PROP (category theory)In , a branch of mathematics, a PROP is a strict whose objects are the natural numbers n identified with the finite sets and whose tensor product is given on objects by the addition on numbers. Because of “symmetric”, for each n, the symmetric group on n letters is given as a subgroup of the of n. The name PROP is an abbreviation of "PROduct and ". The notion was introduced by Adams and MacLane; the topological version of it was later given by Boardman and Vogt. Following them, J. P.
OperadIn mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad , one defines an algebra over to be a set together with concrete operations on this set which behave just like the abstract operations of . For instance, there is a Lie operad such that the algebras over are precisely the Lie algebras; in a sense abstractly encodes the operations that are common to all Lie algebras.
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.
Timeline of category theory and related mathematicsThis is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: of abstract algebraic structures including representation theory and universal algebra; Homological algebra; Homotopical algebra; Topology using categories, including algebraic topology, categorical topology, quantum topology, low-dimensional topology; Categorical logic and set theory in the categorical context such as algebraic set theory; Foundations of mathematics building on categories, for instance topos theory; Abstract geometry, including algebraic geometry, categorical noncommutative geometry, etc.
