In category theory, the notion of ''final functor'' is a generalization of the notion of cofinal set from order theory.
A functor <math>F: C \to D</math> is called ''final'' if, for any set-valued functor <math>G: D \to \textbf{Set}</math>, the colimit of ''G'' is the same as the colimit of <math>G \circ F</math>. The inclusion function of a cofinal subset into a partially ordered set is a final functor.
The notion of ''initial functor'' is defined as above, replacing ''final'' by ''initial'' and ''colimit'' by ''limit''.
==Terminology== The prefix "co-" in the term "cofinal" is used in the sense of "together with", similar to its use in the word "covariant". However, the prefix "co" in category theory has a conflicting meaning, indicating the dualized form of a concept obtained by passing to the opposite category, and so the word "cofinal" could easily be misinterpreted to mean "dual to final". The term "final" as it is used in this article historically arises from modifying the term "cofinal" to avoid confusion over the meaning of the prefix. ==References== {{refbegin}} *{{citation|at=Definition 2.12, p. 24|title=Algebraic Theories: A Categorical Introduction to General Algebra|volume=184|series=Cambridge Tracts in Mathematics|first1=J.|last1=Adámek|first2=J.|last2=Rosický|authorlink2=Jiří Rosický (mathematician)|first3=E. M.|last3=Vitale|publisher=Cambridge University Press|year=2010|isbn=9781139491884|url=https://books.google.com/books?id=siNlAn8Bm30C&pg=PA24}}. *{{citation|page=37|title=Shape Theory: Categorical Methods of Approximation|series=Dover Books on Mathematics|first1=J. M.|last1=Cordier|first2=T.|last2=Porter|publisher=Courier Corporation|year=2013|isbn=9780486783475|url=https://books.google.com/books?id=6w3CAgAAQBAJ&pg=PA37}}. *{{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}}. {{refend}}
==External links== * http://ncatlab.org/nlab/show/final+functor
Category:Functors
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