nLab
full subcategory

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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→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→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)
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