Subcategory

A subcategory $𝖣$ of a category $𝖢$ is a category containing subclasses of objects and morphisms in $𝖢$ , i.e. for any $𝑋, 𝑌 \in 𝖣$ also $𝑋, 𝑌 \in 𝖢$ and $𝖣 (𝑋, 𝑌) \subseteq 𝖢 (𝑋, 𝑌)$ . #m/def/cat It comes with the natural inclusion $1 : 𝖣 \to 𝖢$ which is clearly a Faithful functor.

A full subcategory $𝖣$ is a subcategory of $𝖢$ for which the inclusion functor is fully faithful, i.e. $𝖣 (𝑋, 𝑌) = 𝖢 (𝑋, 𝑌)$ for all $𝑋, 𝑌 \in 𝖣 (𝑋, 𝑌)$ .
A wide subcategory $𝖣$ is a subcategory of $𝖢$ containing all objects of $𝖢$ .
An essentially wide subcategory $𝖣$ is a subcategory containing a skeleton category of $𝖢$ .

Examples

$𝖠 𝖻$ is a full subcategory of $𝖦 𝗋 𝗉$

#state/tidy | #lang/en | #SemBr