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Makkai's conceptual completeness and Lurie's generalization to infinity-pretopoi

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In the first hour I am going to review the basics: construction of syntactic categories, obtaining internal language as a left adjoint to syntactic construction, coherent categories, category of models, and Morita equivalence of theories. I particularly aim to give examples of the latter; for instance I will show why the theory of posets in which every element is less than a maximal element is not Morita equivalent to any coherent or in fact a geometric theory.

For the second hour of the talk, I will introduce Booleanization construction , and use if to characterize all Boolean coherent categories. This characterization will avail us in deducing Gödel’s completeness theorems from Deligne’s completeness theorems.

One of the main point I want to hammer home in all this is that while in the case of locales, the points of a locale (aka models of corresponding propositional theory) form a topological space, in the case of coherent categories and pretoposes they form categories with very nice structures.


  • Makkai, M. & Reyes, G. (1977), ‘First order categorical logic’, Lecture Notes in Math. 611. Springer, Berlin .
  • Johnstone, P. (2002), ‘Sketches of an elephant: A topos theory compendium’, Oxford Logic Guides Vol.1(no.44, Oxford University Press).

This talk is part of the Cargo series.

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