I would like to pose a question, first in the concrete situation where it arose, and then more abstractly. Let n-MAN be the category of (topological) n-manifolds and continuous maps. My question is, what are the endofunctors of n-MAN ? To start with, there is no shortage. Any functor F: n-MAN --> Set gives rise to an endofunctor F': n-MAN --> n-MAN as follows. On objects M \in n-MAN , F'(M)=M x F(M) (where F(M) is given the discrete topology) and for morphisms f \in n-MAN , F'(f)(p,x)=(p,f(x)) . Let us call these endofunctors of set type. The endofunctors of set type form a proper class, since for any cardinal k there is a set valued functor taking M to the underlying set of M x k. So one may ask if there are any others? If there are, then one can ask a more refined question. The endofunctors of n-MAN form a (proper class based) monoid. It still makes sense to consider the monoid ideal generated by the class of endofunctors of set type. One might then try to form the quotient monoid and as a measure of how close the class of set type endofunctors comes to exhausting the class of all endofunctors. Is this construction familiar? If one considers instead the category of pointed n-manifolds, then one has the universal covering space endofunctor, which is not of set type; there the quotient monoid will be non-trivial. A few more thoughts: The construction above makes sense in any category with arbitrary coproducts, or even just copowers, if that is the right term. If instead of the category of n-manifolds, I had considered the category of connected n-manifolds (and continuous maps), then the set type endofunctors are no longer available - so is the monoid of endomorphism of this category related the quotient monoid alluded to above (same question for pointed connected n-manifolds.) In the case of the category of n-manifolds (but not pointed n-manifolds) there is a large supply of constant maps. Given an n-manifold M, the lattice of subobjects of M. Fixing another manifold N, consider all the constant maps from N to M or from N to a subobject of N. If an endofunctor F preserves constant maps (and nontrivial endofunctors of set type don't) then considering all this structure, I don't expect that it will be difficult to show that F is trivial. I would appreciate any reactions or references to similar considerations in the literature. Thanks, David Feldman University of New Hampshire =========================================================================