Free products of higher operad algebras

Theory and Applications of Categories, 28:24-65, 2013

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], [4] 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 a normalised 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.

[1] M. Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories, Adv. in Math., 136:39–103, 1998.

[2] M. Batanin, D-C. Cisinski, and M. Weber, The lifting theorem for multitensors.

[3] M. Batanin and M. Weber, Algebras of higher operads as enriched categories.

[4] M. Weber, Multitensors and monads on categories of enriched graphs.