Category Of Functors Research Articles

On the lattices of exact and weakly exact structures

We initiate in this article the study of weakly exact structures, a generalisation of Quillen exact structures. We introduce weak counterparts of one-sided exact structures and show that a left and a right weakly exact structure generate a weakly exact structure. We further define weakly extriangulated structures on an additive category and characterize weakly exact structures among them.We investigate when these structures on A form lattices. We prove that the lattice of substructures of a weakly extriangulated structure is isomorphic to the lattice of topologizing subcategories of a certain functor category. In the weakly idempotent complete case, we characterise the lattice of all weakly exact structures and we prove the existence of a unique maximal weakly exact structure.We study in detail the situation when A is additively finite, giving a module-theoretic characterization of closed sub-bifunctors of Ext1 among all additive sub-bifunctors.

Tilting and Cotilting in Functor Categories

In this paper, we introduce the notion of n-tilting (resp. n-cotilting) objects in functor categories and give some characterizations of n-tilting objects and n-tilting classes (resp. n-cotilting objects and n-cotilting classes).

Explicit presentations of topological categories of gestures

Thanks to Mazzola's notion of gestures on topological categories we can appreciate how the notion of gesture transcends its manifestation as the movement of the body's limbs and takes a more abstract form that blends diagrammatic (discrete gesturality) and bodily aspects (continuous gesturality). These two aspects are strongly related to two main branches of mathematical music theory, namely, a discrete branch and a continuous branch. The discrete branch corresponds to the diagrams of transformational theory, that is, to networks in musical analysis. The continuous branch corresponds to the movement of the musical performer's body. Informally, a gesture on a topological category is a diagram of continuous paths of morphisms in the category. This definition amounts to that of topological category of gestures, whose structure we study in this article. Specifically, we study the presentation of topological categories of gestures as suitable categories of topological functors and as suitable categories of sequences, and the explicit presentation of morphisms of a typical topological category of gestures. In particular, we present an exhaustive study, not included in previous publications, of the topological category of continuous paths of an arbitrary digraph. This article can be regarded as a continuation of a previous publication, in a previous issue of this journal, on the presentation of spaces of gestures as function spaces. We include an application of the theory to the variations in Mozart's Piano Sonata K. 331. We provide an Online Supplement, in which we include some technical passages.

A category of quantum posets

We investigate a category of quantum posets that generalizes the category of posets and monotone functions. Up to equivalence, its objects are hereditarily atomic von Neumann algebras equipped with quantum partial orders in Weaver’s sense. We show that this category is complete, cocomplete and symmetric monoidal closed. As a consequence, any discrete quantum family of maps from a discrete quantum space to a partially ordered set is canonically equipped with a quantum preorder. In particular, the quantum power set of a quantum set is canonically a quantum poset. We show that each quantum poset embeds into its quantum power set in complete analogy with the classical case.

Torsion and torsion-free classes from objects of finite type in Grothendieck categories

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of FPn-injective objects to be a torsion class. By doing so, we propose a notion of n-hereditary categories. We also define and study the class of FPn-flat objects in Grothendieck categories with a generating set of small projective objects, and provide several equivalent conditions for this class to be torsion-free. At the end, we present several applications and examples of n-hereditary categories in the contexts modules over a ring, chain complexes of modules and categories of additive functors from an additive category to the category of abelian groups. Concerning the latter setting, we find a characterization of when these functor categories are n-hereditary in terms of the domain additive category.

Inverse Laplace Transform of the Incomplete H-Function

Special functions have enormous applications in theoretical and applied mathematics. This field is constantly expanding and addressing new problems in engineering applications and applied sciences. H-function one and more variables have been applied in large range of areas, such as electronics and communication, astrophysics, fractional differential equations, super statistics, diffusion, reaction–diffusion, and other fields like probability theory, biology, and theoretical physics. Some situations of heat conduction and astrophysics problems are not adequately addressed by basic category of special functions. To address such problems, incomplete special functions found the way in the literature. Incomplete H-functions are attracting researchers working on heat conduction and astrophysics problems. The paper aims to transform the products of algebraic powers and incomplete H-function in another domain using inverse laplace transform. Results are obtained in terms of the incomplete H-function as compact form. Further, some particular cases are mentioned by giving special values to the incomplete H-function’s parameters.

