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We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient A → B satisfying some finiteness conditions, the Frobenius tensor category D of graded B-comodules with its stable model structure induces a monoidal model structure on C . We consider the corresponding homotopy quotient γ : C → H o C and the induced quotient T → H o T for the tensor category T of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in H o T . We apply these results in the Rep(GL(m|n))-case and study its homotopy category H o T associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of H o T by the negligible morphisms is again the representation category of a supergroup scheme. [ABSTRACT FROM AUTHOR]

We show that the category of non-counital conilpotent dg-coalgebras and the category of non-unital dg-algebras carry model structures compatible with their closed non-unital monoidal and closed non-unital module category structures respectively. Furthermore, we show that the Quillen equivalence between these two categories extends to a non-unital module category Quillen equivalence, i.e. providing an enriched form of Koszul duality. [ABSTRACT FROM AUTHOR]

Categories enriched in the opposite poset of non-negative reals can be viewed as generalizations of metric spaces, known as Lawvere metric spaces. In this article, we develop model structures on the categories R + - Cat and R + - Cat sym of Lawvere metric spaces and symmetric Lawvere metric spaces, each of which captures different features pertinent to the study of metric spaces. More precisely, in the three model structures we construct, the fibrant–cofibrant objects are the extended metric spaces (in the usual sense), the Cauchy complete Lawvere metric spaces, and the Cauchy complete extended metric spaces, respectively. Finally, we show that two of these model structures are unique in a similar way to the canonical model structure on Cat . [ABSTRACT FROM AUTHOR]

It is well-known that DG-enhancements of the unbounded derived category D qc (X) of quasi-coherent sheaves on a scheme X are all equivalent to each other. Here we present an explicit model which leads to applications in deformation theory. In particular, we shall describe three models for derived endomorphisms of a quasi-coherent sheaf F on a finite-dimensional Noetherian separated scheme (even if F does not admit a locally free resolution). Moreover, these complexes are endowed with DG-Lie algebra structures, which we prove to control infinitesimal deformations of F . [ABSTRACT FROM AUTHOR]

We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield–Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality. [ABSTRACT FROM AUTHOR]

This paper contains an elementary proof of the existence of the classical model structure on the category of unbounded DG-Lie algebras over a field of characteristic zero, with an emphasis on the properties of free and semifree extensions, which are particularly nice cofibrations. The cobar construction of a locally conilpotent cocommutative coalgebra is shown to be an example of semifree DG-Lie algebra. We also give an example of a non-cofibrant DG-Lie algebra whose underlying graded Lie algebra is free; this cannot occur in the bounded above case, where DG-Lie algebras of this form are always cofibrant.

We show that if the canonical left semi-model structure on the category of Grothendieck n-groupoids exists, then it satisfies the homotopy hypothesis, i.e. the associated (∞, 1)-category is equivalent to that of homotopy n-types, thus generalizing a result of the first-named author. As a corollary of the second named author's proof of the existence of the canonical left semi-model structure for Grothendieck 3-groupoids, we obtain a proof of the homotopy hypothesis for Grothendieck 3-groupoids. [ABSTRACT FROM AUTHOR]

We prove that any right Quillen functor between arbitrary model categories admits non trivial functorial factorizations that are similar to those of a model structure. Given a monad, operad or a PROP(erad) O , if we apply one of the factorizations to the forgetful functor U : O -Alg (M) ⟶ M , we extend the theory of Quillen–Segal O -algebras initiated in Bacard (Higher Struct 4(1):57–114, 2020), without the hypothesis of M being a combinatorial model category. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a theory of (∞ , 1) -stacks, in the sense of Hirschowitz–Simpson's 'Descent Pour Les n–Champs', using the language of quasi-category theory and the author's local Joyal model structure. The main result is a characterization of (∞ , 1) -stacks in terms of mapping space presheaves. An important special case of this theorem gives a sufficient condition for the presheaf of quasi-categories associated to a presheaf of model categories to be a higher stack. In the final section, we apply this result to construct the higher stack of unbounded complexes associated to a ringed site. [ABSTRACT FROM AUTHOR]

