query: subclass_of enriched category
supports: In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance. Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. A small quantity is sometimes referred to as a quantulum. In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor
  that is associative up to a natural isomorphism, and an object "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 diagrams commute. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single object. Thus, they capture the idea of function composition within a set. In fact, all functions from a set into itself form naturally a monoid with respect to function composition. Monoids are also commonly used in computer science, both in its foundational aspects and in practical programming. The set of strings built from a given set of characters is a free monoid. The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history monoids provide a foundation for process calculi and concurrent computing. Some of the more important results in the study of monoids are the KrohnRhodes theorem and the star height problem. The history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups. In category theory , a branch of mathematics , an enriched category generalizes the idea of a category by replacing hom - sets with objects from a general monoidal category . It is motivated by the observation that , in many practical applications , the hom - set often has additional structure that should be respected , e.g. , that of being a vector space of morphisms , or a topological space of morphisms . In an enriched category , the set of morphisms ( the hom - set ) associated with every pair of objects is replaced by an opaque object in some fixed monoidal category of `` hom - objects '' . In order to emulate the ( associative ) composition of morphisms in an ordinary category , the hom - category must have a means of composing hom - objects in an associative manner : that is , there must be a binary operation on objects giving us at least the structure of a monoidal category , though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint ( i.e. , making the category symmetric monoidal or even symmetric closed monoidal , respectively ) . Enriched category theory thus encompasses within the same framework a wide variety of structures including ordinary categories where the hom - set carries additional structure beyond being a set . That is , there are operations on , or properties of morphisms that need to be respected by composition ( e.g. , the existence of 2 - cells between morphisms and horizontal composition thereof in a 2 - category , or the addition operation on morphisms in an abelian category ) category - like entities that do n't themselves have any notion of individual morphism but whose hom - objects have similar compositional aspects ( e.g. , preorders where the composition rule ensures transitivity , or Lawvere 's metric spaces , where the hom - objects are numerical distances and the composition rule provides the triangle inequality ) . In the case where the hom - object category happens to be the category of sets with the usual cartesian product , the definitions of enriched category , enriched functor , etc. . . reduce to the original definitions from ordinary category theory . An enriched category with hom - objects from monoidal category M is said to be an enriched category over M or an enriched category in M , or simply an M - category . Due to Mac Lane 's preference for the letter V in referring to the monoidal category , enriched categories are also sometimes referred to generally as V - categories . In mathematics, and particularly category theory a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the essence of a term) and extensional definitions (which proceed by listing the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. Mathematics (from Greek  "máthma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. In mathematics, the phrase up to appears in discussions about the elements of a set (say "S"), and the conditions under which subsets of those elements may be considered equivalent. The statement "elements "a" and "b" of set S are equivalent up to "X"" means that "a" and "b" are equivalent if criterion "X" (such as rotation or permutation) is ignored. That is, "a" and "b" can be transformed into one another if a transform corresponding to "X" (rotation, permutation etc.) is applied. Category theory formalizes mathematical structure and its concepts in terms of a collection of "objects" and of "arrows" (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups.
candidates: ability algebra binary operation branch capture category composition computer computer science concept concurrent computing condition data definition direction element essence function general greek height history idea learning light mathematical object mathematics measurement natural object part phrase position problem quantity range relation science semigroup set single space state statement structure tensor theory time value word
answer: mathematical object