GROUPOID Meaning and
Definition
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A groupoid is a mathematical structure that lies between a group and a category. It can be thought of as a generalization of a group where not all elements necessarily have inverses, and composition is not necessarily well-defined for all elements.
Formally, a groupoid consists of a set of objects and a set of morphisms. Each morphism connects a pair of objects and possesses an identity element associated with each object. Furthermore, each morphism has an inverse morphism, such that the composition of a morphism with its inverse yields the identity morphism.
The concept of a groupoid arises in various areas of mathematics, including algebraic topology, dynamical systems, and differential geometry. In algebraic topology, for example, a groupoid can be used to study the fundamental group of a space, allowing for a more flexible treatment of spaces that have non-trivial symmetries.
One of the distinguishing features of a groupoid is that it allows for a broader notion of equivalence between objects. In a groupoid, two objects are considered equivalent if there exists a morphism between them, whereas in a group, equivalence is solely based on the existence of an isomorphism. Consequently, groupoids provide a more flexible framework for capturing symmetries and transformations.
Common Misspellings for GROUPOID
- froupoid
- vroupoid
- broupoid
- hroupoid
- yroupoid
- troupoid
- geoupoid
- gdoupoid
- gfoupoid
- gtoupoid
- g5oupoid
- g4oupoid
- griupoid
- grkupoid
- grlupoid
- grpupoid
- gr0upoid
- gr9upoid
- groypoid
- grohpoid
Etymology of GROUPOID
The word "groupoid" is derived from the combination of two words: "group" and "-oid".
The term "group" comes from the Old English word "ġerāp", which means "to grasp or hold". It entered the English language in the 15th century and initially referred to a social or political alliance. In mathematics, a group is a set equipped with an operation that combines any two elements to form a third element, satisfying certain axioms such as associativity, identity, and inverses.
The suffix "-oid" is derived from the Greek suffix "-oeidēs", which means "resembling or having the form or likeness of". It is commonly used in English to denote similarity or resemblance to something. For example, the term "humanoid" indicates something resembling a human.
