Lattices of subgroup and subsystem functors

Lattices of subgroup and subsystem functors are investigated. In particular, it is proved that for the case where X is a formation of finite groups and width of the lattice F₀(X) is at most |π(X)|, the formation X is metanilpotent and |π(X)| ⩽ 3. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 710–730, November–December, 2006.     

Relative left derived functors of tensor product functors

We introduce and study the relative left derived functor Tor_n^(F,F') (-,-) in the module category, which unifies several related left derived functors. Then we give some criteria for computing the F-resolution dimensions of modules in terms of the properties of Tor_n^(F,F') (-,-). We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.     

A classification of Taylor towers of functors of spaces and spectra

We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which we construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this comonad.     

Symmetric homotopy construction

Homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system. Computations will be performed more efficiently if the symmetric structure of the system can be exploited. This paper presents the construction of a symmetric homotopy. Using this homotopy, only the paths according to the generating solutions have to be traced during continuation.     

On homotopy varieties

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties.     

An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators

We investigate the connection between the sign of $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)$ and the monotone behavior of $t\mapsto f\left(t\right)$. In particular, given a function $f:{\mathbb{N}}_a\to \mathbb{R}$ we consider the homotopy $G_{\nu }:\mathbb{R}\times [0,1]\to \mathbb{R}$ defined by $G_{\nu }\left(f\left(a\right),\eta \right)≔\eta \cdot \frac{1}{2}(1-\nu )\left(\nu \right)f\left(a\right)$, and we analyze the effect of changing $\eta \in [0,1]$ in the condition $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)\ge G_{\nu }\left(f\left(a\right),\eta \right)$. We show that as $\eta \to 1^{-}$ there is an increasingly strong connection between the sign of $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)$ and the monotone behavior of $f$.     

Reflection functors and symplectic reflection algebras for wreath products

We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of wreath product groups.     

Gorenstein derived functors

Over any associative ring it is standard to derive using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product using Gorenstein flat modules.

     

A model for the homotopy theory of homotopy theory

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory', or more precisely that the category of such models has a well-behaved internal hom-object.

     

Noncommutative Mackey functors and Hopf-cyclic Homology

We show that a duality of the Hopf-cyclic homology and cohomology can be explained in terms of functors defined on a PROP for Hopf algebras.     

Exactness of homotopy functors of spaces

We will provide an analysis of the generalized Atiyah-Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called n-exact functors, which are analogous to Goodwillie’s n-excisive functors. In the study of these functors, we introduce a new spectral sequence, the homological Barratt-Goerss spectral sequence (HBGSS), which has properties similar to those of the classical Barratt-Goerss Spectral Sequence on homotopy. We close by giving an identification of the E² term of the GAHSS in the case of 2-exact functors on Moore spaces.     

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.     

Calculus II: Analytic functors

Homotopy functors (for example, from spaces to spaces) are called analytic if, when evaluated on certain n-cubical diagrams, they satisfy certain connectivity estimates. Tools for verifying these estimates include certain generalizations of the triad connectivity theorem. Waldhausen's functor A is analytic. Analyticity has strong consequences, when combined with the concept derivative of a homotopy functor that was introduced in the previous article in this series. In particular, any analytic functor with derivative zero is, in a sense, locally constant.Research partially supported by NSF grant DMS-8806444 and a Sloan Fellowship.     

Some pairs of adjoint functors involving track homotopy categories

K. A. Hardie and K. H. Kamps investigated the track homotopy categoryH _B over a fixed spaceB ([1]). They have introduced two pairs of adjoint functors:P _B N _B andm _* m ^*, whereP _B :H _B →H ^B , andm _*:H _A →H _B for a fixed mapm:A→B. We have introduced a split fibration of categoriesL:H _b →H _B and provedL J, J L in [2]. This paper first extendsP _B N _B to for any fixed mapb: . Moreover we also extend these results to obtain two pairs of adjoint functors involving track homotopy categoriesH _b andH ^b whereH ^b is the dual ofH _b . One of our results isN _b P _b . This differs fromP _B N _B . Supported by National Natural Science Foundation of China     

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.