Module structures and the derived functors of iterated loop functors on unstable modules over the Steenrod algebra

The calculation of the iterated loop functors and their left derived functors on the category of unstable modules over the Steenrod algebra is a non-trivial problem; Singer constructed an explicit and functorial chain complex to calculate these functors. The results of Singer are analysed to give information on the behaviour of these functors with respect to the nilpotent filtration of the category of unstable modules.We show that, if an unstable module M supports an action of an unstable algebra K, then the derived functors of the iterated loop functors applied to M support actions of iterated doubles of K. This allows the finiteness results of Henn on unstable modules which support actions of unstable algebras to be applied to deduce structural results on the derived functors of iterated loops on such modules.     

Catégories de Möbius et fonctorialités: un cadre général pour l'inversion de Möbius

This paper proposes a general setting for Möbius inversion which includes the cases of locally finite partially ordered sets and monoids with the finite decomposition property (f.d.p.). The unifying concept is that of categories with the f.d.p., called Möbius categories, which we characterize in four different ways. The established theory of incidence algebras, Möbius functions, product formulas, reduced algebras, etc., is carried over. Furthermore, the study of functors between Möbius categories yields results in two directions: firstly, inspired from the substitution of formal power series is the construction of a homomorphism between incidence algebras which allows the transfer of Möbius inversions; secondly, surjective functors often give rise to a reduced incidence algebra, thus shedding new light on many important reduced algebras in combinatorial theory.     

The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras

The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map of compact Hausdorff spaces induces a continuous affine map extending f. Together with the canonical embedding associating to every point its Dirac measure and the barycentric map β associating to every probability measure on PK its barycenter, we obtain a monad (P,ε,β). The Eilenberg-Moore algebras of this monad have been characterised to be the compact convex sets embeddable in locally convex topological vector spaces by Swirszcz [T. Swirszcz, Monadic functors and convexity, Bul. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 22 (1974) 39-42].We generalise this result to compact ordered spaces in the sense of Nachbin [L. Nachbin, Topology and Order, Von Nostrand, Princeton, NJ, 1965. Translated from the 1950 monograph “Topologia e Ordem” (in Portugese). Reprinted by Robert E. Kreiger Publishing Co., Huntington, NY, 1967]. The probability measures form again a compact ordered space when endowed with the stochastic order. The maps ε and β are shown to preserve the stochastic orders. Thus, we obtain a monad over the category of compact ordered spaces and order preserving continuous maps. The algebras of this monad are shown to be the compact convex ordered sets embeddable in locally convex ordered topological vector spaces.This result can be seen as a step towards the characterisation of the algebras of the monad of probability measures on the category of stably compact spaces (see [G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl., vol. 93, Cambridge University Press, 2003, Section VI-6]).     

Abelian categories of modules over a (lax) monoidal functor

Crane and Yetter (Deformations of (bi)tensor categories, Cahier de Topologie et Géometrie Differentielle Catégorique, 1998) introduced a deformation theory for monoidal categories. The related deformation theory for monoidal functors introduced by Yetter (in: E. Getzler, M. Kapranov (Eds.), Higher Category Theory, American Mathematical Society Contemporary Mathematics, Vol. 230, American Mathematical Society, Providence, RI, 1998, pp. 117-134.) is a proper generalization of Gerstenhaber's deformation theory for associative algebras (Ann. Math. 78(2) (1963) 267; 79(1) (1964) 59; in: M. Hazewinkel, M. Gerstenhaber (Eds.), Deformation Theory of Algebras and Structure and Applications, Kluwer, Dordrecht, 1988, pp. 11-264). In the present paper we solidify the analogy between lax monoidal functors and associative algebras by showing that under suitable conditions, categories of functors with an action of a lax monoidal functor are abelian categories. The deformation complex of a monoidal functor is generalized to an analogue of the Hochschild complex with coefficients in a bimodule, and the deformation complex of a monoidal natural transformation is shown to be a special case. It is shown further that the cohomology of a monoidal functor F with coefficients in an F,F-bimodule is given by right derived functors.     

Symmetrisation of n-operads and compactification of real configuration spaces

It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym₁ given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author's theory of higher operads, the nonsymmetric operads are 1-operads and Sym₁ is the first term of the infinite series of left adjoint functors Symn, called symmetrisation functors, from n-operads to symmetric operads with the property that the category of one object, one arrow, …, one (n−1)-arrow algebras of an n-operad A is isomorphic to the category of algebras of Symn(A).In this paper we consider some geometrical and homotopical aspects of the symmetrisation of n-operads. We follow Getzler and Jones and consider their decomposition of the Fulton-Macpherson operad of compactified real configuration spaces. We construct an n-operadic counterpart of this compactification which we call the Getzler-Jones operad. We study the properties of Getzler-Jones operad and find that it is contractible and cofibrant in an appropriate model category. The symmetrisation of the Getzler-Jones operad turns out to be exactly the operad of Fulton and Macpherson. These results should be considered as an extension of Stasheff's theory of 1-fold loop spaces to n-fold loop spaces n?2. We also show that a space X with an action of a contractible n-operad has a natural structure of an algebra over an operad weakly equivalent to the little n-disks operad. A similar result holds for chain operads. These results generalise the classical Eckman-Hilton argument to arbitrary dimension.Finally, we apply the techniques to the Swiss-Cheese type operads introduced by Voronov and prove analogous results in this case.     

