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Categorical homotopy theory
 HOMOLOGY, HOMOTOPY APPL
, 2006
"... This paper is an exposition of the ideas and methods of Cisinksi, in the context of Apresheaves on a small ..."
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Cited by 322 (9 self)
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This paper is an exposition of the ideas and methods of Cisinksi, in the context of Apresheaves on a small
The homotopy theory of fusion systems
"... The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group con ..."
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Cited by 132 (19 self)
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The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group consists, roughly speaking, of a finite pgroup S and fusion data on subgroups of S, encoded in a way explained below. Our starting point is our earlier paper [BLO] on pcompleted classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig [Pu], [Pu2] of systems of fusion among subgroups of a given pgroup. The pcompletion of a space X is a space X ∧ p which isolates the properties of X at the prime p, and more precisely the properties which determine its mod p cohomology. For example, a map of spaces X f −− → Y induces a homotopy equivalence
Combinatorial model categories have presentations
 Adv. in Math. 164
, 2001
"... Abstract. We show that every combinatorial model category is Quillen equivalent to a localization of a diagram category (where ‘diagram category’ means diagrams of simplicial sets). This says that every combinatorial model ..."
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Cited by 102 (10 self)
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Abstract. We show that every combinatorial model category is Quillen equivalent to a localization of a diagram category (where ‘diagram category’ means diagrams of simplicial sets). This says that every combinatorial model
Spaces over a category and assembly maps in isomorphism conjectures in Kand Ltheory’, KTheory 15
, 1998
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Embeddings from the point of view of immersion theory: Part I
, 1999
"... Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M,N) should come from an analysis of the cofunctor V ↦ → emb(V,N) from the poset O of open subsets of M to spaces. We therefore abstract some of the properties of t ..."
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Cited by 86 (13 self)
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Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M,N) should come from an analysis of the cofunctor V ↦ → emb(V,N) from the poset O of open subsets of M to spaces. We therefore abstract some of the properties of this cofunctor, and develop a suitable calculus of such cofunctors, Goodwillie style, with Taylor series and so on. The terms of the Taylor series for the cofunctor V ↦ → emb(V,N) are explicitly determined. In a sequel to this paper, we introduce the concept of an analytic cofunctor from O to spaces, and show that the Taylor series of an analytic cofunctor F converges to F. Deep excision theorems due to Goodwillie and Goodwillie–Klein imply that the cofunctor V ↦ → emb(V,N) is analytic when dim(N) − dim(M) ≥ 3.
The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres’, Invent
 Math
, 1999
"... Abstract. We investigate Goodwillie’s \Taylor tower " of the identity functor from spaces to spaces. More specically, we reformulate Johnson’s description of the Goodwillie derivatives of the identity, and prove that in the case of an odddimensional sphere the only layers in the tower that are ..."
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Cited by 56 (9 self)
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Abstract. We investigate Goodwillie’s \Taylor tower " of the identity functor from spaces to spaces. More specically, we reformulate Johnson’s description of the Goodwillie derivatives of the identity, and prove that in the case of an odddimensional sphere the only layers in the tower that are not contractible are those indexed by powers of a prime. Moreover, in the case of a sphere the tower is nite in vkperiodic homotopy. Contents
Universal homotopy theories
 Adv. Math
"... Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy the ..."
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Cited by 55 (3 self)
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Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents
DG coalgebras as formal stacks
 J. Pure Appl. Algebra
"... 1.1. In this paper we provide the category of unital (dg, unbounded) coalgebras dgcu(k) ..."
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Cited by 54 (4 self)
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1.1. In this paper we provide the category of unital (dg, unbounded) coalgebras dgcu(k)
Descent of Deligne groupoids
 Int. Math. Res. Notices
, 1997
"... Abstract. To any nonnegatively graded dg Lie algebra g over a field k of characteristic zero we assign a functor Σg: art/k → Kan from the category of commutative local artinian kalgebras with the residue field k to the category of Kan simplicial sets. There is a natural homotopy equivalence betwee ..."
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Cited by 49 (3 self)
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Abstract. To any nonnegatively graded dg Lie algebra g over a field k of characteristic zero we assign a functor Σg: art/k → Kan from the category of commutative local artinian kalgebras with the residue field k to the category of Kan simplicial sets. There is a natural homotopy equivalence between Σg and the Deligne groupoid corresponding to g. The main result of the paper claims that the functor Σ commutes up to homotopy with the ”total space ” functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman [S1, S2, HS3] which implies that if a deformation problem is described “locally ” by a sheaf of dg Lie algebras g on a topological space X then the global deformation problem is described by the homotopy Lie algebra RΓ(X, g). 1.