1) Fields have algebraic closures unique up to a non-unique isomorphism. 2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism. 3) Modules have ...
A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I. The question is then, does Fields have a weakly ...
Since there is no "free field generated by a set", it would seem that 1) there is no monad on Set whose algebras are exactly the fields and 2) there is no Lawvere theory whose models in Set are ...