Quillen Model Category Research Articles

Combinatorial and accessible weak model categories

In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to weak and semi-model categories. We also provide general existence theorems for both left and right Bousfield localization of combinatorial and accessible weak model structures, which combined with the results above gives existence results for left and right Bousfield localization of combinatorial and accessible left and right semi-model categories, generalizing previous results of Barwick. Surprisingly, we show that any left or right Bousfield localization of an accessible or combinatorial Quillen model category always exists, without properness assumptions, and is simultaneously both a left and a right semi-model category, without necessarily being a Quillen model category itself.

Algebraically cofibrant and fibrant objects revisited

We extend all known results about transferred model structures on algebraically cofibrant and fibrant objects by working with weak model categories. We show that for an accessible weak model category there are always Quillen equivalent transferred weak model structures on both the categories of algebraically cofibrant and algebraically fibrant objects. Under additional assumptions, these transferred weak model structures are shown to be left, right or Quillen model structures. By combining both constructions, we show that each combinatorial weak model category is connected, via a chain of Quillen equivalences, to a combinatorial Quillen model category in which all objects are fibrant.

Semantics of higher inductive types

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.

Higher cyclic operads

We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $\Xi$ of trees, which carries a tight relationship to the Moerdijk-Weiss category of rooted trees $\Omega$. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on $\Xi$ which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad.

A criterion for existence of right‐induced model structures

Suppose that F : N → M is a functor whose target is a Quillen model category. We give a succinct sufficient condition for the existence of the right-induced model category structure on N in the case when F admits both adjoints. We give several examples, including change-of-rings, operad-like structures, and anti-involutive structures on infinity categories. For the last of these, we explore anti-involutive structures for several different models of ( ∞ , 1 ) -categories and show that known Quillen equivalences between base model categories lift to equivalences.

Path Categories and Propositional Identity Types

Connections between homotopy theory and type theory have recently attracted a lot of attention, with Voevodsky’s univalent foundations and the interpretation of Martin-Löf’s identity types in Quillen model categories as some of the highlights. In this article, we establish a connection between a natural weakening of Martin-Löf’s rules for the identity types that has been considered by Cohen, Coquand, Huber and Mörtberg in their work on a constructive interpretation of the univalence axiom on the one hand and the notion of a path category, a slight variation on the classic notion of a category of fibrant objects due to Brown, on the other. This involves showing that the syntactic category associated to a type theory with weak identity types carries the structure of a path category, strengthening earlier results by Avigad, Lumsdaine, and Kapulkin. In this way, we not only relate a well-known concept in homotopy theory with a natural concept in logic but also provide a framework for further developments.

Rigidity in equivariant stable homotopy theory

For any finite group G, we show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy cofiber sequences and the stable Burnside category determine all "higher order structure" of the 2-local G-equivariant stable homotopy category, such as the equivariant homotopy types of function G-spaces. The theorem can be seen as an equivariant version of Schwede's rigidity theorem at the prime 2.

On equivariant homotopy theory for model categories

We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological presheaves indexed by the orbit category of a fixed topological group $G$ and the category of $G$-spaces can be endowed with Quillen equivalent model category structures. We prove an analogous result for any cofibrantly generated model category and discrete group $G$, under certain conditions on the fixed point functors of the subgroups of $G$. These conditions hold in many examples, though not in the category of chain complexes, where we nevertheless establish and generalize to collections an equivariant Whitehead Theorem \`{a} la Kropholler and Wall for the normalized chain complexes of simplicial $G$-sets.

Skew structures in 2-category theory and homotopy theory

We study Quillen model categories equipped with a monoidal skew closed structure that descends to a genuine monoidal closed structure on the homotopy category. Our examples are 2-categorical and include permutative categories and bicategories. Using the skew framework, we adapt Eilenberg and Kelly’s theorem relating monoidal and closed structure to the homotopical setting. This is applied to the construction of monoidal bicategories arising from the pseudo-commutative 2-monads of Hyland and Power.

The hammock localization preserves homotopies

The hammock localization provides a model for a homotopy function complex in any Quillen model category. We prove that a homotopy between a pair of morphisms induces a homotopy between the maps induced by taking the hammock localization. We describe applications of this fact to the study of homotopy algebras over monads and homotopy idempotent functors. Among other things, we prove that, under Vop\v{e}nka's principle, every homotopy idempotent functor in a cofibrantly generated model category is determined by simplicial orthogonality with respect to a set of morphisms. We also give a new proof of the fact that left Bousfield localizations with respect to a class of morphisms always exist in any left proper combinatorial model category under Vop\v{e}nka's principle.

Almost free modules and Mittag-Leffler conditions

Drinfeld recently suggested to replace projective modules by the flat Mittag-Leffler ones in the definition of an infinite dimensional vector bundle on a scheme X (Drinfeld, 2006 [8]). Two questions arise: (1) What is the structure of the class D of all flat Mittag-Leffler modules over a general ring? (2) Can flat Mittag-Leffler modules be used to build a Quillen model category structure on the category of all chain complexes of quasi-coherent sheaves on X?We answer (1) by showing that a module M is flat Mittag-Leffler, if and only if M is ℵ1-projective in the sense of Eklof and Mekler (2002) [10]. We use this to characterize the rings such that D is closed under products, and relate the classes of all Mittag-Leffler, strict Mittag-Leffler, and separable modules. Then we prove that the class D is not deconstructible for any non-right perfect ring. So unlike the classes of all projective and flat modules, the class D does not admit the homotopy theory tools developed recently by Hovey (2002) [26]. This gives a negative answer to (2).

A Quillen model category structure on some categories of comonoids

We prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure.

A Cartan–Eilenberg approach to homotopical algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan–Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.

Simplicial homotopy in semi-abelian categories

AbstractWe study Quillen's model category structure for homotopy of simplicial objects in the context of Janelidze, Márki and Tholen's semi-abelian categories. This model structure exists as soon as is regular Mal'tsev and has enough regular projectives; then the fibrations are the Kan fibrations of S. When, moreover, is semi-abelian, weak equivalences and homology isomorphisms coincide.

Homotopy limits for 2-categories

AbstractWe study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

Homotopy theory of well-generated algebraic triangulated categories

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.

HZ -algebra spectra are differential graded algebras

We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that basically any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects).

The Ganea and Whitehead Variants of the Lusternik–Schnirelmann Category

AbstractThe Lusternik–Schnirelmann category has been described in different ways. Two major ones, the first by Ganea, the second by Whitehead, are presented here with a number of variants. The equivalence of these variants relies on the axioms of Quillen's model category, but also sometimes on an additional axiom, the so-called “cube axiom”.

Lax factorization algebras

It is shown that many weak factorization systems appearing in functorial Quillen model categories, including all those that are cofibrantly generated, come with a rich computational structure, defined by a certain lax algebra with respect to the “squaring monad” on CAT. This structure largely facilitates natural choices for left or right liftings once certain basic natural choices have been made. The use of homomorphisms of such lax algebras is also discussed, with focus on “lax freeness”.

CLAPP-PUPPE TYPE LUSTERNIK-SCHNIRELMANN (CO)CATEGORY IN A MODEL CATEGORY

We introduce Clapp-Puppe type generalized Lusternik-Schnirelmann (co)category in a Quillen model category. We establish some of their basic properties and give various characterizations of them. As the first application of these characterizations, we show that our generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category is invariant under Quillen modelization equivalences. In particular, generalized (co)category of spaces and simplicial sets coincide. Another application of these characterizations is to define and study rational cocategory. Various other applications are also given.