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Dependent products and 1-inaccessible universes

Theory and applications of categories, 2021-01, Vol.37 (13), p.107 [Peer Reviewed Journal]

Copyright R. Rosebrugh 2021 ;EISSN: 1201-561X

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  • Title:
    Dependent products and 1-inaccessible universes
  • Author: Monaco, Giulio Lo
  • Subjects: Algebraic group theory ; Axioms ; Classification ; Classifiers ; Geometry ; Theoretical mathematics ; Universe
  • Is Part Of: Theory and applications of categories, 2021-01, Vol.37 (13), p.107
  • Description: The purpose of this writing is to explore the exact relationship running between geometric ∞-toposes and Mike Shulman's proposal for the notion of elementary ∞-topos, and in particular we will focus on the set-theoretical strength of Shulman's axioms, especially on the last one dealing with dependent sums and products, in the context of geometric ∞-toposes. Heuristically, we can think of a collection of morphisms which has a classifier and is closed under these operations as a well-behaved internal universe in the ∞-category under consideration. We will show that this intuition can in fact be made to a mathematically precise statement, by proving that, once fixed a Grothendieck universe, the existence of such internal universes in geometric ∞-toposes is equivalent to the existence of smaller Grothendieck universes inside the bigger one. Moreover, a perfectly analogous result can be shown if instead of geometric ∞-toposes our analysis relies on ordinary sheaf toposes, although with a slight change due to the impossibility of having true classifiers in the 1-dimensional setting. In conclusion, it will be shown that, under stronger assumptions positing the existence of intermediate-size Grothendieck universes, examples of elementary ∞-toposes with strong universes which are not geometric can be found.
  • Publisher: Sackville: R. Rosebrugh
  • Language: English
  • Identifier: EISSN: 1201-561X
  • Source: Open Access: Freely Accessible Journals by multiple vendors
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