In this paper we provide another argument to support the recently reinvigorated interest in treating Ramsey-type phenomena categorically. Using purely categorical strategies we construct a Ramsey expansion for every category of finite objects with finite small Ramsey degrees. Our construction is based on the relationship between small Ramsey degrees, weak amalgamation, and recent results about weak Fraïssé categories. Starting from the fact that weak Fraïssé categories allow for certain model-theoretic properties to be reflected in the free ω-cocompletion of the category, we show that classes with finite Ramsey degrees have weak amalgamation and then invoke the machinery of weak Fraïssé categories to perform the construction. This improves previous similar results where an analogous construction was carried out under the assumption that everything sits comfortably in a bigger class with enough infrastructure, and that in this wider context there is an ultrahomogeneous structure under whose umbrella the construction takes place.
Keywords: Fraïssé categories, free ω-cocompletion, Ramsey degrees, Ramsey expansion, weak amalgamation
2020 MSC: 18A35, 05C55
Theory and Applications of Categories, Vol. 41, 2024, No. 43, pp 1513-1535.
Published 2024-10-11.
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