Stable homotopy theoryIn mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space , the homotopy groups stabilize for sufficiently large. In particular, the homotopy groups of spheres stabilize for . For example, In the two examples above all the maps between homotopy groups are applications of the suspension functor.
Mapping cylinderIn mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient where the denotes the disjoint union, and ∼ is the equivalence relation generated by That is, the mapping cylinder is obtained by gluing one end of to via the map . Notice that the "top" of the cylinder is homeomorphic to , while the "bottom" is the space . It is common to write for , and to use the notation or for the mapping cylinder construction.
Strict 2-categoryIn , a strict 2-category is a with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category over Cat (the , with the structure given by ). The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou.
Peano axiomsIn mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.
Tarski's axiomsTarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory (i.e., that part of Euclidean geometry that is formulable as an elementary theory). Other modern axiomizations of Euclidean geometry are Hilbert's axioms and Birkhoff's axioms. Early in his career Tarski taught geometry and researched set theory.
Leray spectral sequenceIn mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence. Let be a continuous map of topological spaces, which in particular gives a functor from sheaves of abelian groups on to sheaves of abelian groups on .
Kan fibrationIn mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. For each n ≥ 0, recall that the , , is the representable simplicial set Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard -simplex: the convex subspace of Rn+1 consisting of all points such that the coordinates are non-negative and sum to 1.
Homotopy extension propertyIn mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations. Let be a topological space, and let . We say that the pair has the homotopy extension property if, given a homotopy and a map such that then there exists an extension of to a homotopy such that .
Chain complexIn mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology.
Separation axiomIn topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff. The separation axioms are not fundamental axioms like those of set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces.
