Hereditary Uniserial Categories with Serre Duality

An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A _n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.     

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.     

Morita Duality for Grothendieck Categories with Applications to Coalgebras

Finiteness conditions of reflexive objects of a Morita duality of Grothendieck category is studied. It is observed that the relation between coproduct and product is a fundamental fact. We show that for coalgebras with a duality, the class of reflexive objects coincides with the class of quasi-finite comodules.     

Tannaka Duality,Coclosed Categories and Reconstruction for Nonarchimedean Bialgebras

Applied Categorical Structures - The topic of this paper is a generalization of Tannaka duality to coclosed categories. As an application we prove reconstruction theorems for coalgebras...     

New Turaev Braided Group Categories and Group Schur–Weyl Duality

For a group G, we describe a new construction of a Turaev braided G-category with a particular braided monoidal subcategory and then we study the structure of a Hopf algebra in this subcategory. As an application, we establish a generalized G-Schur–Weyl duality between certain Turaev G-algebra and the symmetric group algebra.     

Tropical Duality in $$(d+2)$$ ( d + 2 ) -Angulated Categories

Applied Categorical Structures - Let $$\mathscr {C}$$ be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories $$\mathscr {T}$$ and $$\mathscr {U}$$ . A result from...     

Gr-Morita对偶与Morita对偶

设G为有单位元e的群R=(?)R_x和A=(?)A_x都是有单位元的G型强分次环,U=(?)U_z是分次(R,A)一双模.本文主要证明了_RU_A导出一个Gr—Morita对偶当且仅当R_eU_(eAc)导出一个Morita对偶.     

Strange Duality and Polar Duality

A relation is described between Arnold's strange duality anda polar duality between the Newton polytopes which is mostlydue to M. Kobayashi. It is shown that this relation continuesto hold for the extension of Arnold's strange duality foundby C. T. C. Wall and the author. By a method of Ehlers–Varchenko,the characteristic polynomial of the monodromy of a hypersurfacesingularity can be computed from the Newton diagram. This methodis generalized to the isolated complete intersection singularitiesembraced in the extended duality. This is used to explain theduality of characteristic polynomials of the monodromy discoveredby K. Saito for Arnold's original strange duality and extendedby the author to the other cases.     

Subfactor Categories of Triangulated Categories

Let 𝒳 ? 𝒜 be subcategories of a triangulated category 𝒯, and 𝒳 a functorially finite subcategory of 𝒜. If 𝒜 has the properties that any 𝒳-monomorphism of 𝒜 has a cone and any 𝒳-epimorphism has a cocone, then the subfactor category 𝒜/[𝒳] forms a pretriangulated category in the sense of [4 Beligiannis , A. , Reiten , I. ( 2007 ). Homological and Homotopical Aspects of Torsion Theories . Memoirs of the AMS 883 : 426 – 454 . [Google Scholar]]. Moreover, the above pretriangulated category 𝒜/[𝒳] with 𝒯(𝒳, 𝒳[1]) = 0 becomes a triangulated category if and only if (𝒜, 𝒜) forms an 𝒳-mutation pair and 𝒜 is closed under extensions.     

Cartesian Differential Categories as Skew Enriched Categories

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

     

Recollements of Singularity Categories and Monomorphism Categories

We generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of Cohen–Macaulay modules.     

One-Side Derived Categories of Pre-Strict $P$-Semi-Abelian Categories

In this paper, we study a class of $P$-semi-abelian categories, as well as left and right cohomological functors. Then we establish the corresponding one-side derived categories.     

Fenchel-Lagrange Duality Versus Geometric Duality in Convex Optimization

We present a new duality theory to treat convex optimization problems and we prove that the geometric duality used by Scott and Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions to achieve strong duality are considered and optimality conditions are derived. Next, we apply our approach to some problems considered by Scott and Jefferson, determining their duals. We give weaker sufficient conditions to achieve strong duality and the corresponding optimality conditions. Finally, posynomial geometric programming is viewed also as a particular case of the duality approach that we present. Communicated by V. F. Demyanov The first author was supported in part by Gottlieb Daimler and Karl Benz Stiftung 02-48/99. The second author was supported in part by Karl und Ruth Mayer Stiftung.     

Directed Algebraic Topology,Categories and Higher Categories

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applications have been mostly developed in the theory of concurrency. Unexpected links with noncommutative geometry and the modelling of biological systems have emerged. Work partially supported by MIUR Research Projects.     

Approximate Duality

We extend the Lagrangian duality theory for convex optimization problems to incorporate approximate solutions. In particular, we generalize well-known relationships between minimizers of a convex optimization problem, maximizers of its Lagrangian dual, saddle points of the Lagrangian, Kuhn–Tucker vectors, and Kuhn–Tucker conditions to incorporate approximate versions. As an application, we show how the theory can be used for convex quadratic programming and then apply the results to support vector machines from learning theory.     

Flag Duality

We introduce a duality on complex flag manifolds that extendsthe usual point-hyperplane duality of complex projective spaces. Thishas consequences for the structure of the linear cycle spaces of flagdomains, especially when those flag domains are not measurable.