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The locally $\mathcal{U}$-small Dedekind real numbers for a universe $\mathcal{U}$ is defined as the Archimedean ordered integral domain $\mathbb{R}_\mathcal{U}$ with a strictly monotonic function? $i:\mathbb{I}_\mathcal{U} \to \mathbb{R}_\mathcal{U}$ from the locally $\mathcal{U}$-small Dedekind real unit interval to $\mathbb{R}_\mathcal{U}$ such that $i(0) = 0$ and $i(1) = 1$.
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Auke B. Booij, Extensional constructive real analysis via locators, (abs:1805.06781)