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14
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 colimits  comparison lemmas for combinatorial applications
, 1997
"... We provide a "toolkit " of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used on quite different fields of applications. We demonstrate this with respect to 1. Bjorner's " ..."
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Cited by 18 (2 self)
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We provide a &quot;toolkit &quot; of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used on quite different fields of applications. We demonstrate this with respect to 1. Bjorner's "Generalized Homotopy Complementation Formula" [4], 2. the topology of toric varieties, 3. the study of homotopy types of arrangements of subspaces, 4. the analysis of homotopy types of subgroup complexes.
New perspectives in self linking
 Adv. Math
"... Abstract. We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified npoint configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth ma ..."
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Abstract. We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified npoint configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integervalued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finitetype invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot. Contents
Centric Maps and Realization of Diagrams in the Homotopy Category
 Proc. Amer. Math. Soc
, 1992
"... Introduction Let D be a small category. Suppose that ¯ X is a Ddiagram in the homotopy category (in other words, a functor from D to the homotopy category of simplicial sets). The question of whether or not ¯ X can be realized by a Ddiagram of simplicial sets has been treated by [5]. The purpose ..."
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Cited by 10 (2 self)
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Introduction Let D be a small category. Suppose that ¯ X is a Ddiagram in the homotopy category (in other words, a functor from D to the homotopy category of simplicial sets). The question of whether or not ¯ X can be realized by a Ddiagram of simplicial sets has been treated by [5]. The purpose of this note is to study a special situation in which the treatment can be simplified quite a bit. We look at two examples to which this simplified treatment is applicable; both examples involve homotopy decomposition diagrams for compact Lie groups. Our results show that in at least one of these examples ([13]) the decomposition diagram is completely determined by its underlying diagram in the homotopy category. It is possible that this "rigidity" result will eventually contribute to a general homotopy theoretic characterization theorem for
Comparison lemmas and applications for diagrams of spaces
 PREPRINT NO. 11, INSTITÜT FUR EXPERIMENTELLE MATHEMATIK
, 1995
"... We provide a "toolkit " of basic lemmas for the comparison of homotopy types of (homotopy) limits of diagrams of spaces over finite partially ordered sets, among them several new ones. In the setting of this paper, we obtain simple inductive proofs that provide explicit homotopy eq ..."
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Cited by 4 (2 self)
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We provide a &quot;toolkit &quot; of basic lemmas for the comparison of homotopy types of (homotopy) limits of diagrams of spaces over finite partially ordered sets, among them several new ones. In the setting of this paper, we obtain simple inductive proofs that provide explicit homotopy equivalences. (In an appendix we provide the link to the general setting of diagrams of spaces over an arbitrary small category.) We show how this toolkit of old and new diagram lemmas can be used on quite different fields of applications. In this paper we demonstrate this with respect to ffl the &quot;generalized homotopycomplementation formula &quot; by Bjorner [4], ffl the topology of toric varieties (which turn out to be homeomorphic to homotopy limits, and for which the homotopy limit construction provides a suitable spectral sequence), ffl in the study of homotopy types of arrangements of subspaces, where we establish a new, general combinatorial formula for the homotopy types of &quot;Grassmannian &quot; arrangements,
Localization with respect to a class of maps I – Equivariant localization of diagrams of spaces
"... Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3, 12, 20, 29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model cat ..."
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Cited by 4 (4 self)
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Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3, 12, 20, 29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces [11]. This model category is not cofibrantly generated [7]. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the class of maps which induces ordinary localizations on the generalized fixedpoints sets.
Codescent theory II: Cofibrant approximations
, 2003
"... Abstract. We establish a general method to produce cofibrant approximations in the model category US(C, D) of Svalued Cindexed diagrams with Dweak equivalences and Dfibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial mode ..."
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Cited by 2 (2 self)
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Abstract. We establish a general method to produce cofibrant approximations in the model category US(C, D) of Svalued Cindexed diagrams with Dweak equivalences and Dfibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial model category and D ⊂ C are small categories. An application to the notion of homotopy colimit is presented. 1.
CODESCENT THEORY I: FOUNDATIONS
, 2003
"... C to S with a model structure, defining weak equivalences and fibrations objectwise but only on D. Our first concern is the effect of moving C, D and S. The main notion introduced here is the “Dcodescent ” property for objects in S C. Our longterm program aims at reformulating as codescent stateme ..."
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C to S with a model structure, defining weak equivalences and fibrations objectwise but only on D. Our first concern is the effect of moving C, D and S. The main notion introduced here is the “Dcodescent ” property for objects in S C. Our longterm program aims at reformulating as codescent statements the Conjectures of BaumConnes and FarrellJones, and at tackling them with new methods. Here, we set the grounds of a systematic theory of codescent, including pullbacks, pushforwards and various invariance properties. 1.
Equivariant Simplicial Cohomology With Local Coefficients and its Classification
, 905
"... We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of BredonIllman cohomology with local coefficients is isomorp ..."
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We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of BredonIllman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients.
DERIVATIVES OF EMBEDDING FUNCTORS I: THE STABLE CASE
, 2007
"... For smooth manifolds M and N, let Emb(M, N) be the homotopy fiber of the map Emb(M, N) − → Imm(M, N). Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula V ↦ → Σ ∞ Emb(M, N × V). In this paper, we describe the derivatives of this functor, in ..."
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For smooth manifolds M and N, let Emb(M, N) be the homotopy fiber of the map Emb(M, N) − → Imm(M, N). Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula V ↦ → Σ ∞ Emb(M, N × V). In this paper, we describe the derivatives of this functor, in the sense of M. Weiss ’ orthogonal calculus. Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in M, and a certain “homotopy bundle of spectra ” over this space of trees. The nth derivative is then described as the “spectrum of restricted sections ” of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor Emb(M, N × V), involving a similar construction with certain spaces