A note on the non-existence of functors

We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. In addition, we consider questions about nonexistence of subfunctors and quotients of the identity functor on the category of groups (or abelian groups). For example, we show that there are no natural non-trivial abelian subgroup of a group, nor a natural perfect quotient group of a group. More generally, we prove that natural subgroups, i.e. values of a sub-functor of the identity functor on groups, cannot all belong to a proper reflective subcategory. As an auxiliary result we prove that, for any non-trivial subfunctor F of the identity functor on the category of groups, any group can be embedded into a simple group that lies in the essential image of F.

The paper concludes with a few questions regarding the non-existence of certain (co-)augmented functors in the ∞-category of spaces.