Smash Products for Non-cartesian Internal Prestacks

The smash product construction (or the Grothendieck construction) takes a functor (or prestack) $F\text{\hspace{0.17em}⁣}:B^{op}\to \mathbf{C}\mathbf{a}\mathbf{t}$ and returns a fibration $p\text{\hspace{0.17em}⁣}:A\to B$. In this paper, we develop an analogue of the smash product for prestacks internal to a non-cartesian monoidal category. Our construction simultaneously generalizes the Grothendieck construction for prestacks and smash products for $B$-module algebras over a bialgebra $B$. Further, taking fibers or coinvariants allows one to recover the original prestack.