A Duality Between Ω-categories and Algebraic Ω-categories

Weak ω-Categories as ω-Hypergraphs

In this paper, firstly, we introduce a higher-dimensional analogue of hypergraphs, namely ω-hypergraphs. This notion is thoroughly flexible because unlike ordinary ω-graphs, an n-dimensional edge called an n-cell has many sources and targets. Moreover, cells have polarity, with which pasting of cells is implicitly defined. As examples, we also give some known structures in terms of ω-hypergraph...

Symmetric Bimonoidal Intermuting Categories and ω × ω Reduced Bar Constructions

A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for ω×ω-indexed family of iterated reduced bar constructions based on such a category. Mathematics Subject Classification (2000): 18D10, 55P47

The Multitopic Ω–category of All Multitopic Ω–categories

Weak Ω-categories from Intensional Type Theory

We show that for any type in Martin-Löf Intensional Type Theory, the terms of that type and its higher identity types form a weak ω-category in the sense of Leinster. Precisely, we construct a contractible globular operad PMLId of definable “composition laws”, and give an action of this operad on the terms of any type and its identity types.

Strict Ω-categories Are Monadic over Polygraphs

We give a direct proof that the category of strict ω-categories is monadic over the category of polygraphs.