Full functor

This article defines a functor property: a property that can be evaluated to true/false given a functor between two categories.
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Definition

Symbol-free definition

A full functor is a functor such that the induced mapping on the collection of morphisms between any two objects is surjective.

Definition with symbols

Suppose ${\mathcal {F}}:{\mathcal {C}}\to {\mathcal {D}}$ is a functor between two categories. ${\mathcal {F}}$ is termed full if for any $A,B\in \operatorname {Ob} {\mathcal {C}}$, the induced map ${\mathcal {F}}:{\mathcal {C}}(A,B)\to {\mathcal {D}}(FA,FB)$ is surjective.

Relation with other properties

Related properties

• Essentially surjective functor
• Faithful functor