Finite models of sketches

AbstractFor every sketch with countable limit specifications and countable colimit specifications we prove that there exists a finitary sketch (i.e., one with finite limit and colimit specifications) with the same category of finite models. The sketch is even coherent, i.e., describable by the finitary first-order logic. Assuming the non-existence of measurable cardinals, we also prove that for every geometric sketch there exists a coherent sketch with the same category of finite models. The latter result is, in fact, equivalent to the assumption of non-existence of measurable cardinals