# Category theory

Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the source and the target of the morphism. One often says that a morphism is an arrow that maps its source to its target. Morphisms can be composed if the target of the first morphism equals the source of the second one, and morphism composition has similar properties as function composition (associativity and existence of identity morphisms). Morphisms are often some sort of function, but this is not always the case. For example, a monoid may be viewed as a category with a single object, whose morphisms are the elements of the monoid. The second fundamental concept of category is the concept of a functor, which plays the role of a morphism between two categories and it maps objects of to objects of and morphisms of to morphisms of in such a way that sources are mapped to sources and targets are mapped to targets (or, in the case of a contravariant functor, sources are mapped to targets and vice-versa). A third fundamental concept is a natural transformation that may be viewed as a morphism of functors. (Wikipedia).

Category Theory 1.2: What is a category?

What is a Category?

From playlist Category Theory

Category Theory 1.1: Motivation and Philosophy

Motivation and philosophy

From playlist Category Theory

Category Theory 2.1: Functions, epimorphisms

Functions, epimorphisms

From playlist Category Theory

Category Theory 3.1: Examples of categories, orders, monoids

Examples of categories, orders, monoids.

From playlist Category Theory

Category Theory: The Beginner’s Introduction (Lesson 1 Video 2)

Lesson 1 is concerned with defining the category of Abstract Sets and Arbitrary Mappings. We also define our first Limit and Co-Limit: The Terminal Object, and the Initial Object. Other topics discussed include Duality and the Opposite (or Mirror) Category. Follow me on Twitter: @mjmcodr

From playlist Category Theory: The Beginner’s Introduction

Category Theory 9.1: Natural transformations

Natural transformations

From playlist Category Theory

PNWS 2014 - What every (Scala) programmer should know about category theory

By, Gabriel Claramunt Aren't you tired of just nodding along when your friends starts talking about morphisms? Do you feel left out when your coworkers discuss a coproduct endofunctor? From the dark corners of mathematics to a programming language near you, category theory offers a compac

From playlist PNWS 2014

The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories - Emily Riehl

Vladimir Voevodsky Memorial Conference Topic: The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories Speaker: Emily Riehl Affiliation: Johns Hopkins University Date: September 12, 2018 For more video please visit http://video.ias.edu

From playlist Mathematics

David Ben-Zvi: Geometric Langlands correspondence and topological field theory - Part 2

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Algebraic and Complex Geometry

Lecture 1: Invitation to topos theory

This talk introduces the motivating question for this semester of the Curry-Howard seminar, which is how to organise mathematical knowledge using topoi. The approach sketched out in the talk is via first-order theories, their associated classifying topoi, and adjoint pairs of functors betw

From playlist Topos theory seminar

David Ben-Zvi: Geometric Langlands correspondence and topological field theory - Part 1

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Algebraic and Complex Geometry

Shadows of Computation - Lecture 1 - Making subtle ideas apparent

Welcome to Shadows of Computation, an online course taught by Will Troiani and Billy Snikkers, covering the foundations of category theory and how it is used by computer scientists to abstract computing systems to reveal their intrinsic mathematical properties. In the first lecture Will in

Laurent Lafforgue - 3/4 Classifying toposes of geometric theories

Course at the school and conference “Toposes online” (24-30 June 2021): https://aroundtoposes.com/toposesonline/ Slides: https://aroundtoposes.com/wp-content/uploads/2021/07/LafforgueSlidesToposesOnline.pdf The purpose of these lectures will be to present the theory of classifying topose

From playlist Toposes online

Huawei Young Talents Programme - Laurent Lafforgue

The online ceremony celebrating the official launch of the Huawei Young Talents Program at the Institut des Hautes Etudes Scientifiques was held on 6 November 2020. This program aims to support the work of talented researchers in mathematics and theoretical physics at the beginning of thei

From playlist Huawei Young Talents Program - November 2020

Laurent Lafforgue - 1/4 Classifying toposes of geometric theories

Course at the school and conference “Toposes online” (24-30 June 2021): https://aroundtoposes.com/toposesonline/ Slides: https://aroundtoposes.com/wp-content/uploads/2021/07/LafforgueSlidesToposesOnline.pdf The purpose of these lectures will be to present the theory of classifying topose

From playlist Toposes online

A Sensible Introduction to Category Theory

Remember when I used a video with a coconut in the thumbnail to drive a stake through the heart of mathematical structure? Today, in this introduction to the basics of category theory, I attempt to remove it. 27 Unhelpful Facts About Category Theory: https://www.youtube.com/watch?v=H0Ek86

From playlist Mathematics

David Ben-Zvi: Boundary conditions and hamiltonian actions in geometric Langlands

SMRI Algebra and Geometry Online: ‘Boundary conditions and hamiltonian actions in geometric Langlands’ David Ben-Zvi (University of Texas at Austin)

From playlist SMRI Algebra and Geometry Online