Finitary Sketches

Finitary sketches, i.e., sketches with nite-limit and nite-colimit specii-cations, are proved to be as strong as geometric sketches, i.e., sketches with nite-limit and arbitrary colimit speciications. Categories sketchable by such sketches are fully characterized in the innnitary rst-order logic: they are axiomatizable by-coherent theories, i.e., basic theories using nite conjunctions, countable disjunctions, and-nite quantiications. The latter result is absolute; the equivalence of geometric and nitary sketches requires (in fact, is equivalent to) the non-existence of measurable cardinals.