Related papers: Data Base Mappings and Monads: (Co)Induction
Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in…
We give a general categorical construction that yields several monads of measures and distributions as special cases, alongside several monads of filters. The construction takes place within a categorical setting for generalized functional…
Nominal automata models serve as a formalism for data languages, and in fact often relate closely to classical register models. The paradigm of name allocation in nominal automata helps alleviate the pervasive computational hardness of…
Regular languages -- the languages accepted by deterministic finite automata -- are known to be precisely the languages recognized by finite monoids. This characterization is the origin of algebraic language theory. In this paper, we…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
In this paper, we propose a fresh perspective on argumentation semantics, to view them as a relational database. It offers encapsulation of the underlying argumentation graph, and allows us to understand argumentation semantics under a…
In this paper we present a new family of Intensional RDBs (IRDBs) which extends the traditional RDBs with the Big Data and flexible and 'Open schema' features, able to preserve the user-defined relational database schemas and all…
Relational data sources are still one of the most popular ways to store enterprise or Web data, however, the issue with relational schema is the lack of a well-defined semantic description. A common ontology provides a way to represent the…
Faceted classifications define dimensions for the types of entities included. In effect, the facets provide an "ontological commitment". We compare a faceted thesaurus, the Art and Architecture Thesaurus (AAT), with ontologies derived from…
Given a Hopf algebra in a symmetric monoidal category with duals, the category of modules inherits the structure of a monoidal category with duals. If the notion of algebra is replaced with that of monad on a monoidal category with duals…
Quantitative algebras are $\Sigma$-algebras acting on metric spaces, where operations are nonexpanding. Mardare, Panangaden and Plotkin introduced 1-basic varieties as categories of quantitative algebras presented by quantitative equations.…
Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category…
Large language models (LLMs) are emerging as few-shot learners capable of handling a variety of tasks, including comprehension, planning, reasoning, question answering, arithmetic calculations, and more. At the core of these capabilities is…
We introduce indexing of tables referencing complex structures such as digraphs and spatial objects, appearing in genetics and other data intensive analysis. The indexing is achieved by extracting dimension schemas from the referenced…
We study finite-rank left-translation invariant algebraic $D$-modules on complex affine algebraic groups. Using the standard description of these objects as left-invariant flat algebraic connections on the trivial vector bundle, modulo…
A significant number of novel database architectures and data models have been proposed during the last decade. While some of these new systems have gained in popularity, they lack a proper formalization, and a precise understanding of the…
The theory of presentations of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to subcategories of arities in recent work of Bourke-Garner and the authors. We…
The human ability to flexibly reason using analogies with domain-general content depends on mechanisms for identifying relations between concepts, and for mapping concepts and their relations across analogs. Building on a recent model of…
The execution logs that are used for process mining in practice are often obtained by querying an operational database and storing the result in a flat file. Consequently, the data processing power of the database system cannot be used…
Tables have gained significant attention in large language models (LLMs) and multimodal large language models (MLLMs) due to their complex and flexible structure. Unlike linear text inputs, tables are two-dimensional, encompassing formats…