scholarly journals Quasi-categories vs. Segal spaces: Cartesian edition

2021 ◽  
Author(s):  
Nima Rasekh
Keyword(s):  
List Type ◽  
Model Structure ◽  
Simplicial Sets ◽  
Segal Spaces ◽  
Quillen Equivalent

AbstractWe prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On bisimplicial spaces (due to deBrito [12]), On bisimplicial sets, On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.

scholarly journals SYMMETRIC OPERADS IN ABSTRACT SYMMETRIC SPECTRA

2018 ◽  
Vol 18 (4) ◽  
pp. 707-758 ◽  
Author(s):  
Dmitri Pavlov ◽  
Jakob Scholbach
Keyword(s):  
Algebraic Geometry ◽  
Commutative Ring ◽  
Model Category ◽  
Obstruction Theory ◽  
Model Structure ◽  
Simplicial Sets ◽  
Ring Spectra ◽  
Symmetric Spectra ◽  
Derived Algebraic Geometry ◽  
Quillen Equivalent

This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and $\text{E}_{\infty }$-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of $\text{E}_{\infty }$-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.

scholarly journals AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

2018 ◽  
Vol 107 (02) ◽  
pp. 181-198
Author(s):  
JAMES GILLESPIE
Keyword(s):  
The Other ◽  
Homotopy Category ◽  
Gorenstein Ring ◽  
Model Structure ◽  
Projective Modules ◽  
Stable Module Category ◽  
Stable Module ◽  
Coherent Rings ◽  
Quillen Equivalent ◽  
Compactly Generated

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

scholarly journals Props in model categories and homotopy invariance of structures

2010 ◽  
Vol 17 (1) ◽  
pp. 79-160
Author(s):  
Benoit Fresse
Keyword(s):  
General Setting ◽  
Model Category ◽  
Topological Spaces ◽  
Algebra Structure ◽  
Model Structure ◽  
Homotopy Invariance ◽  
Model Categories ◽  
Arbitrary Ring ◽  
General Argument ◽  
Simplicial Sets

Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴∞-algebras.

scholarly journals Dwyer–Kan homotopy theory for cyclic operads

2021 ◽  
pp. 1-30
Author(s):  
Gabriel C. Drummond-Cole ◽  
Philip Hackney
Keyword(s):  
Homotopy Theory ◽  
Forgetful Functor ◽  
General Definition ◽  
Model Structure ◽  
Simplicial Sets ◽  
Explicit Formulae

Abstract We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.

scholarly journals A Novel Analytical Framework for Dynamic Quaternion Ship Domains

2012 ◽  
Vol 66 (2) ◽  
pp. 265-281 ◽  
Author(s):  
Ning Wang
Keyword(s):  
Traffic Congestion ◽  
Mental States ◽  
Analytical Framework ◽  
List Type ◽  
Model Structure ◽  
Simulation Studies ◽  
Motion Model ◽  
Type Number ◽  
Wind Force ◽  
Input Variables

In this paper, a novel analytical framework for Dynamic Quaternion Ship Domain (DQSD) models has been initially proposed via the Quaternion Ship Domain (QSD) model structure. Unlike previous ship domains, the proposed DQSD model is able to capture essential subjectivity and objectivity of ship domains. To be specific, the significant characteristics are as follows: (1)The proposed DQSD model is integrated by three independent submodels of ship, human and circumstance, which are determined by ship manoeuvrability, navigator's states, and navigation circumstance, respectively.(2)The ship manoeuvrability derived from the MMG-type ship motion model is employed to establish the ship submodel which identifies the DQSD scale.(3)A novel navigator reliability model is proposed to realize the human submodel which defines the ship domain shape with navigator ability, physical and mental states being input variables.(4)In addition, visibility, wind force, wave and traffic congestion are incorporated into the circumstance submodel which is employed to zoom in or out of the DQSD-type ship domain.Finally, the well-known Esso Osaka tanker model is used to conduct simulation studies on various typical stationary and dynamic situations, and comparative investigations with each other have been comprehensively analysed. Simulation results demonstrate that the DQSD model can capture critical dynamics of ship domains and undoubtedly be effective and superior to previous ship domains in terms of performance and accuracy.

