In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically in
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, 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 essentially surjective if each object
of
is isomorphic to an object of the form
for some object
of
.
Any functor that is part of an
equivalence of categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of cate ...
is essentially surjective. As a partial converse, any
full and faithful functor
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a fully faithful functor.
Formal definitions
Explicitly, let ''C'' and ' ...
that is essentially surjective is part of an equivalence of categories.
[Mac Lane (1998), Theorem IV.4.1]
Notes
References
*
*
External links
*
Functors
{{Categorytheory-stub