Category Of Functors Research Articles

The spectrum of derived Mackey functors

We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum of derived Mackey functors can be understood as the space obtained from the spectrum of the Burnside ring by “ungluing” closed points. In order to compute the spectrum, we provide a new description of Kaledin’s category, as the derived category of an equivariant ring spectrum, which may be of independent interest. In fact, we clarify the relationship between several different categories, establishing symmetric monoidal equivalences and comparisons between the constructions of Kaledin, the spectral Mackey functors of Barwick, the ordinary derived category of Mackey functors, and categories of modules over certain equivariant ring spectra. We also illustrate an interesting feature of the ordinary derived category of Mackey functors that distinguishes it from other equivariant categories relating to the behavior of its geometric fixed points.

Representation of bifinite domains by BF-closure spaces

Abstract In this paper, we propose the notion of BF-closure spaces as concrete representation of bifinite domains. We prove that every bifinite domain can be obtained as the set of F-closed sets of some BF-closure space under set inclusion. Furthermore, we obtain that the category of bifinite domains and Scott-continuous functions is equivalent to that of BF-closure spaces and F-morphisms.

The geometry of Bayesian programming

AbstractWe give two geometry of interaction models for a typed λ-calculus with recursion endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The models are based on the category of measurable spaces and partial measurable functions, and the category of measurable spaces and s-finite kernels, respectively. The former is proved adequate with respect to both a distribution-based and a sampling-based operational semantics, while the latter is proved adequate with respect to a sampling-based operational semantics.

Adenosine triphosphate energy-independently controls protein homeostasis with unique structure and diverse mechanisms.

Proteins function in the crowded cellular environments with high salt concentrations, thus facing tremendous challenges of misfolding/aggregation which represents a pathological hallmark of aging and an increasing spectrum of human diseases. Recently, intrinsically disordered regions (IDRs) were recognized to drive liquid-liquid phase separation (LLPS), a common principle for organizing cellular membraneless organelles (MLOs). ATP, the universal energy currency for all living cells, mysteriously has concentrations of 2-12 mM, much higher than required for its previously-known functions. Only recently, ATP was decoded to behave as a biological hydrotrope to inhibit protein LLPS and aggregation at mM. We further revealed that ATP also acts as a bivalent binder, which not only biphasically modulates LLPS driven by IDRs of human and viral proteins, but also bind to the conserved nucleic-acid-binding surfaces of the folded proteins. Most unexpectedly, ATP appears to act as a hydration mediator to antagonize the crowding-induced destabilization as well as to enhance folding of proteins without significant binding. Here, this review focuses on summarizing the results of these biophysical studies and discussing their implications in an evolutionary context. By linking triphosphate with unique hydration property to adenosine, ATP appears to couple the ability for establishing hydrophobic, π-π, π-cation and electrostatic interactions to the capacity in mediating hydration of proteins, which is at the heart of folding, dynamics, stability, phase separation and aggregation. Consequently, ATP acquired a category of functions at ~mM to energy-independently control protein homeostasis with diverse mechanisms, thus implying a link between cellular ATP concentrations and protein-aggregation diseases.

On solutions of fractional multi-term sequential problems via some special categories of functions and (AEP)-property

The main intention of this article is that new techniques of existence theory are used to derive some required criteria pertinent to a given fractional multi-term problem and its inclusion version. In such an approach, we do our research on a fractional integral equation corresponding to the mentioned BVPs. In more precise words, by virtue of this integral equation, we construct new operators which belong to a special category of functions named α-admissible and α-ψ-contraction maps coupled with operators having (AEP)-property. Next, by considering some new properties on the existing Banach space having properties (B) and (C_{alpha }), our argument for ensuring the existence of solutions is completed. In addition, we also add two simulative examples to review our findings by a numerical view.

Lifting functors from to

Let (correspondingly ) denote the abelian category of functors (strict polynomial functors in the sense of Friedlander and Suslin) from finite dimensional vector spaces over Fp to vector spaces over Fp . These two categories are related via the exact forgetful functor The category is strongly related to topology and representation theory of symmetric and general linear groups but the homological algebra in is rather mysterious. The category is easier for cohomological calculations. The known calculations are obtained only for functors which belong to the image of ι and are performed using comparison of - and -groups induced by ι. The aim of the following note is to find cohomological conditions which guarantee that a given functor comes from via ι.

