Results 1  10
of
24
Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence
, 139
"... 1.2. The axioms ..."
(Show Context)
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 26 (0 self)
 Add to MetaCart
(Show Context)
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 ...
Homology and cohomology of E∞ ring spectra
 MATHEMATISCHE ZEITSCHRIFT
, 2005
"... Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads. ..."
Abstract

Cited by 25 (1 self)
 Add to MetaCart
(Show Context)
Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads.
OPERADS, ALGEBRAS, MODULES, AND MOTIVES
"... Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
(Show Context)
Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and conversion theorems III Derived categories from a topological point of view IV Rational derived categories and mixed Tate motives V Derived categories of modules over E ∞ algebras In differential algebra, operads are systems of parameter chain complexes for multiplication on various types of differential graded algebras “up to homotopy”, for example commutative algebras, nLie algebras, nbraid algebras, etc. Our primary focus is the development of the concomitant theory of modules up to homotopy and the study of both classical derived categories of modules over DGA’s and derived categories of modules up to homotopy over DGA’s up to homotopy. Examples of such derived categories provide the appropriate setting for one approach to mixed Tate motives in algebraic geometry, both rational and integral.
Homomorphisms of higher categories
 U.U.D.M. REPORT 2008:47
, 2008
"... We describe a construction that to each algebraically specified notion of higherdimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associativ ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
We describe a construction that to each algebraically specified notion of higherdimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricategories; and here we do not obtain the trihomomorphisms defined by Gordon, Power and Street, but rather something which is equivalent in a suitable sense. The second application is to Batanin’s weak ωcategories.
TYPE THEORY AND HOMOTOPY
"... The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
(Show Context)
The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy
Weighted limits in simplicial homotopy theory
 Journal of Pure and Applied Algebra vol
, 2010
"... Abstract. We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functo ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor. 1.
Parametrized spaces model locally constant homotopy sheaves
 Topology Appl
, 2008
"... Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopytheoretic version of the correspondence between covering spaces and π1sets. We then use these two equivalences to study base change functors for parametrized spaces. Contents
Categorical aspects of toric topology
 Contemporary Mathematics series
"... Abstract. We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces XK, such as momentangle complexes ZK, quasitoric manifolds M, and DavisJanuszkiewicz spaces DJ(K). We first exhibit XK a ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces XK, such as momentangle complexes ZK, quasitoric manifolds M, and DavisJanuszkiewicz spaces DJ(K). We first exhibit XK as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as algebraic models for XK. Such models encode many standard algebraic invariants, and their existence is assured by the Quillen structure. We provide several illustrative calculations, often over the rationals, including proofs that quasitoric manifolds (and various generalisations) are rationally formal; that the rational Pontrjagin ring of the loop space ΩDJ(K) is isomorphic to the quadratic dual of the StanleyReisner algebra Q[K] for flag complexes K; and that DJ(K) is coformal precisely when K is flag. We conclude by describing algebraic models for the loop space ΩDJ(K) for any complex K, which mimic our previous description as a homotopy colimit of topological monoids. 1.