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 [math]\displaystyle{ F: C \to D }[/math] is called final if, for any set-valued functor [math]\displaystyle{ G: D \to \textbf{Set} }[/math], the colimit of G is the same as the colimit of [math]\displaystyle{ G \circ F }[/math]. Note that an object d ∈ Ob(D) is a final object in the usual sense if and only if the functor [math]\displaystyle{ \{*\} \xrightarrow{d} D }[/math] 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, https://books.google.com/books?id=siNlAn8Bm30C&pg=PA24 .
• Cordier, J. M.; Porter, T. (2013), Shape Theory: Categorical Methods of Approximation, Dover Books on Mathematics, Courier Corporation, p. 37, ISBN 9780486783475, https://books.google.com/books?id=6w3CAgAAQBAJ&pg=PA37 .
• Riehl, Emily (2014), Categorical Homotopy Theory, New Mathematical Monographs, 24, Cambridge University Press, Definition 8.3.2, p. 127, https://books.google.com/books?id=6xpvAwAAQBAJ&pg=PA127 .

See also

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