---- > [!definition] Definition. ([[subcategory]]) > A **subcategory** $\mathsf{C'}$ of a [[category]] $\mathsf{C}$ consists of a collection of objects of $\mathsf{C}$ with sets of morphisms $\text{Hom}_{\mathsf{C'}}(A,B) \subset \text{Hom}_{\mathsf{C}}(A,B)$ for all objects $A,B$ in $\text{Obj}(\mathsf{C}')$, such that inheriting identities and the composition [[binary operation|operation]] from $\mathsf{C}$ makes $\mathsf{C'}$ into a [[category]]. > \ > A subcategory is called **full** if in fact $\text{Hom}_{\mathsf{C'}}(A,B)=\text{Hom}_{\mathsf{C}}(A,B)$ for all $A,B$ in $\text{Obj}(\mathsf{C'})$; in other words, if the [[inclusion functor]] is a [[full functor]]. ^definition > [!basicexample] We can construct a category of *infinite sets* by defining the objects to be objects in $\mathsf{Set}$ whose [[cardinality]] is not finite, and the morphism set * for a given pair of objects to be all functions between them. Such a category is a full subcategory of $\mathsf{Set}$. If we instead constrained the objects to be vector spaces over $\mathbb{R}$ and the morphisms to be linear maps between them, this would still be a subcategory, but it would no longer be full. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch02