We develop a theory of weak omega categories that will be accessible to anyone who is familiar with the language of categories and functors and who has encountered the definition of a strict 2-category.
The most remarkable feature of this theory is its simplicity. We build upon an idea due to Jacques Penon by defining a weak omega category to be a span of omega magmas with certain properties. (An omega magma is a reflexive, globular set with a system of partially defined, binary composition operations which respects the globular structure.)
Categories, bicategories, strict omega categories and Penon's weak omega categories are all instances of our weak omega categories. We offer a heuristic argument to justify the claim that Batanin's weak omega categories also fit into our framework.
We show that the Baez-Dolan stabilization hypothesis is a direct consequence of our definition of weak omega categories.
We define a natural notion of a pseudo-functor between weak omega categories and show that it includes the classical notion of a homomorphism between bicategories. In any weak omega category the operation of composition with a fixed 1-cell defines such a pseudo-functor.
Finally, we define a notion of weak equivalence between weak omega categories which generalizes the standard definition of an equivalence between ordinary 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 omega-categories, showing that the universal and canonical cofibrant replacement of the operad for strict omega-categories is precisely Leinster's operad for weak omega-categories.
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 homotopy theory and the Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier and Porter's homotopy coherent nerve, and show that this faithfully represents complicial Gray-categories as weak complicial sets. The category of weak complicial sets may itself be canonically enriched to a compl…
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 construct an equivalence between the categories of strict omega-categories and of sets with complical identities.
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 equivalences. In particular we give a conceptual proof that the nerves of omega-categories are complicial sets, and a direct proof that complicial sets are sets with complicial identities.
We study abelian group objects in $ω$-categories and discuss the well-known Dold-Kan correspondence from the perspective of $ω$-categories as a model for strict $\infty$-categories. The first part of the paper is intended to compile results from the existing literature and to fill some gaps therein. We go on to consider a parameterized Dold-Kan correspondence, i.e. a Dold-Kan correspondence for presheaves of $ω$-categories. The main result is to describe the descent or sheaf condition in terms of a glueing condition that is familiar for 1 and 2-stacks.
We revisit Kapranov and Voevodsky's idea of spaces modelled on combinatorial pasting diagrams, now as a framework for higher-dimensional rewriting and the basis of a model of weak omega-categories. In the first part, we elaborate on Steiner's theory of directed complexes as a combinatorial foundation. We individuate convenient classes of directed complexes and develop the theory of diagrammatic sets relative to one such class. We study a notion of equivalence internal to a diagrammatic set, and single out as models of weak omega-categories those diagrammatic sets whose every composable diagram…
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 omega-category in the sense of Leinster. Precisely, we construct a contractible globular operad of definable composition laws, and give an action of this operad on the terms of any type and its identity types.
We establish a model structure on the category of strict omega-categories. The constructions leading to the model structure in question are expressed entirely within the scope of omega-categories, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while cofibrant objects are exactly the free ones. Our model structure transfers to n-categories along right-adjoints, for each n, thus recovering the known cases n = 1 and n = 2.
A short introduction to Grothendieck weak omega-groupoids is given. Our aim is to give evidence that, in certain contexts, this simple language is a convenient one for constructing globular weak omega-groupoids. To this end, we give a short reworking of van den Berg and Garner's construction of a Batanin weak omega-groupoid from a type using the language of Grothendieck weak omega-groupoids.
We define a notion of weak omega-category internal to a model of Martin-Löf type theory, and prove that each type bears a canonical weak omega-category structure obtained from the tower of iterated identity types over that type. We show that the omega-categories arising in this way are in fact omega-groupoids.
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 sequence of strict omega-categories, called orientals, which are associated to simplexes. In this paper we give a detailed algebraic description of the morphisms between orientals. The aim is to describe complicial sets algebraically, by operators and equational axioms.
Clemens Berger showed that Weak Omega Categories of Michael Batanin can be defined as model of a certain kind of theories that he called "homogeneous theories". By using the work of Mark Weber on the Abstract Nerves for the specific case of the $n$-Transformations ($n\in {\mathbb{N}^*}$), we show that we can also define Weak Omega Functors, Weak Omega Natural Transformations, and so on, as models of certain kind of colored theories which are homogeneous as well.
We discuss the dendroidal notion of weak higher categories introduced by Moerdijk and Weiss and we prove that dendroidal weak 2-categories are equivalent to bicategories.
Developing an idea of Kapranov and Voevodsky, we introduce a model of weak omega-categories based on directed complexes, combinatorial presentations of pasting diagrams. We propose this as a convenient framework for higher-dimensional rewriting. We define diagrammatic sets to be presheaves on a category of directed complexes presenting pasting diagrams with spherical boundaries. Diagrammatic sets have structural face and degeneracy operations, but no structural composition. We define a notion of equivalence cell in a diagrammatic set, and say that a diagrammatic set is representable if all pas…
We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories. This provides a more flexible way to realize E-2 spaces categorically.
