# Smash Products for Non-cartesian Internal Prestacks

## TL;DR

This paper generalizes the smash product construction to prestacks internal to non-cartesian monoidal categories, unifying the Grothendieck construction and smash products for B-module algebras.

## Contribution

It introduces a new smash product analogue for internal prestacks in non-cartesian monoidal categories, extending existing constructions.

## Key findings

- Provides a unified framework for internal prestacks and smash products.
- Recovers original prestacks via fibers or coinvariants.
- Generalizes classical constructions to a broader categorical context.

## Abstract

The smash product construction (or the Grothendieck construction) takes a functor (or prestack) $F \colon B^{op} \to \mathbf{Cat}$ and returns a fibration $p \colon A \to B$. 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.