Related papers: Relevant Categories and Partial Functions
Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made…
Reverse differentiation is an essential operation for automatic differentiation. Cartesian reverse differential categories axiomatize reverse differentiation in a categorical framework, where one of the primary axioms is the reverse chain…
One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is a…
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…
Given an algebraic theory which can be described by a (possibly symmetric) operad $P$, we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for $P$-algebras hold…
We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues…
The study of complex systems through the lens of category theory consistently proves to be a powerful approach. We propose that cognition deserves the same category-theoretic treatment. We show that by considering a highly-compact cognitive…
A modular tensor category provides the appropriate data for the construction of a three-dimensional topological field theory. We describe the following analogue for two-dimensional conformal field theories: a 2-category whose objects are…
We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group $G$ and the same for…
If a compact closed category has finite products or finite coproducts then it in fact has finite biproducts, and so is semi-additive.
It is proved that MacLane's coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in…
An (additive) functor F from an additive category A to an additive category B is said to be objective, provided any morphism f in A with F(f) = 0 factors through an object K with F(K) = 0. In this paper we concentrate on triangle functors…
Fong developed `decorated cospans' to model various kinds of open systems: that is, systems with inputs and outputs. In this framework, open systems are seen as the morphisms of a category and can be composed as such, allowing larger open…
A relevant collection is a collection, $F$, of sets, such that each set in $F$ has the same cardinality, $\alpha(F)$. A Konig Egervary (KE) collection is a relevant collection $F$, that satisfies $|\bigcup F|+|\bigcap F|=2\alpha(F)$. An hke…
The central focus is on clarifying the distinction between sets and proper classes. To this end we identify several categories of concepts (surveyable, definite, indefinite), and we attribute the classical set theoretic paradoxes to a…
This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through *-autonomous monoidal categories and related structures.
We prove that a category which is symmetric (relaxed) monoidal closed, (small) complete, well-powered and has a small cogenerating family, is cocomplete.
We provide a complete description of the category of pseudo-categories (including pseudo-functors, natural and pseudo-natural transformations and pseudo modifications). A pseudo-category is a non strict version of an internal category. It…
Disentangling the factors of variation in data is a fundamental concept in machine learning and has been studied in various ways by different researchers, leading to a multitude of definitions. Despite the numerous empirical studies, more…
We investigate monoidal categories of formal contexts, in which states correspond to formal concepts. In particular we examine the category of bonds or Chu correspondences between contexts, which is known to be equivalent to the…