Related papers: Formulating Categorical Concepts using Classes

We present in this paper a reformulation of the usual set-theoretical semantics of the description logic $\mathcal{ALC}$ with general TBoxes by using categorical language. In this setting, $\mathcal{ALC}$ concepts are represented as…

We are interested in aligning how people think about objects and what machines perceive, meaning by this the fact that object recognition, as performed by a machine, should follow a process which resembles that followed by humans when…

This article serves as a preliminary introduction to the design of a new, open-source applied and computational category theory framework, named Categorica, built on top of the Wolfram Language. Categorica allows one to configure and…

The notion of a simplicial set originated in algebraic topology, and has also been utilized extensively in category theory, but until relatively recently was not used outside of those fields. However, with the increasing prominence of…

Category theory provides an alternative to Hilbert's Formal Axiomatic method and goes beyond Mathematical Structuralism

Algebra objects in $\infty$-categories of spans admit a description in terms of $2$-Segal objects. We introduce a notion of span between $2$-Segal objects and extend this correspondence to an equivalence of $\infty$-categories.…

Acquiring lexical information is a complex problem, typically approached by relying on a number of contexts to contribute information for classification. One of the first issues to address in this domain is the determination of such…

We define the notion of 2-filtered 2-category and give an explicit construction of the bicolimit of a category valued 2-functor. A category considered as a trivial 2-category is 2-filtered if and only if it is a filtered category, and our…

The focus of these lecture notes is on abstract models and basic ideas and results that relate to the operational semantics of programming languages largely conceived. The approach is to start with an abstract description of the computation…

We extend the logical categories framework to first order modal logic. In our modal categories, modal operators are applied directly to subobjects and interact with the background factorization system. We prove a Joyal-style representation…

Due to the increased complexity of software development projects more and more systems are described by models. The sheer size makes it impractical to describe these systems by a single model. Instead many models are developed that provide…

Constellations are partial algebras that are one-sided generalisations of categories. It has previously been shown that the category of inductive constellations is isomorphic to the category of left restriction semigroups. Here we consider…

\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…

While many different models for $(\infty,1)$-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for $(\infty,…

This article is an introduction to formal languages from the point of view of combinatorial group theory. Group theoretic applications are included and language classes are defined algebraically.

Traditional treatments of formal logic provide: 1. A syntax for formulas. 2. An inference relation between sets of formulas. 3. A rule for assigning meaning to formulas (semantics) that is sound with respect to the inference relation. First…

A situation calculus is presented that provides a solution to the frame problem for hierarchical situations, that is, situations that have a modular structure in which parts of the situation behave in a relatively independent manner. This…

This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck…

Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term `(S,*)-multicategory', subject to…

In nature, one observes that a K-theory of an object is defined in two steps. First a "structured" category is associated to the object. Second, a K-theory machine is applied to the latter category to produce an infinite loop space. We…