64 Final Values of Functors

The conjecture (and associated speculation) that I would like to present below is motivated by ideas I learnt from Goodwillie and Weiss. I present it here because I think it is also very much in the spirit of Mislin’s approach to mathematics. All of my suggestions can be discussed for categories of a certain sort and covariant functors of appropriate kinds, and dualized to contravariant functors (changing the title of this note to “initial values” of functors) and can lead to interesting objects to compute and interpret. But it will still be too general, even when we restrict attention to groups and injections, a category I’ll call G , and various of its full subcategories (which we’ll then denote by adding on an adjective). To get a feeling for this project, start with G(finite) , the category of finite groups and injections. Consider first, the functor of group homology, H . Now we can form the limit of H ( ) over the category G . Of course this category has no final finite group (under injections !). If there were, the limit considered would be H (final object) . Nevertheless, we can compute this limit easily :