# Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces

## TL;DR

This paper develops a new framework for multiplicative parametrized homotopy theory using symmetric spectra in retractive spaces, enabling better constructions of ring spectra and comparisons of Thom spectrum models.

## Contribution

It introduces a point-set level convolution smash product that aligns with infinity-categorical products, facilitating the study of multiplicative phenomena in twisted (co)homology.

## Key findings

- Convolution smash product descends to infinity-categorical product
- Enables construction of commutative parametrized ring spectra
- Provides tools for comparing Thom spectrum models

## Abstract

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding infinity-categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted K-theory.