Sketches Dedicated to Michael Barr on the occasion of his 60th birthday

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 model and prove that any sketch has a free model on it. This is all done with enrichment in any monoidal biclosed category that is locally finitely presentable as a closed category. The leading example is the category of small categories together with the monad for small categories with binary products: we then recover the usual notion of binary product sketch; and that is typical. This generalises many of the extant notions of sketch.