  We prove the existence of a power structure over the Grothendieck ring of
geometric dg categories. We show that a conjecture by Galkin and Shinder
(proved recently by Bergh, Gorchinskiy, Larsen, and Lunts) relating the motivic
and categorical zeta functions of varieties can be reformulated as a
compatibility between the motivic and categorical power structures. Using our
power structure we show that the categorical zeta function of a geometric dg
category can be expressed as a power with exponent the category itself. We give
applications of our results for the generating series associated with Hilbert
schemes of points, categorical Adams operations, and series with exponent a
linear algebraic group.
