A functor $F\colon C \to D$ is essentially surjective, or essentially surjective on objects (sometimes abbreviated to eso), if it is surjective on objects “up to isomorphism”.
$F\colon C \to D$ is essentially surjective if for every object $y$ of $D$, there exists an object $x$ of $C$ and an isomorphism $F(x) \cong y$ in $D$.
A functor $F : C \to D$ is essentially surjective if for all $y:D$ there merely exists an $x:C$ such that $F(x) \cong y$.
A functor between discrete categories (or, more generally, skeletal categories) is essentially surjective iff it is a surjective function between the classes of objects.
Any bijective-on-objects functor or surjective-on-objects functor is essentially surjective.
A composition of any two essentially surjective functors is essentially surjective.
If $g f$ is essentially surjective, then $g$ is essentially surjective.
An essentially surjective functor is additionally fully faithful precisely when it is an equivalence of categories.
The inclusion functor of a subcategory is essentially surjective precisely when the subcategory is essentially wide.
Every split essentially surjective functor is essentially surjective. The converse is true for strict functors in the presence of the axiom of choice.
Strengthening the last example, there is an orthogonal factorization system (in the up-to-isomorphism strict sense) on $Cat$, in which eso functors are the left class and fully faithful functors are the right class.
This is an “up-to-isomorphism” version of the bo-ff factorization system, which is a 1-categorical orthogonal factorization system on $Cat$ in which the left class consists of bijective-on-objects functors. Thus the notion of essential surjectivity is a version of “bijective on objects” which does respect the principle of equivalence, i.e. the version which views Cat as a bicategory.
In particular, while a functor factors uniquely-up-to-isomorphism as a b.o. functor followed by a fully faithful one, it factors only uniquely-up-to-equivalence as an e.s.o. functor followed by a fully faithful one. Since b.o. functors are also e.s.o., any (eso,ff) factorization of some functor is equivalent to its (bo,ff) factorization.
In any 2-category there is a notion of eso morphism which generalizes the essentially surjective functors in Cat. In a regular 2-category, these form a factorization system in a 2-category together with the ff morphisms.
basic properties of…
Last revised on May 1, 2023 at 11:04:04. See the history of this page for a list of all contributions to it.