Category-theoretic Properties Research Articles

Boolean Algebras with Semigroup Operators: Free Product and Free Objects

Two important algebraic structures are S -acts and Boolean algebras. Combining these two structures, one gets S -Boolean algebras, equipped with a compatible right action of a monoid S which is a special case of Boolean algebras with operators. In this article, we considered some category-theoretic properties of the category Boo-S of all S -Boolean algebras with action-preserving maps between them which also preserve Boolean operations. The purpose of the present article is to study certain categorical and algebraical concepts of the category Boo-S, such as congruences, indecomposable objects, coproducts, pushouts, and free objects.

Many-one reductions and the category of multivalued functions

Multivalued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the monikersearch problemsleading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees).A more general understanding is possible, if the category-theoretic properties of multivalued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multivalued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch).Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multivalued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multivalued functions.

Monomorphisms in categories of log schemes

In the present paper, we study category-theoretic properties of monomorphisms in categories of log schemes. This study allows one to give a purely category-theoretic reconstruction of the log scheme that gave rise to the category under consideration. We also obtain analogous results for categories of schemes of locally finite type over the ring of rational integers that are equipped with archimedean structures. Such reconstructions were discussed in two previous papers by the author, but these reconstructions contained some errors, which were pointed out to the author by C. Nakayama and Y. Hoshi. These errors revolve around certain elementary combinatorial aspects of fan decompositions of two-dimensional rational polyhedral cones—i.e., of the sort that occur in the classical theory of toric varieties—and may be repaired by applying the theory developed in the present paper.

The category of soft sets

Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. In this paper, we consider some category-theoretic properties of the category of all soft sets, with morphisms between them. We also characterize several kinds of epimorphisms and monomorphisms.

Category theory of symbolic dynamics

We study the central objects of symbolic dynamics, that is, subshifts and block maps, from the perspective of basic category theory, and present several natural categories with subshifts as objects and block maps as morphisms. Our main goals are to find universal objects in these symbolic categories, to classify their block maps based on their category theoretic properties, to prove category theoretic characterizations for notions arising from symbolic dynamics, and to establish as many natural properties (finite completeness, regularity etc.) as possible. Existing definitions in category theory suggest interesting new problems in symbolic dynamics. Our main technical contributions are the solution to the dual problem of the Extension Lemma and results on certain types of conserved quantities, suggested by the concept of a coequalizer.

A general Galois theory for operations and relations in arbitrary categories

In this paper, we generalize the notions of polymorphisms and invariant relations to arbitrary categories. This leads us to a Galois connection that coincides with the classical case from universal algebra if the underlying category is the category of sets, but remains applicable no matter how the category is changed. In analogy to the situation in the classical case, we characterize the Galois closed classes by local closures of clones of operations and local closures of what we will introduce as clones of (generalized) relations. Since the approach is built on purely category-theoretic properties, we will also discuss the dualization of our notions.

Toy Quantum Categories (Extended Abstract)

We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general categorytheoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set in FRel. Keywords: Quantum category, dagger symmetric monoidal category, finite dimensional Hilbert spaces, FdHilb, FRel

Approach Merotopological Spaces and their Completion

This paper introduces the concept of an approach merotopological space and studies its category‐theoretic properties. Various topological categories are shown to be embedded into the category whose objects are approach merotopological spaces. The order structure of the family of all approach merotopologies on a nonempty set is discussed. Employing the theory of bunches, bunch completion of an approach merotopological space is constructed. The present study is a unified look at the completion of metric spaces, approach spaces, nearness spaces, merotopological spaces, and approach merotopological spaces.

Category theoretic aspects of chain-valued frames: Part I: Categorical and presheaf theoretic foundations

This paper is Part I of a two-part series dealing with category theoretic aspects of chain-valued frames. A new categorical motivation for lattice-valued frames is given from presheaves, and then, under the assumption that L be a complete chain, it is established that standard category–theoretic properties (completeness, cocompleteness, factorization structures) hold for L - Frm (the category of L -frames and L -frame morphisms), properties which are a platform for the “upper” free functor L and “lower” free functor R used in Part II to give a broad range of applications of chain-valued frames to lattice-valued topology.

The Category of S-Posets

In this paper, we consider some category-theoretic properties of the category Pos-S of all S-posets (posets equipped with a compatible right action of a pomonoid S), with monotone action-preserving maps between them. We first discuss some general category-theoretic ingredients of Pos-S; specifically, we characterize several kinds of epimorphisms and monomorphisms. Then, we present some adjoint relations of Pos-S with Pos, Set, and Act-S. In particular, we discuss free and cofree objects. We also examine other category-theoretic properties, such as cartesian closedness and monadicity. Finally, we consider projectivity in Pos-S with respect to regular epimorphisms and show that it is the same asprojectivity, although projectives are not generally retracts of free objects over posets.

Exotic groups and quotients of loop groups

A study is made of various category-theoretic properties of exotic groups. Exotic groups that are non-commutative and non-metrizable are constructed for the first time. A proof is given of a theorem on the construction of exotic groups by means of groups of continuous maps (or maps of smoothness ) from a real complete space (respectively, a locally compact manifold) to a locally compact group (respectively, a Lie group) via factorization. It is shown that quotients of loop groups or generalized loop groups with respect to their closed normal subgroups are either commutative exotic groups, or else non-exotic groups.

Category theoretic properties of fuzzy topological spaces

In this paper we establish some important category theoretic properties, in particular smallness conditions, for categories of fuzzy topological spaces. We also obtain a non-existence of compactifications.