category theory

# Contents

## 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 $S↪C$ 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 $F$).

Revised on September 27, 2012 00:22:00 by Aaron F (66.90.167.53)