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Quasitoposes as elementary quotient completions

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arxiv 2111.15299 v4 pith:4MDI3U7X submitted 2021-11-30 math.LO math.CT

Maria Emilia Maietti , Fabio Pasquali , Giuseppe Rosolini This is my paper · ORCID
classification math.LOmath.CT
keywords elementarycompletionsexactquotientcharacterisationcompletionwhencarboni
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The elementary quotient completion of an elementary doctrine in the sense of Lawvere was introduced in previous work by the first and third authors. It generalises the exact completion of a category with finite products and weak equalisers. In this paper we characterise when an elementary quotient completion is a quasi-topos. We obtain as a corollary a complete characterisation of when an elementary quotient completions is an elementary topos. As a byproduct we determine also when the elementary quotient completion of a tripos is equivalent to the doctrine obtained via the tripos-to-topos construction. Our results are reminiscent of other works regarding exact completions and put those under a common scheme: in particular, Carboni and Vitale's characterisation of exact completions in terms of their projective objects, Carboni and Rosolini's characterisation of locally cartesian closed exact completions, also in the revision by Emmenegger, and Menni's characterisation of the exact completions which are elementary toposes.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Projective covers, doctrines of algebras and the relational quotient completion

    math.CT 2026-08 conditional novelty 8.0 of 10

    In relational doctrines with quotients, the extensional quotient completion is characterized by the existence of a projective cover, and this transfers to doctrines of algebras for quotient-preserving monads.

  2. The Relational Quotient Completion

    math.CT 2024-12 conditional novelty 7.0 of 10

    A new categorical framework, relational doctrines, yields universal quotient and extensionality completions that unify exact completion, setoids, and quantitative metric quotients.

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