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We prove the conjecture that any Grothendieck $(\infty,1)$-topos can bepresented by a Quillen model category that interprets homotopy type theory withstrict univalent universes. Thus, homotopy type theory can be used as a formallanguage for reasoning internally to $(\infty,1)$-toposes, just as higher-orderlogic is used for 1-toposes. As part of the proof, we give a new, moreexplicit, characterization of the fibrations in injective model structures onpresheaf categories. In particular, we show that they generalize the coflexiblealgebras of 2-monad theory.

All $(\infty,1)$-toposes have strict univalent universes