teaches the basics of category theory, in a way that is accessible and
relevant to computer scientists. The emphasis is on gaining a good
understanding the basic definitions, examples, and techniques, so that
students are equipped for further study on their own of more advanced topics
Hutton, School of Computer Science, University of Nottingham
explains the following topics related to Category Theory:Foundations, Graphs,
Monoids, Categories, Constructions on categories, Functors, Special types of
functors, Natural transformations, Representable functors and the Yoneda Lemma,
Terminal and initial objects, The extension principle, Isomorphisms,
Monomorphisms and epimorphisms, Products, Adjoint functors and monads.
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: Monoidal categories, The pentagon axiom, Basic properties of unit
objects in monoidal categories, monoidal categories, Monoidal functors,
equivalence of monoidal categories, Morphisms of monoidal functors, MacLane's
strictness theorem, The MacLane coherence theorem, Invertible objects,
Exactness of the tensor product, Semisimplicity of the unit object, Groupoids,
Finite abelian categories and exact faithful functors, Fiber functors, Hopf
algebras, Pointed tensor categories and pointed Hopf algebras, Chevalley's
theorem, The Andruskiewitsch-Schneider conjecture, The Cartier-Kostant
theorem, Pivotal categories and dimensions, Spherical categories and
Grothendieck rings of semisimple tensor categories.
Etingof, S. Gelaki, D. Nikshych, and V. Ostrik
These notes are targeted to a student
with significant mathematical sophistication and a modest amount of specific
knowledge. Covered topics are: Mathematics in Categories, Constructing
Categories, Functors and Natural Transformations, Universal Mapping
Properties, Algebraic Categories, Cartesian Closed Categories, Monoidal
Categories, Enriched Category Theory, Additive and Abelian Categories,
2-Categories and Fibered Categories.
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
explains the following topics related to Category Theory: Duality, Universal and
couniversal properties, Limits and colimits, Biproducts in Vect and Rel,
Functors, Natural transformations, Yoneda'a Lemma, Adjoint Functors, Cartesian
Closed Categories, The Curry-Howard-Lambek Isomorphism, Induction and
Coinduction, Stream programming examples and Monads.
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.
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.