Multitensor lifting and strictly unital higher category theory

Michael Batanin (1) , Denis-Charles Cisinski (2) , Mark Weber (1)

1 Macquarie University
2 IMT - Institut de Mathématiques de Toulouse UMR5219

In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result - the lifting theorem for multitensors - enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray categories as part of this framework. We define weak n-categories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak n-categories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)-categories with strict units.

Mots clés: higher categories; enriched categorie; lax monoidal structures

Domaines: Catégories et ensembles [math.CT]; Topologie algébrique [math.AT]

Michael Batanin, Denis-Charles Cisinski, Mark Weber. Multitensor lifting and strictly unital higher category theory. Theory and Applications of Categories, 2013, 28 (25), pp.804-856. ⟨hal-00606274v2⟩