Dependent products and 1-inaccessible universes

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F21%3A00122244" target="_blank" >RIV/00216224:14310/21:00122244 - isvavai.cz</a>

Výsledek na webu

<a href="http://www.tac.mta.ca/tac/volumes/37/5/37-05abs.html" target="_blank" >http://www.tac.mta.ca/tac/volumes/37/5/37-05abs.html</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Dependent products and 1-inaccessible universes

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Dependent products and 1-inaccessible universes

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

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Návaznosti

S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Theory and Applications of Categories

ISSN

1201-561X

e-ISSN

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Svazek periodika

37

Číslo periodika v rámci svazku

2021

Stát vydavatele periodika

CA - Kanada

Počet stran výsledku

37

Strana od-do

107-143

Kód UT WoS článku

000674967700005

EID výsledku v databázi Scopus

2-s2.0-85100847954