Model Category Structure Research Articles

Simplicial presheaves of coalgebras

The category of simplicial R-coalgebras over a presheaf of commutative unital rings on a small Grothendieck site is endowed with a left proper, simplicial, cofibrantly generated model category structure where the weak equivalences are the local weak equivalences of the underlying simplicial presheaves. This model category is naturally linked to the R-local homotopy theory of simplicial presheaves and the homotopy theory of simplicial R-modules by Quillen adjunctions. We study the comparison with the R-local homotopy category of simplicial presheaves in the special case where R is a presheaf of algebraically closed (or perfect) fields. If R is a presheaf of algebraically closed fields, we show that the R-local homotopy category of simplicial presheaves embeds fully faithfully in the homotopy category of simplicial R-coalgebras.

Dendroidal sets and simplicial operads

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by A. Joyal and J. Lurie. We also explain how this theory applies to the usual notion of operad (i.e. with a single colour) in the category of spaces.

On some new model category structures from old, on the same underlying category

Abstract We 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.

Maurer–Cartan moduli and models for function spaces

We set up a formalism of Maurer–Cartan moduli sets for L∞ algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential graded coalgebras). Among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of Chevalley–Eilenberg and Harrison cohomology. We apply the developed technology to construct rational homotopy models for function spaces.

Homotopy colimits in stable representation theory

We study the problem of existence and uniqueness of homotopy colimits in stable representation theory, where one typically does not have model category structures to guarantee that these homotopy colimits exist or have good properties. We get both negative results (homotopy cofibers fail to exist if there exist any objects of positive finite projective dimension!) and positive results (reasonable conditions under which homotopy colimits exist and are unique, even when model category structures fail to exist). Along the way, we obtain relative-homological-algebraic generalizations of classical theorems of Hilton-Rees and Oort. We describe some applications to Waldhausen $K$-theory and to deformation-theoretic methods in stable representation theory.

Model category structures arising from Drinfeld vector bundles

We present a general construction of model category structures on the category C(Qco(X)) of unbounded chain complexes of quasi-coherent sheaves on a semi-separated scheme X. The construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of X. It does not require closure under direct limits as previous methods. We apply it to describe the derived category D(Qco(X)) via various model structures on C(Qco(X)). As particular instances, we recover recent results on the flat model structure for quasi-coherent sheaves. Our approach also includes the case of (infinite-dimensional) vector bundles, and restricted Drinfeld vector bundles. Finally, we prove that the unrestricted case does not induce a model category structure as above in general.

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 model structure approach to the finitistic dimension conjectures

AbstractWe explore the interlacing between model category structures attained to classes of modules of finite \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {X}$\end{document}‐dimension, for certain classes of modules \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {X}$\end{document}. As an application we give a model structure approach to the finitistic dimension conjectures and present a new conceptual framework in which these conjectures can be studied.

Algebraic models for higher categories

We introduce the notion of algebraic fibrant objects in a general model category and establish a (combinatorial) model category structure on algebraic fibrant objects. Based on this construction, we propose algebraic Kan complexes as an algebraic model for ∞ -groupoids and algebraic quasi-categories as an algebraic model for ( ∞ , 1 ) -categories. We furthermore give an explicit proof of the homotopy hypothesis.

Units of equivariant ring spectra

It is well known that very special $\Gamma$-spaces and grouplike $\E_\infty$ spaces both model connective spectra. Both these models have equivariant analogues. Shimakawa defined the category of equivariant $\Gamma$-spaces and showed that special equivariant $\Gamma$-spaces determine positive equivariant spectra. Costenoble and Waner showed that grouplike equivariant $\E_\infty$-spaces determine connective equivariant spectra. We show that with suitable model category structures the category of equivariant $\Gamma$-spaces is Quillen equivalent to the category of equivariant $\E_\infty$ spaces. We define the units of equivariant ring spectra in terms of equivariant $\Gamma$-spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.

Dendroidal sets as models for homotopy operads

The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure whose fibrant objects are the infinity-operads (i.e. dendroidal inner Kan complexes). This extends the theory of infinity-categories in the sense that the Joyal model category structure on simplicial sets whose fibrant objects are the infinity-categories is recovered from the model category structure on dendroidal sets by simply slicing over the point.

Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence

This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived categories of CDG-comodules, and the contraderived categories of CDG-contramodules. The equivalence between the latter two categories (the comodule-contramodule correspondence) is established. Nonhomogeneous Koszul duality or "triality" (an equivalence between exotic derived categories corresponding to Koszul dual (C)DG-algebra and CDG-coalgebra) is obtained in the conilpotent and nonconilpotent versions. Various $A_\infty$-structures are considered, and a number of model category structures are described. Homogeneous Koszul duality and $D$-$\Omega$ duality are discussed in the appendices.

Bar constructions and Quillen homology of modules over operads

We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical proof after showing that certain homotopy colimits in algebras and modules over operads can be easily understood. A key result here, which lies at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. We also prove analogous results for algebras and modules over operads in unbounded chain complexes.

A cartesian presentation of weakn–categories

We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Theta_n. This notion is a generalization of that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian.

Homotopy theory of modules over operads and non- [formula omitted] operads in monoidal model categories

We establish model category structures on algebras and modules over operads and non- Σ operads in monoidal model categories. The results have applications in algebraic topology, stable homotopy theory, and homological algebra.

Homotopy theory of modules over operads in symmetric spectra

We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads.

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.

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.

Deformation theory of representations of prop(erad)s II

We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.

Fibrant Presheaves of Spectra and Guillén–Navarro Extension

In this remark we prove that the Guillén–Navarro extension of a presheaf of spectra in the category of algebraic varieties over a field of characteristic zero, when exists, coincides up to weak equivalence with the fibrant replacement of the presheaf in the injective model category structure with the cd-topology of abstract blow-ups.