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.     

Many homotopy categories are homotopy categories

We show that any category that is enriched, tensored, and cotensored over the category of compactly generated weak Hausdorff spaces, and that satisfies an additional hypothesis concerning the behavior of colimits of sequences of cofibrations, admits a Quillen closed model structure in which the weak equivalences are the homotopy equivalences. The fibrations are the Hurewicz fibrations and the cofibrations are a subclass of the Hurewicz cofibrations. This result applies to various categories of spaces, unbased or based, categories of prespectra and spectra in the sense of Lewis and May, the categories of L-spectra and S-modules of Elmendorf, Kriz, Mandell and May, and the equivariant analogues of all the afore-mentioned categories.     

A Cartan-Eilenberg approach to homotopical algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0. The first author was partially supported by NSERC research grant. Received December 12, 2001; in revised form September 7, 2002 Published online February 28, 2003     

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.     

A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M[∗] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F(EF₊,M[∗]) is the F-adic completion of M[∗], in the sense that the map M[∗]→F(EF₊,M[∗]) is the universal map into an algebraically F-adically complete pro-spectrum. Here, F(EF₊,M[∗]) denotes the pro-G-spectrum , where runs over the finite subcomplexes of EF₊.     

Stable model categories are categories of modules

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.     

Mixing model structures

We prove that if a category has two Quillen closed model structures (W₁,F₁,C₁) and (W₂,F₂,C₂) that satisfy the inclusions W₁⊆W₂ and F₁⊆F₂, then there exists a “mixed model structure” (Wm,Fm,Cm) for which Wm=W₂ and Fm=F₁. This shows that there is a model structure for topological spaces (and other topological categories) for which Wm is the class of weak equivalences and Fm is the class of Hurewicz fibrations. The cofibrant spaces in this model structure are the spaces that have CW homotopy type.     

Models for homotopy n-types in diagram categories

The classical Mac Lane-Whitehead equivalence showing that crossed modules of groups are algebraic models of connected homotopy 2-types has found a corresponding equivariant version by Moerdijk and Svensson ([22]). In this paper we show that this equivariant result has a higher-dimensional version which gives an equivalence between the homotopy category of diagrams of certain objects indexed by the orbit category of a group H and H-equivariant homotopy n-types for n1.Supported by DGICYT:PS90-0226     

Rational equivalences between classifying spaces

The paper contains a homotopy classification of rational equivalences between classifying spaces of compact connected Lie groups with an application to genus sets of such spaces. Received: 22 June 1992; in final form: 30 August 1993/ Published online: 6 August 2002     

The loop orbifold of the symmetric product

Using the loop orbifold of the symmetric product, we give a formula for the Poincaré polynomial of the free loop space of the Borel construction of the symmetric product. We also show that the Chas-Sullivan orbifold product structure in the homology of the free loop space of the Borel construction of the symmetric product induces a ring structure in the homology of the inertia orbifold of the symmetric product. For a general almost complex orbifold, we define a new ring structure on the cohomology of its inertia orbifold which we call the virtual intersection ring. Finally we show that under Poincaré duality in the case of the symmetric product orbifold, both ring structures are isomorphic.     

Continuous group actions on profinite spaces

For a profinite group, we construct a model structure on profinite spaces and profinite spectra with a continuous action. This yields descent spectral sequences for the homotopy groups of homotopy fixed point spaces and for stable homotopy groups of homotopy orbit spaces. Our main example is the Galois action on profinite étale topological types of varieties over a field. One motivation is to understand Grothendieck’s section conjecture in terms of homotopy fixed points.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

Localization and completion of nilpotent groups of automorphisms

We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.     

On the algebras over equivariant little disks

We describe the structure present in algebras over the little disks operads for various representations of a finite group G, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in more detail what happens for $G=C_2$, describing the structure on algebras over the little disks operad for the sign representation. Here we can also describe the resulting structure in Bredon homology. Finally, we produce a stable splitting of coinduced spaces analogous to the stable splitting of the product, and we use this to determine the homology of the signed James construction.     

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 Σn-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)].     

Representability theorems for presheaves of spectra

Suppose that M is a simplicial model category and that F is a contravariant simplicial functor defined on M which takes values in pointed simplicial sets. This note displays conditions on the simplicial model category M and the functor F such that F is representable up to weak equivalence. The conditions on F are homotopy coherent versions of the classical conditions for Brown representability, while M should have the fundamental properties of the stable model structure for presheaves of spectra on a Grothendieck site.     

Models for the Maclaurin tower of a simplicial functor via a derived Yoneda embedding

We prove that the Goodwillie tower of a weak equivalence preserving functor from spaces to spectra can be expressed in terms of the tower for stable mapping spaces. Our proof is motivated by interpreting the functors P_n and D_n as pseudo-differential operators which suggests certain ‘integral’ presentations based on a derived Yoneda embedding. These models allow one to extend computational tools available for the tower of stable mapping spaces. As an application we give a classical expression for the derivative over the basepoint.