# 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 is called final if, for any set-valued functor , the colimit of *G* is the same as the colimit of . Note that an object *d∈Ob(D)* is a final object in the usual sense if and only if the functor 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

- Adámek, J.; Rosický, J.; Vitale, E. M. (2010),
*Algebraic Theories: A Categorical Introduction to General Algebra*, Cambridge Tracts in Mathematics,**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,**24**, Cambridge University Press, Definition 8.3.2, p. 127.

## See also

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