Sketches

Abstract

We generalise the notion of sketch. For any locally finitely presentable category, one can speak of algebraic structure on the category, or equivalently, a finitary monad on it. For any such finitary monad, we define 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 finitely presentable as a closed category. Restricting 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 categories together with the monad for small categories with finite products: we then recover the usual notions of finite product sketch and model; and that is typical. This generalises many of the extant notions of sketch.