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Let $\mathfrak{X}$ X$ and $\mathfrak{Y}$ Y$ be Artin algebraic stacks, and $f:\mathfrak{X}\to\mathfrak{Y}$ f:X\to Y$ a morphism. I am interested in a pullback morphism $Lf^\ast : D^-_{qcoh}(\mathfrak{Y})\to D^-_{qcoh}(\mathfrak{X})$D^-_{qcoh}(Y)\to D^-_{qcoh}(X)$.

In his paper Sheaves on Artin stacks, Martin Olsson uses the lisse-etale site to define $D_{qcoh}(\mathfrak{X})$ D_{qcoh}(X)$ and $D_{qcoh}(\mathfrak{Y})$, D_{qcoh}(Y)$, and remarks that because $f:\mathfrak{X}\to\mathfrak{Y}$ f:X\to Y$ does not induce a morphism of lisse-etale topoi, he can define $Lf^\ast$ only on a category of projective systems of derived category objects.

Question: Is this the best one can expect in the world of Artin stacks, or can we do better, for example, by replacing the lisse-etale site with the big etale site. I would hope so, because $f$ does induce a morphism of big etale topoi.
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Derived pullback of quasi-coherent complexes between algebraic stacks

Let $\mathfrak{X}$ and $\mathfrak{Y}$ be Artin algebraic stacks, and $f:\mathfrak{X}\to\mathfrak{Y}$ a morphism. I am interested in a pullback morphism $Lf^\ast : D^-_{qcoh}(\mathfrak{Y})\to D^-_{qcoh}(\mathfrak{X})$.

In his paper Sheaves on Artin stacks, Martin Olsson uses the lisse-etale site to define $D_{qcoh}(\mathfrak{X})$ and $D_{qcoh}(\mathfrak{Y})$, and remarks that because $f:\mathfrak{X}\to\mathfrak{Y}$ does not induce a morphism of lisse-etale topoi, he can define $Lf^\ast$ only on a category of projective systems of derived category objects.

Question: Is this the best one can expect in the world of Artin stacks, or can we do better, for example, by replacing the lisse-etale site with the big etale site. I would hope so, because $f$ does induce a morphism of big etale topoi.