Results 1  10
of
20
Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
Abstract

Cited by 47 (0 self)
 Add to MetaCart
Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
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 ..."
Abstract

Cited by 14 (0 self)
 Add to MetaCart
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 ...
On ∞topoi
, 2003
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X, G); we will single out three of them for discussion here. First of all, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of Gvalu ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X, G); we will single out three of them for discussion here. First of all, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of Gvalued singular cochains. Alternatively, one may regard H n (•, G) as a representable functor on the homotopy category of topological spaces, and thereby define H n rep(X, G) to be the set of homotopy classes of maps from X into an EilenbergMacLane space K(G, n). A third possibility is to use the sheaf cohomology H n sheaf (X, G) of X with coefficients in the constant sheaf G on X. If X is a sufficiently nice space (for example, a CW complex), then all three of these definitions agree. In general, however, all three give different answers. The singular cohomology of X is constructed using continuous maps from simplices ∆k into X. If there are not many maps into X (for example if every path in X is constant), then we cannot expect H n sing (X, G) to tell us very much about X. Similarly, the cohomology group H n rep(X, G) is defined using maps from X into a simplicial complex, which (ultimately) relies on the existence of continuous realvalued functions on X. If X does not admit many realvalued functions, we should not expect H n rep (X, G) to be a useful invariant. However, the sheaf cohomology of X seems to be a good invariant for arbitrary spaces: it has excellent formal properties in general and sometimes yields
TQFTs from Homotopy ntypes
, 1995
"... : Using simplicial methods developed in [22], we construct topological quantum field theories using an algebraic model of a homotopy ntype as initial data, generalising a construction of Yetter in [23] for n=1 and in [24] for n=2 Introduction In [23], Yetter showed how to construct a topological q ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
: Using simplicial methods developed in [22], we construct topological quantum field theories using an algebraic model of a homotopy ntype as initial data, generalising a construction of Yetter in [23] for n=1 and in [24] for n=2 Introduction In [23], Yetter showed how to construct a topological quantum field theory with coefficients in a finite group. In [24], he showed that his construction could be extended to handle coefficients in a finite categorical group, or cat 1 group. These objects are algebraic models for certain homotopy 2types. The topological quantum field theories thus constructed are (2+1) TQFTs, but the methods used do not depend on the manifolds being surfaces, except to avoid possible irregularities related to problems of triangulations in low dimensions. Yetter ended that second note with some open questions, the third of which was: can one carry out the same sort of construction for algebraic models of higher homotopy types? In this note we will show that a ...
Joins for (Augmented) Simplicial Sets
, 1998
"... We introduce a notion of join for (augmented) simnplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category \Delta. 1991 Math. Subj. Class.: 18G30 1 Introduction The theory of joins of (geomet ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We introduce a notion of join for (augmented) simnplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category \Delta. 1991 Math. Subj. Class.: 18G30 1 Introduction The theory of joins of (geometric) simplicial complexes as given by Brown, [2], or Spanier, [13], reveals the join operation to be a basic geometric construction. It is used in the development of several areas of geometric topology (cf. Hudson, [8]) whilst also being applied to the basic properties of polyhedra relating to homology. The theories of geometric and abstract simplicial complexes run in a largely parallel way and when describing the theory, expositions often choose which aspect  abstract combinatorial or geometric  to emphasise at each step. Historically in algebraic topology geometric simplicial complexes, as tools, were largely replaced by CW complexes whilst the combinatorial abstract complex became pa...
On the homotopy theory of ntypes
 Homology, Homotopy Appl
"... Abstract. An ntruncated model structure on simplicial (pre)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine’s intermediate model structures we construct such an ntype model structure via Bo ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. An ntruncated model structure on simplicial (pre)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine’s intermediate model structures we construct such an ntype model structure via BousfieldFriedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called nhyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analogous ntruncations on simplicial groupoids and prove a Quillen equivalence between these settings. We achieve a classification of ntypes of simplicial presheaves in terms of (n −1)types of presheaves of simplicial groupoids. Our classification holds for general n. Therefore this can also be viewed as the homotopy theory of (pre)sheaves of (weak) higher groupoids. Contents
Internal categorical structure in homotopical algebra
 Proceedings of the IMA workshop ?nCategories: Foundations and Applications?, June 2004, (to appear). CROSSED MODULES AND PEIFFER CONDITION 135 [Ped95] [Por87
, 1995
"... Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1. ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1.
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
. 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...
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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.
SOME REMARKS ON PROFINITE COMPLETION OF SPACES
"... Abstract. We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of ArtinMazur and Sullivan which extends also to nonconnected spaces. Another new aspect is an equivariant profinite completion functor and equivari ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of ArtinMazur and Sullivan which extends also to nonconnected spaces. Another new aspect is an equivariant profinite completion functor and equivariant fibrant replacement functor for a profinite group acting on a space. This is crucial for applications where, for example, Galois groups are involved, or for profinite Teichmüller theory where equivariant completions are applied. Along the way we collect and survey the most important known results about profinite completion of spaces. 1.