On the Limitations of Sketches

Call a category "sketchable" if it is the category of models in sets of some sketch. This paper explores the subtle boundary between sketchable and nonsketchable categories. We show that the category of small categories that have at least one initial object and functors that take an initial object to an initial object is sketchable. The same is true for weak initial objects, but is false for subinitial objects (that every object has at most one arrow to). Analogous results hold if we substitute finite limits for terminal object. We also show that the category of groups and center-preserving homomorphisms is not sketchable. We describe briefly how "higher-order" sketches can fill these gaps. 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 describe structures. This raises the question of exactly what can be sketched with an ordinary sketch. A theorem of Lair [1981] (rediscovered by Makkai and Paré [1990]) says that a category is sketchable if and only if it is accessible. This means that for some cardinal K the category has colimits of all K filtered diagrams and that every object is a K filtered colimit of K presentable objects. An object C of a category is n presentable if the functor Hom(C, —) preserves the colimits of K filtered diagrams. Since in practice one can usually decide quite easily whether 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 In the preparation of this paper, the first author has been assisted by a grant from the NSERC of Canada and the second by NSF grant CCR-8702425. The authors would also like to thank McGill University and Case Western Reserve University, respectively, for sabbatical leaves and the University of Pennsylvania for a very congenial setting in which to spend those leaves. Received by the editors 3 April, 1990; revised 19 June, 1990. AMS subject classification: 18C10,18A10.