Category theory; homological algebra Definition

Essentially surjective functor

Builds on Functor Morphism Full subcategory Vector space

Definition

Let $F \colon \mathcal{C} \to \mathcal{D}$ be a functor. The functor $F$ is essentially surjective if, for every object $D$ of $\mathcal{D}$ , there is an object $A$ of $\mathcal{C}$ with $F(A) \cong D$ . Here $\cong$ means that there is an isomorphism in $\mathcal{D}$ .

Remarks

Example: let $K$ be a field and let $\mathcal{C}$ be the full subcategory of finite-dimensional vector spaces over $K$ spanned by the spaces $K^n$ , $n \ge 0$ . Its inclusion into all finite-dimensional vector spaces over $K$ is essentially surjective, since a space of dimension $n$ is isomorphic to $K^n$ . Non-example: the inclusion of the full subcategory of finite sets into the category of sets is not essentially surjective. An infinite set is not isomorphic, that is, not in bijection with, any finite set. Every functor that is surjective on objects is essentially surjective.

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Full subcategory concept
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Essentially surjective functor definition

Tag	`AYQ3VQ4B0V`
URI	`m2://AYQ3VQ4B0V`
Kind	definition
Writing	W0 stub
MSC	18A22
Defines	Essentially surjective functor
Uses	Functor, Morphism, Full subcategory, Vector space

Essentially surjective functor

Definition

Remarks

Reading path

Further reading

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