Derived categories of functors and quiver sheaves

Let [Formula: see text] be small category and [Formula: see text] an arbitrary category. Consider the category [Formula: see text] whose objects are functors from [Formula: see text] to [Formula: see text] and whose morphisms are natural transformations. Let [Formula: see text] be another category, and again, consider the category [Formula: see text]. Now, given a functor [Formula: see text] we construct the induced functor [Formula: see text]. Assuming [Formula: see text] and [Formula: see text] to be abelian categories, it follows that the categories [Formula: see text] and [Formula: see text] are also abelian. We have two main goals: first, to find a relationship between the derived category [Formula: see text] and the category [Formula: see text]; second relate the functors [Formula: see text] and [Formula: see text]. We apply the general results obtained to the special case of quiver sheaves.

UNITARY FUNCTOR CALCULUS WITH REALITY

AbstractWe construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study ‘functors with reality’ such as the Real classifying space functor, . The calculus produces a Taylor tower, the n-th layer of which is classified by a spectrum with an action of . We further give model categorical considerations, producing a zigzag of Quillen equivalences between spectra with an action of and a model structure on the category of input functors which captures the homotopy theory of the n-th layer of the Taylor tower.

Representability and autoequivalence groups

AbstractFor a finite dimensional algebra A, the bounded homotopy category of projective A-modules and the bounded derived category of A-modules are dual to each other via certain categories of locally-finite cohomological functors. We prove that the duality gives rise to a 2-categorical duality between certain strict 2-categories involving bounded homotopy categories and bounded derived categories, respectively. We apply the 2-categorical duality to the study of triangle autoequivalence groups.

A Monad for Probabilistic Point Processes

A point process on a space is a random bag of elements of that space. In this paper we explore programming with point processes in a monadic style. To this end we identify point processes on a space X with probability measures of bags of elements in X. We describe this view of point processes using the composition of the Giry and bag monads on the category of measurable spaces and functions and prove that this composition also forms a monad using a distributive law for monads. Finally, we define a morphism from a point process to its intensity measure, and show that this is a monad morphism. A special case of this monad morphism gives us Wald's Lemma, an identity used to calculate the expected value of the sum of a random number of random variables. Using our monad we define a range of point processes and point process operations and compositionally compute their corresponding intensity measures using the monad morphism.

Models for functor categories over self-injective quivers

Let [Formula: see text] be a ring, [Formula: see text] a small [Formula: see text]-preadditive category, and let [Formula: see text] be the category of [Formula: see text]-linear functors from [Formula: see text] to the left [Formula: see text]-module category [Formula: see text]. Given a cotorsion pair [Formula: see text] in [Formula: see text], we construct four cotorsion pairs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in [Formula: see text] and investigate when these cotorsion pairs are hereditary and complete. Moreover, under few assumptions, we show that there exist a recollement of homotopy categories induced by these cotorsion pairs. Finally, some applications are given in the category of [Formula: see text]-periodic chain complexes of [Formula: see text]-modules.

Composing dinatural transformations: Towards a calculus of substitution

Dinatural transformations, which generalise the ubiquitous natural transformations to the case where the domain and codomain functors are of mixed variance, fail to compose in general; this has been known since they were discovered by Dubuc and Street in 1970. Many ad hoc solutions to this remarkable shortcoming have been found, but a general theory of compositionality was missing until Petric, in 2003, introduced the concept of g-dinatural transformations, that is, dinatural transformations together with an appropriate graph: he showed how acyclicity of the composite graph of two arbitrary dinatural transformations is a sufficient and essentially necessary condition for the composite transformation to be in turn dinatural. Here we propose an alternative, semantic rather than syntactic, proof of Petric's theorem, which the authors independently rediscovered with no knowledge of its prior existence; we then use it to define a generalised functor category, whose objects are functors of mixed variance in many variables, and whose morphisms are transformations that happen to be dinatural only in some of their variables. We also define a notion of horizontal composition for dinatural transformations, extending the well known version for natural transformations, and prove it is associative and unitary. Horizontal composition embodies substitution of functors into transformations and vice-versa, and is intuitively reflected from the string-diagram point of view by substitution of graphs into graphs.

