full subcategory



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

A subcategory SS of a category CC is a full subcategory if for any xx and yy in SS, every morphism f:xyf : x \to y in CC is also in SS (that is, the inclusion functor SCS \hookrightarrow C is full).

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

Notice that to specify a full subcategory SS of CC, it is enough to say which objects belong to SS. Then SS must consist of all morphisms whose source and target belong to SS (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:SCF : S \to C exhibits SS as a full subcategory of CC precisely if it is a full and faithful functor. (SS is the essential image of FF).

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