Über assoziierte lineare und lokalkonvexe Topologien

Certain properties E of linear topological or locally convex spaces induce a functor in the corresponding category, which assigns to every space (X,F) an associated topologyF ^E. The well-known notions of the coarsest barrelled topology stronger than a given locally convex topology or of the strongest locally convex topology weaker than a given linear topology are examples of this concept. In the first two parts of this paper we consider the problem, whether the above functors commute with other processes, such as forming products, linear and locally convex direct sums, inductive limits and completions. With help of two technical lemmas we prove in the third part, that every separated locally convex space is a quotient of a complete locally convex space, in which every bounded set has a finite dimensional linear span. This sharpens results of Y. Kōmura [12], M. Valdivia [18] and W.J. Wilbur [20].     

Weak compactness of the integration map associated with a spectral measure

It is shown that the integration map I_P associated with a closed, equicontinuous spectral measure P in a locally convex space X is weakly compact, if and only if, P has finite range. The proof is based on a mixture of techniques from vector-valued integration theory and locally convex algebras. The same characterization is valid if the equicontinuity requirement on P is relaxed, provided now that P is σ-additive for the topology of uniform convergence on bounded sets in X.     

A Morita theorem for dual operator algebras

We prove that two dual operator algebras are weak^∗ Morita equivalent in the sense of [D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401-2412] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak^∗-continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak^∗ Morita equivalence bimodule. We also develop the theory of the W^∗-dilation, which connects the non-selfadjoint dual operator algebra with the W^∗-algebraic framework. In the case of weak^∗ Morita equivalence, this W^∗-dilation is a W^∗-module over a von Neumann algebra generated by the non-selfadjoint dual operator algebra. The theory of the W^∗-dilation is a key part of the proof of our main theorem.     

A complex semigroup approach to group algebras of infinite dimensional Lie groups

A host algebra of a topological group G is a C ^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C ^*-algebra C ^*(S,α), and we describe a setting which permits us to conclude that this C ^*-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future. To K.H. Hofmann on the occasion of his 75th birthday     

A note on stable equivalences of Morita type

We investigate when an exact functor --Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ, it is possible to replace Λ by a Morita equivalent k-algebra Δ such that Γ is a subring of Δ and the induction and restriction functors induce inverse stable equivalences.     

Weighted algebras of continuous functions

We give conditions under which the weighted spaces CV(X) and CV₀ (X) are locally convex algebras of a certain type and proceed to a systematic study of these algebras. We show, in particular, that the spectrum of CV₀ (X) is always homeomorphic to X and we detemine the spectrum of CV(X).     

Deformation theory of objects in homotopy and derived categories I: General theory

This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module E over a DG category we define four deformation functors Defh(E), coDefh(E), Def(E), coDef(E). The first two functors describe the deformations (and co-deformations) of E in the homotopy category, and the last two - in the derived category. We study their properties and relations. These functors are defined on the category of artinian (not necessarily commutative) DG algebras.     

Strict Topologies as Topological Algebras

Let X be a completely regular Hausdorff space, C_b(X) the space of all scalar-valued bounded continuous functions on X with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally m-convex.     

A categorical approach to Weyl modules

Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.     

Quantified functional analysis and seminormed spaces: A dual adjunction

In this work we investigate the natural algebraic structure that arises on dual spaces in the context of quantified functional analysis. We show that the category of absolutely convex modules is obtained as the category of Eilenberg-Moore algebras induced by the dualization functor [−,R] on locally convex approach spaces. We also establish a dual adjunction between the latter category and the category of seminormed spaces.     

La conjecture de Karoubi pour la K-théorie lisse

Smooth K-functors are constructed on the category of locally convex real (resp. complex) algebras. The Karoubi's conjecture for the smooth K-theory on the isomorphism of algebraic and smooth K-functors is confirmed on the category of real (resp. complex) quasi-stable Fréchet algebras. To cite this article: H. Inassaridzé, T. Kandélaki, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Galois G-covering of quotients of linear categories

In this paper, we introduce the notion of G-liftable ideals, which extends the liftable ideas defined by Assem and Le Meur. We characterize the G-liftable ideals and construct the Galois G-coverings of quotients of categories associated to the G-liftable ideals. In particular, we study the behavior of G-liftable admissible ideals under Galois G-coverings. Furthermore, we show that the ideals generated by finite dimensional projective modules over a locally bounded linear categories are admissible G-liftable ideals. As an application, we provide a reduction technique for dealing with the existence of Serre functors in the stable categories of Gorenstein projective objects.     

An equivariant tensor product on Mackey functors

For all subgroups H of a cyclic p-group G we define norm functors that build a G-Mackey functor from an H-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects.