Spectrum (topology)In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory,there exist spaces such that evaluating the cohomology theory in degree on a space is equivalent to computing the homotopy classes of maps to the space , that is.Note there are several different of spectra leading to many technical difficulties, but they all determine the same , known as the stable homotopy category.
Monoidal categoryIn mathematics, a monoidal category (or tensor category) is a equipped with a bifunctor that is associative up to a natural isomorphism, and an I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant s commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples.
Postnikov systemIn homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov. A Postnikov system of a path-connected space is an inverse system of spaces with a sequence of maps compatible with the inverse system such that The map induces an isomorphism for every .
MKS system of unitsThe MKS system of units is a physical system of measurement that uses the metre, kilogram, and second (MKS) as base units. The modern International System of Units (SI) was originally created as a formalization of the MKS system, and although the SI has been redefined several times since then and is now based entirely on fundamental physical constants, it still closely approximates the original MKS system for most practical purposes. By the mid-19th century, there was a demand by scientists to define a coherent system of units.
Coherence (units of measurement)A coherent system of units is a system of units of measurement used to express physical quantities that are defined in such a way that the equations relating the numerical values expressed in the units of the system have exactly the same form, including numerical factors, as the corresponding equations directly relating the quantities. It is a system in which every quantity has a unique unit, or one that does not use conversion factors.
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.
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.
Category of setsIn the mathematical field of , the category of sets, denoted as Set, is the whose are sets. The arrows or morphisms between sets A and B are the total functions from A to B, and the composition of morphisms is the composition of functions. Many other categories (such as the , with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind.
∞-groupoidIn , a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard ). It is an generalization of a groupoid, a category in which every morphism is an isomorphism. The homotopy hypothesis states that ∞-groupoids are equivalent to spaces up to homotopy. Alexander Grothendieck suggested in Pursuing Stacks that there should be an extraordinarily simple model of ∞-groupoids using globular sets, originally called hemispherical complexes.
Monad (category theory)In , a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a in the of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.
Weber (unit)In physics, the weber (ˈveɪb-,_'wEb.ər ; symbol: Wb) is the unit of magnetic flux in the International System of Units (SI), whose units are volt-second. A magnetic flux density of one Wb/m2 (one weber per square metre) is one tesla. The weber is named after the German physicist Wilhelm Eduard Weber (1804–1891). The weber may be defined in terms of Faraday's law, which relates a changing magnetic flux through a loop to the electric field around the loop.
Fundamental groupIn the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups.
