Quillen Equivalent Research Articles

Quillen equivalences inducing Grothendieck duality for unbounded chain complexes of sheaves

Quillen equivalences inducing Grothendieck duality for unbounded chain complexes of sheaves

Enriched Koszul duality

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.

A homotopy coherent nerve for (∞,n)-categories

In the case of (∞,1)-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams of (∞,1)-categories can equivalently be defined as functors of quasi-categories or as simplicially enriched functors out of the homotopy coherent categorifications.In this paper, we construct a homotopy coherent nerve for (∞,n)-categories. We show that it realizes a right Quillen equivalence between the models of categories strictly enriched in (∞,n−1)-categories and of Segal category objects in (∞,n−1)-categories. This similarly enables us to define homotopy coherent diagrams of (∞,n)-categories equivalently as functors of Segal category objects or as strictly enriched functors out of the homotopy coherent categorifications.

A Quillen adjunction between globular and complicial approaches to (∞,n)-categories

We prove the compatibility between the suspension construction and the complicial nerve of ω-categories. As a motivating application, we produce a Quillen pair between the models of (∞,n)-categories given by Rezk's complete Segal Θn-spaces and Verity's n-complicial sets.

Equivalence of cubical and simplicial approaches to (∞,n)-categories

We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for (n-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for (n-trivial, saturated) complicial sets is a Quillen equivalence. Our proof is based on the theory of cones, previously developed by the first two authors together with Lindsey and Sattler.

Strictification theorems for the homotopy time-slice axiom

It is proven that the homotopy time-slice axiom for many types of algebraic quantum field theories (AQFTs) taking values in chain complexes can be strictified. This includes the cases of Haag–Kastler-type AQFTs on a fixed globally hyperbolic Lorentzian manifold (with or without time-like boundary), locally covariant conformal AQFTs in two spacetime dimensions, locally covariant AQFTs in one spacetime dimension, and the relative Cauchy evolution. The strictification theorems established in this paper prove that, under suitable hypotheses that hold true for the examples listed above, there exists a Quillen equivalence between the model category of AQFTs satisfying the homotopy time-slice axiom and the model category of AQFTs satisfying the usual strict time-slice axiom.

Homotopy theory of net representations

The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore, a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell [Formula: see text]-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.

Torsion models for tensor-triangulated categories: the one-step case

Given a suitable stable monoidal model category $\mathscr{C}$ and a specialization closed subset $V$ of its Balmer spectrum one can produce a Tate square for decomposing objects into the part supported over $V$ and the part supported over $V^c$ spliced with the Tate object. Using this one can show that $\mathscr{C}$ is Quillen equivalent to a model built from the data of local torsion objects, and the splicing data lies in a rather rich category. As an application, we promote the torsion model for the homotopy category of rational circle-equivariant spectra from [18] to a Quillen equivalence. In addition, a close analysis of the one step case highlights important features needed for general torsion models which we will return to in future work.

The mathbb {R}-local homotopy theory of smooth spaces

Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. There is a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We consider the localisation of the (projective or injective) model category of smooth spaces at the morphisms which become weak equivalences under the singular complex functor. We prove that this localisation agrees with a motivic-style mathbb {R}-localisation of the model category of smooth spaces. Further, we exhibit the singular complex functor for smooth spaces as one of several Quillen equivalences between model categories for spaces and the above mathbb {R}-local model category of smooth spaces. In the process, we show that the singular complex functor agrees with the homotopy colimit functor up to a natural zig-zag of weak equivalences. We provide a functorial fibrant replacement in the mathbb {R}-local model category of smooth spaces and use this to compute mapping spaces in terms of singular complexes. Finally, we explain the relation of our fibrant replacement to the concordance sheaf construction introduced recently by Berwick-Evans, Boavida de Brito and Pavlov.

Algebraic models of change of groups functors in (co)free rational equivariant spectra

Free and cofree equivariant spectra are important classes of equivariant spectra which represent equivariant cohomology theories on free equivariant spaces. Greenlees-Shipley [24], [26] and Pol and the author [45] have given an algebraic model for rational (co)free equivariant spectra. In this paper, we extend this framework by proving that the Quillen functors of induction-restriction-coinduction between categories of (co)free rational equivariant spectra correspond to Quillen functors between the algebraic models in the case of connected compact Lie groups. This is achieved using new abstract techniques regarding correspondences of Quillen functors along Quillen equivalences, which we expect to be of use in other applications.

Frobenius functors, stable equivalences and K-theory of Gorenstein projective modules

Owing to the difference in K-theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition for the Frobenius pair of faithful functors between two abelian categories to be a Quillen equivalence, which is also equivalent to that the Frobenius functors induce mutually inverse equivalences between stable categories of Gorenstein projective objects.We show that the category of Gorenstein projective objects is a Waldhausen category, then Gorenstein K-groups are introduced and characterized. As applications, we show that stable equivalences of Morita type preserve Gorenstein K-groups, CM-finiteness and CM-freeness. Two specific examples of path algebras are presented to illustrate the results, for which the Gorenstein K0 and K1-groups are calculated.

Categorical Koszul duality

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. We show that the normalized chain complex functor transforms the Quillen equivalence between quasicategories and simplicial categories into this Koszul duality. This allows us to give a conceptual interpretation of the dg nerve of a dg category and its adjoint. As an application, we prove that the category of representations of a quasicategory K is equivalent to the coderived category of comodules over C⁎(K), the chain coalgebra of K. A corollary of this is a characterization of the category of constructible dg sheaves on a stratified space as the coderived category of a certain dg coalgebra.

Unitary calculus: model categories and convergence

We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra—a variation of orthogonal spectra—as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the n-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.

Equivariant dendroidal sets and simplicial operads

We establish a Quillen equivalence between the homotopy theories of equivariant Segal operads and equivariant simplicial operads with norm maps. Together with previous work, we further conclude that the homotopy coherent nerve is a right-Quillen equivalence from the model category of equivariant simplicial operads with norm maps to the model category structure for equivariant-$\infty$-operads in equivariant dendroidal sets.

Model category structures on multicomplexes

We present a family of model structures on the category of multicomplexes. There is a cofibrantly generated model structure in which the weak equivalences are the morphisms inducing an isomorphism at a fixed stage of an associated spectral sequence. Corresponding model structures are given for truncated versions of multicomplexes, interpolating between bicomplexes and multicomplexes. For a fixed stage of the spectral sequence, the model structures on all these categories are shown to be Quillen equivalent.

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.

Homotopy theory of Moore flows (II)

This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.

Homotopy theory of Moore flows (I)

Erratum, 11 July 2022: This is an updated version of the original paper \cite{Moore1} in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix.A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the biclosed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed d-spaces and the q-model structure of flows.

Quasi-categories vs. Segal spaces: Cartesian edition

We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]),On bisimplicial spaces (due to deBrito [12]),On bisimplicial sets,On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.

2–Segal objects and the Waldhausen construction

In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an $S_\bullet$-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known $S_\bullet$-constructions.