In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically 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 ...
, 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 (or dense) 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 categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...
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 full and faithful functor.
Formal definitions
Explicitly, let ''C'' ...
that is essentially surjective is part of an equivalence of categories.
[Mac Lane (1998), Theorem IV.4.1]
Notes
References
*
External links
*
{{Functors
Functors