full sub-2-category in nLab

Contents

Contents

Definition

A 2-functor F:C→DF \,\colon\, C \to D exhibits the 2-category CC as a full sub-2-category of DD if for all objects c 1,c 2∈Cc_1,c_2 \in C the component functor F c 1,c 2F_{c_1, c_2} is an equivalence of categories

F c 1,c 2:C(c 1,c 2)→≃D(F(c 1),F(c 2)), F_{c_1, c_2} \;\colon\; C(c_1,c_2) \xrightarrow{\;\; \simeq \;\;} D\big( F(c_1), F(c_2) \big) \,,

hence if FF is a 2-fully-faithful 2-functor.

Properties

CC and DD can be considered as (1-)categories by forgetting their 2-morphisms, and FF can be considered as a (1-)functor via decategorification. As a result, every full sub-2-category is also a full subcategory.

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

References

• Math Overflow, “When is a full sub-2-category not a full subcategory?”, web