For a diagram of simplicial combinatorial model categories, we show that the associated lax limit, endowed with the projective model structure, is a presentation of the lax limit of the underlying ∞-categories. Our approach can also allow for the indexing category to be simplicial, as long as the diagram factors through its homotopy category. Analogous results for the associated homotopy limit (and other intermediate limits) directly follow. [ABSTRACT FROM AUTHOR]

We introduce a notion of \weak model category" which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences, and most of the usual constructions of categorical homotopy theory. Both left and right semi-model categories are weak model categories, and the opposite of a weak model category is again a weak model category. The main advantage of weak model categories is that they are easier to construct than Quillen model categories. In particular we give some simple criteria on two weak factorization systems for them to form a weak model category. The theory is developed in a very weak constructive, even predicative, framework and we use it to give constructive proofs of the existence of weak versions of various standard model categories, including the Kan{Quillen model structure, Lurie's variant of the Joyal model structure on marked simplicial sets, and the Verity model structure for weak complicial sets. We also construct semi-simplicial versions of all these. [ABSTRACT FROM AUTHOR]

We construct combinatorial model category structures on the categories of (marked) categories and (marked) preadditive categories, and we characterize (marked) additive categories as fibrant objects in a Bousfield localization of preadditive categories. These model category structures are used to present the corresponding 1-categories obtained by inverting equivalences. We apply these results to explicitly calculate limits and colimits in these 1-categories. The motivating application is a systematic construction of the equivariant coarse algebraic K-homology with coefficients in an additive category from its non-equivariant version. [ABSTRACT FROM AUTHOR]

Let X be a Noetherian separated and finite dimensional scheme over a field K of characteristic zero. The goal of this paper is to study deformations of X over a differential graded local Artin K -algebra by using local Tate–Quillen resolutions, i.e., the algebraic analogous of the Palamodov's resolvent of a complex space. The above goal is achieved by describing the DG-Lie algebra controlling deformation theory of a diagram of differential graded commutative algebras, indexed by a direct Reedy category. [ABSTRACT FROM AUTHOR]

We generalize the operadic approach to algebraic quantum field theory (arXiv:1709.08657) to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful quantization prescription for linear gauge theories. [ABSTRACT FROM AUTHOR]

We construct model category structures on various types of (marked) *-categories. These structures are used to present the infinity categories of (marked) *-categories obtained by inverting (marked) unitary equivalences. We use this presentation to explicitly calculate the ∞-categorical G-fixed points and G-orbits for G-equivariant (marked) *-categories. [ABSTRACT FROM AUTHOR]

In this note we study the local projective model structure on presheaves of complexes on a site, that is, we describe its classes of cofibrations, fibrations, and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties. [ABSTRACT FROM AUTHOR]

Motivated by gauge theory, we develop a general framework for chain complex-valued algebraic quantum field theories. Building upon our recent operadic approach to this subject, we show that the category of such theories carries a canonical model structure and explain the important conceptual and also practical consequences of this result. As a concrete application, we provide a derived version of Fredenhagen's universal algebra construction, which is relevant e.g. for the BRST/BV formalism. We further develop a homotopy theoretical generalization of algebraic quantum field theory with a particular focus on the homotopy-coherent Einstein causality axiom. We provide examples of such homotopy-coherent theories via (1) smooth normalized cochain algebras on ∞ -stacks, and (2) fiber-wise groupoid cohomology of a category fibered in groupoids with coefficients in a strict quantum field theory. [ABSTRACT FROM AUTHOR]

