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

Let $$k$$ to be an area of ​​character $$0$$and 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 $$G$$?