Results 11  20
of
21
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 3 (0 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.
M A Mandell, Homology and cohomology of E1 ring spectra
 Math. Z
"... Abstract. We show that every homology or cohomology theory on a category of E1 ring spectra is Topological Andre{Quillen homology or cohomology with appropriate coecients. We show that the cotangent complex of MU is MU ^ bu. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We show that every homology or cohomology theory on a category of E1 ring spectra is Topological Andre{Quillen homology or cohomology with appropriate coecients. We show that the cotangent complex of MU is MU ^ bu.
A folk model structure on omegacat
, 2009
"... The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generat ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while free objects are cofibrant. We further exhibit model structures of this type on ncategories for arbitrary n ∈ N, as specialisations of the ωcategorical one along right adjoints. In particular, known cases for n = 1 and n = 2 nicely fit into the scheme.
MODULES IN MONOIDAL MODEL CATEGORIES
, 2006
"... This paper studies the existence of and compatibility between derived change of ring, balanced product, and function module derived functors on module categories in monoidal model categories. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
This paper studies the existence of and compatibility between derived change of ring, balanced product, and function module derived functors on module categories in monoidal model categories.
PARAMETRIZED SPACES ARE LOCALLY CONSTANT HOMOTOPY SHEAVES
, 706
"... 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 1 (1 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
Bicategorical fibration structures and stacks
"... Abstract. In this paper we introduce two notions —systems of fibrant objects and fibration structures — which will allow us to associate to a bicategory B a homotopy bicategory Ho(B) in such a way that Ho(B) is the universal way to add pseudoinverses to weak equivalences in B. Furthermore, Ho(B) is ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. In this paper we introduce two notions —systems of fibrant objects and fibration structures — which will allow us to associate to a bicategory B a homotopy bicategory Ho(B) in such a way that Ho(B) is the universal way to add pseudoinverses to weak equivalences in B. Furthermore, Ho(B) is locally small when B is and Ho(B) is a 2category when B is. We thereby resolve two of the problems with known approaches to bicategorical localization. As an important example, we describe a fibration structure on the 2category of prestacks on a site and prove that the resulting homotopy bicategory is the 2category of stacks. We also show how this example can be restricted to obtain algebraic, differentiable and topological (respectively) stacks as homotopy categories of algebraic, differential and topological (respectively) prestacks.
A Notion of Homotopy for the Effective Topos
, 2010
"... We define a notion of homotopy in the effective topos. AMS Subject Classification (2000): 18B25 (Topos Theory),55U35 (Abstract and axiomatic homotopy theory) ..."
Abstract
 Add to MetaCart
(Show Context)
We define a notion of homotopy in the effective topos. AMS Subject Classification (2000): 18B25 (Topos Theory),55U35 (Abstract and axiomatic homotopy theory)
Abstract Simulations as Homotopies
"... We exhibit a model structure on 2Cat, obtained by transfer from sSet across the adjunction C2 ◦ Sd 2 ⊣ Ex 2 ◦ N2. A certain class of homotopies in this model structure turns out to be in 1to1 correspondence with strong simulations among labeled transitions systems, formalising the geometric intui ..."
Abstract
 Add to MetaCart
(Show Context)
We exhibit a model structure on 2Cat, obtained by transfer from sSet across the adjunction C2 ◦ Sd 2 ⊣ Ex 2 ◦ N2. A certain class of homotopies in this model structure turns out to be in 1to1 correspondence with strong simulations among labeled transitions systems, formalising the geometric intuition of simulations as deformations. The correspondence still holds in the cubical setting, characterising simulations of higherdimensional transition systems (HDTS). 1
Formal groups and stable homotopy of commutative rings
, 2004
"... We explain a new relationship between formal group laws and ring spectra in stable homotopy theory. We study a ring spectrum denoted DB which depends on a commutative ring B and is closely related to the topological André–Quillen homology of B. We present an explicit construction which to every 1–di ..."
Abstract
 Add to MetaCart
(Show Context)
We explain a new relationship between formal group laws and ring spectra in stable homotopy theory. We study a ring spectrum denoted DB which depends on a commutative ring B and is closely related to the topological André–Quillen homology of B. We present an explicit construction which to every 1–dimensional and commutative formal group law F over B associates a morphism of ring spectra F∗: HZ − → DB from the Eilenberg–MacLane ring spectrum of the integers. We show that formal group laws account for all such ring spectrum maps, and we identify the space of ring spectrum maps between HZ and DB. That description involves formal group law data and the homotopy units of the ring spectrum DB.