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Mike kindly wrote model of type theory in an (infinity,1)-topos (homotopytypetheory) with an explicit statement of how univalent Tarskian types-of-types have semantics by object classifiers in general $\infty$-toposes.
I have added a brief pointer to this to some relevant $n$Lab entries: to homotopy type theory, to relation between category theory and type theory, to (infinity,1)-topos and to elementary (infinity,1)-topos.
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