Narrow Results

A Lie groupoid is a groupoid with additional smooth manifold structures on the object set and the morphism set that makes various maps arise from the groupoid structure smooth. In this paper, we describe the properties of the category Rep(${𝒢}$) whose objects are the classical representations of the Lie groupoid ${𝒢}$ and morphisms are the base preserving vector bundle morphisms that respects the representations. It is shown that this category is an additive category which is monoidal with subobjects. Also we have discussed the kernel and cokernel for certain types of morphisms.

It is observed that the proof of strong normalizability of typed terms of typed λ-calculus looks like the process of the model construction. The observation is clarified by the works of Hyland and Ong [8], Altenkirch [2], Stefanova and Geuvers [15], and so on. The model of Hyland and Ong is based on PL-category [14] and the model of Altenkirch or Stefanova and Geuvers is a non-categorical model that is specially designed for this purpose. In this paper, we will construct yet another categorical model for the second order λ-calculus — a non-extensional PL-category. To demonstrate the significance of the non-extensional model, we will give a model in which we can carry out the proof of the uniqueness of the β-normal form. The non-extensionality comes from the use of semi-adjoint for interpretation of abstraction and application (uncurry).

Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned ‘rate constants’, any reaction network gives rise to a nonlinear dynamical system called its ‘rate equation’. Here we generalize these ideas to ‘open’ reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks as morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding ‘open dynamical system’. This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a ‘black-boxing’ functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes.

We give a description of the category structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.

SUSTAIN (Supervised and Unsupervised STratified Adaptive Incremental Network) is a network model of human category learning. SUSTAIN initially assumes a simple category structure. If simple solutions prove inadequate and SUSTAIN is confronted with a surprising event (e.g. it is told that a bat is a mammal instead of a bird), SUSTAIN recruits an additional cluster to represent the surprising event. Newly recruited clusters are available to explain future events and can themselves evolve into prototypes/attractors/rules. SUSTAIN has expanded the scope of findings that models of human category learning can address. This paper extends SUSTAIN to account for both supervised and unsupervised learning data through a common mechanism. The modified model, uSUSTAIN (unified SUSTAIN), is successfully applied to human learning data that compares unsupervised and supervised learning performances.¹⁸

We propose a new method to categorize continuous numeric percepts for Q-learning, where percept vectors are classified into categories on the basis of fuzzy ART and Q-learning uses categories as states to acquire rules for agent behavior. For efficient learning, we modify fuzzy ART to reduce the number of categories without deteriorating the efficiency of reinforcement learning. In our modification, a vigilance parameter is defined for each category in order to control the size of a category and it is updated during learning. The method to update a vigilance parameter is based on category integration, which contributes to reducing the number of categories. Here, we define the similarity for any category pair to judge whether category integration should be performed or not. When two categories are integrated into a new category, a vigilance parameter for the category is calculated and categories used for integration are discarded, so that the number of categories is reduced without restricting the number of categories. Experimental results show that Q-learning with modified fuzzy ART acquires good rules for agent behavior more efficiently than Q-learning with ordinary fuzzy ART, although the number of categories generated by modified fuzzy ART is much less than that generated by ordinary one.

In universal algebraic geometry (UAG), some primary notions of classical algebraic geometry are applied to an arbitrary variety of algebras Θ and an arbitrary algebra H ∈ Θ. We consider an algebraic geometry in Θ over the distinguished algebra H and we also analyze H from the point of view of its geometric properties. This insight leads to a system of new notions and stimulates a number of new problems. They are new with respect to algebra, algebraic geometry and even with respect to the classical algebraic geometry.

In our approach, there are two main aspects: the first one is a study of the algebra H and its geometric properties, while the second is focused on studying algebraic sets and algebraic varieties over a "good", particular algebra H.

Considering the subject from the second standpoint, the main goal is to get forward as far as possible in a classification of algebraic sets over the given H. The first approach does not require such a classification which is itself an independent and extremely difficult task. We also consider some geometric relations between different H₁ and H₂ in Θ.

The present paper should be viewed as a brief review of what has been done in universal algebraic geometry. We also give a list of unsolved problems for future work.

Let be a variety of universal algebras. We suggest an approach for describing automorphisms of a category of free -algebras. In particular, this approach allows us to answer the question: is an automorphism of such a category inner? Most of the results actually deal with arbitrary categories supplied with a represented forgetful functor.

We show how one can use the partialization functor to obtain several recently defined inverse monoids, and use this functor to define new objects, which we call the inverse braid-permutation monoids. A presentation for this monoid is obtained. Finally, we study some abstract properties of the partialization functor and its iterations. This leads to a categorification of a monoid of all order-preserving maps, and series of orthodox generalizations of the symmetric inverse semigroup.

Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. We characterize the categories that correspond to locally inverse semigroups in the realm of Nambooripad’s cross-connection theory. Further, we specialize our cross-connection description of locally inverse semigroups to inverse semigroups and completely $0$-simple semigroups, obtaining structure theorems for these classes. In particular, we show that the structure theorem for inverse semigroups can be obtained using only one category, quite analogous to the Ehresmann–Schein–Nambooripad Theorem; for completely $0$-simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools.

