Exact colimits and fixed points

Abstract:

In this paper we shall give details of some work sketched in [6] on the exactness of the functor colim: ${\text {Ab}}^\mathcal {\text {C}} \to {\text {Ab}}$. We shall also investigate the connection between this work and a paper of J. Adámek and J. Reiterman [1] characterizing those categories $\mathcal {\text {C}}$ with the property that every endomorphism of an indecomposable functor $\mathcal {\text {C}} \to$ Sets has a fixed point. Exactness of colim implies the fixed point property, and in some cases (such as when $\mathcal {\text {C}}$ has only finitely many objects) both conditions turn out to be equivalent to the components of $\mathcal {\text {C}}$ being filtered. We do not expect that the two conditions are equivalent in general, although we have no example. However the category of finite ordinals and order preserving injections is an example of a connected, nonfiltered category relative to which colim is exact. This was conjectured by Mitchell, and is proved by Isbell in [5].