Grothendieck Universe

In set theory , a Grothendieck universe (according to Alexander Grothendieck ) is a set (of sets) in which the usual set operations on the elements do not lead out, that is, it is a model of Zermelo-Fraenkel set theory , of which Set-theoretic operations (element relation, power set formation) agree with those of the Zermelo-Fraenkel set theory, in which they are defined. The universe axiom , which requires that every quantity is an element of a Grothendieck universe, is used in category theory and algebraic geometry and extends the Zermelo-Fraenkel set theory to Tarski-Grothendieck set theory . ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$

Formal definition

A set is called a Grothendieck universe if it satisfies the following axioms : ${\displaystyle U}$

• ${\displaystyle x\in U\Longrightarrow x\subseteq U}$: If an element of , then all elements of  themselves are also elements of  ( transitivity ). ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle U}$
• ${\displaystyle x\in U\Longrightarrow {\mathcal {P}}(x)\in U}$, where the power set operator denotes: If an element of , then the power set of is also an element of , and thus all subsets of  . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle x}$
• ${\displaystyle x\in U\Longrightarrow \left\{x\right\}\in U}$: If an element of , the single-element set is  also an element of  . ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle \{x\}}$ ${\displaystyle U}$
• The following applies to every family with and :: Associations of elements of are again elements of  . ${\displaystyle \left\{x_{i}\right\}_{i\in I}}$ ${\displaystyle I\in U}$ ${\displaystyle x_{i}\in U\;\forall \,i\in I}$ ${\displaystyle \bigcup \left\{x_{i}:i\in I\right\}\in U}$ ${\displaystyle U}$ ${\displaystyle U}$
• ${\displaystyle U}$ is not empty.

This definition corresponds to that of Father Gabriel, cf. Literature . Sometimes the empty set is also admitted as a Grothendieck universe, for example in the SGA .

In other words, a Grothendieck Universe is a model of the form of the two-tier version of ZFC (that is, the substitution axiom scheme is replaced by a single axiom in second-tier logic with quantification via functions). ${\displaystyle (U,\in )}$

Unattainable cardinal numbers

A cardinal number is called (strongly) unreachable if: ${\displaystyle \kappa }$

• ${\displaystyle \mathrm {card} \left(\cup \left\{x_{i}:i\in I\right\}\right)<\kappa }$for any set of sets with and ${\displaystyle \left\{x_{i}\right\}_{i\in I}}$ ${\displaystyle \,\mathrm {card} (I)<\kappa }$ ${\displaystyle \mathrm {card} (x_{i})<\kappa \quad \forall \,i\in I}$
• ${\displaystyle \alpha ^{\beta }<\kappa \quad \forall \,\alpha ,\beta <\kappa }$

The only unreachable cardinal number known in the Zermelo-Fraenkel set theory ZFC is . The existence of further unreachable cardinal numbers cannot be proven within the framework of this theory (once the consistency of these is assumed), but must be postulated by a new axiom. ${\displaystyle \aleph _{0}}$

The connection between unreachable cardinal numbers and Grothendieck universes is now established by the following sentence:

For a set , the following properties are equivalent: ${\displaystyle \,U}$

• ${\displaystyle \,U}$ is a Grothendieck universe
• There is an unreachable cardinal number such that one and therefore all of the following equivalent properties hold: ${\displaystyle \kappa }$
  • ${\displaystyle \kappa \subseteq U}$and for every quantity applies: ${\displaystyle X\subseteq U}$ ${\displaystyle X\in U\Longleftrightarrow \mathrm {card} (X)<\kappa }$
  • ${\displaystyle U=V_{\kappa }}$(see Von Neumann hierarchy )
  • ${\displaystyle U=H_{\kappa }=\{x\mid \mathrm {card} (TC(x))<\kappa \}}$(see transitive quantity )

This is just the cardinality of the . ${\displaystyle \kappa }$ ${\displaystyle \,U}$

The existence of Grothendieck universes (except those with , which contain only finite sets and are therefore not rated as interesting ) can generally not be proven in the context of ZFC set theory, but only relatively weak additional requirements are necessary, namely the existence of further ones unreachable cardinal numbers. ${\displaystyle \mathrm {card} (U)=\aleph _{0}=\mathrm {card} (\mathbb {N} )}$

Application in category theory

Assuming the existence of a real class of unreachable cardinal numbers, statements about all sets can be made with the help of Grothendieck universes in category theory.

It is possible to assign a Grothendieck universe to every unreachable cardinal number. In order to be able to make a statement about all sets, a corresponding unreachable cardinal number is required for each set, which is genuinely greater than the cardinality of the set so that a suitable Grothendieck universe exists in which the desired constructions can be carried out.

literature

• Andreas Blass: The interaction between Category theory and Set theory. In: John Walker Gray (Ed.): Mathematical Applications of Category Theory (= Contemporary Mathematics. Vol. 30). American Mathematical Society, Providence RI 1984, ISBN 0-8218-5032-6 , pp. 5-29, online (PDF; 3.6 MB) .
• N. Bourbaki: Univers. Appendix to Exposé I by M. Artin, A. Grothendieck, JL Verdier (ed.): Théorie des Topos et Cohomologie Étale des Schémas (SGA 4). 2nd Edition. Springer-Verlag, Heidelberg 1972, ISBN 3-540-05896-6 .
• NH Williams: On Grothendieck universes. In: Compositio Mathematica. Vol. 21, No. 1, ISSN  0010-437X , 1969, pp. 1-3, online (PDF; 261 kB) .
• AH Kruse: Grothendieck universes and the super-complete models of Shepherdson. In: Compositio Mathematica. Vol. 17, 1965/1966, pp. 96-101, online (PDF; 550 kB) .
• P. Gabriel: Des catégories abéliennes. In: Bulletin de la Société Mathématique de France. Vol. 90, 1962, ISSN  0037-9484 , pp. 323-448, online (PDF; 10.45 MB) .
• M. Kühnrich: About the concept of the universe. In: Journal for Mathematical Logic and Fundamentals of Mathematics. Vol. 12, 1966, ISSN  0044-3050 , pp. 37-59.
• Saunders Mac Lane : Categories for the Working Mathematician (= Graduate Texts in Mathematics. Vol. 5). 2nd Edition. Springer, New York NY et al. 1998, ISBN 0-387-98403-8 , I.6.
• Michael D. Potter: Sets. An Introduction. Clarendon Press, Oxford et al. 1990, ISBN 0-19-853388-8 , 3.3

Individual evidence

1. ↑ Akihiro Kanamori : The Higher Infinite . Large Cardinals in Set Theory from Their Beginnings. 2nd Edition. Springer , 2009, ISBN 978-3-540-88867-3 , pp. 19 , doi : 10.1007 / 978-3-540-88867-3 .
2. ↑ Kanamori, p. 299.