Quillen Equivalent Research Articles

Differential graded versus simplicial categories

We establish a connection between differential graded and simplicial categories by constructing a three-step zig-zag of Quillen adjunctions relating the homotopy theories of the two. In an intermediate step, we extend the Dold–Kan correspondence to a Quillen equivalence between categories enriched over non-negatively graded complexes and categories enriched over simplicial modules. As an application, we obtain a simple calculation of Simpson's homotopy fiber, which is known to be a key step in the construction of a moduli stack of perfect complexes on a smooth projective variety.

Chains on suspension spectra

Some background is needed to understand why this functor deserves attention. There is a much simpler functor called N (normalised simplicial chains) from simplicial sets to integral chain complexes that computes integral homology, and one can just tensor with Q to compute rational homology. There is a dual complex N that calculates integral cohomology. This is equipped with a natural product N .X / N .X /! N .X / which is commutative up to homotopy but not on the nose. The theory of Steenrod operations shows that if we work integrally then neither N .X / nor any reasonable replacement can be given a strictly commutative product (even with the usual signs). Rationally, however, the situation is better: in [10] Sullivan developed a rational and simplicial version of de Rham theory giving a cochain complex  .X / with a strictly commutative product that computes the ordinary rational cohomology of X . This can be used as a starting point for the rich and powerful theory of rational homotopy (originally introduced by Quillen [8] using slightly different machinery). One can then stabilise and consider the category SQ of rational spectra, which makes things considerably simpler: it is well-known that the homotopy category of SQ is equivalent to the category of graded rational vector spaces. However, we can make things harder again by considering rational spectra with a ring structure or a group action. To handle these, we need to improve the homotopy classification of rational spectra to some kind of monoidal Quillen equivalence of SQ with a suitable model category ChQ of rational chain complexes.

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.

Classifying rational $G$-spectra for finite $G$

We give a new proof that for a finite group G, the category of rational Gequivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model. A G-equivariant cohomology theory E ∗ is said to be rational if E ∗ (X) is a rational vector space for every G-space X. For G, a finite group, we want to describe the category of rational G-equivariant cohomology theories in terms of a simple algebraic model. In particular, we want to understand those cohomology theories with a multiplication and the modules over such a theory. To do so, we give a particular construction of a model category of G-spectra whose homotopy category is (equivalent to) the category of rational G-equivariant cohomology theories. We show that this model category is symmetric monoidally Quillen equivalent to an explicit algebraic model: the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Since our Quillen equivalences are symmetric monoidal, the category of ring spectra is Quillen equivalent to the category of monoids in the algebraic model. Let WGH be the Weyl group of H in G, the quotient of the normaliser of H in G by H, then a monoid in the category of chain complexes of modules over the rational group ring of WGH, is a differential graded Q-algebra with an action (through algebra maps) of the group WGH. The category of modules over a ring spectrum will then be Quillen equivalent to modules over a monoid in the algebraic model. So, if one has a ring spectrum R that one wishes to study, one can look at its image ˜ R

Splitting monoidal stable model categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [ S , S ] C . An idempotent e of this ring will split the homotopy category: [ X , Y ] C ≅ e [ X , Y ] C ⊕ ( 1 − e ) [ X , Y ] C . We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to L e S C × L ( 1 − e ) S C and [ X , Y ] L e S C ≅ e [ X , Y ] C . This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.

On the homotopy theory of $n$-types

An n-truncated model structure on simplicial (pre-)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine's intermediate model structures we construct such an n-type model structure via Bousfield-Friedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called n-hyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analogous n-truncations on simplicial groupoids and prove a Quillen equivalence between these settings. We achieve a classification of n-types of simplicial presheaves in terms of (n-1)-types of presheaves of simplicial groupoids. Our classification holds for general n. Therefore this can also be viewed as the homotopy theory of (pre-)sheaves of (weak) higher groupoids.

Diagrams indexed by Grothendieck constructions

Let I be a small indexing category, G: I op ! Cat be a functor and BG 2 Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (resp. groupoids) on a stack M and presheaves of simplicial sets (resp. groupoids) over M. The motivation for this paper was the study of homotopy theory of (pre)sheaves on a stack. Since the site associated to a stack M is a Grothedieck construction this led us to an investigation of the homotopy theory of diagrams indexed on a category which is itself a Grothendieck construction (of a diagram of small categories). The body of the paper is concerned with analyzing various Quillen pairs between diagram categories. These adjunctions are of general interest and we present some examples not related to the theory of stacks. We conclude the paper with the applications to stacks. Stacks were introduced in algebraic geometry in order to parametrize families of objects when the presence of automorphisms prevented representability by a scheme or even a sheaf [A, DM, Gi]. Recently stacks have come to play an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and in certain situations, conversely, stacks over the moduli stack of formal groups give rise to spectra [G, R2, GHMR, B]. One fundamental example is the spectrum of topological modular forms [Hp] which is associated to the moduli stack of elliptic curves. Classically, stacks were defined as categories fibered in groupoids over a site C which satisfy descent [DM, Definition 4.1]. In [H] we show that a category fibered in groupoids F over C is a stack if and only if the assignment satisfies the homotopy sheaf

A model category structure for equivariant algebraic models

In the equivariant category of spaces with an action of a finite group, algebraic ‘minimal models’ exist which describe the rational homotopy for G G -spaces which are 1-connected and of finite type. These models are diagrams of commutative differential graded algebras. In this paper we prove that a model category structure exists on this diagram category in such a way that the equivariant minimal models are cofibrant objects. We show that with this model structure, there is a Quillen equivalence between the equivariant category of rational G G -spaces satisfying the above conditions and the algebraic category of the models.

