Related papers: An Introduction to Categorical Proof Theory
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain…
These Course Notes provide an introduction to mathematical proofs for undergraduate students transitioning from computational calculus to abstract mathematics. Topics include propositional logic, proof techniques, mathematical induction,…
In mathematical applications, category theory remains a contentious issue, with enthusiastic fans and a skeptical majority. In a muted form this split applies to the authors of this note. When we learned that the only mathematically sound…
We give a short introduction to category theory aimed at philosophers. We emphasize methodological issues and philosophical ramifications.
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…
These lecture notes survey the emerging area of Universal Proof Theory, which investigates general questions about the existence, equivalence, and characterization of good proof systems for broad classes of logics. In particular, the notes…
This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…
This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…
The importance of category theory in recent developments in both mathematics and in computer science cannot be overstated. However, its abstract nature makes it difficult to understand at first. Graphical languages have been developed to…
Category theory can be used to state formulas in First-Order Logic without using set membership. Several notable results in logic such as proof of the continuum hypothesis can be elegantly rewritten in category theory. We propose in this…
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
In these lecture notes, we give a brief introduction to some elements of category theory. The choice of topics is guided by applications to functional programming. Firstly, we study initial algebras, which provide a mathematical…
We consider (finitary, propositional) logics through the original use of Category Theory: the study of the "sociology of mathematical objects", aligning us with a recent, and growing, trend of study logics through its relations with other…
Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…
Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this paper, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using…
We present a domain-specific type theory for constructions and proofs in category theory. The type theory axiomatizes notions of category, functor, profunctor and a generalized form of natural transformations. The type theory imposes an…
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced…
In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a…
The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design…
In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules.…