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.

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 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 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 Homotopy theory of well-generated algebraic triangulated categories

2008 ◽  
Vol 3 (1) ◽  
pp. 53-75 ◽  
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Regular Cardinal ◽  
Homotopy Theory ◽  
Model Category ◽  
Triangulated Categories ◽  
Model Structure ◽  
Quillen Model Category ◽  
Quillen Model

AbstractFor every regular cardinal α, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially dg categories which are stable under suspensions, cosuspensions, cones and α-small sums.Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.

scholarly journals Model structure on the universe of all types in interval type theory

2020 ◽  
pp. 1-32
Author(s):  
Simon Boulier ◽  
Nicolas Tabareau
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Model Structure ◽  
Model Categories ◽  
Model Interpretation ◽  
Slight Generalization ◽  
Cubical Sets ◽  
Definition Of ◽  
Interval Type ◽  
The Universe

Abstract Model categories constitute the major context for doing homotopy theory. More recently, homotopy type theory (HoTT) has been introduced as a context for doing syntactic homotopy theory. In this paper, we show that a slight generalization of HoTT, called interval type theory (⫿TT), allows to define a model structure on the universe of all types, which, through the model interpretation, corresponds to defining a model structure on the category of cubical sets. This work generalizes previous works of Gambino, Garner, and Lumsdaine from the universe of fibrant types to the universe of all types. Our definition of ⫿TT comes from the work of Orton and Pitts to define a syntactic approximation of the internal language of the category of cubical sets. In this paper, we extend the work of Orton and Pitts by introducing the notion of degenerate fibrancy, which allows to define a fibrant replacement, at the heart of the model structure on the universe of all types. All our definitions and propositions have been formalized using the Coq proof assistant.

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.

scholarly journals Pure exact structures and the pure derived category of a scheme

2016 ◽  
Vol 163 (2) ◽  
pp. 251-264 ◽  
Author(s):  
SERGIO ESTRADA ◽  
JAMES GILLESPIE ◽  
SINEM ODABAŞI
Keyword(s):  
Derived Category ◽  
Category Structure ◽  
Model Category ◽  
Grothendieck Category ◽  
Coherent Sheaves ◽  
Monoidal Structure ◽  
Chain Complexes

AbstractLet$\mathcal{C}$be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the categoryC($\mathcal{C}$) of unbounded chain complexes in$\mathcal{C}$. We use λ-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi–coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi–coherent sheaves.

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.

New model structures and projective (injective) cotorsion pairs

2021 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
Positive Integer ◽  
Triangulated Categories ◽  
Model Structure ◽  
Cotorsion Pair ◽  
Flat Dimension ◽  
Gorenstein Projective Module ◽  
Chain Complexes ◽  
Cotorsion Pairs ◽  
Gorenstein Injective ◽  
Model Structures

Let [Formula: see text] be either the category of [Formula: see text]-modules or the category of chain complexes of [Formula: see text]-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer [Formula: see text], and hence, one can get new algebraic triangulated categories. Second, it is shown that any [Formula: see text]-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let [Formula: see text] be an [Formula: see text]-module with finite flat dimension and [Formula: see text] a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then [Formula: see text] is injective.

scholarly journals The projective stable category of a coherent scheme

2018 ◽  
Vol 149 (1) ◽  
pp. 15-43 ◽  
Author(s):  
Sergio Estrada ◽  
James Gillespie
Keyword(s):  
Homotopy Category ◽  
Model Structure ◽  
Stable Category ◽  
Coherent Sheaves ◽  
Chain Complexes

We define the projective stable category of a coherent scheme. It is the homotopy category of an abelian model structure on the category of unbounded chain complexes of quasi-coherent sheaves. We study the cofibrant objects of this model structure, which are certain complexes of flat quasi-coherent sheaves satisfying a special acyclicity condition.

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