The category of models of any theory T in any first-order language L has the surprising property that any small category that is elementarily equivalent with it, already embeds in it. The proof uses an abstract argument via ultrapowers, leading one to wonder which concrete categorical axioms, depending on T and L, are responsible for this embedding result.
Keywords: universal categories
2020 MSC: 18A15, 03C52
Theory and Applications of Categories, Vol. 45, 2026, No. 52, pp 2214-2229.
Published 2026-10-27.
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