Lecture

Introduction to Category Theory: Examples

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Description

This lecture introduces the theory of categories through examples, such as defining a category CR based on a relation R on a set X, exploring the composition and identity properties. It also discusses the construction of a category BG from a group G, where the objects are elements of G and the morphisms are given by the group operation. Furthermore, it explains the product of categories C and D, denoted as C x D, detailing how objects and morphisms are defined and composed in this product category.

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Mathematical logic: Set theory

Related concepts (32)

Binary relation

In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y. It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation.

Product (category theory)

In , the product of two (or more) in a is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

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. Category theory is used in almost all areas of mathematics. In particular, numerous constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories.

Product category

In the mathematical field of , the product of two C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors. The product category C × D has: as : pairs of objects (A, B), where A is an object of C and B of D; as arrows from (A1, B1) to (A2, B2): pairs of arrows (f, g), where f : A1 → A2 is an arrow of C and g : B1 → B2 is an arrow of D; as composition, component-wise composition from the contributing categories: (f2, g2) o (f1, g1) = (f2 o f1, g2 o g1); as identities, pairs of identities from the contributing categories: 1(A, B) = (1A, 1B).

Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a ) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the , whose objects are sets and whose arrows are functions. is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent.

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