# Free Products of Higher Operad Algebras

## TL;DR

This paper develops a systematic construction of tensor products for higher operad algebras, generalizing the Gray tensor product to higher categories using operads, and establishing a symmetric monoidal closed structure.

## Contribution

It introduces a new tensor product for algebras of higher operads, extending Gray's construction to higher categorical structures and providing a foundational framework.

## Key findings

- Defines a tensor product for higher operad algebras.
- Establishes the symmetric monoidal closed structure on these categories.
- Generalizes Gray's tensor product to higher dimensions.

## Abstract

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2-categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray's little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable by an n-operad in the sense of Batanin [1], an analogous tensor product which forms a symmetric monoidal closed structure on the category of algebras of the operad.