scholarly journals Morita homotopy theory for (∞,1)-categories and ∞-operads

2019 ◽  
Vol 31 (3) ◽  
pp. 661-684 ◽  
Author(s):  
Giovanni Caviglia ◽  
Javier J. Gutiérrez
Keyword(s):  
Homotopy Theory ◽  
Model Structure ◽  
Simplicial Sets ◽  
Simplicial Presheaves ◽  
Model Structures ◽  
Simplicial Algebras ◽  
Bousfield Localization

Abstract We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of {(\infty,1)} -categories and {\infty} -operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski–Moerdijk model structure on simplicial operads, respectively.

scholarly journals A Quillen model structure for Gray-categories

2010 ◽  
Vol 8 (2) ◽  
pp. 183-221 ◽  
Author(s):  
Stephen Lack
Keyword(s):  
Full Subcategory ◽  
Model Structure ◽  
Theoretic Proof ◽  
Quillen Model ◽  
Quillen Equivalent

AbstractA Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to provide a functorial and model-theoretic proof of the unpublished theorem of Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model structure on Gray-Cat is conjectured to be Quillen equivalent to a model structure on the category Tricat of tricategories and strict homomorphisms of tricategories.

Model structures for pro-simplicial presheaves

2011 ◽  
Vol 7 (3) ◽  
pp. 499-525 ◽  
Author(s):  
J.F. Jardine
Keyword(s):  
Type Structure ◽  
Model Structure ◽  
Simplicial Sets ◽  
Postnikov Tower ◽  
Simplicial Presheaves ◽  
Model Structures

AbstractThis paper displays model structures for the category of pro-objects in simplicial presheaves on an arbitrary small Grothendieck site. The first of these is an analogue of the Edwards-Hastings model structure for pro-simplicial sets, in which the cofibrations are monomorphisms and the weak equivalences are specified by comparisons of function complexes. Other model structures are built from the Edwards-Hastings structure by using Bousfield-Friedlander localization techniques. There is, in particular, an n-type structure for pro-simplicial presheaves, and also a model structure in which the map from a pro-object to its Postnikov tower is formally inverted.

scholarly journals Another approach to the Kan–Quillen model structure

2019 ◽  
Vol 15 (1) ◽  
pp. 143-165
Author(s):  
Sean Moss
Keyword(s):  
Careful Analysis ◽  
Combinatorial Proof ◽  
Topological Spaces ◽  
Model Structure ◽  
Simplicial Sets ◽  
Simplicial Set ◽  
New Construction ◽  
Weak Homotopy ◽  
Basic Facts ◽  
Quillen Model

Abstract By careful analysis of the embedding of a simplicial set into its image under Kan’s $$\mathop {\mathop {\mathsf {Ex}}^\infty }$$Ex∞ functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about $$\mathop {\mathop {\mathsf {Ex}}^\infty }$$Ex∞ and hence provide a new construction of the Kan–Quillen model structure on simplicial sets, one which avoids the use of topological spaces or minimal fibrations.

scholarly journals Homotopy theory of Moore flows (I)

2021 ◽  
Vol 3 ◽  
pp. 3
Author(s):  
Philippe Gaucher
Keyword(s):  
Tensor Product ◽  
Commutative Semigroup ◽  
Homotopy Theory ◽  
Model Category ◽  
Model Structure ◽  
Q Model ◽  
Quillen Equivalent

A reparametrization category is a small topologically enriched symmetric semimonoidal category such that the semimonoidal structure induces a structure of a commutative semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the closed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed d-spaces and the q-model structure of flows.

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