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. Examples include quotient spaces, direct products, completion, and duality. Many areas of computer science also rely on category theory, such as functional programming and semantics. A is formed by two sorts of objects: the s 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 theory 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. Category (mathematics) and Morphism A category C consists of the following three mathematical entities: A class ob(C), whose elements are called objects; A class hom(C), whose elements are called morphisms or maps or arrows.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (10)

Related people (2)

Related concepts (243)

Related courses (15)

Related lectures (191)

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

MATH-688: Reading group in applied topology I

The subject of this reading group is Ginerstra Bianconi's book "Higher-order networks - an introduction to simplicial complexes". Participants will take turns presenting chapters, then leading a discu

Homotopy theory of chain complexes

Explores the homotopy theory of chain complexes, focusing on retractions and model category structures.

Products: Theory of Categories

Covers the concept of products in category theory, focusing on their properties, uniqueness, and isomorphisms.

Natural Transformations in Group Theory

Introduces natural transformations in group theory and category theory, emphasizing their definition and significance.

Computational tools for twisted topological Hochschild homology of equivariant spectra

Kathryn Hess Bellwald, Inbar Klang

Twisted topological Hochschild homology of Cn-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on nor

ELSEVIER2022

A Shadow Perspective on Equivariant Hochschild Homologies

Kathryn Hess Bellwald, Inbar Klang

Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids i

OXFORD UNIV PRESS2022

This thesis is part of a program initiated by Riehl and Verity to study the category theory of (infinity,1)-categories in a model-independent way. They showed that most models of (infinity,1)-categori

EPFL2017

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.

Category theory

Morphism

In mathematics, particularly in , a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in analysis and topology, continuous functions, and so on.