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

• 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

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