Dear Bob: How do you feel about whole papers. Charles and I have a joint paper of which I attach the title and (most of) the introduction. Perhaps it is just as well to distribute this much and say that anyone who wants the whole paper, either by email or pmail should write to me. Regards, Mike =========================================== On the limitations of sketches Michael Barr and Charles Wells Introduction Sketches, as described for example in ?Barr and Wells, 1985?, can be used to describe many, but not all, kinds of mathematical structure. Recently Wells ?1990? has described an extension of the notion to allow more powerful constructors than those given by limits and colimits to be used to described structures. This raises the question of exactly what can be sketched with an ordinary sketch. A remarkable theorem of Makkai and Par\'e ?1990? (and discovered independently by Lair) says that a category is sketchable if and only if it is accessible. This means that for some cardinal $\kappa$ the category has colimits of all $\kappa$ filtered diagrams and that every object is a $\kappa$ filtered colimit of $\kappa$ presentable objects. An object $C$ of a category is $\kappa$ presentable if the functor $\Hom(C,-)$ preserves the colimits of $\kappa$ filtered diagrams. Since in practice one can usually decide quite easily if a category is accessible, this gives a usable criterion for sketchability, without, unfortunately, giving any idea how to sketch certain theories. Consider the category of categories with finite limits and functors that preserve them. We are not supposing canonical finite limits; the functors are merely required to take a finite limit diagram in the source to some finite limit diagram over the same base in the target. At first, it would seem that a theory to describe the set of all finite limit cones in a category would require a universal quantifier, and thus would not be sketchable. On the other hand, it is easy to see that the category of these categories with finite limits and functors that preserve them is $\aleph_0$ accessible and therefore from the theorem mentioned previously is sketchable. In this paper we actually exhibit a simple sketch for that category. A similar argument works for the category of categories with weak finite limits (a cone is a weak limit if every other cone has at least one arrow to it). We indicate the minor change needed in the argument for the case of weak terminal objects. On the other hand, the related category of categories with sublimits (a cone is a sublimit if every other cone has at most one arrow to it) is not accessible and hence not sketchable. In this case, a universal quantifier or some higher order construct is needed. We show that a universal quantifier suffices. Among other things this shows that there is a class of first order sketches that has more expressive power than that of ordinary sketches.