Concerning question 1 of France Dacar:
Q1: In "Algebraic Theories", there are several propositions (say 4.22, 4.23, 4.24) which include the following assumption: "Let (K,E,M) be a regular category _such that every equalizer is in in M_." This mystifies me, because an equalizer is _always_ an M-monic, for any image-factorization system (E,M). In any category, if an equalizer f factors as f = ge (composition to the left...) with e epi, then e is an isomorphism. This is quite trivial; since I cannot imagine that this could have been overlooked for as long as image-factorization systems were around up to the Manes' book, I am starting to doubt my ability to spot a mistake in my reasoning...
I don't have my copy of Manes' AT handy, but it seems to me that the assumption that E consists of epis is not warranted. After all, E = all morphisms and M = all isos constitute an image-factorization system, and not all equalizers are isos. Regards, J"urgen -- J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) +++++++++++++++++++++