nLab
full sub-2-category

Definition

A 2-functor $F:C\to D$ exhibits the 2-category $C$ as a full sub-2-category of $D$ if for all objects $c_1,c_2\in C$ the component functor $F_{c_1,c_2}$ is an equivalence of categories

$$F_{c_1,c_2}:C(c_1,c_2)\stackrel{\simeq }{\to }D(F(c_1),F(c_2)),$$F_{c_1, c_2} : C(c_1,c_2) \stackrel{\simeq}{\to} D(F(c_1), F(c_2)) \,,

hence if $F$ is a 2-fully-faithful 2-functor.

Properties

If $C$ and $D$ are ordinaray categories regarded as 2-categories, a full sub 2-category $F:C\hookrightarrow D$ is equivalently a full subcategory of $D$.

If $D$ is a (2,1)-category a full sub-2-category is equivalently a full sub-(∞,1)-category.

Related concepts

• locally full and faithful 2-functor