Categorical models of dependent type theory with sums, products, identity types and univalent universes present elementary ∞-toposes via ∞-localization, which also possess small subobject classifiers.
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Elementary $\infty$-toposes from type theory
Categorical models of dependent type theory with sums, products, identity types and univalent universes present elementary ∞-toposes via ∞-localization, which also possess small subobject classifiers.