We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (EQ, TQ) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the ‘free operad’ monad construction on suitable monads. We use this to give an explicit description of the relationship betwe...

We develop a theory of Spanier–Whitehead duality in categories with cofibrations and weak equivalences (Waldhausen categories, for short). This includes L–theory, the involution on K–theory introduced by [Vo] in a special case, and a map Ξ relating L–theory to the Tate spectrum of Z/2 acting on K–theory. The map Ξ is a distillation of the long exact Rothenberg sequences [Sha], [Ra1], [Ra2], inc...

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...

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.

We generalise the concepts introduced by Baez and Dolan to define opetopes constructed from symmetric operads with a category, rather than a set, of objects. We describe the category of 1-level generalised multicategories, a special case of the concept introduced by Hermida, Makkai and Power, and exhibit a full embedding of this category in the category of symmetric operads with a category of o...

We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R ⊗ R.