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Final Functor
In category theory, the notion of ''final functor'' (resp. ''initial functor'') is a generalization of the notion of Initial and terminal objects, final object (resp. initial object) in a Category (mathematics), category. A functor F: C \to D is called ''final'' if, for any set-valued functor G: D \to \textbf, the colimit of ''G'' is the same as the colimit of G \circ F. Note that an Object (category theory), object ''d'' ∈ Ob(''D'') is a final object in the usual sense if and only if the functor \ \xrightarrow D is a final functor as defined here. The notion of ''initial functor'' is defined as above, replacing ''final'' by ''initial'' and ''colimit'' by ''limit''. References *. *. *{{citation, at=Definition 8.3.2, p. 127, title=Categorical Homotopy Theory, volume=24, series=New Mathematical Monographs, first=Emily, last=Riehl, publisher=Cambridge University Press, year=2014, url=https://books.google.com/books?id=6xpvAwAAQBAJ&pg=PA127. See also * http://ncatlab.org/n ... [...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] |
