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Center (category Theory)
In category theory, a branch of mathematics, the center (or Drinfeld center, after Soviet-American mathematician Vladimir Drinfeld) is a variant of the notion of the center of a monoid, group, or ring to a category. Definition The center of a monoidal category \mathcal = (\mathcal,\otimes,I), denoted \mathcal, is the category whose objects are pairs ''(A,u)'' consisting of an object ''A'' of \mathcal and an isomorphism u_X:A \otimes X \rightarrow X \otimes A which is natural isomorphism, natural in X satisfying : u_ = (1 \otimes u_Y)(u_X \otimes 1) and : u_I = 1_A (this is actually a consequence of the first axiom). An arrow from ''(A,u)'' to ''(B,v)'' in \mathcal consists of an arrow f:A \rightarrow B in \mathcal such that :v_X (f \otimes 1_X) = (1_X \otimes f) u_X. This definition of the center appears in . Equivalently, the center may be defined as :\mathcal Z(\mathcal C) = \mathrm_(\mathcal C), i.e., the endofunctors of ''C'' which are compatible with the left and righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
