Dependent products and 1-inaccessible universes
The result's identifiers
Result code in 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>
Result on the web
<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
—
Alternative languages
Result language
angličtina
Original language name
Dependent products and 1-inaccessible universes
Original language 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.
Czech name
—
Czech description
—
Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
—
Continuities
S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Theory and Applications of Categories
ISSN
1201-561X
e-ISSN
—
Volume of the periodical
37
Issue of the periodical within the volume
2021
Country of publishing house
CA - CANADA
Number of pages
37
Pages from-to
107-143
UT code for WoS article
000674967700005
EID of the result in the Scopus database
2-s2.0-85100847954