcubes of fibrations, where n is a nonnegative integer, in his study of spaces with finitely many nontrivial homotopy groups [4]. His main result is the construction of an algebraic category equivalent to the weak homotopy category of pathconnected spaces Z with TI^Z = 0 for /> w+1 [4, 1.7]. One step in
CONTINUOUS CONTROL AND THE ALGEBRAIC LTHEORY ASSEMBLY MAP
, 2003
"... Abstract. In this work, the assembly map in Ltheory for the family of finite subgroups is proven to be a split injection for a class of groups. Groups in this class, including virtually polycyclic groups, have universal spaces that satisfy certain geometric conditions. The proof follows the method ..."
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Abstract. In this work, the assembly map in Ltheory for the family of finite subgroups is proven to be a split injection for a class of groups. Groups in this class, including virtually polycyclic groups, have universal spaces that satisfy certain geometric conditions. The proof follows the method developed by CarlssonPedersen to split the assembly map in the case of torsion free groups. Here, the continuously controlled techniques and results are extended to handle groups with torsion. 1.
Pro objects in simplicial presheaves
, 2009
"... This paper describes model structures for the category of pro objects in simplicial presheaves on an arbitrary small Grothendieck site, all based on the injective model structure for simplicial presheaves. The most fundamental of these structures is an analogue of the Edwards ..."
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This paper describes model structures for the category of pro objects in simplicial presheaves on an arbitrary small Grothendieck site, all based on the injective model structure for simplicial presheaves. The most fundamental of these structures is an analogue of the Edwards
CALCULATING LIMITS AND COLIMITS IN PROCATEGORIES
, 2001
"... Abstract. We present some constructions of limits and colimits in procategories. These are critical tools in several applications. In particular, certain technical arguments concerning strict promaps are essential for a theorem about étale homotopy types. Also, we show that cofiltered limits in pr ..."
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Abstract. We present some constructions of limits and colimits in procategories. These are critical tools in several applications. In particular, certain technical arguments concerning strict promaps are essential for a theorem about étale homotopy types. Also, we show that cofiltered limits in procategories commute with finite colimits.