We introduce a notion of "weak model category" which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences and most of the usual construction of categorical homotopy theory. Both left and right semi-model categories are weak model categories, and the opposite of a weak model category is again a weak model category. The main advantages of weak model categories is that they are easier to construct than Quillen model categories. In particular we give some simple criteria on two weak factorization systems f…
We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $\infty$-categories if $n \geq 2$, and the homotopy $n$-categories of presentable stable $\in…
In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to we…
We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of "homotopy category", that freely adjoins a section for every "acyclic fibration" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of …
In [BaSc2] the authors introduced a much weaker homotopical structure than a model category, called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The purpose of this paper is to serve as a companion to the papers above, proving results which say that if a certain property or structure exists in the weak cofibration category, then the same property or structure also holds in the induced model structure on the …
Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.
We study a new type of higher categorical structure, called weakly globular n-fold category, previously introduced by the author. We show that this structure is a model of weak n-categories by proving that it is suitably equivalent to the Tamsamani-Simpson model. We also introduce groupoidal weakly globular n-fold categories and show that they are algebraic models of n-types.
This paper introduces the notion of weakly globular double categories, a particular class of strict double categories, as a way to model weak 2-categories; it explores its use in defining a double category of fractions, and shows that the sub-2-category of groupoidal weakly globular double categories forms an algebraic model of homotopy 2-types.
The Euler characteristic was defined for finite strict n-categories by Leinster using the theory of enriched categories. This was an extension of some of his earlier work, which defined Euler characteristic for finite categories. Building on Leinster's work, we extend the notion of Euler characteristic to certain finite weak 2-categories and present a sketch of a similar extension to weak n-categories. We also discuss the extension of the Euler characteristic to certain finite $\left(\infty,1\right)$-categories.
Classical definitions of weak higher-dimensional categories are given inductively; for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak $n$-categories for all natural numbers $n$, or of weak $ω$-categories, take more sophisticated approaches, and the nature of the "hom" is often not immediate from the definitions. In this paper, we focus on Leinster's definition of weak $ω$-category based on an earlier definition by Batanin, and construct for each weak $ω$-category $\mathcal{A}$,…
This paper answers the following question: what algebraic structure on a category corresponds to an $A_n$ structure (in the sense of Stasheff) on the geometric realization of its nerve?
The orientals are the free strict $ω$-categories on the simplices introduced by Street. The aim of this paper is to show that they are also the free weak $ω$-categories on the same generating data. More precisely, we exhibit the Street nerves of the orientals as fibrant replacements of the simplices in Verity's model structure for complicial sets.
We investigate the notion of involutive weak globular $ω$-categories via Jacque Penon's approach. In particular, we give the constructions of a free self-dual globular $ω$-magma, of a free strict involutive globular $ω$-category, over an $ω$-globular set, and a contraction between them. The monadic definition of involutive weak globular $ω$-categories is given as usual via algebras for the monad induced by a certain adjunction. In our case, the adjunction is obtained from the "free functor" that associates to every $ω$-globular set the above contraction. Some examples of involutive weak globul…
In this work we discuss a new type of factorisation systems for \textbf{Ord}-enriched categories. We start by defining the new notion of lax weak orthogonality, which involves the existence of lax diagonal morphisms for lax squares. Using the usual theory of factorisation systems as a blueprint, we introduce a lax version of weak and functorial factorisation systems. We provide a characterisation of lax functorial weak factorisation systems, namely lax weak factorisation systems such that their factorisations are lax functorial. Then, we present a particular case of such lax functorial weak fa…
Higher categorical structures are often defined by induction on dimension, which a priori produces only finite-dimensional structures. In this paper we show how to extend such definitions to infinite dimensions using the theory of terminal coalgebras, and we apply this method to Trimble's notion of weak n-category. Trimble's definition makes explicit the relationship between n-categories and topological spaces; our extended theory produces a definition of Trimble infinity-category and a fundamental infinity-groupoid construction. Furthermore, terminal coalgebras are often constructed as limits…
Lack described a Quillen model structure on the category GrayCat of Gray-categories and Gray-functors, for which the weak equivalences are the weak 3-equivalences. In this note, we adapt the technique of Gurski, Johnson, and Osorno to show the localization of GrayCat at the weak equivalences is equivalent to the category of algebraic tricategories and pseudo-natural equivalence classes of weak 3-functors.
We introduce a weakening of the notion of fusion 2-category given in arXiv:1812.11933. Then, we establish a number of properties of (multi)fusion 2-categories. Finally, we describe the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories.
A combinatorial category Disks was introduced by André Joyal to play a role in his definition of weak omega-category. He defined the category Theta to be dual to Disks. In the ensuing literature, a more concrete description of Theta was provided. In this paper we provide another proof of the dual equivalence and introduce various categories equivalent to Disk or Theta, each providing a helpful viewpoint. In this second version the paper's contents have been reorganized with the goal of a more readable presentation. We define augmented categories and their reduced counterparts (which lack a sin…
Computads with invertible generators for weak ω-categories
Laboratoire Méthodes Formelles, CNRS, ENS Paris-Saclay, Université Paris-Saclay（形式方法实验室，法国国家科学研究中心，巴黎萨克雷高等师范学校，巴黎萨克雷大学）；Department of Computer Science and Technology, University of Cambridge（剑桥大学计算机科学与技术系）
