In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, the notion of ''final functor'' (resp. ''initial functor'') is a generalization of the notion of
final object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
(resp. initial object) in a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
.
A
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
is called ''final'' if, for any set-valued functor
, the
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
of ''G'' is the same as the colimit of
. Note that an
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
''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
*.
*.
*{{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.
See also
* http://ncatlab.org/nlab/show/final+functor
Functors