Coring Categories and Villamayor–Zelinsky Sequence for Symmetric Finite Tensor Categories

Applied Categorical Structures - In the preceeding paper we constructed an infinite exact sequence a la Villamayor–Zelinsky for a symmetric finite tensor category. It consists of cohomology...     

Möbius Categories as Reduced Standard Division Categories of Combinatorial Inverse Monoids

The aim of this paper is to characterize inverse monoids such that their reduced standard division categories C_F(S) are Möbius categories and special Möbius categories. We give also a general technique for the evaluation of the Möbius function of C_F(S).     

V-modules and V-additive Categories

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Λ-Cofibration Categories and the Homotopy Categories of Global Actions and Simplicial Complexes

We investigate the homotopy category of a -cofibration category and compare the homotopy categories of Global Actions, Simplicial Complexes and Topological Spaces.     

Spanier–Whitehead Categories of Resolving Subcategories and Comparison with Singularity Categories

Algebras and Representation Theory - Let $\mathcal {A}$ be an abelian category with enough projective objects, and let $\mathcal {X}$ be a quasi-resolving subcategory of $\mathcal {A}$ . In this...     

Cotorsion Pairs and Cartan–Eilenberg Categories

Cartan–Eilenberg categories were recently introduced by Guillén Santos, Navarro Aznar, Pascual and Roig. In the present paper, we give a method of constructing Cartan–Eilenberg categories for abelian categories, based on cotorsion pairs. In particular, we recover the left Cartan–Eilenberg structure for (un)bounded below chain complexes of modules (Pascual, Collect Math 63(2):203–216, 2012) and extend it to more general categories, including the category of (quasi-coherent)sheaves over a projective scheme.     

The Cuboid Lemma and Mal’tsev Categories

We prove that a regular category ? is a Mal’tsev category if and only if a strong form of the denormalised 3 × 3 Lemma holds true in ?. In this version of the 3 × 3 Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal’tsev categories (= 2-permutable) is similar to the one previously obtained for Goursat categories (= 3-permutable). We also analyse the “relative” version of the Cuboid Lemma and extend our results to that context.     

Congruence Distributivity in Goursat and Mal’cev Categories

We characterize the congruence distributive property for the Goursat and regular Malcev categories in terms of preservation of the intersection by direct images. This splits a previous characterization for the exact Malcev categories in two cases: plain congruence distributive property and weak congruence distributive property.Mathematics Subject Classifications (2000) Primary: 08B10, 08C05, 18C05; secondary: 08A30, 18D05.     

Constructions and Cohomology of Hom–Lie Color Algebras

The main purpose of this paper is to define representations and a cohomology of Hom–Lie color algebras and to study some key constructions and properties. We describe Hartwig–Larsson–Silvestrov Theorem in the case of Γ-graded algebras, study one-parameter formal deformations, discuss α ^k -generalized derivations and provide examples.     

Effective Constructions in Plethysms and Weintraub’s Conjecture

We give a short proof of Weintraub’s conjecture (Weintraub J Algebra 129:103–114, 1990), first proved in Bürgisser et al. (J Algebra 328:322–329, 2011), by constructing explicit highest weight vectors in the plethysms S ^p (?∧?^2q W).     

Symmetric and quasi-symmetric functions associated to polymatroids

To every subspace arrangement X we will associate symmetric functions ℘[X] and ℋ[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant ℋ[X] specializes to the Tutte polynomial . Billera, Jia and Reiner recently introduced a quasi-symmetric function ℱ[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function for polymatroids which has this property as well. Moreover, specializes to ℘[X], ℋ[X], and ℱ[X]. The author is partially supported by the NSF, grant DMS 0349019.     

Symmetric Duality and Self-duality for Nonsmooth Mathematical Programming

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Külshammer Ideals of Graded Categories and Hochschild Cohomology

We generalize the notion of Külshammer ideals to the setting of a graded category. This allows us to define and study some properties of Külshammer type ideals in the graded center of a triangulated category and in the Hochschild cohomology of an algebra, providing new derived invariants. Further properties of Külshammer ideals are studied in the case where the category is d-Calabi-Yau.     

Determinants and Symmetries in ‘Yetter—Drinfeld’ Categories

A Yetter—Drinfeld category over a Hopf algebra H with a bijective antipode, is equipped with a braiding which may be symmetric for some of its subcategories (e.g. when H is a triangular Hopf algebra). We prove that under an additional condition (which we term the u-condition) such symmetric subcategories completely resemble the category of vector spaces over a field k, with the ordinary flip map. Consequently, when Char k=0, one can define well behaving exterior algebras and non-commutative determinant functions.     

Constructions and Dévissage in Hermitian K-Theory

We first give the good definition of the Hermitian K-theory of an exact category with duality. Then we prove a Dévissage theorem which allows us to compare the Hermitian K-theory of a number field with the Hermitian K-theory of its integers.     

Reflected Symmetric α-Stable Processes and Regional Fractional Laplacian

In this paper we investigate the reflected symmetric α-stable processes and their generators. We show that the generators are regional fractional Laplacians on the closed region. In the case of 1≤α<2 their existence requires that on the boundary. Among other things we obtain the integration by parts formula of the regional fractional Laplacian and the semi-martingale decomposition of the reflected symmetric α-stable processes.     

On the Locally Symmetric and Cosympectic Bochner Flat Manifolds

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Ultrafilters on ω and atoms in the lattice of uniformities I

This is the first part of a paper devoted to atoms in the lattice of uniformities (on a given set). It is shown that there are two types of atoms: proximally discrete and proximally non-discrete. Both are related to ultrafilters. Proximally non-discrete atoms are completely described; they correspond to ultrafilters in a one-to-one way. This correspondence is used to classify ultrafilters on a countable set according as the corresponding atoms are, or are not, proximally fine.