Coherence for monoidal globular categories

Michael Batanin – 30 October 1996

In my previous talk I introduced a notion of monoidal globular category which is a sequence of categories together with source and target functors satisfying the same conditions as the wellknown source and target functions for globular sets. We can also compose (in the sense of a $k$-th composition) a pair of objects if the $k$-source of one object coincides with $k$-target of another. There are also the functors which play the role of units for these compositions. The compositions must be `weakly associative' and `inerchangeable`. There are also the coherence laws for these isomorphisms. A monoidal globular category is said to be strict if all these isomorphisms are identities. I give in this talk the constructions of free monoidal and strict free monoidal globular categories. For this I consider a special strict monoidal globular category $Tr$ called the category of multistage trees. I proved that the discretisation of $Tr$ is the free $\omega$-category generated by one object, one enddomorphism of this object, one endomotphism of this endomorphism and so on. This discrete monoidal category will play a central role in the definition of weak $\omega$-category. The coherence theorem is proved, which asserts that a natural strict monoidal functor from the free monoidal globular category generated by a globular category to the free strict monoidal globular category generated by this globular category is an equivalence. This theorem generalizes MacLane's coherence theorem for monoidal and symmetric monoidal categories, and the coherence theorem for bicategories and for braided monoidal categories of Joyal-Street. On the other hand it contains as a special case a sort of pasting theorem for $n$-categories. The applications of this theorem to the theory of weak $\omega$-categories will be given in my next talk.