Using purely syntactical arguments, it is shown that every nontrivial pseudovariety of monoids contained in DO whose corresponding variety of languages is closed under unambiguous product, for instance DA, is local in the sense of Tilson.

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics.

A compact quantum metric space is a complete order unit space endowed with a Lip-norm. We introduce a metric on the state space and give several equivalent conditions to characterize the Lipschitz morphisms and Lipschitz isomorphisms between two compact quantum metric spaces.

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form ${ℒ}_{\infty \lambda }$. We prove that many naturally defined classes are anti-elementary, including the following:

• the class of all lattices of finitely generated convex $\ell$-subgroups of members of any class of $\ell$-groups containing all Archimedean $\ell$-groups;
• the class of all semilattices of finitely generated $\ell$-ideals of members of any nontrivial quasivariety of $\ell$-groups;
• the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem;
• the class of all semilattices of finitely generated two-sided ideals of rings;
• the class of all semilattices of finitely generated submodules of modules;
• the class of all monoids encoding the nonstable K₀-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero;
• (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large $4$-frame.

The main underlying principle is that under quite general conditions, for a functor $\mathrm{\Phi }:{𝒜}\to {ℬ}$, if there exists a noncommutative diagram $\overrightarrow{D}$ of ${𝒜}$, indexed by a common sort of poset called an almost join-semilattice, such that

• $\mathrm{\Phi }{\overrightarrow{D}}^I$ is a commutative diagram for every set $I$,
• $\mathrm{\Phi }\overrightarrow{D}\ncong \mathrm{\Phi }\overrightarrow{X}$ for any commutative diagram $\overrightarrow{X}$ in ${𝒜}$,

then the range of $\mathrm{\Phi }$ is anti-elementary.

Considering co-well-powered and cocomplete categories, replacing closure under taking quotients by closure under taking strong ones and doing similarly for closure under domains of epis, we get what we call weakly behavioral weak (Horn, quasi) covarieties and characterize minimal such classes as far as Horn covarieties are concerned. This enables us to define atomic (weakly) behavioral (weak) quasi covarieties and covarieties and characterize them too. We show that minimal (weakly) behavioral (weak) Horn covarieties form a basis of open classes for a topology on the class of objects of the category for which open classes include all (weakly) behavioral (weak) Horn covarieties. Dualizing these results, we characterize minimal classes of objects closed under domains and codomains of (strong) monos and nonempty products and some variations thereof and investigate the particular case of the category of rings with unit and unit-preserving ring homomorphisms.

In general, topological quantum field theory (TQFT) is studied in detail from the n-dimensional cobordism (nCob) to the Hilbert vector space. However, we study the TQFT in the different way in this paper, that is, the TQFT is studied from the Hilbert vector space to nCob. To do this, the theory called the ϕ-mapping topological current theory is used. The relation between the objects and zero points of the Hilbert states in the Hilbert vector space is studied in this frame. The relation between the morphism and topological current is revealed too.

A U-abundant semigroup whose subset U satisfies a permutation identity is said to be U^σ-abundant. In this paper, we consider the minimum Ehresmann congruence δ on a U^σ-abundant semigroup and explore the relationship between the category of U^σ-abundant semigroups (S,U) and the category of Ehresmann semigroups (S,U)/δ. We also establish a structure theorem of U^σ-abundant semigroups by using the concept of quasi-spined product of semigroups. This generalizes a result of Yamada for regular semigroups in 1967 and a result of Guo for abundant semigroups in 1997.

With the purpose of providing a categorical treatment of weak multiplier bialgebras (introduced by Böhm, Gómez-Torrecillas and López-Centella in 2015), an appropriate notion of morphism for these algebraic objects is proposed. This allows us to define a category wmb of (regular) weak multiplier bialgebras (with a right full comultiplication), containing as a full subcategory the category wba of weak bialgebras defined by Böhm, Gómez-Torrecillas and López-Centella in 2014. We present a great source of examples of these morphisms proving that, under some assumption, a functor between small categories induces a morphism of this kind between the natural weak multiplier bialgebra structures carried by the linear spans of the arrow sets of the categories. We explore the notion of elements of group-like type in a weak multiplier bialgebra, proposing a definition in the line of the one by the aforementioned authors for weak bialgebras. We show a big number of its properties and provide more general versions of many results known in the context of weak bialgebras. In particular, in analogy with the classical bialgebra setting (where the set of group-like elements is a monoid), we prove that the set of these elements possesses a structure of category.

Using the idea of changing the basis-lattice, we investigate in this article the impact of change-of-basis to the notion of stratified lattice-valued generalized convergence group by the so-called functorial mechanism. We discuss here some of the subcategories of the category of stratified lattice-valued generalized convergence groups, such as, stratified lattice-valued Kent convergence groups, stratified lattice-valued limit groups, and look into the possible link between these objects with stratified lattice-valued neighborhood groups and stratified lattice-valued neighborhood topological groups when bases are changed. Moreover, with the help of the notion of stratified lattice-valued filter attributed to U. Höhle and A. Šostak, we introduce a category >HŠ-SL-FilGrp, of stratified lattice-valued filter groups, and study its relationship with other categories so far achieved in this paper.