Monoidal globular categories as a natural environment for the theory of weak n-categories

Michael Batanin – 23 October 1996

The subject of this talk was a construction of some categorical base for the development of the theory of omega-operads and weak omega-categories. This work was motivated by the necessity to clarify the definition of weak omega-category and to have a possibility to develop a sort of "internal" weak omega-category theory. For this one can try to define a sort of natural environment where the further constructions will be developed. It turns out that an appropriate enviroment is the theory of monoidal globular categories. A monoidal globular category 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 "weak units" for these compositions. The compositions must be `weakly associative' and `weakly inerchangeable`. There are also the coherence laws for these isomorphisms. A globular monoidal category is said to be strict if all these isomorphisms are identities. We have also the constructions of free monoidal and strict free monoidal categories. The examples of monoidal glogular categories were given that include the strict omega-categories, bicategories, braided monoidal and symmetric monoidal categories. There is also a monoidal globular category of generalized spans, which will play an important role in further development. A coherence theorem for monoidal globular categories and its applicatins to higher order weak category theory will be the subject of my talk on the next seminar.