scholarly journals The Quillen model category of topological spaces

2019 ◽  
Vol 37 (1) ◽  
pp. 2-24
Author(s):  
Philip S. Hirschhorn
Keyword(s):  
Model Category ◽  
Topological Spaces ◽  
Quillen Model Category ◽  
Quillen Model

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 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 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 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.

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