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 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 Semantics of higher inductive types

2019 ◽  
Vol 169 (1) ◽  
pp. 159-208 ◽  
Author(s):  
PETER LEFANU LUMSDAINE ◽  
MICHAEL SHULMAN
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Model Category ◽  
Suitable Model ◽  
Simplicial Sets ◽  
James Construction ◽  
Quillen Model Category ◽  
Wide Range ◽  
Inductive Types ◽  
Cartesian Closed

AbstractHigher inductive typesare a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings.We introduce the notion ofcell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category hasweakly stable typal initial algebrasfor any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations.Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

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.

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 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.

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