Projective Model Structure Research Articles

Homotopy in Exact Categories

In this monograph we develop various aspects of the homotopy theory of exact categories. We introduce different notions of compactness and generation in exact categories, and use these to study model structures on categories of chain complexes C h ∗ ( E ) Ch_{*}(\mathcal {E}) which are induced by cotorsion pairs on E \mathcal {E} . As a special case we show that under very general conditions the categories C h + ( E ) Ch_{+}(\mathcal {E}) , C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) , and C h ( E ) Ch(\mathcal {E}) are equipped with the projective model structure, and that a generalisation of the Dold-Kan correspondence holds. We also establish conditions under which categories of filtered objects in exact categories are equipped with natural model structures. When E \mathcal {E} is monoidal we also examine when these model structures are monoidal and conclude by studying some homotopical algebra in such categories. In particular we provide conditions under which C h ( E ) Ch(\mathcal {E}) and C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) are homotopical algebra contexts, thus making them suitable settings for derived geometry.

A model structure for Grothendieck fibrations

We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category C. For the discrete case, we build a model structure on the slice Cat/C, Quillen equivalent to the projective model structure on [Cop,Set] via the classical category of elements construction. The cartesian case requires the use of markings, and we define a model structure on the slice Cat/C+, Quillen equivalent to the projective model structure on [Cop,Cat] via a marked version of the Grothendieck construction.We further show that both of these model structures have the expected interactions with their ∞-counterparts; namely, with the contravariant model structure on sSet/NC and with Lurie's cartesian model structure on sSet/NC+.

Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle

Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle

MODEL CATEGORIES AND PRO-p IWAHORI–HECKE MODULES

AbstractLet G denote a possibly discrete topological group admitting an open subgroup I which is pro-p. If H denotes the corresponding Hecke algebra over a field k of characteristic p, then we study the adjunction between H-modules and k-linear smooth G-representations in terms of various model structures. If H is a Gorenstein ring, we single out a full subcategory of smooth G-representations which is equivalent to the category of all Gorenstein projective H-modules via the functor of I-invariants. This applies to groups of rational points of split connected reductive groups over finite and over non-Archimedean local fields, thus generalizing a theorem of Cabanes. Moreover, we show that the Gorenstein projective model structure on the category of H-modules admits a right transfer. On the homotopy level, the right derived functor of I-invariants then admits a right inverse and becomes an equivalence when restricted to a suitable subcategory.

On notions of compactness, object classifiers, and weak Tarski universes

AbstractWe prove a correspondence between $\kappa$ -small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $\kappa$ -compact maps in their underlying quasi-categories for suitably large regular cardinals $\kappa$ . We thus obtain a transition result between weakly universal small fibrations in the (type-theoretic) injective Dugger–Rezk-style standard presentations of model toposes and object classifiers in Grothendieck $\infty$ -toposes in the sense of Lurie.

Noncommutative CW-spectra as enriched presheaves on matrix algebras

Motivated by the philosophy that C^* -algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of C^* -algebras. We focus on C^* -algebras which are noncommutative CW-complexes in the sense of Eilers et al. (1998). We construct the stable \infty -category of noncommutative CW-spectra, which we denote by \mathtt{NSp} . Let \mathcal{M} be the full spectral subcategory of \mathtt{NSp} spanned by “noncommutative suspension spectra” of matrix algebras. Our main result is that \mathtt{NSp} is equivalent to the \infty -category of spectral presheaves on \mathcal{M} . To prove this, we first prove a general result which states that any compactly generated stable \infty -category is naturally equivalent to the \infty -category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an \infty -categorical version of a result by Schwede and Shipley (2003). In proving this, we use the language of enriched 1-categories as developed recently by Hinich. We end by presenting a “strict” model for \mathcal{M} . That is, we define a category \mathcal{M}_s strictly enriched in a certain monoidal model category of spectra \mathtt{Sp}^\mathtt{M} . We give a direct proof that the category of \mathtt{Sp}^\mathtt{M} -enriched presheaves \mathcal{M}_s^{\mathtt{op}}\to\mathtt{Sp}^{\mathtt{M}} with the projective model structure models \mathtt{NSp} and conclude that \mathcal{M}_s is a strict model for \mathcal{M} .

Arrow categories of monoidal model categories

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.

