In
mathematics, specifically in
category theory, a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
:
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 ...
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'' a ...
that is essentially surjective is part of an equivalence of categories.
[Mac Lane (1998), Theorem IV.4.1]
Notes
References
*
External links
*
{{Functors
Functors