Quillen Equivalent Research Articles

Homotopy theory of modules over diagrams of rings

Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M ( s ) \mathcal {M}(s) (as s s runs through the diagram), we consider the category of diagrams where the object X ( s ) X(s) at s s comes from M ( s ) \mathcal {M}(s) . We develop model structures on such categories of diagrams and Quillen adjunctions that relate categories based on different diagram shapes. Under certain conditions, cellularizations (or right Bousfield localizations) of these adjunctions induce Quillen equivalences. As an application we show that a cellularization of a category of modules over a diagram of ring spectra (or differential graded rings) is Quillen equivalent to modules over the associated inverse limit of the rings. Another application of the general machinery here is given in work by the authors on algebraic models of rational equivariant spectra. Some of this material originally appeared in the preprint “An algebraic model for rational torus-equivariant stable homotopy theory”, arXiv:1101.2511, but has been generalized here.

Derived equivalences induced by big cotilting modules

We prove that given a Grothendieck category G with a tilting object of finite projective dimension, the induced triangle equivalence sends an injective cogenerator of G to a big cotilting module. Moreover, every big cotilting module can be constructed like that in an essentially unique way. We also prove that the triangle equivalence is at the base of an equivalence of derivators, which in turn is induced by a Quillen equivalence with respect to suitable abelian model structures on the corresponding categories of complexes.

Group actions on Segal operads

We give a Quillen equivalence between model structures for simplicial operads, described via the theory of operads, and Segal operads, thought of as certain reduced dendroidal spaces. We then extend this result to give an Quillen equivalence between the model structures for simplicial operads equipped with a group action and the corresponding Segal operads.

Fixed point adjunctions for equivariant module spectra

We consider the Quillen adjunction between fixed points and inflation in the context of equivariant module spectra over equivariant ring spectra, and give numerous examples including some based on geometric fixed points and some on the Eilenberg-Moore spectral sequence. These results were originally presented as part of our equivalence between rational torus-equivariant spectra and an algebraic model in arXiv:1101.2511. However, the present results apply in many other interesting cases explored here, which are not rational and where the ambient group is not a torus. The material in arXiv:1101.2511v3 will be revised to refer to this paper.

Moduli spaces of algebras over nonsymmetric operads

We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.

Conjugate pairs of categories and Quillen equivalent stable model categories of functors

We describe the notion of a conjugate pair(B,A) of small categories, wherein maps in B admit a factorization by maps in the subcategory A, much in the spirit of a “two-sided” calculus of fractions. When (B,A) is a conjugate pair, we prove that for any cofibrantly generated model category C there is an induced Quillen adjunction between the functor categories [Bop,C] and [Aop,C]. When C is a left proper stable model category, this adjunction is a Quillen equivalence. Finally, we demonstrate that minor modifications of our arguments give the analogous result when C is instead assumed to be an Ab-category.

Comparison of models for (∞,n)–categories, I

While many different models for $(\infty,1)$-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for $(\infty, n)$-categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as $(\infty,n)$-categories. Furthermore, we establish Quillen equivalences between them.

Preorientations of the derived motivic multiplicative group

We provide a proof in the language of model categories and symmetric spectra of Lurie's theorem that topological complex $K$-theory represents orientations of the derived multiplicative group. Then we generalize this result to the motivic situation. Along the way, a number of useful model structures and Quillen adjunctions both in the classical and in the motivic case are established.

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.

Model categories for orthogonal calculus

We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)-action. This improves upon the classification of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.

Group completion and units in I-spaces

The category of I-spaces is the diagram category of spaces indexed by finite sets and injections. This is a symmetric monoidal category whose commutative monoids model all E-infinity spaces. Working in the category of I-spaces enables us to simplify and strengthen previous work on group completion and units of E-infinity spaces. As an application we clarify the relation to Gamma-spaces and show how the spectrum of units associated with a commutative symmetric ring spectrum arises through a chain of Quillen adjunctions.

