Final functor

In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, the notion of final functor (resp. initial functor) is a generalization of the notion of <a href="/facts/Initial_and_terminal_objects/oeckvXRj">final object</a> (resp. initial object) in a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a>.
A <a href="/facts/Functor/l2u3rIBE">functor</a> 
 
 
 
 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 <a href="/facts/Colimit/JoLugLs3">colimit</a> of G is the same as the colimit of 
 
 
 
 G
 ∘
 F
 
 
 {\displaystyle G\circ F}
 
. Note that an <a href="/facts/Object_(category_theory)/7WzxUThf">object</a> 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.

<ul><li>Adámek, J.; <a href="/facts/Ji%C5%99%C3%AD_Rosick%C3%BD_(mathematician)/Pwyn8T9t">Rosický, J.</a>; Vitale, E. M. (2010), <a href="https://books.google.com/books?id=siNlAn8Bm30C&pg=PA24">Algebraic Theories: A Categorical Introduction to General Algebra</a>, Cambridge Tracts in Mathematics, vol. 184, Cambridge University Press, Definition 2.12, p. 24, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781139491884.</li>
<li>Cordier, J. M.; Porter, T. (2013), <a href="https://books.google.com/books?id=6w3CAgAAQBAJ&pg=PA37">Shape Theory: Categorical Methods of Approximation</a>, Dover Books on Mathematics, Courier Corporation, p. 37, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780486783475.</li>
<li>Riehl, Emily (2014), <a href="https://books.google.com/books?id=6xpvAwAAQBAJ&pg=PA127">Categorical Homotopy Theory</a>, New Mathematical Monographs, vol. 24, Cambridge University Press, Definition 8.3.2, p. 127.</li></ul>

Final functor open-in-new

Final functor