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Action Groupoid
In mathematics, an action groupoid or a transformation groupoid is a groupoid that expresses a group action. Namely, given a (right) group action :X \times G \to X, we get the groupoid \mathcal (= a category whose morphisms are all invertible) where *objects are elements of X, *morphisms from x to y are the actions of elements g in G such that y = xg, *compositions for x \overset\to y and y \overset\to z is x \overset\to z. A groupoid is often depicted using two arrows. Here the above can be written as: :X \times G \,\overset\underset\rightrightarrows\, X where s, t denote the source and the target of a morphism in \mathcal; thus, s(x, g) = x is the projection and t(x, g) = xg is the given group action (here the set of morphisms in \mathcal is identified with X \times G). In an ∞-category Let C be an ∞-category and G a groupoid object in it. Then a group action or an action groupoid on an object ''X'' in ''C'' is the simplicial diagram :\cdots \, \underset\rightrightarrows \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoid
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 equippe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |