Editing Free structure (section)

==== Free structure functors ====

One possible objection to the above description (even a more formal version thereof) is that the characterization of "simple" is somewhat vague. [[Category theory]] gives a somewhat better solution. Generally, structures-over-a-set will form a category, with arrows being structure-preserving homomorphisms. "Simplest" (in the sense we want) structures in that category will then either be [http://en.wikipedia.org/wiki/Initial_and_terminal_objects initial or terminal], [1] and thus, freeness can be defined in terms of such universal constructions.

In its [http://en.wikipedia.org/wiki/Free_object full categorical generality], freeness isn't necessarily categorized by underlying set structure, either. Instead, one looks at "forgetful" functors [2] from the category of structures to some other category. For our free monoids above, it'd be:

* <math>U : Mon \to Set</math>

The functor taking monoids <math>(M, e, *)</math> to their underlying set <math>M</math>. Then, the relevant universal property is given by finding an [http://en.wikipedia.org/wiki/Adjunction adjoint functor]:

* <math>F : Set \to Mon</math>, <math> F</math> ⊣ <math>U </math>

<math>F</math> being the functor taking sets to the free monoids over those sets. So, free structure functors are left adjoint to forgetful functors. It turns out this categorical presentation also has a dual: cofree structure functors are right adjoint to forgetful functors.