nLab
full subcategory

Definition

If one accepts the notion of subcategory without any qualification (as discussed there), then:

A subcategory $S$ of a category $C$ is a full subcategory if for any $x$ and $y$ in $S$, every morphism $f : x \to y$ in $C$ is also in $S$ (that is, the inclusion functor $S \hookrightarrow C$ is full).

This inclusion functor is often called a full embedding or a full inclusion.

Notice that to specify a full subcategory $S$ of $C$, it is enough to say which objects belong to $S$. Then $S$ must consist of all morphisms whose source and target belong to $S$ (and no others). One speaks of the full subcategory on a given set of objects.

This means that equivalently we can say:

A functor $F : S \to C$ exhibits $S$ as a full subcategory of $C$ precisely if it is a full and faithful functor. ($S$ is the essential image of $F$).

Properties

Example

A fully faithful functor (hence a full subcategory inclusion) reflects all limits and colimits.

This is evident from inspection of the defining universal property.

Related concepts

• reflective subcategory

• coreflective subcategory

• fully faithful functor

• sub-(infinity,1)-category