$n$-quasi-abelian categories vs $n$-tilting torsion pairs

It is a well established fact that the notions of quasi-abelian categories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of t -structures. Firstly, we extend this picture into a hierarchy of n -quasi-abelian categories and n -tilting torsion classes. We prove that any n -quasi-abelian category \mathcal{E} admits a "derived" category D(\mathcal{E}) endowed with a n -tilting pair of t -structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t -structures as quotient categories of coherent functors, generalizing Auslander's Formula. Thirdly, we apply our results to Bridgeland's theory of perverse coherent sheaves for flop contractions. In Bridgeland's work, the relative dimension 1 assumption guaranteed that f_\ast -acyclic coherent sheaves form a 1 -tilting torsion class, whose associated heart is derived equivalent to D(Y) . We generalize this theorem to relative dimension 2 .

BFO-based firefly algorithm for multi-objective optimal allocation of generation by integrating renewable energy sources

Among the rapid evolution of modernisation of alternative energy, the electric power system can be made out of a few Renewable Energy Resources (RES). This paper presents a modern and proficient technique for clearing up the ELD issue. To resolve this issue we have amalgamated two meta-heuristic optimisation algorithms, e.g., the Bacterial Foraging Optimisation (BFO) algorithm and the Firefly Optimisation Algorithm (FA) by incorporating both the renewable energies, such as solar and wind power. The quality of the proposed methodology is tried and approved on the standard IEEE 3, 6 and the 10-unit systems by solving some cases as the fuel cost minimisation, whole generation cost minimisation, emission minimisation, and at the same time the system transmission loss. The attained results are contrasted and the MOPSO and the hybrid BOA algorithms. The results show that the proposed methodology gives an accurate solution for some category of objective functions.

Coalgebras on Digital Images

In this article, we investigate the fundamental properties of coalgebras with coalgebra comultiplications, counits, and coalgebra homomorphisms of coalgebras over a commutative ring R with identity 1R based on digital images with adjacency relations. We also investigate a contravariant functor from the category of digital images and digital continuous functions to the category of coalgebras and coalgebra homomorphisms based on digital images via the category of unitary R-modules and R-module homomorphisms.

The Auslander–Reiten Formula on Finitely Presented Functor Categories

Let R be an artin algebra. The Auslander–Reiten formula for the category (mod-R)-mod of finitely presented functors is presented, which is extended from the Auslander–Reiten formula for the category mod-R of finitely presented modules.

Quantum sets

A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum sets are certain binary relations that can be characterized in terms of this dagger compact structure, and the resulting category of quantum sets and functions generalizes the category of ordinary sets and functions in the manner of noncommutative mathematics. In particular, this category is dual to a subcategory of von Neumann algebras. The basic properties of quantum sets are presented thoroughly, with the noncommutative dictionary in mind, and with an eye to convenient application. As a motivating example, a notion of quantum graph coloring is derived within this framework, and it is shown to be equivalent to the notion that appears in the quantum information theory literature.

The nominal/FM Yoneda Lemma

AbstractThis paper explores versions of the Yoneda Lemma in settings founded upon FM sets. In particular, we explore the lemma for three base categories: the category of nominal sets and equivariant functions; the category of nominal sets and all finitely supported functions, introduced in this paper; and the category of FM sets and finitely supported functions. We make this exploration in ordinary, enriched and internal settings. We also show that the finite support of Yoneda natural transformations is a theorem for free.

Multiplication-invariant operators and the classification of LCA group frames

In this paper we study the properties of multiplication invariant (MI) operators acting on subspaces of the vector-valued space L2(X;H). We characterize such operators in terms of range functions by showing that there is an isomorphism between the category of MI spaces (with MI operators as morphisms) and the category of measurable range functions whose morphisms are measurable range operators. We investigate how global properties of an MI operator are reflected by local pointwise properties of its corresponding range operator. We also establish several results about frames generated by multiplications in L2(X;H). This includes the classification of frames of multiplications with respect to unitary equivalence by measurable fields of Gramians. Finally, we show applications of our results in the study of abelian group frames and translation-invariant (TI) operators acting on subspaces of L2(G), where G is a locally compact group.

Exponential Functions in Cartesian Differential Categories

In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function e^x from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of e^0 = 1, e^{x+y} = e^x e^y, and frac{partial e^x}{partial x} = e^x all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. In particular, differential exponential maps can be defined without the need of limits, converging power series, or unique solutions of certain differential equations—which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, the split complex exponential function, and the dual numbers exponential function. As another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category.