Higher order operads and their algebras from the point of view of the theory of monoidal globular categories
Michael Batanin – 6 November 1996
The theory of monoidal globular categories gives me the necessary tools for defining the notion of weak n-category.
I call an $\omega$-collection of spans a globular functor from the globular category of trees Tr to Span. The coherence theorem allows me to introduce, on the category of $\omega$-collections, a sort of monoidal structure (in some weak sense). This monoidal structure generalizes a monoidal structures on the nonsymmetric collections used in the definition of the (nonsymmetric) operads.
So I define an n-operad to be a monoid in the category of n-collections.
One can also define an n-operad of endomorphisms generated by an $\omega$-globular set (or more generally by a globular object ( i.e. a globular functor from U to C, where U is a terminal globular category) in a monoidal globular category C). Finally, a weak $\omega$-category is an $\omega$-globular set together with a map from a contractible (in a suitable sense) $\omega$-operad to the operad of endomorphisms of this globular set.
For example, the strict $\omega$-category are the algebras of a terminal omega-operad. The weak 2-categories are exactly the bicategories and the category of contractible 2-operads is isomorphic to the category of non-symmetric chaotic Cat-operads (this means that we have in every dimension a category which is equivalent to one-object one-arrow category). There exists a universal $\omega$-operad which acts on every weak n-category (including n=$\omega$.) Recently, Clemens Berger suggested to me a construction of a contractible 3-operad whose algebras are Gray-categories.
The theory of monoidal globular categories allows also the development of the theory of internal weak $\omega$-category and cocategory objects that will be useful for the problem of modelling of homotopy types.