scholarly journals Locally constant functors

2009 ◽  
Vol 147 (3) ◽  
pp. 593-614 ◽  
Author(s):  
DENIS–CHARLES CISINSKI
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Model Categories ◽  
Combinatorial Model ◽  
Constant Coefficients ◽  
Kan Extensions ◽  
Bousfield Localization

AbstractWe study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. Finally, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors.

scholarly journals Arrow categories of monoidal model categories

2019 ◽  
Vol 125 (2) ◽  
pp. 185-198
Author(s):  
David White ◽  
Donald Yau
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Topological Spaces ◽  
Model Structure ◽  
Model Categories ◽  
Projective Model ◽  
Monoidal Structure ◽  
Chain Complexes

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, in the course of his work on Smith ideals. As a corollary, we prove that the projective model structure in cubical homotopy theory is a monoidal model structure. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, pro-categories, and topological spaces.

Equivariant Topology and Derived Algebra

2021 ◽  
Keyword(s):  
Homotopy Theory ◽  
Algebraic Models ◽  
Research Papers ◽  
Model Categories ◽  
Homotopy Limits ◽  
Equivariant Topology ◽  
Mackey Functors ◽  
Chromatic Homotopy ◽  
Equivariant Spectra ◽  
Bousfield Localization

This volume contains eight research papers inspired by the 2019 'Equivariant Topology and Derived Algebra' conference, held at the Norwegian University of Science and Technology, Trondheim in honour of Professor J. P. C. Greenlees' 60th birthday. These papers, written by experts in the field, are intended to introduce complex topics from equivariant topology and derived algebra while also presenting novel research. As such this book is suitable for new researchers in the area and provides an excellent reference for established researchers. The inter-connected topics of the volume include: algebraic models for rational equivariant spectra; dualities and fracture theorems in chromatic homotopy theory; duality and stratification in tensor triangulated geometry; Mackey functors, Tambara functors and connections to axiomatic representation theory; homotopy limits and monoidal Bousfield localization of model categories.

scholarly journals Homotopy theoretic models of identity types

2009 ◽  
Vol 146 (1) ◽  
pp. 45-55 ◽  
Author(s):  
STEVE AWODEY ◽  
MICHAEL A. WARREN
Keyword(s):  
Mathematical Logic ◽  
Type Theory ◽  
Theoretical Basis ◽  
Homotopy Theory ◽  
Model Category ◽  
Proof Assistants ◽  
Model Categories ◽  
Wide Range ◽  
Abstract Framework ◽  
Computer Proof

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11,12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such asCoqandAgda(cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature.

scholarly journals A generalization of Ohkawa’s theorem

2014 ◽  
Vol 150 (5) ◽  
pp. 893-902 ◽  
Author(s):  
Carles Casacuberta ◽  
Javier J. Gutiérrez ◽  
Jiří Rosický
Keyword(s):  
Equivalence Classes ◽  
Model Categories ◽  
Combinatorial Model

AbstractA theorem due to Ohkawa states that the collection of Bousfield equivalence classes of spectra is a set. We extend this result to arbitrary combinatorial model categories.

scholarly journals On some new model category structures from old, on the same underlying category

2015 ◽  
Vol 27 (3) ◽  
Author(s):  
Alexandru E. Stanculescu
Keyword(s):  
Homotopy Theory ◽  
Existence Result ◽  
Model Category ◽  
New Model

AbstractWe make a study of ℓℓ-extensions of model category structures. We prove an existence result of ℓℓ-extensions, present some specific and some rather formal results about them and give an application of the existence result to the homotopy theory of categories enriched over a monoidal model category.

Model Structures for the 𝔸1-homotopy Category

2019 ◽  
pp. 175-212
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Full Subcategory ◽  
Homotopy Category ◽  
Symmetric Power ◽  
Model Structure ◽  
Model Categories ◽  
Simplicial Presheaves ◽  
Basic Ideas ◽  
Quillen Model ◽  
Model Structures ◽  
Bousfield Localization

This chapter provides the 𝔸1-local projective model structure on the categories of simplicial presheaves and simplicial presheaves with transfers. These model categories, written as Δ‎opPshv(Sm)𝔸1 and Δ‎op PST(Sm)𝔸1, are first defined. Their respective homotopy categories are Ho(Sm) and the full subcategory DM eff nis ≤0 of DM eff nis. Afterward, this chapter introduces the notions of radditive presheaves and ̅Δ‎-closed classes, and develops their basic properties. The theory of ̅Δ‎-closed classes is needed because the extension of symmetric power functors to simplicial radditive presheaves is not a left adjoint. This chapter uses many of the basic ideas of Quillen model categories, which is a category equipped with three classes of morphisms satisfying five axioms. In addition, much of the material in this chapter is based upon the technique of Bousfield localization.

scholarly journals Quillen model structures for relative homological algebra

2002 ◽  
Vol 133 (2) ◽  
pp. 261-293 ◽  
Author(s):  
J. DANIEL CHRISTENSEN ◽  
MARK HOVEY
Keyword(s):  
Homotopy Theory ◽  
Homological Algebra ◽  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Category Structure ◽  
Model Category ◽  
Projective Class ◽  
Chain Complexes ◽  
Model Structures

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.

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.

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