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We construct new six-functor formalisms capturing cohomological invariants ofvarieties with potentials. Starting from any six-functor formalism $C$, encodedas a coefficient system, we associate a new six-functor formalism$C_{\text{exp}}$. This requires in particular constructing the convolutionproduct symmetric monoidal structure at the $\infty$-categorical level. Westudy $C_{\text{exp}}$ and how it relates to $C$. We also define motives in$C_{\text{exp}}$ attached to varieties with potential and study theirproperties.

Exponentiation of coefficient systems and exponential motives