| Popis: |
Algebras for globular operads share a strong formal similarity with higher categories. While significant progress has been made with this approach, the current literature offers no way to find the globular operad corresponding to any given notion of higher category, nor a proof that such an operad must exist. To rectify this, we develop the theory of presentations for globular operads, and demonstrate how this provides a way to find and describe the globular operad for any variety of higher category. %We demonstrate this by constructing the globular operads arising from several key theories of $n$-category for $n \leqslant 4$.Using presentations, we also give a concrete definition of slices for globular operads, which were previously only conjectured to exist. These slices allow us to isolate the internal algebraic structure of our higher categories in each dimension. Finally, we outline how it is likely possible to use slices to show that two different notions of semi-strict $n$-category, namely $n$-categories with weak units in low dimensions and $n$-categories with weak interchange laws, respectively, are both equivalent to the fully weak variety. |