Triangular matrix categories II: Recollements and functorially finite subcategories

In this paper we continue the study of triangular matrix categories $\mathbf {{\varLambda }}=\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M & \mathcal {U} \end {smallmatrix}\right ]$ initiated in León-Galeana et al. (2022). First, given a additive category $\mathcal {C}$ and an ideal $\mathcal {I}_{{\mathscr{B}}}$ in $\mathcal {C}$ , we prove a well known result that there is a canonical recollement We show that given a recollement between functor categories we can induce a new recollement between triangular matrix categories, this is a generalization of a result given by Chen and Zheng in (J. Algebra, 321 (9), 2474–2485 2009, [Theorem 4.4]). In the case of dualizing K-varieties we can restrict the recollement we obtained to the categories of finitely presented functors. Given a dualizing variety $\mathcal {C}$ , we describe the maps category of $\text {mod}(\mathcal {C})$ as modules over a triangular matrix category and we study its Auslander-Reiten sequences and contravariantly finite subcategories, in particular we generalize several results from Martínez-Villa and Ortíz-Morales (Inter. J Algebra, 5 (11), 529–561 2011). Finally, we prove a generalization of a result due to Smalø (2011, [Theorem 2.1]), which give us a way of construct functorially finite subcategories in the category $\text {Mod}\Big (\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M & \mathcal {U} \end {smallmatrix}\right ]\Big )$ from those of $\text {Mod}(\mathcal {T})$ and $\text {Mod}(\mathcal {U})$ .

From reversible programming languages to reversible metalanguages

During the past decade reversible programming languages have been formalized using various established semantics frameworks. However, these semantics fail to effectively specify the distinct properties of reversible languages at the metalevel, even including the central question of whether the defined language is reversible. In this paper, we build a metalanguage foundation for reversible languages from categorical principles, based on the category of sets and partial injective functions. We exemplify our approach by a step-by-step development of the full semantics of an r-Turing complete reversible while-language with recursive procedures. The use of the metalanguage leads to a formalization of the reversible semantics. A language defined in the metalanguage is guaranteed to have reversibility and inverse semantics. Also, program inverters for this language are obtained for free. We further discuss applications and directions for reversible semantics.

What do Abelian categories form?

Abstract Given two finitely presentable Abelian categories and , we outline a construction of an Abelian category of functors from to , which has nice 2-categorical properties and provides an explicit model for a stable category of stable functors between the derived categories of and . The construction is absolute, so it makes it possible to recover not only Hochschild cohomology but also Mac Lane cohomology. Bibliography: 29 titles.

Quantum information effects

We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate (noniterative) quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.

Fully abstract models for effectful λ-calculi via category-theoretic logical relations

We present a construction which, under suitable assumptions, takes a model of Moggi’s computational λ-calculus with sum types, effect operations and primitives, and yields a model that is adequate and fully abstract. The construction, which uses the theory of fibrations, categorical glueing, ⊤⊤-lifting, and ⊤⊤-closure, takes inspiration from O’Hearn & Riecke’s fully abstract model for PCF. Our construction can be applied in the category of sets and functions, as well as the category of diffeological spaces and smooth maps and the category of quasi-Borel spaces, which have been studied as semantics for differentiable and probabilistic programming.

Shifts in microbial community structure and function in polycyclic aromatic hydrocarbon contaminated soils at petrochemical landfill sites revealed by metagenomics

