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14
A Cellular Nerve for Higher Categories
, 2002
"... ... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there ..."
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... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of Aalgebras for each ooperad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of Aalgebras (i.e. weak ocategories) is
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Stability for holomorphic spheres and Morse theory
 in Geometry and topology: Aarhus
, 1998
"... Abstract. In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (X, ω) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operations, the space of based holomorp ..."
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Abstract. In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (X, ω) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operations, the space of based holomorphic maps from a sphere to X, becomes homotopy equivalent to the space of all continuous maps, lim − → Holx0 (P1, X) ≃ Ω 2 X. This limit will be viewed as a kind of stabilization of Holx0 (P1, X). We conjecture that this stability property holds if and only if an evaluation map E: lim Holx0 (P1, X) → X is a quasifibration. In this paper we will prove that in the presence of this quasifibration condition, then the stability property holds if and only if the Morse theoretic flow category (defined in [4]) of the symplectic action functional on the Z cover of the loop space, ˜ LX, defined by the symplectic form, has a classifying space that realizes the homotopy type of ˜ LX. We conjecture that in the presence of this quasifibration condition, this Morse theoretic condition always holds. We will prove this in the case of X a homogeneous space, thereby giving an alternate proof of the stability theorem for holomorphic spheres for a projective homogeneous variety originally due to Gravesen [7].
Pointed torsors
, 2010
"... This paper gives a characterization of homotopy fibres over trivial torsors of the inverse image maps π ∗ : B(H − Tors) → B(π ∗ H − tors) of torsor categories which are induced by geometric morphisms π: Shv(C) → ..."
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This paper gives a characterization of homotopy fibres over trivial torsors of the inverse image maps π ∗ : B(H − Tors) → B(π ∗ H − tors) of torsor categories which are induced by geometric morphisms π: Shv(C) →
FROM MAPPING CLASS GROUPS TO AUTOMORPHISM GROUPS OF FREE GROUPS
, 2005
"... It is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of ..."
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It is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.
A Simplicial Description Of The Homotopy Category Of Simplicial Groupoids
, 2000
"... . In this paper we use Quillen's model structure given by DwyerKan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characteri ..."
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. In this paper we use Quillen's model structure given by DwyerKan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicial identities and give a simplicial description of the homotopy category of simplicial groupoids. Finally, we show loop and suspension functors in the pointed case. 1. Introduction 1.1. Summary. A wellknown and quite powerful context in which an abstract homotopy theory can be developed is supplied by a category with a closed model structure in the sense of Quillen [16]. The category Simp(Gp) of simplicial groups is a remarkable example of what a closed model category is, and the homotopy theory in Simp(Gp) developed by Kan [12] occurs as the homotopy theory associated to this closed model structure. According to the t...
Diagonal model structures
, 2010
"... The original purpose of this note was to display a model structure for the category s 2 Set of bisimplicial sets whose cofibrations are the monomorphisms and whose weak equivalences are the diagonal weak equivalences, and then show that it is cofibrantly generated in a very precise way. The project ..."
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The original purpose of this note was to display a model structure for the category s 2 Set of bisimplicial sets whose cofibrations are the monomorphisms and whose weak equivalences are the diagonal weak equivalences, and then show that it is cofibrantly generated in a very precise way. The project grew to include
DÉCALAGE AND KAN’S SIMPLICIAL LOOP GROUP FUNCTOR
"... Abstract. Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios an ..."
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Abstract. Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak homotopy equivalence for any bisimplicial set. In this paper we will give a new, elementary proof of this result. As an application, we will revisit Kan’s simplicial loop group functor G. We will give a simple formula for this functor, which is based on a factorization, due to Duskin, of Eilenberg and Mac Lane’s classifying complex functor W. We will give a new, short, proof of Kan’s result that the unit map for the adjunction G ⊣ W is a weak homotopy equivalence for reduced simplicial sets. 1.
The homotopy classification of gerbes
, 2006
"... Gerbes are locally connected presheaves of groupoids on a small Grothendieck site C. Gerbes are classified up to local weak equivalence by path components of a cocycle category taking values in the diagram Grp(C) of 2groupoids consisting of all sheaves of groups, their isomorphisms and homotopies. ..."
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Gerbes are locally connected presheaves of groupoids on a small Grothendieck site C. Gerbes are classified up to local weak equivalence by path components of a cocycle category taking values in the diagram Grp(C) of 2groupoids consisting of all sheaves of groups, their isomorphisms and homotopies. If F is a full subpresheaf of Grp(C) then the set [∗, BF] of morphisms in the homotopy category of simplicial presheaves classifies gerbes locally equivalent to objects of F up to weak equivalence. If St(πF) is the stack completion of the fundamental groupoid πF of F, if L is a global section of St(πF), and if FL is the homotopy fibre over L of the canonical map BF → B St(πF), then [∗, FL] is in bijective correspondence with Giraud’s nonabelian cohomology object H 2 (C, L) of equivalence classes of gerbes with band L.
DIAGONAL FIBRATIONS ARE POINTWISE FIBRATIONS
"... On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrat ..."
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On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration. 1. Introduction and