Weak ω-Categories as ω-Hypergraphs
نویسندگان
چکیده
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-hypergraphs. Then we specify a special type of ω-hypergraph, namely directed ω-hypergraphs, which are made of cells with direction. Finally, besed on them, we construct our weak ωcategories. It is an ω-dimensional variant of the weak n-categoreis given by Baez and Dolan [2]. We introduce ω-identical, ω-invertible and ωuniversal cells instead of universality and balancedness in [2]. The whole process of our definition is in parallel with the way of regarding categories as graphs with composition and identities.
منابع مشابه
A Homotopy-theoretic Universal Property of Leinster’s Operad for Weak Ω-categories
We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster’s operad for weak ω-categories.
متن کاملA type-theoretical definition of weak ω-categories
We introduce a dependent type theory whose models are weak ω-categories, generalizing Brunerie’s definition of ω-groupoids. Our type theory is based on the definition of ω-categories given by Maltsiniotis, himself inspired by Grothendieck’s approach to the definition of ω-groupoids. In this setup, ω-categories are defined as presheaves preserving globular colimits over a certain category, calle...
متن کاملA Type-Theoretical Definition of Weak {\omega}-Categories
We introduce a dependent type theory whose models are weak ω-categories, generalizing Brunerie’s definition of ω-groupoids. Our type theory is based on the definition of ω-categories given by Maltsiniotis, himself inspired by Grothendieck’s approach to the definition of ω-groupoids. In this setup, ω-categories are defined as presheaves preserving globular colimits over a certain category, calle...
متن کاملWeak Complicial Sets A Simplicial Weak ω-Category Theory Part I: Basic Homotopy Theory
This paper develops the foundations of a simplicial theory of weak ω-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ω-categories, Kan complexes and Joyal’s quasi-categories. We generalise a number of results due to the...
متن کامل00 8 Types Are Weak Ω - Groupoids
We define a notion of weak ω-category internal to a model of Martin-Löf type theory, and prove that each type bears a canonical weak ω-category structure obtained from the tower of iterated identity types over that type. We show that the ω-categories arising in this way are in fact ω-groupoids.
متن کامل