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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] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
