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Enough Projectives
In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in Abelian category, abelian Category (mathematics), categories are used in homological algebra. The Duality (mathematics), dual notion of a projective object is that of an injective object. Definition An Object (category theory), object P in a category \mathcal is ''projective'' if for any epimorphism e:E\twoheadrightarrow X and morphism f:P\to X, there is a morphism \overline:P\to E such that e\circ \overline=f, i.e. the following diagram Commutative diagram, commutes: That is, every morphism P\to X mathematical jargon#factor through, factors through every epimorphism E\twoheadrightarrow X. If ''C'' is Category_(mathematics)#Small_and_large_categories, locally small, i.e., in particular \operatorname_C(P, X) is a Set (mathematics), set for any object ''X'' in ''C'', this definition is equivalent to the condition that the hom functor (also known as representab ... [...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 com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
