Direct limits and fixed point sets

For which groups G is it true that whenever one forms a direct limit of left G-sets, lim −→ i∈I Xi, the set of its fixed points, (lim −→I Xi) , can be obtained as the direct limit lim −→I (X i ) of the fixed point sets of the given G-sets? An easy argument shows that this holds 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 concept of fixed point set with that of the limit of a functor, a criterion of a similar nature is obtained. Simplified statements are noted for the cases where E has only finitely many objects, and where E is a 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 gets a vast area for further investigation.