2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130507For which groups G is it true that whenever we form a direct limit of G-sets, dirlim_{i\in I} X_i, the set of its fixed points, (dirlim_I X_i)^G, can be obtained as the direct limit dirlim_I(X_i^G) 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. Replacing our group or monoid with a small category E, the concept of 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. The case where E is a partially ordered set leads to a condition on partially ordered sets which I have not seen before (pp.23-24, Def. 12 and Lemma 13). If one allows the {\em codomain} category Set to be replaced with other categories, and/or allows direct limits to be replaced with other kinds of colimits, one gets a vast area for further investigation.28 pages. Notes on 1 Aug.'05 revision: Introduction added; Cor.s 9 and 10 strengthened and Cor.10 added; section 9 removed and section 8 rewritten; source file re-formatted for Elsevier macros. To appear, J.AlgCategory TheoryGroup Theory18A30 (primary) 06A06, 20M30 (secondary)text