We show that the category of simplicial sets is a co-reective subcategory of the category of cubical sets with connections, with the inclusion given by a version of the straightening functor. We show that using the co-reector, one can transfer any cofibrantly generated model structure in which cofibrations are monomorphisms to cubical sets, thus obtaining cubical analogues of the Quillen and Joyal model structures. [ABSTRACT FROM AUTHOR]

The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal's Γ-spaces. The formalism of topological operads generalises well to different categories yielding such notions as $\mathbb E_n$ -algebras in chain complexes, while the Γ-space approach faces difficulties. In this paper we discuss how, by attempting to extend the Segal approach to arbitrary categoires, one arrives to the problem of understanding 'weak' sections of a homotopical Grothendieck fibration. We propose a model for such sections, called derived sections, and study the behaviour of homotopical categories of derived sections under the base change functors. The technology developed for the base-change situation is then applied to a specific class of 'resolution' base functors, which are inspired by cellular decompositions of classifying spaces. For resolutions, we prove that the inverse image functor on derived sections is homotopically full and faithful. [ABSTRACT FROM AUTHOR]

In [Barnea, Schlank, 2015(2)], the author and Tomer Schlank studied a much weaker homotopical structure than a model category, which we called a \weak cofibration category". We showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The main purpose of this paper is to give sufficient intrinsic conditions on a weak cofibration category for this two out of three property to hold. We consider an application to the category of compact metrizable spaces, and thus obtain a model structure on its ind-category. This model structure is defined on a category that is closely related to a category of topological spaces and has many convenient formal properties. A more general application of these results, to the (opposite) category of separable C*-algebras, appears in a paper by the author, Michael Joachim and Snigdhayan Mahanta [Barnea, Joachim, Mahanta, 2017]. [ABSTRACT FROM AUTHOR]

We compute the Bousfield localizations and Bousfield colocalizations of discrete model categories, including the homotopy categories and the algebraic K -groups of these localizations and colocalizations. We prove necessary and sufficient conditions for a subcategory of a category to appear as the subcategory of fibrant objects for some such model structure. We also prove necessary and sufficient conditions for a monad to be the fibrant replacement monad of some such model structure. [ABSTRACT FROM AUTHOR]

For a connected pasting scheme G, under reasonable assumptions on the underlying category, the category of C-colored G-props admits a cofibrantly generated model category structure. In this paper, we show that, if G is closed under shrinking internal edges, then this model structure on G-props satisfies a (weaker version) of left properness. Connected pasting schemes satisfying this property include those for all connected wheeled graphs (for wheeled properads), wheeled trees (for wheeled operads), simply connected graphs (for dioperads), unital trees (for symmetric operads), and unitial linear graphs (for small categories). The pasting scheme for connected wheel-free graphs (for properads) does not satisfy this condition. We furthermore prove, assuming G is shrinkable and our base categories are nice enough, that a weak symmetric monoidal Quillen equivalence between two base categories induces a Quillen equivalence between their categories of G-props. The final section gives illuminating examples that justify the conditions on base model categories. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Model categories have long been a useful tool in homotopy theory, allowing many generalizations of results in topological spaces to other categories. Giving a localization of a model category provides an additional model category structure on the same base category, which alters what objects are being considered equivalent by increasing the class of weak equivalences. In some situations, a model category where the class of weak equivalences is restricted from the original one could be more desirable. In this situation we need the notion of a delocalization. In this paper, right Bousfield delocalization is defined, we provide examples of right Bousfield delocalization as well as an existence theorem. In particular, we show that given two model category structures M1 and M2 we can define an additional model category structure M1 ∩M2 by defining the class of weak equivalences to be the intersection of the M1 and M2 weak equivalences. In addition we consider the model category on diagram categories over a base category (which is endowed with a model category structure) and show that delocalization is often preserved by the diagram model category structure. [ABSTRACT FROM AUTHOR]

