Section (category theory) - Wikipedia
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of . [1]
Category theory - Wikipedia
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. [1] Category theory is …
Many categories can be found living inside the category of sets { objects are sets equipped with certain structures, and morphisms are still functions. Here are a few: Each of these is what we call a subcategory of the category of sets. De nition 1.3 (Subcategory).
Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction,
Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject.
The material is divided into three sections. Section 0.1 gives an introduction to category theory, and in particular the categories Setof sets and functions, and Rel of sets and relations. Section 0.2 introduces the mathematical formalism of Hilbert spaces that underlies quantum mechanics, and defines the categories
An object i of a category C is said to be initial, if the set of morphisms from i to any object c consists of a single morphism. We are generally not assuming that all objects in a category are equipped with the identity object, so the single endomorphism i: i !i (5) is not necessary the identity idi. Exercise 9 Show that (5) is a right identity. 5
This section introduces basic category theory concepts that are required for later sections, including categories, functors, natural transformations and notions of equivalence between categories.
So we will settle for the theory of 1-categories, which is good enough to amply display the di erence with set theory and illustrate the salient features of the new theory. The main distinction with set theory is that equality is not a meaningful notion in the new setting. We cannot say that two objects x and y are equal.
Category theory is the mathematical study of universal properties: it brings to light, makes explicit, and abstracts out the relevant structure, often hidden by traditional approaches;