Transfer of algebras over operads along Quillen adjunctions

Let 𝒱 be a cofibrantly generated monoidal model category and ℳ be a monoidal 𝒱-model category. If C is any set, then we denote by ℳC the product category of copies of ℳ indexed by C. Given a cofibrant C-coloured operad 𝒫 in 𝒱, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in ℳC to preserve 𝒫-algebra structures. In particular, we show how 𝒫-algebra structures can be transferred along Quillen adjunctions of monoidal 𝒱-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on ℳ. As an application, we prove that in the category of symmetric spectra the n-connective cover functor preserves A∞ and E∞ module spectra over connective ring spectra, for every n∈ℤ.

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.

Rational ℤp–equivariant spectra

We find a simple algebraic model for rational G-equivariant spectra, where G is the p-adic integers, via a series of Quillen equivalences. This model, along with an Adams short exact sequence, will allow us to easily perform constructions and calculations.

On the structure of simplicial categories associated to quasi-categories

AbstractThe homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint are important to the study of (∞, 1)-categories because they provide a means for comparing two models of their respective homotopy theories, giving a Quillen equivalence between the model structures for quasi-categories and simplicial categories. The functor also gives a cofibrant replacement for ordinary categories, regarded as trivial simplicial categories. However, the hom-spaces of the simplicial category X arising from a quasi-category X are not well understood. We show that when X is a quasi-category, all Λ21 horns in the hom-spaces of its simplicial category can be filled. We prove, unexpectedly, that for any simplicial set X, the hom-spaces of X are 3-coskeletal. We characterize the quasi-categories whose simplicial categories are locally quasi, finding explicit examples of 3-dimensional horns that cannot be filled in all other cases. Finally, we show that when X is the nerve of an ordinary category, X is isomorphic to the simplicial category obtained from the standard free simplicial resolution, showing that the two known cofibrant “simplicial thickenings” of ordinary categories coincide, and furthermore its hom-spaces are 2-coskeletal.

Mapping spaces in quasi-categories

We apply the Dwyer-Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [DS1], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way.

Lifting of model structures to fibred categories

A consists of a functor $p: \mathbf{N} \to \mathbf{M}$ between categories $\mathbf{N}$ and $\mathbf{M}$ such that objects of $\mathbf{N}$ may be pulled back along any arrow of $\mathbf{M}$. Given a $p: \mathbf{N} \to \mathbf{M}$ and a model structure on the category $\mathbf{M}$, we show that there exists a lifting of the model structure on $\mathbf{M}$ to a model structure on $\mathbf{N}$. We will refer to such a system as a fibred model category and give several examples of such structures. We show that, under certain conditions, right homotopies of maps in the base $\mathbf{M}$ may be lifted to right homotopic maps in the category. Further, we show that these lifted model structures are well behaved with respect to Quillen adjunctions and Quillen equivalences. Finally, we show that if $\mathbf{N}$ and $\mathbf{M}$ carry compatible closed monoidal structures and the functor $p$ commutes with colimits, then a Quillen pair on $\mathbf{M}$ lifts to a Quillen pair on $\mathbf{N}$.

Generalized spectral categories, topological Hochschild homology and trace maps

We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to the homotopy theory of Eilenberg-Mac Lane spectral categories. Finally, we show that over the rationals two dg categories are topological equivalent if and only if they are quasi-equivalent.

Homotopy theory of posets

This paper studies the category of posets Pos as a model for the homotopy theory of spaces. We prove that: (i) Pos admits a (cofibrantly generated and proper) model structure and the inclusion functor Pos → Cat into Thomason's model category is a right Quillen equivalence, and (ii) there is a proper class of different choices of cofibrations for a model structure on Pos or Cat where the weak equivalences are defined by the nerve functor. We also discuss the homotopy theory of posets from the viewpoint of Alexandroff T0-spaces, and we apply a result of McCord to give a new proof of the classification theorems of Moerdijk and Weiss in the case of posets.