Weak n-categories: comparing opetopic foundations
نویسندگان
چکیده
منابع مشابه
Weak n-categories: comparing opetopic foundations
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...
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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...
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We address the fact that composition in an opetopic weak n-category is in general not unique and hence is not a well-defined operation. We define composition with a given k-cell in an n-category by a span of (n− k)-categories. We characterise such a cell as universal if its composition span gives an equivalence of (n− k)-categories.
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The theory of operads, defined through categories of labeled graphs, is generalized to suit definitions of higher categories with arbitrary basic shapes. Constructions of cubical, globular and opetopic weak higher categories are obtained as examples.
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We propose a notion of weak (n + k, n)-category, which we call (n + k, n)Θ-spaces. The (n + k, n)-Θ-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal’s category Θn. This notion is a generalization of that of complete Segal spaces (which are precisely the (∞, 1)-Θ-spaces). Our main result is that the above mod...
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ژورنال
عنوان ژورنال: Journal of Pure and Applied Algebra
سال: 2004
ISSN: 0022-4049
DOI: 10.1016/s0022-4049(03)00140-3