It is shown that the category of enriched functors [ C , V ] is Grothendieck whenever V is a closed symmetric monoidal Grothendieck category and C is a category enriched over V . Localizations in [ C , V ] associated to collections of objects of C are studied. Also, the category of chain complexes of generalized modules Ch ( C R ) is shown to be identified with the Grothendieck category of enriched functors [ mod R , Ch ( Mod R ) ] over a commutative ring R , where the category of finitely presented R -modules mod R is enriched over the closed symmetric monoidal Grothendieck category Ch ( Mod R ) as complexes concentrated in zeroth degree. As an application, it is proved that Ch ( C R ) is a closed symmetric monoidal Grothendieck model category with explicit formulas for tensor product and internal Hom-objects. Furthermore, the class of unital algebraic almost stable homotopy categories generalizing unital algebraic stable homotopy categories of Hovey–Palmieri–Strickland [14] is introduced. It is shown that the derived category of generalized modules D ( C R ) over commutative rings is a unital algebraic almost stable homotopy category which is not an algebraic stable homotopy category. [ABSTRACT FROM AUTHOR]

We prove that a Quillen adjunction of model categories (of which we do not require functorial factorizations and of which we only require finite bicompleteness) induces a canonical adjunction of underlying quasicategories. [ABSTRACT FROM AUTHOR]

We develop a cofibrantly generated model category structure in the category of topological spaces in which weak equivalences are A -weak equivalences and such that the generalized CW( A )-complexes are cofibrant objects. With this structure the exponential law turns out to be a Quillen adjunction. [ABSTRACT FROM AUTHOR]

We give a short proof of the existence of a cofibrantly generated Quillen model structure on the category of small topologically enriched categories, obtained by a transfer from Bergner's model structure on simplicially enriched categories. [ABSTRACT FROM AUTHOR]

Résumé: Le but de cet article est d'exposer quelques réflexions en vue d'obtenir une structure de catégorie de modèles sur la catégorie des petites n-catégories strictes généralisant celle obtenue par Thomason dans le cas des catégories ordinaires. En s'inspirant d'idées de Grothendieck et de Cisinski, nous obtenons un ű théorème de Thomason abstrait Ƈ dont le théorème de Thomason classique est une conséquence facile. Nous en déduisons un théorème de Thomason 2-catégorique dont une preuve incorrecte a été publiée par K. Worytkiewicz, K. Hess, P. Parent et A. Tonks. Pour , nous isolons des conditions suffisantes pour obtenir un théorème de Thomason n-catégorique. L'étude de ces conditions fera l'objet de travaux ultérieurs. [ABSTRACT FROM AUTHOR]

Abstract: In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsion pairs and deconstructible classes, and our constructions are based on more general results about localization and transfer of abelian model structures. We indicate how recollements of triangulated categories can be obtained model categorically, discussing in detail Krauseʼs recollement . In the special case of curved mixed -graded complexes, we show that one of our singular models is Quillen equivalent to Positselskiʼs contraderived model for the homotopy category of matrix factorizations. [Copyright &y& Elsevier]

Abstract: In this article we establish the foundations of the Morita homotopy theory of -categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure (denoted by ) on the category of small unital -categories. The weak equivalences are the Morita equivalences and the cofibrations are the ⁎-functors which are injective on objects. As an application, we obtain an elegant description of Brown–Green–Rieffelʼs Picard group in the associated homotopy category . We then prove that is semi-additive. By group completing the induced abelian monoid structure at each Hom-set we obtain an additive category and a composite functor which is characterized by two simple properties: inversion of Morita equivalences and preservation of all finite products. Finally, we prove that the classical Grothendieck group functor becomes co-represented in by the tensor unit object. [Copyright &y& Elsevier]

Abstract: We discuss the monoidal structure on Frankeʼs algebraic model for the -local stable homotopy category at odd primes and show that its Picard group is isomorphic to the integers. [Copyright &y& Elsevier]

Abstract: We show that Quillenʼs small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jørgensen, Neeman, Murfet, Prest, Trlifaj and possibly others. [Copyright &y& Elsevier]