A new method to construct model structures from a cotorsion pair

Let be a complete and hereditary cotorsion pair in an abelian category . We show that if is closed under kernels of epimorphisms, then is a strong left Frobenius pair, which induces a unique projective exact model structure on . Some examples are given. Furthermore, we construct additional abelian model structures from and .

Homotopy theory of dg sheaves

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.

Injective and projective model structures on enriched diagram categories

In the enriched setting, the notions of injective and projective model structures on a category of enriched diagrams also make sense. In this paper, we prove the existence of these model structures on enriched diagram categories under local presentability, accessibility, and "acyclicity" conditions, using the methods of lifting model structures from an adjunction introduced by Garner, Hess, Kedziorek, Riehl, and Shipley.

Frobenius pairs in abelian categories

In this work, we revisit Auslander-Buchweitz Approximation Theory and find some relations with cotorsion pairs and model category structures. From the notions of relatives generators and cogenerators in Approximation Theory, we introduce the concept of left Frobenius pairs $(\mathcal{X},\omega)$ in an abelian category $\mathcal{C}$. We show how to construct from $(\mathcal{X},\omega)$ a projective exact model structure on $\mathcal{X}^\wedge$, as a result of Hovey-Gillespie Correspondence applied to two compatible and complete cotorsion pairs in $\mathcal{X}^\wedge$. These pairs can be regarded as examples of what we call cotorsion pairs relative to a thick subcategory of $\mathcal{C}$. We establish some correspondences between Frobenius pairs, relative cotorsion pairs, exact model structures and Auslander-Buchweitz contexts. Finally, some applications of these results are given in the context of Gorenstein homological algebra by generalizing some existing model structures on the categories of modules over Gorenstein and Ding-Chen rings, and by encoding the stable module category of a ring as a certain homotopy category. We also present some connections with perfect cotorsion pairs, covering classes, and cotilting modules.

Positive Model Structures for Abstract Symmetric Spectra

We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorn’s sense, of a certain positive projective model structure on spectra, where positivity basically means the truncation of the zero level. The localization is by the set of stabilizing morphisms or their truncated version.

Derived Moduli of Complexes and Derived Grassmannians

In the first part of this paper we construct a model structure for the category of filtered cochain complexes of modules over some associative ring R and explain how the classical Rees construction relates this to the usual projective model structure over cochain complexes. The second part of the paper is devoted to the study of derived moduli of sheaves: we give a new proof of the representability of the derived stack of perfect complexes over a proper scheme and then use the new model structure for filtered complexes to tackle moduli of filtered derived modules. As an application, we construct derived versions of Grassmannians and flag varieties.

A projective model structure on pro-simplicial sheaves, and the relative étale homotopy type

In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study of étale homotopy. Using this model structure we define a pro-space associated with a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur's étale homotopy type [3] in the relevant special case. Our definition extends naturally to a relative notion, namely, a pro-object associated with a map of topoi. This relative notion lifts the relative étale homotopy type that was used in [18] for the study of obstructions to the existence of rational points. This relative notion enables to generalize these homotopical obstructions from fields to general base schemas and general maps of topoi.Our model structure is constructed using a general theorem that we prove. Namely, we introduce a much weaker structure than a model category, which we call a “weak fibration category”. Our theorem says that a weak fibration category can be “completed” into a full model category structure on its pro-category, provided it satisfies some additional technical requirements. Our model structure is obtained by applying this result to the weak fibration category of simplicial sheaves over a Grothendieck site, where the weak equivalences and the fibrations are local in the sense of Jardine [23].

Corrigendum to “Homotopy theory of modules over operads in symmetric spectra”

It is well known that the cofibrations in S , equipped with the projective model structure, are precisely the monomorphisms such that G acts freely on the simplices of the codomain not in the image. One way to verify this is to (i) argue that the image of such a map is a subcomplex of the codomain (ie the codomain can be built from the image by attaching G–cells), and (ii) note that every monomorphism is isomorphic to its image, hence verifying that such maps are cofibrations, (iii) conversely, to note that every generating cofibration is such a map, and (iv) hence conclude that every cofibration is such a map, by using the fact that every cofibration is a retract of a (possibly transfinite) composition of pushouts of the generating cofibrations. The problem with our argument for the cofibration description in [2, Proposition 6.3] was a cavalier application of the subcomplex argument (i) above; we ignored the fact that

