# Boardman--Vogt tensor products of absolutely free operads

## TL;DR

This paper develops a combinatorial model for the Boardman--Vogt tensor product of absolutely free operads, showing it remains free as an $S$-module, contrasting with previous results on non-unital associative operads.

## Contribution

It introduces a new combinatorial framework for understanding tensor products of absolutely free operads, demonstrating their freeness as $S$-modules.

## Key findings

- Tensor product of absolutely free operads is always a free $S$-module.
- Contrasts with previous results showing hidden commutativity in certain operad tensor squares.
- Provides a combinatorial model for these tensor products.

## Abstract

We establish a combinatorial model for the Boardman--Vogt tensor product of several absolutely free operads, that is free symmetric operads that are also free as $\mathbb{S}$-modules. Our results imply that such a tensor product is always a free $\mathbb{S}$-module, in contrast with the results of Kock and Bremner--Madariaga on hidden commutativity for the Boardman--Vogt tensor square of the operad of non-unital associative algebras.