Final functor

In category theory, the notion of final functor (resp. initial functor) is a generalization of the notion of final object (resp. initial object) in a category.

A functor F : C → D {\displaystyle F:C\to D} is called final if, for any set-valued functor  G : D → Set {\displaystyle G:D\to {\textbf {Set}}} , the colimit of G is the same as the colimit of  G ∘ F {\displaystyle G\circ F} . Note that an object d ∈ Ob(D) is a final object in the usual sense if and only if the functor  { ∗ } → d D {\displaystyle \\{*\\}\xrightarrow {d} 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

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* Adámek, J.; Rosický, J.; Vitale, E. M. (2010), Algebraic Theories: A Categorical Introduction to General Algebra, Cambridge Tracts in Mathematics, vol. 184, Cambridge University Press, Definition 2.12, p. 24, ISBN 9781139491884.
* Cordier, J. M.; Porter, T. (2013), Shape Theory: Categorical Methods of Approximation, Dover Books on Mathematics, Courier Corporation, p. 37, ISBN 9780486783475.
* Riehl, Emily (2014), Categorical Homotopy Theory, New Mathematical Monographs, vol. 24, Cambridge University Press, Definition 8.3.2, p. 127.

## External links

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* http://ncatlab.org/nlab/show/final+functor

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