Free Products of Higher Operad Algebras

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.

💡 Research Summary

The paper tackles a long‑standing problem in higher category theory: how to systematically construct higher‑dimensional analogues of the Gray tensor product of 2‑categories. While the Gray tensor provides a rich, non‑symmetric monoidal structure for 2‑categories, extending it to arbitrary higher categorical structures has remained elusive. The authors build on the framework of Batanin’s n‑operads, which encode higher‑dimensional operations in a single algebraic object, and they show that for any n‑operad O there is a natural tensor product on the category Alg(O) of O‑algebras that behaves like the “funny tensor product” of ordinary categories.

The construction proceeds by defining a binary operation ⊗₊ on Alg(O) as a pushout of free O‑algebras. Given two O‑algebras A and B, one first applies the free‑algebra functor F to embed them into the ambient category of O‑algebras, then forms the pushout of the diagram F(A) ← I → F(B), where I is the initial O‑algebra. The resulting object A ⊗₊ B is precisely the free product of A and B in the operadic sense. This definition automatically yields a symmetric monoidal structure: the unit is I, associativity follows from the universal property of iterated pushouts, and the symmetry is built into the pushout construction because the diagram is symmetric in A and B.

To obtain a closed structure, the authors construct an internal hom functor