Related papers: Category theory for scientists (Old version)
We introduce a theory of "patterns" in order to study geodesics in a certain class of group presentations. Using patterns we show that there does not exist a geodesic automatic structure for certain group presentations, and that certain…
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and…
With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for…
We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between…
We explain the sense in which a warping on a monoidal category is the same as a pseudomonad on the corresponding one-object bicategory, and we describe extensions of this to the setting of skew monoidal categories: these are a…
The paper proposes a text-mining based analytical framework aiming at the cognitive organization of complex scientific discourses. The approach is based on models recently developed in science mapping, being a generalization of the…
We introduce a categorical language in which it is possible to talk about DNA sequencing, alignment methods, CRISPR, homologous recombination, haplotypes, and genetic linkage. This language takes the form of a class of limit-sketches whose…
We implement a novel representation of model search spaces as diagrams over a category of models, where we have restricted attention to a broad class of models whose structure is presented by \C-sets. (Co)limits in these diagram categories…
Conscious experience permeates our daily lives, yet general consensus on a theory of consciousness remains elusive. In the face of such difficulty, an alternative strategy is to address a more general (meta-level) version of the problem for…
Operads may be represented as symmetric monoidal functors on a small symmetric monoidal category. We discuss the axioms which must be imposed on a symmetric monoidal functor in order that it give rise to a theory similar to the theory of…
Categorization axioms have been proposed to axiomatizing clustering results, which offers a hint of bridging the difference between human recognition system and machine learning through an intuitive observation: an object should be assigned…
Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy…
Let $\mathcal C$ be a category of a set of (small) categories. This paper concerns with the ${\mathbf {Cat}}$-valued presheaves and sieves over category $\mathcal C.$ Since ${\mathbf {Cat}}$ is not a concrete category, existing definition…
We introduces a category-theoretic framework for modelling trust as applied to trusted computation systems and remote attestation. By formalizing elements, claims, results, and decisions as objects within a category, and the processes of…
What is a time-varying graph, a time-varying topological space, or, more generally, a mathematical structure that evolves over time? In this work, we lay the foundations for a general theory of temporal data by introducing categories of…
We firstly introduce some key concepts in category theory, such as quotient category, completion of limits, $\mathrm{Mor}$ category, and so on; then give the concept of topology algebras and sheaves, and discuss how to restore the structue…
\emph{Approximation Theory} uses nicely-behaved subcategories to understand entire categories, just as projective modules are used to approximate arbitrary modules in classical homological algebra. We use set-theoretic \emph{elementary…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
The standard assumptions that underlie many conceptual and quantitative frameworks do not hold for many complex physical, biological, and social systems. Complex systems science clarifies when and why such assumptions fail and provides…
Understanding how people represent categories is a core problem in cognitive science. Decades of research have yielded a variety of formal theories of categories, but validating them with naturalistic stimuli is difficult. The challenge is…