Related papers: Orientals as free weak $\omega$-categories
The orientals or oriented simplexes are a family of strict omega-categories constructed by Ross Street. We show that the category of orientals is isomorphic to a subcategory of the category of chain complexes. This leads to a very simple…
The nerve of a strict omega-category is a simplicial set with additional structure, making it into a so-called complicial set, and strict omega-categories are in fact equivalent to complicial sets. The nerve functor is represented by a…
The aim of this paper is to give an alternative construction of Street's cosimplicial object of orientals, based on an idea of Burroni that orientals are free algebras for some algebraic structure on strict $\omega$-categories. More…
We show that the nerve of a strict omega-category can be described algebraically as a simplicial set with additional operations subject to certain identities. The resulting structures are called sets with complicial identities. We also…
This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of…
This paper develops the foundations of a simplicial theory of weak omega-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…
We show that the Street nerve of a strict $\omega$-category $C$ is a Kan complex (respectively a quasi-category) if and only if the $n$-cells of $C$ for $n\geq 1$ (respectively $n> 1$) are weakly invertible. Moreover, we equip…
In this paper, firstly, we introduce a higher-dimensional analogue of hypergraphs, namely $\omega$-hypergraphs. This notion is thoroughly flexible because unlike ordinary $\omega$-graphs, an n-dimensional edge called an n-cell has many…
The primary purpose of this work is to characterise strict \omega-categories as simplicial sets with structure. We prove the Street-Roberts conjecture which states that they are exactly the ``complicial sets'' defined and named by John…
Thick simplices are the nerves of the contractible groupoids obtained by inverting the arrows in the categories [n]. Using explicit expansions of simplicial subsets of the thick simplices, we present a new approach to results of Rezk and of…
We construct a left semi-model category of "marked strict $\infty$-categories" for which the fibrant objects are those whose marked arrows satisfy natural closure properties and are weakly invertible. The canonical model structure on strict…
In this dissertation, we compare the "classical" homology of an $\omega$-category (defined as the homology of its Street nerve) with its polygraphic homology. More precisely, we prove that both homologies generally do not coincide and call…
The aim of these notes is to introduce the intuition motivating the notion of a "complicial set", a simplicial set with certain marked "thin" simplices that witness a composition relation between the simplices on their boundary. By varying…
We study $\omega$-equifibrations between weak $\omega$-categories in the sense of Batanin--Leinster. We define $\omega$-equifibrations as a natural weak $\omega$-categorical analogue of isofibrations between categories, and show that they…
It is known that strict omega-categories are equivalent through the nerve functor to complicial sets and to sets with complicial identities. It follows that complicial sets are equivalent to sets with complicial identities. We discuss these…
The category of strict omega-categories has an important full subcategory whose objects are the simple omega-categories freely generated by planar trees or by globular cardinals. We give a simple description of this subcategory in terms of…
We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an $\omega$-category that is suitably free and loop-free. An instance of the result recovers the fact that…
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes…
We introduce a notion of normalised oplax $3$-functor suitable for the elementary homotopy theory of strict $3$-categories, following the combinatorics of orientals. We show that any such morphism induces a morphism of simplicial sets…
Strong Steiner $\omega$-categories are a class of $\omega$-categories that admit algebraic models in the form of chain complexes, whose formalism allows for several explicit computations. The conditions defining strong Steiner…