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.
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.
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
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.
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.
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
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.