The homotopy category of parametrized spectra

A homotopy category of spectra parametrized over some space B is constructed, which has useful properties for applications. It is a symmetric monoidal category with multiplication given by the smash product. In the original construction the objects are coordinate free spectra indexed by inner product spaces. A translation of the results to a category whose objects are the more familiar spectra indexed by natural numbers is also given.     

Algebras and Modules in Monoidal Model Categories

In recent years the theory of structured ring spectra (formerlyknown as A- and E-ring spectra) has been simplified by the discoveryof categories of spectra with strictly associative and commutativesmash products. Now a ring spectrum can simply be defined asa monoid with respect to the smash product in one of these newcategories of spectra. In this paper we provide a general methodfor constructing model category structures for categories ofring, algebra, and module spectra. This provides the necessaryinput for obtaining model categories of symmetric ring spectra,functors with smash product, Gamma-rings, and diagram ring spectra.Algebraic examples to which our methods apply include the stablemodule category over the group algebra of a finite group andunbounded chain complexes over a differential graded algebra.1991 Mathematics Subject Classification: primary 55U35; secondary18D10.     

Monoidal Uniqueness of Stable Homotopy Theory

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, -spaces, orthogonal spectra, simplicial functors) with symmetric spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence.     

Weak complicial sets I. Basic homotopy theory

This paper develops the foundations of a simplicial theory of weak ω-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ω-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict ω-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to study the weak ω-category theory of these structures in a series of companion papers. In particular, we establish their basic homotopy theory by constructing a Quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets. As a simple corollary of this work we provide an independent construction of Joyal's model structure on simplicial sets for which the fibrant objects are the quasi-categories.     

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.     

A theory of tensor products for module categories for a vertex operator algebra, III

This is the third part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a “complex analogue” of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. In this paper, we focus on a particular element P(z) of a certain moduli space of three-punctured Riemann spheres; in general, every element of this moduli space will give rise to a notion of tensor product, and one must consider all these notions in order to construct a vertex tensor category. Here we present the fundamental properties of the P(z)-tensor product of two modules for a vertex operator algebra. We give two constructions of a P(z)-tensor product, using the results, established in Parts I and II of this series, for a certain other element of the moduli space. The definitions and results in Parts I and II are recalled.     

Algebras of Higher Operads as Enriched Categories

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads (Batanin, Adv Math 136:39–103, 1998) to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)-category is something like a category enriched in weak n-categories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.     

Algebras Graded by Discrete Doi–Hopf Data and the Drinfeld Double of a Hopf Group-Coalgebra

We study Doi–Hopf data and Doi–Hopf modules for Hopf group-coalgebras. We introduce modules graded by a discrete Doi–Hopf datum; to a Doi–Hopf datum over a Hopf group coalgebra, we associate an algebra graded by the underlying discrete Doi–Hopf datum, using a smash product type construction. The category of Doi–Hopf modules is then isomorphic to the category of graded modules over this algebra. This is applied to the category of Yetter–Drinfeld modules over a Hopf group coalgebra, leading to the construction of the Drinfeld double. It is shown that this Drinfeld double is a quasitriangular ${\mathbb{G}}$ -graded Hopf algebra.     

Rings, modules, and algebras in infinite loop space theory

We give a new construction of the algebraic K-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory (elsewhere also called colored operad), a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction. Our method ends up in the Hovey-Shipley-Smith category of symmetric spectra, with an intermediate stop at a category of functors out of a particular wreath product.     

Tensor Products and Support Varieties for Some Noncocommutative Hopf Algebras

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations.     

Superized Troesch complexes and cohomology for strict polynomial superfunctors

We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors. We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors. By an equivalence of categories, this also provides cohomology calculations in the category of left modules over Schur superalgebras.     

Galois Theory for Braided Tensor Categories and the Modular Closure

Given a braided tensor *-category with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory we define a crossed product . This construction yields a tensor *-category with conjugates and an irreducible unit. (A *-category is a category enriched over Vect with positive *-operation.) A Galois correspondence is established between intermediate categories sitting between and and closed subgroups of the Galois group Gal( / )=Aut ( ) of , the latter being isomorphic to the compact group associated with by the duality theorem of Doplicher and Roberts. Denoting by the full subcategory of degenerate objects, i.e., objects which have trivial monodromy with all objects of , the braiding of extends to a braiding of iff . Under this condition, has no non-trivial degenerate objects iff = . If the original category is rational (i.e., has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category ≡ is called the modular closure of since in the rational case it is modular, i.e., gives rise to a unitary representation of the modular group SL(2,  ). If all simple objects of have dimension one the structure of the category can be clarified quite explicitly in terms of group cohomology.     

Model category structures on chain complexes of sheaves

The unbounded derived category of a Grothendieck abelian category is the homotopy category of a Quillen model structure on the category of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently by Beke. However, in most cases of interest, such as the category of sheaves on a ringed space or the category of quasi-coherent sheaves on a nice enough scheme, the abelian category in question also has a tensor product. The injective model structure is not well-suited to the tensor product. In this paper, we consider another method for constructing a model structure. We apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.

     

Fundamental groupoids for simplicial objects in Mal'tsev categories

We show that the category of internal groupoids in an exact Mal'tsev category is reflective, and, moreover, a Birkhoff subcategory of the category of simplicial objects. We then characterize the central extensions of the corresponding Galois structure, and show that regular epimorphisms admit a relative monotone-light factorization system in the sense of Chikhladze. We also draw some comparison with Kan complexes. By comparing the reflections of simplicial objects and reflexive graphs into groupoids, we exhibit a connection with weighted commutators (as defined by Gran, Janelidze and Ursini).     

Trace and Duality in Symmetric Monoidal Categories

Traces taking values in suitable ‘Hochschild complexes’ are defined in the context of symmetric monoidal categories and applied to various categories of chain complexes, simplicial abelian groups, and symmetric spectra. Topological applications to parametrized fixed point theory are given. (Received: October 2003) Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Crossed modules and doi-hopf modules

We prove that crossed modules (or Yetter-Drinfel’d modules) are special cases of Doi’s unified Hopf modules. The category of crossedH-modules is therefore a Grothendieck category (if we work over a field), and the Drinfel’d double appears as a type of generalized smash product. The second and the third author both thank the University of Brussels for its warm hospitality during their visit there. This author was supported partially by the CNCSU nr. 221 and by the Romanian academy, grant nr. 3721.     

One-point suspensions and wreath products of polytopes and spheres

It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way.     

Skew category, Galois covering and smash product of a -category

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.