In a Lawvere theory, one specifies an algebraic theory by giving a smallcategory with finite products. Then a model of the theory is given by a functor that preserves products. A sketch is a generalization of this idea, where we take a small category , and pick some distinguished limit and colimit cones. Then a model of the sketch is a functor that preserves these limits and colimits.
A limit-sketch is a sketch with just limits and no colimits specified, ie. with . Dually, a colimit-sketch is a sketch with .
A model of a sketch is a Set-valued functor preserving the specified limits and colimits.
A category is called sketchable if it is the category of models of a sketch.
A Lawvere theory is a special case of a (limit-)sketch, where the category is one with a distinguished object such that all objects are (isomorphic to) powers of , and and is the set of all product cones.
Relation to accessible and locally representable categories