Whitehead theoremIn homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between CW complexes X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of algebraic topology, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping.
Homotopy category of chain complexesIn homological algebra in mathematics, the homotopy category K(A) of chain complexes in an A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the D(A) of A when A is ; unlike the former it is a , and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence.
Category theoryCategory 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.
IsomorphismIn mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects).
Hom functorIn mathematics, specifically in , hom-sets (i.e. sets of morphisms between ) give rise to important functors to the . These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Let C be a (i.e. a for which hom-classes are actually sets and not proper classes). For all objects A and B in C we define two functors to the as follows: {| class=wikitable |- ! Hom(A, –) : C → Set ! Hom(–, B) : C → Set |- | This is a covariant functor given by: Hom(A, –) maps each object X in C to the set of morphisms, Hom(A, X) Hom(A, –) maps each morphism f : X → Y to the function Hom(A, f) : Hom(A, X) → Hom(A, Y) given by for each g in Hom(A, X).
MorphismIn 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.
Isomorphism theoremsIn mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
Limit (category theory)In , a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as , and inverse limits. The of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, s and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.
Localization of a categoryIn mathematics, localization of a category consists of adding to a inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category.
Set (mathematics)A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics.
Set theorySet theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.
