Sketches

We generalise the notion of sketch. For any locally nitely presentable category, one can speak of algebraic structure on the category, or equivalently, a nitary monad on it. For any such nitary monad, we de ne the notions of sketch and strict model and prove that any sketch has a generic strict model on it. This is all done with enrichment in any monoidal biclosed category that is locally nitely presentable as a closed category. Restricting our attention to enrichment in Cat, we mildly extend the de nition of strict model to give a de nition of model, and we prove that every sketch has a generic model on it. The leading example is the category of small categories together with the monad for small categories with nite products: we then recover the usual notions of nite product sketch and model; and that is typical. This generalises many of the extant notions of sketch. c © 1998 Elsevier Science B.V. All rights reserved. MSC: 18C05; 18C20; 18D20