Investigations of the microbial community structures, potential functions and polycyclic aromatic hydrocarbon (PAH) degradation-related genes in PAH-polluted soils are useful for risk assessments, microbial monitoring, and the potential bioremediation of soils polluted by PAHs. In this study, five soil sampling sites were selected at a petrochemical landfill in Beijing, China, to analyze the contamination characteristics of PAHs and their impact on microorganisms. The concentrations of 16 PAHs were detected by gas chromatography-mass spectrometry. The total concentrations of the PAHs ranged from ND to 3166.52 μg/kg, while phenanthrene, pyrene, fluoranthene and benzo [ghi]perylene were the main components in the soil samples. According to the specific PAH ratios, the PAHs mostly originated from petrochemical wastes in the landfill. The levels of the total toxic benzo [a]pyrene equivalent (1.63–107.73 μg/kg) suggested that PAHs might result in adverse effects on soil ecosystems. The metagenomic analysis showed that the most abundant phyla in the soils were Proteobacteria and Actinobacteria, and Solirubrobacter was the most important genus. At the genus level, Bradyrhizobium, Mycobacterium and Anaeromyxobacter significantly increased under PAH stress. Based on the Kyoto Encyclopedia of Genes and Genomes (KEGG) annotations, the most abundant category of functions that are involved in adapting to contaminant pressures was identified. Ten PAH degradation-related genes were significantly influenced by PAH pressure and showed correlations with PAH concentrations. All of the results suggested that the PAHs from the petrochemical landfill could be harmful to soil environments and impact the soil microbial community structures, while microorganisms would change their physiological functions to resist pollutant stress.

"ADJOINT PREENVELOPES AND ADJOINT PRECOVERS IN THE FUNCTOR CATEGORY"

"Adjoint preenvelopes and adjoint precovers are defined in the category of functors by replacing the functor Hom with ⊗. We investigate the existence and basic properties of adjoint preenvelopes and adjoint precovers. The F-projective (F-injective, F-flat) functors introduced by Mao are characterized in terms of adjoint preenvelopes and adjoint precovers. We obtain relationships among adjoint preenvelopes, adjoint precovers, preenvelopes and precovers."

Generalized persistence and graded structures

We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product. Furthermore, when the indexing set is a poset, we provide a new characterization for a generalized persistence module being finitely presented.

Recollements for derived categories of enriched functors and triangulated categories of motives

We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit into recollements of derived categories of associated Grothendieck categories of Nisnevich sheaves with specific transfers.

Dichotomy between Deterministic and Probabilistic Models in Countably Additive Effectus Theory

Effectus theory is a relatively new approach to categorical logic that can be seen as an abstract form of generalized probabilistic theories (GPTs). While the scalars of a GPT are always the real unit interval [0,1], in an effectus they can form any effect monoid. Hence, there are quite exotic effectuses resulting from more pathological effect monoids. In this paper we introduce sigma-effectuses, where certain countable sums of morphisms are defined. We study in particular sigma-effectuses where unnormalized states can be normalized. We show that a non-trivial sigma-effectus with normalization has as scalars either the two-element effect monoid 0,1 or the real unit interval [0,1]. When states and/or predicates separate the morphisms we find that in the 0,1 case the category must embed into the category of sets and partial functions (and hence the category of Boolean algebras), showing that it implements a deterministic model, while in the [0,1] case we find it embeds into the category of Banach order-unit spaces and of Banach pre-base-norm spaces (satisfying additional properties), recovering the structure present in GPTs. Hence, from abstract categorical and operational considerations we find a dichotomy between deterministic and convex probabilistic models of physical theories.

Character Polynomials and the Restriction Problem

Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.

Existence of n-Auslander–Reiten sequences via a finiteness condition

Xiao and Zhu proved that if is a locally finite triangulated category, then has Auslander–Reiten triangles. Let n be a positive integer. The notion of n-abelian categories is a higher analogue of abelian categories. In this paper, we introduce the notion of locally finite n-abelian categories and we prove that they admit n-Auslander–Reiten sequences. As an application, we obtain more existence conditions based on results of Auslander on functor categories.

Isbell Adjunctions and Kan Adjunctions via Quantale-Enriched Two-Variable Adjunctions

It is shown that every two-variable adjunction in categories enriched in a commutative quantale serves as a base for constructing Isbell adjunctions between functor categories, and Kan adjunctions are precisely Isbell adjunctions constructed from suitable associated two-variable adjunctions. Representation theorems are established for fixed points of these adjunctions.

Tensor products of finitely presented functors

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive functors. Day convolution always yields a closed monoidal structure on the category of all additive functors. In contrast, right exact monoidal structures for finitely presented functor categories are not necessarily closed. We provide a necessary criterion for being closed that relies on the underlying category having weak kernels and a so-called finitely presented prointernal hom structure. Our results are stated in a constructive way and thus serve as a unified approach for the implementation of tensor products in various contexts.