nLab
full image

Full images of functors

Idea

The full image of a functor F:CDF\colon C \to D is a version of its image? that gets its objects from the functor's source CC but its morphisms from the functor's target DD.

You may think of it as (up to equivalence) the full subcategory of DD whose objects lie in the literal image of FF.

We may call it the 11-image of the functor, because it reduces (again, up to equivalence) to the ordinary image for a functor between 00-categories.

Definition

Let CC and DD be categories, and let F:CDF\colon C \to D be a functor. Then the full image of FF is the category im¯F\overline{im} F with:

  • as objects, the objects of CC;
  • as morphisms from xx to yy, the morphisms in DD from F(x)F(x) to F(y)F(y).

If CC is a subcategory of DD, then the full image is the full subcategory of DD whose objects belong to CC.

The full image should be taken as equipped with a functor to DD, which acts as FF on objects and the identity on morphisms. This functor is fully faithful, so im¯F\overline{im} F is always equivalent to a full subcategory of DD.

From in internal point of view, if codisc(S)codisc(S) is the category with object set SS and a unique arrow between any ordered pair of objects (that is, Mor(codisc(S))=S×SMor(codisc(S)) = S\times S), the full image can be defined as a pullback:

im¯F D codisc(Obj(C)) codisc(F 0) codisc(Obj(D)) \begin{matrix} \overline{im} F& \to & D \\ \downarrow&&\, \downarrow \\ codisc(Obj(C))&\underset{codisc(F_0)}{\to} & codisc(Obj(D)) \end{matrix}

in the category Cat. Here F 0F_0 is the object component of FF and codisc(F 0)codisc(F_0) is the obvious functor. This determines im¯F\overline{im} F up to canonical isomorphism as a strict category (or other internal category).

Full images of forgetful functors

Let FF be interpreted as a forgetful functor, so that the objects of CC are thought of as objects of DD with some stuff, structure, property. Then the full image of DD consists of objects of DD with only a property: specifically the property that they are capable of taking the stuff or structure of being an object of CC.

For example, any inhabited set is capable of taking the structure of a group (at least, assuming the axiom of choice). So the full image of the forgetful functor from Grp to Set is equivalent to the category Set{}Set \setminus \{\empty\} of inhabited sets.

Revised on January 28, 2010 22:53:28 by Mike Shulman (128.135.197.48)