Le théorème de Quillen, d'adjonction des foncteurs dérivés, revisité

The aim of this Note is to present a very simple original, purely formal, proof of Quillen's adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an abstract adjunction theorem for ‘absolute’ derived functors. In contrast with all known proofs, the explicit construction of the derived functors is not used. The ‘absolute derived functors’ are a non-additive generalization of Deligne's ‘everywhere defined derived functors’. To cite this article: G. Maltsiniotis, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σ n -action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645–711 (electronic)].

HZ -algebra spectra are differential graded algebras

We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that basically any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects).

Continuous functors as a model for the equivariant stable homotopy category

In this paper, we investigate the properties of the category of equivariant diagram spectra indexed on the category W_G of based G-spaces homeomorphic to finite G-CW-complexes for a compact Lie group G. Using the machinery of Mandell, May, Schwede, and Shipley, we show that there is a "stable model structure" on this category of diagram spectra which admits a monoidal Quillen equivalence to the category of orthogonal G-spectra. We construct a second "absolute stable model structure" which is Quillen equivalent to the "stable model structure". Our main result is a concrete identification of the fibrant objects in the absolute stable model structure. There is a model-theoretic identification of the fibrant continuous functors in the absolute stable model structure as functors Z such that for A in W_G the collection {Z(A smash S^W)} form an Omega-G-prespectrum as W varies over the universe U. We show that a functor is fibrant if and only if it takes G-homotopy pushouts to G-homotopy pullbacks and is suitably compatible with equivariant Atiyah duality for orbit spaces G/H_+ which embed in U. Our motivation for this work is the development of a recognition principle for equivariant infinite loop spaces.

Rigidification of algebras over multi-sorted theories

We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial sets and simplicial categories with a given set of objects.

A discrete model of [formula omitted]-homotopy theory

We construct a discrete model of the homotopy theory of S 1 -spaces. We define a category P with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. P inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between P and the model category of S 1 -spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets.

Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold–Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E ∞ -algebras. We prove that it in fact sends algebras over arbitrary differential graded E ∞ -operads to E ∞ -algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.

Equivalences of monoidal model categories

We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000), 491-511]. As an application we extend the Dold-Kan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [Stable model categories are categories of modules, Topology, 42 (2003) 103-153] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra.

An Algebraic Model for Rational S1-Equivariant Stable Homotopy Theory

Greenlees defined an abelian category A whose derived category is equivalent to the rational S1‐equivariant stable homotopy category whose objects represent rational S1‐equivariant cohomology theories. We show that in fact the model category of differential graded objects in A models the whole rational S1‐equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S1‐equivariant spectra, before the quasi‐isomorphisms or stable equivalences have been inverted. This implies that all of the higher‐order structures such as mapping spaces, function spectra and homotopy (co) limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an Appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category.

Spectra and symmetric spectra in general model categories

We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A 1 -homotopy theory of Morel and Voevodsky (preprint, 1998) and Voevodsky (Proceedings of the International Congress of Mathematicians, Vol. I, Berlin, Doc. Math. Extra Vol. I, 1998, pp. 579–604 (electronic)). One is based on the standard notion of spectra originated by Vogt (Boardman's Stable Homotopy Category, Lecture Notes Series, Vol. 21, Matematisk Institut Aarhus Universitet, Aarhus, 1970). Its input is a well-behaved model category D and an endofunctor T, generalizing the suspension. Its output is a model category Sp N ( D,T) on which T is a Quillen equivalence. The second construction is based on symmetric spectra (Hovey et al., J. Amer. Math. Soc. 13(1) (2000) 149–208) and applies to model categories C with a compatible monoidal structure. In this case, the functor T must be given by tensoring with a cofibrant object K. The output is again a model category Sp Σ( C,K) where tensoring with K is a Quillen equivalence, but now Sp Σ( C,K) is again a monoidal model category. We study general properties of these stabilizations; most importantly, we give a sufficient condition for these two stabilizations to be equivalent that applies both in the known case of topological spaces and in the case of A 1 -homotopy theory.

Sheafifiable homotopy model categories, II

If a Quillen model category is defined via a suitable right adjoint over a sheafifiable homotopy model category (in the sense of part I of this paper), it is sheafifiable as well; that is, it gives rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions. This is chiefly useful in dealing with homotopy theories of algebraic structures defined over diagrams of fixed shape, and unifies a large number of examples.

Simplicial structures on model categories and functors

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.