Category theory; homological algebra Definition

Full subcategory

Builds on Category Subcategory Abelian group Group

Definition

Let $\mathcal{C}$ be a category. A full subcategory of $\mathcal{C}$ is a subcategory $\mathcal{D}$ such that $\Hom_{\mathcal{D}}(A, B) = \Hom_{\mathcal{C}}(A, B)$ for all objects $A$ , $B$ of $\mathcal{D}$ .

Remarks

For every subclass $S$ of $\mathrm{Ob}(\mathcal{C})$ there is exactly one full subcategory with object class $S$ , the full subcategory spanned by $S$ . Example: the abelian groups form a full subcategory of the category of groups. A morphism between two abelian groups in the category of groups is a group homomorphism, and every such homomorphism is a morphism of the subcategory. Non-example: the subcategory of sets with all sets as objects and only the injective functions as morphisms is not full. The constant map $\set{0, 1} \to \set{0}$ is a morphism of the category of sets between two of its objects, but it is not injective.

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Abelian group concept
Category concept
Group concept
Subcategory concept
Full subcategory definition

Tag	`Y1RAX7VVBX`
URI	`m2://Y1RAX7VVBX`
Kind	definition
Writing	W0 stub
MSC	18A05
Defines	Full subcategory
Uses	Category, Subcategory, Abelian group, Group

Full subcategory

Definition

Remarks

Reading path

Further reading

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