[ the first of Peter's messages did not arrive the first time, and the moderator missed that reference in the second. Sorry about that. Bob Rosebrugh ] I sent two messages to the categories net only the second of which seems to have been distributed. The second begins by referring to the first. The following is a slightly edited version of the first message: ******************************************************************** I wonder if Thomas Streicher's question is really as he states it. Take any Heyting Algebra and "ideal", that is, a subset closed under finite meet and hereditary upwards. Then the ideal is a full "sub-left-exact" subcategory and it is an exponential ideal. The bottom element of the Heyting Algebra has a reflection in the ideal iff the ideal is principal. If one wants the subcategory to be closed under the formation of arbitrary limits (not just finite limits) then there are counterexamples but none that I know that can be so easily described. Best thoughts, peter freyd ======================================