Pair Of Adjoint Functors Research Articles

Adjunctions and braided objects

In this paper, we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular, we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behavior of these functors in the case when the base category is braided is also considered.

Closure properties associated to natural equivalences

Given a pair of adjoint functors F:A⇄B:G, we study some closure properties of some full subcategories A¯ and B¯ such that the restrictions F:A¯⇄B¯:G induce an equivalence.

Localizations, colocalizations and non additive ∗-objects

Given two arbitrary categories, a pair of adjoint functors between them induces three pairs of full subcategories, as follows: the subcategories of reflexive objects, that is objects for which the unit (respectively counit) of the adjunction is an isomorphism; the subcategories of local (respectively colocal) objects w.r.t. these adjoint functors; the subcategories of cogenerated (respectively generated) objects w.r.t. this adjoint pair, namely objects for which the unit (counit) of the adjunction is a monomorphism (an epimorphism). We investigate some cases in which the subcategory of reflexive objects coincide with the subcategory of (co)local objects or with the subcategory of (co)generated objects. As an application we define and characterize (weak) ∗-objects in the non additive case, more precisely weak ∗-acts.

The Eilenberg-Moore Category and a Beck-type Theorem for a Morita Context

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a Morita context comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a double adjunction to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the moritability of a pair of functors with a common domain) are derived.

RELATIONAL SPACETIME, MODEL CATEGORIES AND QUANTUM GRAVITY

We propose a mathematically concrete way of modelling the suggestion that in quantum gravity the spacetime manifold disappears. We replace the underlying point set topological space with several apparently different models, which are actually related by pairs of adjoint functors from rational homotopy theory. One is a discrete approximation to the causal null path space derived from the multiple images in the spacetime theory of gravitational lensing, described as an object in the model category of differential graded Lie algebras. Another of our models appears as a thickening of spacetime, which we interpret as a formulation of relational geometry. This model is produced from the finite dimensional differential graded algebra of differential forms which can be transmitted out of a finite region consistent with the Bekenstein bound by another functor, called geometric realisation. The thickening of spacetime, which we propose as a version of relational spacetime, has a surprizingly rich structure. Information which would make up a spin bundle over spacetime is contained in it, making it possible to include fermionic fields in a geometric state sum over it. Avenues toward constructing an actual quantum theory of gravity on our models are given a preliminary exploration.

Category Theory and Universal Models: Adjoints and Brain Functors

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called internalization through a based on universal mapping properties. A recently developed heteromorphic theory of adjoint functors allows the concepts to be more easily applied empirically. This suggests a conceptual structure, albeit abstract, to model a range of universal-selectionist mechanisms (e.g., in the immune system). Closely related to adjoints is the notion of a brain functor which abstractly models structures of cognition and action.

Comatrix Coring Generalized and Equivalences of Categories of Comodules

We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings. We apply our results to corings coming from entwining structures and graded structures, and we obtain new results in the setting of entwining structures and in the graded ring theory.

Constructions of stable equivalences of Morita type for finite-dimensional algebras III

In this paper, we provide a new method to produce stable equivalences of Morita type. Our main results can be stated as follows. Let A and B be two finite-dimensional k-algebras over a field k. Suppose that two bimodules AMB and BNA define a stable equivalence of Morita type between A and B and that R is a generator for A-modules. Then there is a stable equivalence of Morita type defined by X and Y between the endomorphism algebra EndA(R) of the module R and the endomorphism algebra EndB(N⊗AR) of the module N⊗AR. If M and N satisfy the property that both (N⊗A−, M⊗B−) and (M⊗B−, N⊗A−) are adjoint pairs of functors, then so do the modules X and Y. Moreover, we show that the self-injective dimension and the Gorenstein property are invariant under stable equivalences of Morita type with the above-mentioned adjoint property.

Adjoints and emergence: applications of a new theory of adjoint functors

Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called “determination through universals” based on universal mapping properties. A recently developed “heteromorphic” theory about adjoints suggests a conceptual structure, albeit abstract and atemporal, for how new relatively autonomous behavior can emerge within a system obeying certain laws. The focus here is on applications in the life sciences (e.g., selectionist mechanisms) and human sciences (e.g., the generative grammar view of language).

Adjoint and Frobenius Pairs of Functors for Corings

We investigate adjoint and Frobenius pairs between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings, which leads to a reasonable notion of Frobenius coring extension. When applied to corings stemming from entwining structures, we obtain new results in this setting and in graded ring theory.

