### Finitary Sketches

J. Adamek, P. T. Johnstone, J. A. Makowsky, and J. Rosicky
Source: J. Symbolic Logic Volume 62, Issue 3 (1997), 699-707.

#### Abstract

Finitary sketches, i.e., sketches with finite-limit and finite-colimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finite-limit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary first-order logic: they are axiomatizable by $\sigma$-coherent theories, i.e., basic theories using finite conjunctions, countable disjunctions, and finite quantifications. The latter result is absolute; the equivalence of geometric and finitary sketches requires (in fact, is equivalent to) the non-existence of measurable cardinals.

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