Direct limits and fixed point sets

For which groups G is it true that whenever one forms a direct limit of left G-sets, , the set of its fixed points, , can be obtained as the direct limit of the fixed point sets of the given G-sets? An easy argument shows that this is the case if and only if G is finitely generated. If we replace “group G” by “monoid M,” the answer is the less familiar condition that the improper left congruence on M be finitely generated; equivalently, that M be finitely generated under multiplication and “right division.” Replacing our group or monoid with a small category E, the concept of a set on which G or M acts with that of a functor E→Set, and the fixed point set of an action with the limit of a functor, a criterion of a similar nature is proved. Specialized criteria are obtained in the cases where E has only finitely many objects and where E is a (generally infinite) partially ordered set. If one allows the codomain category Set to be replaced with other categories, and/or allows direct limits to be replaced with other classes of colimits, one enters a vast area open to further investigation.