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Let $X$ be a quasicompact quasiseparated scheme. The collection of derivedAzumaya algebras in the sense of To\"en forms a group, which contains theclassical Brauer group of $X$ and which we call $Br^\dagger(X)$ followingLurie. To\"en introduced a map $\phi:Br^\dagger(X)\to H^2_{et}(X,\mathbb G_m)$which extends the classical Brauer map, but instead of being injective, it issurjective. In this paper we study the restriction of $\phi$ to a subgroup$Br(X)\subset Br^\dagger(X)$, which we call the "derived Brauer group", onwhich $\phi$ becomes an isomorphism $Br(X)\simeq H^2_{et}(X,\mathbb G_m)$. Thismap may be interpreted as a derived version of the classical Brauer map whichoffers a way to "fill the gap" between the classical Brauer group and thecohomogical Brauer group. The group $Br(X)$ was introduced by Lurie by makinguse of the theory of prestable $\infty$-categories. There, the mentionedisomorphism of abelian groups was deduced from an equivalence of$\infty$-categories between the "Brauer space" of invertible presentableprestable $\mathcal O_X$-linear categories, and the space $Map(X,K(\mathbbG_m,2))$. We offer an alternative proof of this equivalence of$\infty$-categories, characterizing the functor from the left to the right viagerbes of connective trivializations, and its inverse via connective twistedsheaves. We also prove that this equivalence carries a symmetric monoidalstructure, thus proving a conjecture of Binda an Porta.

The derived Brauer map via twisted sheaves