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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.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland  (1977)  ((especially the article of D.A. Martin on Descriptive set theory))</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Gabriel,   "Des catégories abéliennes"  ''Bull. Soc. Math. France'' , '''90'''  (1962)  pp. 323–448</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Kunen,   "Set theory" , North-Holland  (1980)</TD></TR>
</table>

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