nLab full sub-2-category

Definition

A 2-functor $F \,\colon\, C \to D$ 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

$$F_{c_1, c_2} \;\colon\; C(c_1,c_2) \xrightarrow{\;\; \simeq \;\;} D\big( F(c_1), F(c_2) \big) \,,$$

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

Properties

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

If $D$ 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

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor