We provide a new description of Joyal's arithmetic universes through a characterization of the exact and regular completions of pure existential completions.
We show that the regular and exact completions of the pure existential completion of an elementary doctrine Pare equivalent to the reg/lex and ex/lex-completions, respectively, of the category of predicates of P.
This result generalizes a previous one by the first author with F. Pasquali and G. Rosolini about doctrines equipped with Hilbert's ε-operators.
Thanks to this characterization, each arithmetic universe in the sense of Joyal can be seen as the exact completion of the pure existential completion of the doctrine of predicates of its Skolem theory.
In particular, the initial arithmetic universe in the standard category of ZFC-sets turns out to be the completion with exact quotients of the doctrine of recursively enumerable predicates.
Keywords: Lawvere doctrines, existential completion, quotient completion, arithmetics universes
2020 MSC: 03G30, 18C10, 18C50
Theory and Applications of Categories, Vol. 42, 2024, No. 4, pp 59-83.
Published 2023-07-05.
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