nLab Makkai duality

Idea

Makkai duality is a kind of syntax-semantics duality, due to Michael Makkai, relating pretoposes (categories having to do with the syntax of first-order logic) and ultracategories (which are a way of capturing the semantics of first-order logic). This leads to a proof of conceptual completeness for first-order logic.

Related entries

• Łoś ultraproduct theorem

• Stone duality

References

• Michael Makkai, Stone duality for first order logic, Proceedings of the Herbrand symposium (Marseilles, 1981), 217–232, Stud. Logic Found. Math. 107, North-Holland, Amsterdam, 1982, doi;

• Michael Makkai, Stone duality for first order logic, Adv. Math. 65 (1987) no. 2, 97–170, doi, MR89h:03067;

• Michael Makkai, Duality and definability in first order logic, Mem. Amer. Math. Soc. 105 (1993), no. 503

• Jacob Lurie, Lecture 29X-Makkai Duality, (lecture notes)

• Jacob Lurie, Ultracategories, (pdf)

• Francisco Marmolejo, Ultraproducts and continuous families of models, (thesis; pdf).

Some more general variants are achieved in

• Marek W. Zawadowski, Descent and duality, Annals of Pure and Applied Logic 71, n.2 (1995), 131–188

• Henrik Forssell, First-order logical duality, Ph.D. thesis, Carnegie Mellon U. 2008, pdf