U-Small and U-Locally Small Categories

Coghetto, Roland

doi:10.2478/forma-2024-0006

Abstract

This paper deals with the notions of U-small set, U-small category, and U-locally small category (U is non-empty Grothendieck universe). We reuse the first Mizar formalization of categories contained in CAT_* series of Mizar articles in order to show the expressive power of the Tarski-Grothendieck set theory (which is the base for the Mizar Mathematical Library) in this area. We encode parts of SGA 4 by Nicolas Bourbaki.

References

Michael Artin, Alexander Grothendieck, and Jean-Louis Verdier. Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos (exposés ià iv). In Séminaire de Géométrie Algébrique du Bois Marie, 1963/64, SGA 4, volume 269 of Lecture Notes in Mathematics. Springer, 1972.
Search in Google Scholar Back to article
Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.
Search in Google Scholar Back to article
Chad E. Brown and Karol Pąk. A tale of two set theories. In Cezary Kaliszyk, Edwin Brady, Andrea Kohlhase, and Claudio Sacerdoti Coen, editors, Intelligent Computer Mathematics – 12th International Conference, CICM 2019, CIIRC, Prague, Czech Republic, July 8-12, 2019, Proceedings, volume 11617 of Lecture Notes in Computer Science, pages 44–60. Springer, 2019. doi:10.1007/978-3-030-23250-4 4.
Search in Google Scholar Back to article
Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1 (2):409–420, 1990.
Search in Google Scholar Back to article
Roland Coghetto. Non-trivial universes and sequences of universes. Formalized Mathematics, 30(1):53–66, 2022. doi:10.2478/forma-2022-0005.
Search in Google Scholar Back to article
Masaki Kashiwara and Pierre Schapira. Categories and Sheaves, volume 332 of Grundlehren der Mathematischen Wissenschaften. Springer, 2006. doi:10.1007/3-540-27950-4.
Search in Google Scholar Back to article
Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, Heidelberg, Berlin, 1971.
Search in Google Scholar Back to article
Karol Pąk. Grothendieck universes. Formalized Mathematics, 28(2):211–215, 2020. doi:10.2478/forma-2020-0018.
Search in Google Scholar Back to article
Marco Riccardi. Object-free definition of categories. Formalized Mathematics, 21(3): 193–205, 2013. doi:10.2478/forma-2013-0021.
Search in Google Scholar Back to article
Emily Riehl. Category Theory in Context. Courier Dover Publications, 2017.
Search in Google Scholar Back to article
Zbigniew Semadeni and Antoni Wiweger. Wst¦p do teorii kategorii i funktorów, volume 45 of Biblioteka Matematyczna. PWN, Warszawa, 1978.
Search in Google Scholar Back to article
Alfred Tarski.Über unerreichbare Kardinalzahlen. Fundamenta Mathematicae, 30:68–89, 1938.
Search in Google Scholar Back to article
Alfred Tarski. On well-ordered subsets of any set. Fundamenta Mathematicae, 32:176–183, 1939.
Search in Google Scholar Back to article
Andrzej Trybulec. Categories without uniqueness of cod and dom. Formalized Mathematics, 5(2):259–267, 1996.
Search in Google Scholar Back to article
N. H. Williams. On Grothendieck universes. Compositio Mathematica, 21(1):1–3, 1969.
Search in Google Scholar Back to article

U-Small and U-Locally Small Categories

Abstract

Paradigm

My account