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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] |
Group Action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoid Object
In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined. Definition A groupoid object in a category C admitting finite fiber products consists of a pair of objects R, U together with five morphisms :s, t: R \to U, \ e: U \to R, \ m: R \times_ R \to R, \ i: R \to R satisfying the following groupoid axioms # s \circ e = t \circ e = 1_U, \, s \circ m = s \circ p_1, t \circ m = t \circ p_2 where the p_i: R \times_ R \to R are the two projections, # (associativity) m \circ (1_R \times m) = m \circ (m \times 1_R), # (unit) m \circ (e \circ s, 1_R) = m \circ (1_R, e \circ t) = 1_R, # (inverse) i \circ i = 1_R, s \circ i = t, \, t \circ i = s, m \circ (1_R, i) = e \circ s, \, m \circ (i, 1_R) = e \circ t. Examples Group objects A group object is a special case of a groupoid object, where R = U and s = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Diagram
In mathematics, especially algebraic topology, a simplicial diagram is a diagram indexed by the simplex category (= the category consisting of all = \ and the order-preserving functions). Formally, a simplicial diagram in a category or an ∞-category ''C'' is a contraviant functor from the simplex category to ''C''. Thus, it is the same thing as a simplicial object but is typically thought of as a sequence of objects in ''C'' that is depicted using multiple arrows :\cdots \, \underset\rightrightarrows \, U_2 \, \underset\rightrightarrows \, U_1 \, \rightrightarrows\, U_0 where U_n is the image of /math> from \Delta in ''C''. A typical example is the Čech nerve of a map U \to X; i.e., U_0 = U, U_1 = U \times_X U, \dots. If ''F'' is a presheaf with values in an ∞-category and U_ a Čech nerve, then F(U_) is a cosimplicial diagram and saying F is a sheaf exactly means that F(X) is the limit of F(U_) for each U \to X in a Grothendieck topology. See also: simplicial presheaf. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
