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essentially surjective (Definition)

Let $ \mathcal{C}$ and $ \mathcal{D}$ be categories. A functor $ F\colon \mathcal{C}\to \mathcal{D}$ is essentially surjective if for any object $ A\in\mathcal{OB}(\mathcal{D})$, there exists an object $ X\in\mathcal{OB}(\mathcal{C})$, such that $ F(X)\cong A$. That is, there are morphisms (in $ D$) $ f \colon F(X)\to A$ and $ g\colon A\to F(X)$ such that $ fg=1_A$ and $ gf=1_{F(X)}$.

Remarks.

  • Clearly, if $ F$ is surjective, it is essentially surjective. But the reverse is not true.
  • A functor is an equivalence iff it is full, faithful and essentially surjective.
  • isomorphism-dense subcategory. A full subcategory $ \mathcal{S}$ of a category $ \mathcal{C}$ is said to be isomorphism-dense in $ \mathcal{C}$, if the inclusion functor $ \mathcal{S}\hookrightarrow \mathcal{C}$ is essentially surjective. Since $ \mathcal{S}$ is full, the inclusion functor is full and faithful. As a result, $ \mathcal{S}$ is isomorphism-dense if the inclusion functor is an equivalence.



"essentially surjective" is owned by CWoo.
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See Also: equivalence of categories

Other names:  dense functor, isomorphism-dense in, isomorphism-dense
Also defines:  isomorphism-dense subcategory

Attachments:
a functor is an equivalence iff it is fully faithful and essentially surjective (Derivation) by CWoo
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Cross-references: inclusion functor, full subcategory, iff, surjective, morphisms, object, functor, categories
There are 3 references to this entry.

This is version 10 of essentially surjective, born on 2005-05-22, modified 2007-11-22.
Object id is 7104, canonical name is EssentiallySurjective.
Accessed 2566 times total.

Classification:
AMS MSC: 18A22 (Category theory; homological algebra :: General theory of categories and functors :: Special properties of functors )
Pending Errata and Addenda
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