Abelian model structures and Ding homological dimensions

Let $R$ be an $n$-FC ring. For $0<t\leq n$, we construct a new abelian model structure on $R$-Mod, called the Ding $t$-projective ($t$-injective) model structure. Based on this, we establish a bijective correspondence between $dg$-$t$-projective ($dg$-$t$-injective) $R$-complexes and Ding $t$-projective ($t$-injective) $A$-modules under some additional conditions, where $A=R[x]/(x^2)$. This gives a generalized version of the bijective correspondence established in [14] between $dg$-projective ($dg$-injective) $R$-complexes and Gorenstein projective (injective) $A$-modules. Finally, we show that the embedding functors $K(\mathcal{D} \mathcal{P})\rightarrow K$ ($R$-Mod) and $K(\mathcal{D} \mathcal{J})\rightarrow K$ ($R$-Mod) have right and left adjoints respectively, where $K(\mathcal{D} \mathcal{P})$ ($K(\mathcal{D} \mathcal{J})$) is the homotopy category of complexes of Ding projective (injective) modules, and $K$ ($R$-Mod) denotes the homotopy category.

Left fibrations and homotopy colimits

For a small category \(\mathbf{A}\), we prove that the homotopy colimit functor from the category of simplicial diagrams on \(\mathbf{A}\) to the category of simplicial sets over the nerve of \(\mathbf{A}\) establishes a left Quillen equivalence between the projective (or Reedy) model structure on the former category and the covariant model structure on the latter. We compare this equivalence to a Quillen equivalence in the opposite direction previously established by Lurie. From our results we deduce that a categorical equivalence of simplicial sets induces a Quillen equivalence on the corresponding over-categories, equipped with the covariant model structures. Also, we show that a version of Quillen’s Theorem A for \(\infty \)-categories easily follows.

Bar-cobar duality for operads in stable homotopy theory

We extend bar-cobar duality, defined for operads of chain complexes by Getzler and Jones, to operads of spectra in the sense of stable homotopy theory. Our main result is the existence of a Quillen equivalence between the category of reduced operads of spectra (with the projective model structure) and a new model for the homotopy theory of cooperads of spectra. The crucial construction is of a weak equivalence of operads between the Boardman-Vogt W-construction for an operad P, and the cobar-bar construction on P. This weak equivalence generalizes a theorem of Berger and Moerdijk that says the W- and cobar-bar constructions are isomorphic for operads of chain complexes. Our model for the homotopy theory of cooperads is based on `pre-cooperads'. These can be viewed as cooperads in which the structure maps are zigzags of maps of spectra that satisfy coherence conditions. Our model structure on pre-cooperads is such that every object is weakly equivalent to an actual cooperad, and weak equivalences between cooperads are detected in the underlying symmetric sequences. We also interpret our results in terms of a `derived Koszul dual' for operads of spectra, which is analogous to the Ginzburg-Kapranov dg-dual. We show that the `double derived Koszul dual' of an operad P is equivalent to P (under some finiteness hypotheses) and that the derived Koszul construction preserves homotopy colimits, finite homotopy limits and derived mapping spaces for operads.

Cotorsion pairs and model structures on Ch(R)

AbstractWe show that if the given cotorsion pair $(\mathcal{A},\mathcal{B})$ in the category of modules is complete and hereditary, then both of the induced cotorsion pairs in the category of complexes are complete. We also give a cofibrantly generated model structure that can be regarded as a generalization of the projective model structure.

Gorenstein model structures and generalized derived categories

AbstractIn a paper from 2002, Hovey introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce: the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable module category of R. If such a ring R has finite global dimension, the graded ring R[x]/(x2) is Gorenstein and the three associated Gorenstein model structures on R[x]/(x2)-Mod, the category of graded R[x]/(x2)-modules, are nothing more than the usual projective, injective and flat model structures on Ch(R), the category of chain complexes of R-modules. Although these correspondences only recover these model structures on Ch(R) when R has finite global dimension, we can set R = ℤ and use general techniques from model category theory to lift the projective model structure from Ch(ℤ) to Ch(R) for an arbitrary ring R. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when ℤ[x]/(x2) is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring R and the derived category of R and we give some examples of such generalizations.