Large-<inline-formula><tex-math id="M1">$ N $</tex-math></inline-formula> Limits of Spaces and Structures

Abstract

For a sequence of spaces $ X_N $, with topological, algebraic or measure-theoretic structures, we show how a large-$ N $ limit $ X_\infty $ with corresponding structures is obtained. For example, when each space is a topological group $ G_N $, such as $ G_N = U(N) $, a limiting group $ G_\infty $ with topology results. Using the Weil-Kodaira construction, for compact topological groups $ G_N $ equipped with normalized Haar measures, we obtain a topological structure on $ G_\infty $ that also makes the group operations continuous. When each $ G_N $ is a Lie group we describe a Lie algebra associated to $ G_\infty $.