Category Theory Lecture Notes for ESSLLI (PDF 133P)

Category Theory Lecture Notes for ESSLLI (PDF 133P)

Category Theory Lecture Notes for ESSLLI (PDF 133P)

This note covers the following topics related to
Category Theory: Functional programming languages as categories, Mathematical structures as
categories, Categories of sets with structure, Categories of algebraic
structures, Constructions on categories, Properties of objects and arrows,
Functors, Diagrams and naturality, Products and sums, Cartesian closed
categories, Limits and colimits, Adjoints, Triples, Toposes, Categories with
monoidal structure.

The book is aimed primarily at the beginning graduate
student.It gives the de nition of this notion, goes through the various
associated gadgetry such as functors, natural transformations, limits and
colimits, and then explains adjunctions.

This
note covers the following topics related to Category Theory: Categories,
Functors and Natural Transformations, subcategories, Full and Faithful Functors,
Equivalences, Comma Categories and Slice Categories, Yoneda Lemma, Grothendieck
universes, Limits and Colimits, Adjoint Functors, Adjoint Functor Theorems,
Monads, Elementary Toposes, Cartesian Closed Categories, Logic of Toposes and
Sheaves.

This book explains the following topics: Categories, functors, natural
transformations, String diagrams, Kan extensions, Algebras, coalgebras,
bialgebras, Lambda-calculus and categories.

This book emphasizes
category theory in conceptual aspects, so that category theory has come to be
viewed as a theory whose purpose is to provide a certain kind of conceptual
clarity.

Purpose of this course note
is to prove that category theory is a powerful language for understanding and
formalizing common scientific models. The power of the language will be tested
by its ability to penetrate into taken-for-granted ideas, either by exposing
existing weaknesses or flaws in our understanding, or by highlighting hidden
commonalities across scientific fields.

Category theory, a branch of abstract
algebra, has found many applications in mathematics, logic, and computer
science. Like such fields as elementary logic and set theory, category theory
provides a basic conceptual apparatus and a collection of formal methods useful
for addressing certain kinds of commonly occurring formal and informal problems,
particularly those involving structural and functional considerations. This
course note is intended to acquaint students with these methods, and also
to encourage them to reflect on the interrelations between category theory and
the other basic formal disciplines.

This note covers the following topics related to
Category Theory: Functional programming languages as categories, Mathematical structures as
categories, Categories of sets with structure, Categories of algebraic
structures, Constructions on categories, Properties of objects and arrows,
Functors, Diagrams and naturality, Products and sums, Cartesian closed
categories, Limits and colimits, Adjoints, Triples, Toposes, Categories with
monoidal structure.

This work gives an explanatory introduction to various definitions of
higher dimensional category. The emphasis is on ideas rather than formalities;
the aim is to shed light on the formalities by emphasising the intuitions that
lead there. Covered topics are: Penon, Batanin and Leinster, Opetopic,
Tamsamani and Simpson, Trimble and May.

Higher
dimensional category theory is the study of n categories, operads, braided
monoidal categories, and other such exotic structures. It draws its
inspiration from areas as diverse as topology, quantum algebra, mathematical
physics, logic, and theoretical computer science. This is the first book on
the subject and lays its foundations.

This note covers the following topics: Universal Problems, Basic Notions, Universality, Natural
Transformations and Functor Categories, Colimits, Duality and LKan
Extensions imits, Adjunctions, Preservation of Limits and Colimits, Monads,
Lawvere Theories, Cartesian Closed Categories, Variable Sets and Yoneda
Lemma and 2-Categories.

This note covers the following topics: Categories and Functors, Natural transformations,
Examples of natural transformations, Equivalence of categories, cones and
limits, Limits by products and equalizers, Colimits, A little piece of
categorical logic, The logic of regular categories.