The functor that wouldn’t be. (English) Zbl 0988.18002

Koslowski, Jürgen (ed.) et al., Categorical perspectives. Papers from the international conference held in honor of George E. Strecker on the occasion of his 60th birthday at Kent State University, Kent, OH, USA, August 1998. Boston, MA: Birkhäuser. Trends in Mathematics. 29-35 (2001).
In entertaining Platonic dialogue style the author rekindles questions of “Prague School” type [see V. Koubek, Comment. Math. Univ. Carolinae 12, 175-195 (1971; Zbl 0217.06803)] and proves that there is no endofunctor of the category of sets which maps the empty set to a non-empty set but keeps non-empty sets (of up to three elements) fixed. {Incidently, the bound three is sharp.} In particular, taking injective hulls in the category of sets is not functorial, in fact, hardly in any category, unless every object is already injective, as shown by J. Adámek, H. Herrlich, J. Rosický and W. Tholen [“Injective hulls are not natural”, Algebra Univ., to appear].
For the entire collection see [Zbl 0966.00025].


18B05 Categories of sets, characterizations
18A22 Special properties of functors (faithful, full, etc.)
18A99 General theory of categories and functors


injective hulls


Zbl 0217.06803