Aggressive geometry – Compact generation of quasi-coherent sheaves on a mapping stack

Let k $ to be an area of ​​character $ 0and let $ mathcal {C} = mathbf {Vect} _k ^ { leq 0} $ Be the $ infty $-category of vector spaces concentrated in degrees $ leq $ 0. Consider the category $ mathbf {Pr} ( mathcal {C}): = operatorname {Fun} ( mathbf {CAlg} ( mathcal {C}), mathcal {S}) $ prestacks on k $, or $ mathcal {S} $ is the $ infty $-category of spaces or $ infty $-groupoids.

Suppose we have a similar pre-stack $ G in mathbf {Pr} ( mathcal {C}) $. This is a functor $ G: mathbf {CAlg} ( mathcal {C}) to mathbf {Sp} ^ { text {cn}} $, or $ mathbf {Sp} ^ { text {cn}} $ is the $ infty $-category of connective spectra, conceived as a space functor by composing with the forgetful functor $ mathbf {Sp} ^ { text {cn}} to mathcal {S} $. We can then form the iterated classification spaces $ B ^ nG $.

Suppose we have a pretty pretty stack $ X in mathbf {Pr} ( mathcal {C}) $ (for example a perfect pile). When category $ mathbf {QCoh} ( text {Map} (X, B ^ nG)) $ are quasi-coherent sheaves on the pile of cards generated in a compact way? Does the hypothesis $ X $ to be quite perfect? Should we make assumptions about $ G $?