nLab
full sub-2-category
Contents
Context
2-Category theory
2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

Notions of subcategory
Contents
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 object s $c_1,c_2 \in C$ 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
Last revised on December 10, 2023 at 18:38:25.
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