In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways.
Special cases include: Groupoids are often used to reason about geometrical objects such as manifolds.
[2] A groupoid can be viewed as an algebraic structure consisting of a set with a binary partial function [citation needed].
is not a binary operation because it is not necessarily defined for all pairs of elements of
, Two easy and convenient properties follow from these axioms: A groupoid is a small category in which every morphism is an isomorphism, i.e., invertible.
More generally, one can consider a groupoid object in an arbitrary category admitting finite fiber products.
The algebraic and category-theoretic definitions are equivalent, as we now show.
Conversely, given a groupoid G in the algebraic sense, define an equivalence relation
Sets in the definitions above may be replaced with classes, as is generally the case in category theory.
In that case, all the vertex groups are isomorphic (on the other hand, this is not a sufficient condition for transitivity; see the section below for counterexamples).
Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative.
An important extension of this idea is to consider the fundamental groupoid
for some constant sheaf of abelian groups can be represented as a function
Another commonly studied family of orbifolds are weighted projective spaces
[7] A two term complex of objects in a concrete Abelian category can be used to form a groupoid.
While puzzles such as the Rubik's Cube can be modeled using group theory (see Rubik's Cube group), certain puzzles are better modeled as groupoids.
[8] The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed).
The Mathieu groupoid is a groupoid introduced by John Horton Conway acting on 13 points such that the elements fixing a point form a copy of the Mathieu group M12.
If a groupoid has only one object, then the set of its morphisms forms a group.
Note that the isomorphism just mentioned is not unique, and there is no natural choice.
Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object
In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group.
For example, The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not natural.
As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations.
This is analogous to the fact that the classification of vector spaces with one endomorphism is nontrivial.
embeds Grpd as a full subcategory of the category of simplicial sets.
One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally.
There is a similar composition law for horizontal attachments of squares.
For instance, in Poisson geometry one has the notion of a symplectic groupoid, which is a Lie groupoid endowed with a compatible symplectic form.
Similarly, one can have groupoids with a compatible Riemannian metric, or complex structure, etc.