Complexity and cohomology of cohomological Mackey functors

Let k be a field of characteristic p>0. Call a finite group G a poco group over k if any finitely generated cohomological Mackey functor for G over k has polynomial growth. The main result of this paper is that G is a poco group over k if and only if the Sylow p-subgroups of G are cyclic, when p>2, or have sectional rank at most 2, when p=2.A major step in the proof is the case where G is an elementary abelian p-group. In particular, when p=2, all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra of self extensions of the simple functor , by explicit generators and relations.     

Secondary derived functors and the Adams spectral sequence

Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of “additive groupoid enriched categories”, in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E₃-term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E₃-term explicitly by an algorithm.     

Internal Homs via extensions of dg functors

We provide a simple proof of the existence of internal Homs in the localization of the category of dg categories with respect to all quasi-equivalences and of some of their main properties such as the so-called derived Morita theory. This was originally proved in a seminal paper by Toën.     

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.     

Folding derived categories with Frobenius functors

Following the work [B. Deng, J. Du, Frobenius morphisms and representations of algebras, Trans. Amer. Math. Soc. 358 (2006) 3591-3622], we show that a Frobenius morphism F on an algebra A induces naturally a functor F on the (bounded) derived category D^b(A) of , and we further prove that the derived category D^b(A^F) of for the F-fixed point algebra A^F is naturally embedded as the triangulated subcategory D^b(A)^F of F-stable objects in D^b(A). When applying the theory to an algebra with finite global dimension, we discover a folding relation between the Auslander-Reiten triangles in D^b(A^F) and those in D^b(A). Thus, the AR-quiver of D^b(A^F) can be obtained by folding the AR-quiver of D^b(A). Finally, we further extend this relation to the root categories ?(A^F) of A^F and ?(A) of A, and show that, when A is hereditary, this folding relation over the indecomposable objects in ?(A^F) and ?(A) results in the same relation on the associated root systems as induced from the graph folding relation.     

The fiber of functors between categories of algebras

We investigate the fiber of a functor F:C→D between sketchable categories of algebras over an object D∈D from two points of view: characterizing its classifying space as a universal -space; and parametrizing its objects in cohomological terms.     

Higher order derived functors and the Adams spectral sequence

Classical homological algebra studies chain complexes, resolutions, and derived functors in additive categories. In this paper we define higher order chain complexes, resolutions, and derived functors in the context of a new type of algebraic structure, called an algebra of left cubical balls  . We show that higher order resolutions exist in these algebras, and that they determine higher order Ext-groups. In particular, the _Em$E_m$-term of the Adams spectral sequence (m>2)$(m>2)$ is such a higher Ext-group, providing a new way of constructing its differentials.     

Homology of multiplicative Lie rings

Two homology theories of multiplicative Lie rings are constructed, studied and compared with the usual homology theories of groups and Lie rings. Central extensions of multiplicative Lie rings are introduced. It is shown that the Steinberg multiplicative Lie ring of a ring is the direct product of the Steinberg group (viewed as a multiplicative Lie ring under the commutator bracket) and the Steinberg Lie ring.     

Exact sequences in homology of multiplicative Lie rings and a new version of Stallings' theorem

We show that the non-abelian tensor product of nilpotent, solvable and Engel multiplicative Lie rings is nilpotent, solvable and Engel, respectively. The six term exact sequence in homology of multiplicative Lie rings is obtained. We also prove a new version of Stallings' theorem.     

Limits of small functors

For a small category enriched over a suitable monoidal category , the free completion of under colimits is the presheaf category . If is large, its free completion under colimits is the -category of small presheaves on , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on .     

A normal inverse Gaussian model for a risky asset with dependence

We present a new construction of the normal inverse Gaussian (NIG) fractal activity time model for a risky asset. The construction uses superpositions of diffusion processes and allows for specified exact NIG marginal distributions of the returns and flexible and tractable dependence structure including short or long range dependence. In the case of finite superposition, the fractal activity time is asymptotically self-similar, which is a desired feature seen in practice. The support for the distributional and dependence features of the risky asset model is provided by the data of currency exchange rates.     

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)].     

Polygraphic resolutions and homology of monoids

We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZM-modules.     

A characterization theorem for geometric logic

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.     

A geometric decomposition of spaces into cells of different types II: Homology theory

We develop the homology theory of CW(A)-complexes, generalizing the classical cellular homology theory for CW-complexes. A CW(A)-complex is a topological space which is built up out of cells of a certain core A.     

A remark on fixed points of functors in topological categories

For a topological category over Set we prove that if a functor T: has a fixed cardinal (i.e. for each object K with card (UK)= we have card (UTK)), then T has a least fixed point, and if T has a successive pair of fixed cardinals and ⁺, then T has a greatest fixed point. This extends results of Adámek and Koubek.Partial financial support of the Grant Agency of the Czech Republic under Grant No. 201/93/0950 is gratefully acknowledged.