subclass_of enriched category
In ___MASK94___ ___MASK31___ , a branch of ___MASK12___ , an enriched ___MASK94___ generalizes the idea of a ___MASK94___ by replacing hom - sets with objects from a general monoidal ___MASK94___ . It is motivated by the observation that , in many practical applications , the hom - set often has additional ___MASK17___ that should be respected , e . g . , that of being a vector space of morphisms , or a topological space of morphisms . In an enriched ___MASK94___ , the set of morphisms ( the hom - set ) associated with every pair of objects is replaced by an opaque object in some fixed monoidal ___MASK94___ of `` hom - objects '' . In order to emulate the ( associative ) ___MASK71___ of morphisms in an ordinary ___MASK94___ , the hom - ___MASK94___ 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 ___MASK17___ of a monoidal ___MASK94___ , though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint ( i . e . , making the ___MASK94___ symmetric monoidal or even symmetric closed monoidal , respectively ) . Enriched ___MASK94___ ___MASK31___ thus encompasses within the same framework a wide variety of structures including ordinary categories where the hom - set carries additional ___MASK17___ beyond being a set . That is , there are operations on , or properties of morphisms that need to be respected by ___MASK71___ ( e . g . , the existence of 2 - cells between morphisms and horizontal ___MASK71___ thereof in a 2 - ___MASK94___ , or the addition operation on morphisms in an abelian ___MASK94___ ) ___MASK94___ - 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 ___MASK71___ rule ensures transitivity , or Lawvere ' s metric spaces , where the hom - objects are numerical distances and the ___MASK71___ rule provides the triangle inequality ) . In the case where the hom - object ___MASK94___ happens to be the ___MASK94___ of sets with the usual cartesian product , the definitions of enriched ___MASK94___ , enriched functor , etc . . . reduce to the original definitions from ordinary ___MASK94___ ___MASK31___ . An enriched ___MASK94___ with hom - objects from monoidal ___MASK94___ M is said to be an enriched ___MASK94___ over M or an enriched ___MASK94___ in M , or simply an M - ___MASK94___ . Due to Mac Lane ' s preference for the letter V in referring to the monoidal ___MASK94___ , enriched categories are also sometimes referred to generally as V - categories . In ___MASK12___ , 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 . ___MASK12___ ( from Greek  " máthma ",  knowledge , study , learning ) is the study of topics such as quantity ( numbers ), ___MASK17___ , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics . In ___MASK12___ , a ___MASK17___ on a set is an additional ___MASK84___ that , in some manner , attaches ( or relates ) to that set to endow it with some additional meaning or significance . In ___MASK12___ , a monoidal category ( or ___MASK95___ category ) is a category ___MASK87___ 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 . A ___MASK14___ is a ___MASK91___ 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 . ___MASK15___ is a ___MASK68___ 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 . ___MASK15___ 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 . ___MASK94___ ___MASK31___ formalizes mathematical ___MASK17___ 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 ___MASK31___ has been used to formalize concepts of other high - level abstractions such as sets , rings , and groups . In ___MASK12___ , the ___MASK76___ 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 ___MASK91___ " 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 . In abstract ___MASK96___ , a branch of mathematics , a monoid is an algebraic structure with a ___MASK57___ 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 ___MASK57___ object . Thus , they capture the idea of function ___MASK71___ within a set . In fact , all functions from a set into itself form naturally a monoid with respect to function ___MASK71___ . 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 Krohn  Rhodes 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 . ___MASK42___ is the boundless three - dimensional extent in which objects and events have relative ___MASK79___ and direction . Physical space is often conceived in three linear dimensions , although modern physicists usually consider it , with ___MASK9___ , 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 .