Related papers: Canonical Extensions of Quantale-Enriched Categori…
We enlarge the hom-sets of categories of complete lattices by introducing `state transitions' as generalized morphisms. The obtained category will then be compared with a functorial quantaloidal enrichment and a contextual quantaloidal…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
We introduce enriched notions of purity depending on the left class $\mathcal E$ of a factorization system on the base $\mathcal V$ of enrichment. Ordinary purity is given by the class of surjective mappings in the category of sets. Under…
Building on our previous work on enriched regular logic, we introduce an enriched version of positive logic and relate it to enriched cone-injectivity classes and enriched accessible categories. To do this, we need a factorization system on…
In fairly elementary terms this paper presents how the theory of preordered fuzzy sets, more precisely quantale-valued preorders on quantale-valued fuzzy sets, is established under the guidance of enriched category theory. Motivated by…
A careful study of the classical/quantum connection with the aid of coherent states offers new insights into various technical problems. This analysis includes both canonical as well as closely related affine quantization procedures. The…
Canonical extension of finitary ordered structures such as lattices, posets, proximity lattices, etc., is a certain completion which entirely describes the topological dual of the ordered structure and it does so in a purely algebraic and…
We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with…
A holistic extension of classical propositional logic is introduced in the framework of quantum computation with mixed states. The concepts of tautology and contradiction are investigated in this extensions. A special family of quantum…
Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information.…
We extend the definition of quasi-finite complexes by considering not necessarily countable complexes. We provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications.…
In the framework of Category Theory, we study the association between finite--dimensional representations of a compact quantum group and quantum vector bundles with linear connections for a given quantum principal bundle with a principal…
This paper contains results from two areas -- formal theory of Kan extensions and concrete categories. The contribution to the former topic is based on the extension of the concept of Kan extension to the cones and we prove that limiting…
A coextensive category can be defined as a category $\mathcal{C}$ with finite products such that for each pair $X,Y$ of objects in $\mathcal{C}$, the canonical functor $\times\colon X/\mathcal{C} \times Y/\mathcal{C} \to (X \times…
Conventional canonical quantization procedures directly link various c-number and q-number quantities. Here, we advocate a different association of classical and quantum quantities that renders classical theory a natural subset of quantum…
The arising of central extensions is discussed in two contexts. At first classical counterparts of quantum anomalies (deserving being named as "classical anomalies") are associated with a peculiar subclass of the non-equivariant maps.…
Although classical mechanics and quantum mechanics are separate disciplines, we live in a world where Planck's constant \hbar>0, meaning that the classical and quantum world views must actually {\it coexist}. Traditionally, canonical…
We collect in one place a variety of known and folklore results in enriched model category theory and add a few new twists. The central theme is a general procedure for constructing a Quillen adjunction, often a Quillen equivalence, between…
We study semantic and syntactic properties of spherical orders and their elementary theories, including finite and dense orders and their theories. It is shown that theories of dense $n$-spherical orders are countably categorical and…
We define generalized bialgebras and Hopf algebras and on this basis we introduce quantum categories and quantum groupoids. The quantization of the category of linear (super)spaces is constructed. We establish a criterion for the classical…