Universe

A set $\mathcal{U}$ which is closed under the formation of unions, singletons, subelements, power sets, and pairs; more precisely:

1) $I \in \mathcal{U}$, $X_i \in \mathcal{U}$ implies $\cup_{i\in I}X_i \in \mathcal{U}$;

2) $x \in \mathcal{U}$ implies $\{x\} \in \mathcal{U}$;

3) $x \in X \in \mathcal{U}$ implies $x \in \mathcal{U}$;

4) $X \in \mathcal{U}$ implies $\mathcal{P}X \in \mathcal{U}$;

5) $(x,y) \in \mathcal{U}$ if and only if $x,y \in \mathcal{U}$.

The existence of infinite universes in axiomatic set theory is equivalent to the existence of strongly inaccessible cardinals (cf. Cardinal number). A universe is a model for Zermelo–Fraenkel set theory. Universes were introduced by A. Grothendieck in the context of category theory in order to introduce the "set" of natural transformations of functors between ($\mathcal{U}$-) categories, and in order to admit other "large" category-theoretic constructions.

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