Sketches

We generalise the notion of sketch. For any locally finitely presentable ca tegory, one can speak of algebraic structure on the category, or equivalent ly, a finitary monad on it. For any such finitary monad, we define the noti ons of sketch and strict model and prove that any sketch has a generic stri ct model on it. This is all done with enrichment in any monoidal biclosed c ategory that is locally finitely presentable as a closed category. Restrict ing our attention to enrichment in Cat, we mildly extend the definition of strict model to give a definition of model, and we prove that every sketch has a generic model on it. The leading example is the category of small cat egories together with the monad for small categories with finite products: we then recover the usual notions of finite product sketch and model; and t hat is typical. This generalises many of the extant notions of sketch. (C) 1998 Elsevier Science B.V. All rights reserved. MSC. 18C05; 18C20; 18D20.