Multitensors as monads on categories of enriched graphs

Theory and Applications of Categories, 28:857-932, 2013

In this paper we unify the developments of [1], [3] and [4] into a single framework in which the interplay between multitensors on a category V, and monads on the category of graphs enriched in V, is taken as fundamental. The material presented here is the conceptual background for subsequent work: in [2] the Gray tensor product of 2-categories and the Crans tensor product [5] of Gray categories are exhibited as existing within our framework, and in [6] the explicit construction of the funny tensor product of categories is generalised to a large class of Batanin operads.

[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] E. Cheng, Comparing operadic theories of n-category, preprint, 2008.

[5] S. Crans, A tensor product for Gray categories. Theory and applications of categories, 5:12–69, 1999.

[6] M. Weber, Free products of higher operad algebras.