nLab full subcategory

Idea

A subcategory

$$\mathcal{C} \overset{\iota}{\hookrightarrow} \mathcal{D}$$

of a given category $\mathcal{D}$ is meant to be full, if it includes “some objects but all the morphisms between these objects”.

This means at least that $\iota$ is a fully faithful functor. In fact, that is the most one may demand while respecting the principle of equivalence of category theory and hence constitutes an invariant definition of full subcategory (Def. below).

However, a fully faithful functor need not be an injective function on objects (it is so only up to equivalence of categories). If one insists on defining a subcategory inclusion to involve an injective function on the sets of objects and morphisms, this condition must be added to the condition of the inclusion functor being fully faithful. This leads to a non-invariant definition, discussed below.

See also the discussion at subcategory (here).

Definition

Non-invariant 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 subcategory-inclusion 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 $F$).

Up to equivalence of categories, every fully faithful functor is equivalent to a subcategory-inclusion in these sense of being an injection on the set of objects.

Invariant definition

Therefore, the definition of full subcategory which respects the principle of equivalence is simply this:

Definition

(invariant definition)

A full subcategory-inclusion is a fully faithful functor.

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

• internal full subcategory