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In mathematics, specifically in category theory, a functor

${\displaystyle F:C\to D}$

is essentially surjective if each object ${\displaystyle d}$ of ${\displaystyle D}$ is isomorphic to an object of the form ${\displaystyle Fc}$ for some object ${\displaystyle c}$ of ${\displaystyle C}$.

Any functor that is part of an equivalence of categories is essentially surjective. As a partial converse, any full and faithful functor that is essentially surjective is part of an equivalence of categories.

Notes

1. Mac Lane (1998), Theorem IV.4.1

References

• Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (second ed.). Springer. ISBN 0-387-98403-8.
• Riehl, Emily (2016). Category Theory in Context. Dover Publications, Inc Mineola, New York. ISBN 9780486809038.

External links

• Essentially surjective functor at the nLab

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