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We define natural A_infinity-transformations and construct A_infinity-category of A_infinity-functors. The notion of non-strict units in an A_infinity-category is introduced. The 2-category of (unital) A_infinity-categories, (unital)…
We translate notions and results of decomposition and dimension theories for module categories, into the lattice environment. In particular we translate dimension theory in module categories to complete modular upper-continuous lattices.
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
This note contains some results related to the definitions of toroidal embeddings and toroidal morphisms over non-closed fields of characteristic zero.
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…
We describe a procedure for constructing morphisms in additive categories, combining Auslander's concept of a morphism determined by an object with the existence of flat covers. Also, we show how flat covers are turned into projective…
A notion of general manifolds is introduced. It covers all usual manifolds in mathematics. Essentially, it is a way how to get a bigger 'fibration' over a site which locally coincides with a given one. An enrichment with generalized…
This is an expository article about operads in homotopy theory written as a chapter for an upcoming book. It concentrates on what the author views as the basic topics in the homotopy theory of operadic algebras: the definition of operads,…
Transformations of coherent states of the free particle by bounded and semibounded symmetry operators are considered. Resolution of the identity operator in terms of the transformed states is analyzed. A generalized identity resolution is…
Functionals are an important research subject in Mathematics and Computer Science as well as a challenge in Information Technologies where the current programming paradigm states that only symbolic computations are possible on higher order…
Soft set theory can deal uncertainties in nature by parametrization process. In this paper, we explore the objects and morphisms of category of soft sets, Sset(U) in detail. Also, gives characterizations of monomorphisms and epimorphisms in…
We discuss a class of regions and conformal mappings which are useful in several problems of approximation theory, harmonic analysis and spectral theory.
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
In this paper we study loops, neardomains and nearfields from a categorical point of view. By choosing the right kind of morphisms, we can show that the category of neardomains is equivalent to the category of sharply 2-transitive groups.…
We propose a categorical framework for processes which interact bidirectionally with both an environment and a 'controller'. Examples include open learners, in which the controller is an optimiser such as gradient descent, and an approach…
`Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category…
In this paper we introduce the notion of an operator category and two different models for homotopy theory of $\infty$-operads over an operator category -- one of which extends Lurie's theory of $\infty$-operads, the other of which is…
The intention of this article is to make an attempt of classification of transitive Lie algebroids and on this basis to construct a classifying space. The realization of the intention allows to describe characteristic classes of transitive…
For every finite closure space $X$ one can define a finite topological space $\operatorname{Top} X$ together with a natural projection $\operatorname{Top} X\longrightarrow X$. This could allow to apply the techniques of topological…
Given a non-empty closed convex subset $\mathsf{F}$ of density matrices, we formulate conditions that guarantee the existence of an $\mathsf{F}$-morphism (namely, a completely positive trace-preserving linear map that maps $\mathsf{F}$ into…