Eugenia Cheng
We give an explicit construction of the category Opetope of opetopes. We prove that the category of opetopic sets is equivalent to the category of presheaves over Opetope.
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An elementary definition of opetopic sets
本文提出 opetopes 与 opetopic 集合的初等定义，通过简单结构与公理直接构建，并证明其与多项式单子定义及 Ho Thanh 范畴等价。
From the polynomial approach to the definition of opetopes of Kock et al., we derive a category of opetopes, and show that its set-valued presheaves, or opetopic sets, are equivalent to many-to-one polygraphs. As an immediate corollary, we establish that opetopic sets are equivalent to multitopic sets, introduced and studied by Harnick et al, and we also address an open question of Henry.
Duality for positive opetopes and positive zoom complexes
本文证明正缩放复形范畴与带收缩满同态的正Opetope范畴互为对偶，并将该对偶性推广到Opetopic基数。
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 objects. As an analogy to the Baez-Dolan slice construction, we exhibit a certain multicategory of function replacement as a slice construction in the multitopic setting, and use it to construct multitope…
We continue our previous modifications of the Baez-Dolan theory of opetopes to modify the Baez-Dolan definition of universality, and thereby the category of opetopic n-categories and lax functors. For the case n=2 we exhibit an equivalence between this category and the category of bicategories and lax functors. We examine notions of strictness in the opetopic theory.
We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (E_Q,T_Q) 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 between Baez-Dolan and Leinster opetopes.
We define a family of structures called "opetopic algebras", which are algebraic structures with an underlying opetopic set. Examples of such are categories, planar operads, and Loday's combinads over planar trees. Opetopic algebras can be defined in two ways, either as the algebras of a "free pasting diagram" parametric right adjoint monad, or as models of a small projective sketch over the category of opetopes. We define an opetopic nerve functor that fully embeds each category of opetopic algebras into the category of opetopic sets. In particular, we obtain fully faithful opetopic nerve fun…
We give a simple algebraic description of opetopes in terms of chain complexes, and we show how this description is related to combinatorial descriptions in terms of treelike structures. More generally, we show that the chain complexes associated to higher categories generate graphlike structures. The algebraic description gives a relationship between the opetopic approach and other approaches to higher category theory. It also gives an easy way to calculate the sources and targets of opetopes.
For $Λ$ the category of free opetopic algebras, we construct a model structure \emph{à la Cisinski} on the category of presheaves over $Λ$, and show that is is equivalent to opetopic complete Segal spaces. This generalizes the results of Joyal and Tierney in the simplicial case, and of Cisinski and Moerdijk in the planar dendroidal case.
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.
We show that the category of positive opetopes with contraction morphisms, i.e. all face maps and some degeneracies, forms a test category. The category of positive opetopic sets pOpeSet can be defined as a full subcategory of the category of polygraphs Poly. An object of pOpeSet has generators whose codomains are again generators and whose domains are non-identity cells (i.e. non-empty composition of generators). The category pOpeSet is a presheaf category with the exponent being called the category of positive opetopes pOpe. Objects of pOpe are called positive opetopes and morphisms are face…
Diagrammatic sets as a model of homotopy types
该研究引入源于组合拓扑与高维图重写的图解集合，构建其上余纤维生成的模型结构，证明其与单纯集经典模型结构的两个Quillen等价，且该结构关于Gray积是幺半的，为同伦类型提供新模型。
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