full 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).

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.

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

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

This is evident from inspection of the defining universal property.

Last revised on August 21, 2020 at 03:20:13. See the history of this page for a list of all contributions to it.