Endofunctors of Set
by
Artur Barkhudaryan
Mathematical Institute of the Charles University, Prague

Endofunctors of the category Set of all sets and all mappings between them were studied in a number of papers in early seventies. This field of problems was reopened by Y. T. Rhineghost last year in [1], shortly followed by [2] and [3].

The following definitions are inspired by [1] and [2].

For an endofunctor F of a category K we define W_F to be the class of fix-points of F, i. e.

WF = {x in obj K; F(x) is isomorphic to x}.

We say that a class W of objects of the category K is functorially definable (or uniquely functorially definable ) if there exists an endofunctor F:K --> K (or an endofunctor F:K --> K unique up to a natural equivalence) such that W=W_F.

Under a set-theoretical assumption consistent with ZFC, the (uniquely) functorially definable classes of objects of the category Set will be characterized (this solves Problems 8 and 9 in [2]).

Inspired by [3], we say that a functor F:K --> H between arbitrary categories K and H is Determined by its Values on Objects (abbreviated: DVO-functor ) if F is naturally equivalent to any G:K --> H whenever G(x) is isomorphic to F(x) for every object x of the category K. Under a set-thoretic assumption consistent with ZFC, an infinitary functor Set --> Set is never a DVO-functor. Many examples of finitary DVO-functors Set --> Set (e. g. a DVO-functor with a subfunctor which is not DVO) and results about them will be presented (including solutions of Problems 14, 16, 17 of [3]). Some relevant open problems and conjectures occur in this context.

These problems have been discussed at the Seminar of General Mathematical Structures lead by V. Trnková . The mentioned results are mainly due to R. El Bashir, V. Trnková and the author of this abstract.

References

1.  Y. T. Rhineghost, The functor that wouldn't be , preprint 1999.

2.  Y. T. Rhineghost, The emergence of functors , preprint 2000.

3.  Alois Zmrzlina, Too many functors , preprint 2000.

Papers by J. Adámek, V. Koubek, J. Reiterman, V. Trnková and others.

Paper reference: doi:10.1023/A:1026177620363

Date received: May 28, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-16.