Publication details

Dependent products and 1-inaccessible universes



Year of publication 2021
Type Article in Periodical
Magazine / Source Theory and Applications of Categories
MU Faculty or unit

Faculty of Science

Keywords higher categories; higher toposes; elementary higher toposes; Grothendieck universes; large cardinals; dependent products; dependent sums; classifiers; generic morphisms
Description The purpose of this writing is to explore the exact relationship running between geometric infinity-toposes and Mike Shulman's proposal for the notion of elementary 1-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 infinity-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 infinity-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 infinity-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 infinity-toposes our analysis relies on ordinary sheaf toposes, although with a slight change due to the impossibility of having true classifiers in the infinity-dimensional setting. In conclusion, it will be shown that, under stronger assumptions positing the existence of intermediate-size Grothendieck universes, examples of elementary infinity-toposes with strong universes which are not geometric can be found.
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