Adjoint Functors and Representation Dimensions

We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. This idea is then exploited to compare the representation dimensions of two algebras. In particular, we show that if an Artin algebra is switched from the other, then they have the same representation dimension.

Category-theoretic analysis of the notion of complementarity for quantum systems

In this paper we adopt a category-theoretic viewpoint in order to analyze the semantics of complementarity for quantum systems. Based on the existence of a pair of adjoint functors between the topos of presheaves of the Boolean kind of structure and the category of the quantum kind of structure, we establish a twofold complementarity scheme which constitutes an instance of the concept of adjunction. It is further argued that the established scheme is inextricably connected with a realistic philosophical attitude, although substantially different from the classical one.

Adjoint functors and equivalences of subcategories

For any left R-module P with endomorphism ring S, the adjoint pair of functors P⊗ S − and Hom R ( P,−) induce an equivalence between the categories of P-static R-modules and P-adstatic S-modules. In particular, this setting subsumes the Morita theory of equivalences between module categories and the theory of tilting modules. In this paper we consider, more generally, any adjoint pair of covariant functors between complete and cocomplete Abelian categories and describe equivalences induced by them. Our results subsume the situations mentioned above but also equivalences between categories of comodules.

Induction, Coinduction, and Adjoints

We investigate the reasons for which the existence of certain right adjoints implies the existence of some final coalgebras, and vice-versa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F ⊣ G be a pair of adjoint functors, and suppose that an initial algebra F̂(X) of the functor H(Y) = X + F(Y) exists; then a right adjoint G̃(X) to F̂(X) exists if and only if a final coalgebra Ǧ(X) of the functor K(Y) = X × G(Y) exists. Motivated by the problem of understanding the structures that arise from initial algebras, we show the following: if F is a left adjoint with a certain commutativity property, then an initial algebra of H(Y) = X + F(Y) generates a subcategory of functors with inductive types where the functorial composition is constrained to be a Cartesian product.

Equivalences Induced by Adjoint Functors

Let 𝒜 and ℬ be two Grothendieck categories, R : 𝒜 → ℬ, L : ℬ → 𝒜 a pair of adjoint functors, S ∈ ℬ a generator, and U = L(S). U defines a hereditary torsion class in 𝒜, which is carried by L, under suitable hypotheses, into a hereditary torsion class in ℬ. We investigate necessary and sufficient conditions which assure that the functors R and L induce equivalences between the quotient categories of 𝒜 and ℬ modulo these torsion classes. Applications to generalized module categories, rings with local units and group graded rings are also given here.

FROBENIUS FUNCTORS OF THE SECOND KIND

ABSTRACT A pair of adjoint functors is called a Frobenius pair of the second type if is a left adjoint of for some category equivalences and . Frobenius ring extensions of the second kind provide examples of Frobenius pairs of the second kind. We study Frobenius pairs of the second kind between categories of modules, comodules, and comodules over a coring. We recover the result that a finitely generated projective Hopf algebra over a commutative ring is always a Frobenius extension of the second kind (cf.[27], [17], and prove that the integral spaces of the Hopf algebra and its dual are isomorphic.

A Relationship between Equilogical Spaces and Type Two Effectivity

In this paper I compare two well studied approaches to topological semantics — the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ, and Type Two Effectivity, exemplified by the category of Baire space representations, Rep( B ). These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is how they are related. First, we show that Rep( B ) is equivalent to a full coreflective subcategory of Equ, consisting of the so-called 0-equilogical spaces. This establishes a pair of adjoint functors between Rep( B ) and Equ. The inclusion Rep( B ) → Equ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of Rep( B ) and Equ are essentially different. However, in a second comparison we show that Rep( B ) and Equ do share a common cartesian closed subcategory that contains all countably based T0-spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based T0-spaces. We consider several examples involving the natural numbers and the real numbers to demonstrate how these comparisons make it possible to transfer results from one setting to another.

Absolutely convex modules and Saks spaces

Absolutely or totally convex modules are a canonical generalization of absolutely or countably absolutely convex sets in linear spaces. There are canonical connections between the category of absolutely convex modules and the category of Saks spaces, each of which is given by a pair of adjoint functors. Corresponding results hold for totally convex modules.

Co-Frobenius Hopf Algebras: Integrals, Doi–Koppinen Modules and Injective Objects

We investigate Hopf algebras with non-zero integral from a coalgebraic point of view. Categories of Doi–Koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect, and several pairs of adjoint functors are constructed. As applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of Doi–Koppinen modules.

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 $$P_{\bar B} b_* \dashv N_B b^\# $$ for any fixed mapb: $$B \to \bar B$$ . 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 .