# Homotopy invariance of convolution products

## TL;DR

This paper proves that certain convolution products in homotopy theory are fully homotopical, preserving weak equivalences without cofibrancy assumptions, thus enabling better modeling of symmetric monoidal $$-categories.

## Contribution

It establishes the full homotopical invariance of various convolution products, including those studied by Nikolaus and the authors, without cofibrancy constraints.

## Key findings

- Convolution products preserve weak equivalences in all variables.
- The result applies to diagrams of simplicial sets indexed by finite sets and injections.
- Every presentably symmetric monoidal $$-category can be modeled by a symmetric monoidal model category with a fully homotopical product.

## Abstract

The purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of simplicial sets indexed by the category of finite sets and injections and for tame $M$-simplicial sets, with $M$ the monoid of injective self-maps of the positive natural numbers. We also show that a certain convolution product studied by Nikolaus and the first author is fully homotopical. This implies that every presentably symmetric monoidal $\infty$-category can be represented by a symmetric monoidal model category with a fully homotopical monoidal product.