Related papers: Comparing material and structural set theories
Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids as a new foundation…
We review the concept of well-posedness in the context of evolutionary problems from mathematical physics for a particular subclass of problems from elasticity theory. The complexity of physical phenomena appears as encoded in so called…
This chapter describes interrelations between: (1) algebraic structure on sets of scalars, (2) properties of monads associated with such sets of scalars, and (3) structure in categories (esp. Lawvere theories) associated with these monads.…
In context of efforts of composing category-theoretic and logical methods in the area of knowledge representation we propose the notion of conceptory. We consider intersection/union and other constructions in conceptories as expressive…
Axiomatic Cohesion proposes that the contrast between cohesion and non-cohesion may be expressed by means of a geometric morphism $p :\mathcal{E} \to \mathcal {S}$ (between toposes) with certain special properties that allow to effectively…
A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can…
Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.
We consider notions of metrized categories, and then approximate categorical structures defined by a function of three variables generalizing the notion of $2$-metric space. We prove an embedding theorem giving sufficient conditions for an…
We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples
We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in familiar settings. We show that, in a virtual equipment…
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we…
We study approximations of theories both in general context and with respect to some natural classes of theories. Some kinds of approximations are considered, connections with finitely axiomatizable theories and minimal generating sets of…
Constructor theory is a meta-theoretic approach that seeks to characterise concrete theories of physics in terms of the (im)possibility to implement certain abstract "tasks" by means of physical processes. Process theory, on the other hand,…
As the first part of the treatise on A General Theory of Concept Lattice (I-V), this work develops the general concept lattice for the problem concerning categorization of objects according to their properties. Unlike the conventional…
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 is the second in a series of papers on the relation between algebraic set theory and predicative formal systems. In part I, we introduced the notion of a predicative category of small maps and obtained the result that such categories…
Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of \emph{branching in trees} and of…
In a coherent category, the posets of subobjects have very strong properties. We emphasize the validity of these properties, in general categories, for well-behaved classes of subobjects. As an example of application, we investigate the…
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the…