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Categorical Trace
In category theory, a branch of mathematics, the categorical trace is a generalization of the trace (linear algebra), trace of a matrix (mathematics), matrix. Definition The trace is defined in the context of a symmetric monoidal category ''C'', i.e., a category (mathematics), category equipped with a suitable notion of a product \otimes. (The notation reflects that the product is, in many cases, a kind of a tensor product.) An object (category theory), object ''X'' in such a category ''C'' is called dualizable object, dualizable if there is another object X^\vee playing the role of a dual object of ''X''. In this situation, the trace of a morphism f: X \to X is defined as the composition of the following morphisms: \mathrm(f) : 1 \ \stackrel\ X \otimes X^\vee \ \stackrel\ X \otimes X^\vee \ \stackrel\ X^\vee \otimes X \ \stackrel\ 1 where 1 is the monoidal unit and the extremal morphisms are the coevaluation and evaluation, which are part of the definition of dualizable objects. Th ... [...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] |
