Homotopy theoryIn mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline.
Homotopy categoryIn mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any model category and define its associated homotopy category, with a construction introduced by Quillen in 1967.
Representable functorIn mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.
Natural transformationIn category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category.
Category theoryCategory theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. 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.
Category of setsIn the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are the functions from A to B, and the composition of morphisms is the composition of functions. Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets or restrict the arrows to functions of a particular kind (or both).
Pointed setIn mathematics, a pointed set (also based set or rooted set) is an ordered pair(X,x0){\displaystyle (X,x_{0})} where X{\displaystyle X} is a set and x0{\displaystyle x_{0}} is an element of X{\displaystyle X} called the base point (also spelled basepoint).
Edgar H. BrownEdgar Henry Brown, Jr. (December 27, 1926 – December 22, 2021) was an American mathematician specializing in algebraic topology, and for many years a professor at Brandeis University. LifeBrown was born in Oak Park, Illinois. He completed his bachelor's degree in mathematics at the University of Wisconsin in 1949. He completed his master's degree in mathematics at Washington State University in 1951.CW complexIn mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called cells) of different dimensions in specific ways. The notion generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory.
FunctorIn mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories.CoproductIn category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism.
Necessity and sufficiencyIn logic and mathematics, necessity and sufficiency are concepts used to denote a conditional or implicational relationship between two statements.*As an Amazon Associate I earn from qualifying purchases.