Групоидтар: категория теориясындағы инверттеуге ие морфизмдер жиыны

Category where every morphism is invertible; generalization of a group
groupoids in category theory

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. A groupoid can be seen as a:
Group with a partial function replacing the binary operation;
Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called inverse by analogy with group theory. A groupoid where there is only one object is a usual group. In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed , , say. Composition is then a total function: , so that
Special cases include:
Setoids: sets that come with an equivalence relation,
G sets: sets equipped with an action of a group
Groupoids are often used to reason about geometrical objects such as manifolds. introduced groupoids implicitly via Brandt semigroups.

A groupoid is a small category in which every morphism is an isomorphism, i. e., invertible. It is also true that the category of covering morphisms of a given groupoid is equivalent to the category of actions of the groupoid on sets.

Given a topological space , let be the set The morphisms from the point to the point are equivalence classes of continuous paths from to , with two paths being equivalent if they are homotopic. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of , denoted (or sometimes, ). The usual fundamental group is then the vertex group for the point
The orbits of the fundamental groupoid are the path connected components of Accordingly, the fundamental groupoid of a path connected space is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are equivalent as categories (see the section below for the general theory). An important extension of this idea is to consider the fundamental groupoid where is a chosen set of "base points". Here is a (wide) subgroupoid of , where one considers only paths whose endpoints belong to The set may be chosen according to the geometry of the situation at hand.

Consider the group action of on the finite set which takes each number to its negative, so and The quotient groupoid is the set of equivalence classes from this group action , and has a group action of on it.

Any finite group that maps to gives a group action on the affine space (since this is the group of automorphisms). Then, a quotient groupoid can be of the form , which has one point with stabilizer at the origin. Examples like these form the basis for the theory of orbifolds. Another commonly studied family of orbifolds are weighted projective spaces and subspaces of them, such as Calabi–Yau orbifolds.

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. The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed). This groupoid acts on configurations.

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.

The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, and is denoted by Grpd. The category Grpd is, like the category of small categories, Cartesian closed: for any groupoids we can construct a groupoid whose objects are the morphisms and whose arrows are the natural equivalences of morphisms. Thus if are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids there is a natural bijection

This result is of interest even if all the groupoids are just groups. Another important property of Grpd is that it is both complete and cocomplete.

There is an additional structure which can be derived from groupoids internal to the category of groupoids, double groupoids. Because Grpd is a 2 category, these objects form a 2 category instead of a 1 category since there is extra structure. Essentially, these are groupoids with functorsand an embedding given by an identity functorOne way to think about these 2 groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares and with the same morphism, they can be vertically conjoined giving a diagramwhich can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.

When studying geometrical objects, the arising groupoids often carry a topology, turning them into topological groupoids, or even some differentiable structure, turning them into Lie groupoids. These last objects can be also studied in terms of their associated Lie algebroids, in analogy to the relation between Lie groups and Lie algebras. Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. 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.