相关论文: Fra\"iss\'e sequences: category-theoretic approach…
In the context of classical associations between classes of Banach spaces and classes of compact Hausdorff spaces we survey known results and open questions concerning the existence and nonexistence of universal Banach spaces and of…
We develop a compositional framework for formal synthesis of hybrid systems using the language of category theory. More specifically, we provide mutually compatible tools for hierarchical, sequential, and independent parallel composition.…
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…
Ideas from Hodge theory have found important applications in representation theory. We give a survey of joint work with Ben Elias which uncovers Hodge theoretic structure in the Hecke category ("Soergel bimodules"). We also outline…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
It is well-known that biological phenomena are emergent. Emergent phenomena are quite interesting and amazing. However, they are difficult to be understood. Due to this difficulty, we propose a theory to describe emergence based on a…
We show that the category of symmetric spectra can be used as a model for global equivariant homotopy theory of finite groups.
We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.
We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems…
Category and multicategory theory provide abstract frameworks for describing structures and their compositions, with multicategories extending traditional categories to handle multi-input operations. These theories enable modular reasoning…
The theory of the functional sequences and series is presented; uniformly convergent, convergent in the sense of a mean square and weakly convergent sequences and series are considered. Sequential approach to constructing generalized…
We introduce a new concept called uncertainty spaces which is an extended concept of probability spaces. Then, we express n-layer uncertainty which we call hierarchical uncertainty by a hierarchically constructed sequence of uncertainty…
In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…
Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can…
We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in $R$-linear abelian categories of…
By homotopy linear algebra we mean the study of linear functors between slices of the $\infty$-category of $\infty$-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices…
We introduce and study some general principles and hierarchical properties of expansions and restrictions of structures and their theories The general approach is applied to describe these properties for classes of $\omega$-categorical…
Category theory has been successfully applied in various domains of science, shedding light on universal principles unifying diverse phenomena and thereby enabling knowledge transfer between them. Applications to machine learning have been…
We investigate Riguet congruences and generalized congruences on a category, focusing on their interrelations from both lattice-theoretic and category-theoretic perspectives. We also characterize functors that are full and surjective on…
Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence…