Related papers: Structures in Concrete Categories
The purpose of the present paper is to make a mathematical study of the differences and relations among possible structures inherent in an object, as well as of the whole structure constituted by them (i.e., the structure of structures),…
The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…
We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and…
The exposition of the theory of structure species in Bourbaki's tractate takes only a few pages but still is quite difficult. However, in the exercises, Bourbaki outlines another approach that is based on the notion of structure type rather…
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,…
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…
We introduce some classes of genuine higher categories in homotopy type theory, defined as well-behaved subcategories of the category of types. We give several examples, and some techniques for showing other things are not examples. While…
Category theory has become central to certain aspects of theoretical physics. Bain [Synthese, 190:1621--1635 (2013)] has recently argued that this has significance for ontic structural realism. We argue against this claim. In so doing, we…
We give a categorification of the notion of a mathematical structure originally given by Bourbaki in their set theory textbook. We show that any isomorphism-invariant property of a finite structure can be computed by counting the number of…
Category Theory provides us with a clear notion of what is an internal structure. This will allow us to focus our attention on a certain type of relationship between context and structure.
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…
Expansion of the categorical point of view on many areas of the mathematics and mathematical physics will cause to deeper understanding of genuine features of these problems. New applications of categorical methods are connected with new…
The popular view according to which Category theory provides a support for Mathematical Structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics…
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…
In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…
A quotient construction defines an abstract type from a concrete type, using an equivalence relation to identify elements of the concrete type that are to be regarded as indistinguishable. The elements of a quotient type are…
It has become obvious in the recent development that the structural Ramsey property is a categorical property: it depends not only on the choice of objects, but also on the choice of morphisms involved. In this paper we explicitely put the…
This paper contains results from two areas -- formal theory of Kan extensions and concrete categories. The contribution to the former topic is based on the extension of the concept of Kan extension to the cones and we prove that limiting…
In this note we characterize, within the framework of the theory of finite set, those categories of graphs that are {\em algebraic universal} in the sense that every concrete category embeds in them. The proof of the characterization is…
Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements &…