When I went to read finally what Peter wrote earlier this week, I realized that there was a problem with it. Not that the conclusion, or even the argument, was wrong (although there are two sentences in it I cannot parse), but the tossed off statement, ``The relevant property ?of Set? is that all non-zero objects be injective.'' But the conclusion is false for the opposite of Set and there _every_ object is injective. So what is really involved? The argument breaks into two parts. First that every pre-invariant T-alg (that is one whose structure TA --> A is 1-1) includes a minimal subobject and that is pre-invariant. As a matter of fact, that is obvious; just take the intersection of all the subobjects. It is trivial in fact to show that that is invariant. But Peter adopts a curiously complicated proof. For clarity (it doesn't actually matter), assume that T preserves _all_ monics. Peter then says that you can begin with 0 and take the image of T0 --> TA --> A and so build up a family of subobjects whose union is this minimal subobject. Well, I thought, if it turns him on... The second part of the proof is to show that this minimal invariant object is initial. I am going to reword Peter's argument in a way that I find a bit more congenial, but it is his argument. Let TA --> A be this minimal invariant algebra and TB --> B be any algebra. Let TC --> C be the smallest subalgebra of A x B. Now C can of course be constructed as the intersection of all the subalgebras of A x B or it can be built up from below. But the first argument works in any sufficiently complete category and we know the result is false in most categories, including ones in which every functor preserves monics. (It is also, of course, false for functors in arbitrary categories that preserves monics.) So what is the crucial difference? Well, if you begin with 0 >--> A x B, the composite 0 >--> A x B --> A is also monic. Then so is the composite T0 >--> T(A x B) --> TA = A (= means the structure map is an isomorphism). Thus the image of T0 >--> T(A x B) --> A x B also has the property that its inclusion followed by the projection on A is monic. And so on. When you take the colimit you get a subalgebra of A x B that maps monically (and, it is easy to show, epically) to A. Thus this is the graph of a map A to B. The rest of the argument is routine. The moral is that the basic thing that is working here is that the colimit of monics is monic. In other words, the accessibility of sets is the crucial property. Of course, other properties are also needed. This very much illustrates the principal of versality Peter made in his recent preprint on algebraically complete categories about the importance of things with both universal and couniversal mapping properties. Later: I just realized that there appears to be a minor flaw in Peter's argument. He writes: ``Now let K be the partial map from A to B whose graph is the image of (T0 -> (TA)x(TB) -> AxB).'' The problem is that the composite K >--> A x B --> A isn't necessarily monic, so K isn't necessarily the graph of a partial function A to B. One way around this is to use the argument I gave in my paper on Terminal objects in well-founded set theory that replaces T by a functor T^# that preserves _all_ monics and has the same initial and terminal object as T. Isn't it amazing how often the empty set comes round and stings us! Michael =================================