If one accepts the notion of subcategory without any qualification (as discussed there), then:
A subcategory of a category is a full subcategory if for any and in , every morphism in is also in (that is, the inclusion functor is full).
This inclusion functor is often called a full embedding or a full inclusion.
Notice that to specify a full subcategory of , it is enough to say which objects belong to . Then must consist of all morphisms whose source and target belong to (and no others). One speaks of the full subcategory on a given set of objects.
This means that equivalently we can say:
A functor exhibits as a full subcategory of precisely if it is a full and faithful functor. ( is the essential image of ).
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 September 30, 2016 at 03:58:02. See the history of this page for a list of all contributions to it.