Yuki Maehara
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.
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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 combinatorial description of the morphisms between orientals. We also show that the category of orientals is the closure of the category of simplexes under certain filler operations which represent complicial operations.
Orientals as free algebras
本文基于 Burroni 思想，通过定义严格 ω-范畴上的扩张并证明其遗忘函子是单子的，从空范畴迭代单子构造出余单纯对象，并利用 Steiner 理论证明该对象即为 Street 的 orientals 余单纯对象。
A concrete model of the free skew-monoidal category Fsk on a single generating object is obtained. The situation is clubbable in the sense of G.M. Kelly, so this allows a description of the free skew-monoidal category on any category. As the objects of Fsk are meaningfully bracketed words in the skew unit I and the generating object X, it is necessary to examine bracketings and to find the appropriate kinds of morphisms between them. This leads us to relationships between triangulations of polygons, the Tamari lattice, left and right bracketing functions, and the orientals. A consequence of ou…
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.
We show that the Street nerve of a strict $ω$-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 $\mathcal{N}(C)$ with a structure of saturated complicial set where the $n$-simplices correspond to morphisms from the $n^{th}$ oriental to $C$ sending the unique non-trivial $n$-cell of the domain to a weakly invertible cell of $C$.
Free skew monoidal categories
本文显式刻画单个生成对象上的自由幺半范畴 Fsk，并推广到任意范畴上的自由斜幺半范畴，进而建立相关 operad 间的伴随及与 colax L-代数的对应。
Nerves and cones of free loop-free ω-categories
本文研究自由无环 ω-范畴中复形神经与两类锥构造的同伦相容性，并据此恢复标准 m-单形与 m-oriental 复形神经等价的结论。
This paper is part of a series of papers about homotopy theory of strict $n$-categories. In the first paper of this series, we gave conditions that guarantee the existence of a Thomason model category structure on the category of strict $n$-categories. The main goal of our paper is to show one of these conditions. To do so, we associate to any simplicial complex a strict $\infty$-category generated by a computad. We conjecture that this $\infty$-category has the same homotopy type as the corresponding simplicial complex and we prove this conjecture when the simplicial complex comes from a pose…
An Oriented Street--Roberts Conjecture
提出有向多胞体概念（包括 Street 有向单形和 Gray 有向立方体），证明有向 Street–Roberts 猜想，将 (∞,∞)-范畴表示为合适有向多胞体族上的层，从几何视角理解 (∞,∞)-范畴作为同伦型的有向类比。
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.
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 $\mathcal{A}$,…
Admissible weak factorization systems on extriangulated categories
本文在外三角范畴中引入容许弱分解系统，建立了其与余挠对的双射，并在特定条件下通过该系统证明了遗传余挠对与相容余挠对的等价关系，推广了已有结果。
$ω$-weak equivalences between weak $ω$-categories
研究Batanin-Leinster意义下弱ω-范畴间的ω-弱等价，定义其为各维度本质满射的严格ω-函子（与严格ω-范畴模型范畴弱等价一致），证明其及Garner意义下弱ω-函子推广形式均具2-out-of-3性质。
Involutive Weak Cubical $ω$-categories
本文采用Penon的单子代数方法定义并研究对合弱立方 $ω$-范畴，并提供了若干具体例子。
Doubly weak double categories
本文提出了一种双范畴的定义，其中1-胞腔在两个方向上的复合都是弱的，并通过双计算域、隐式双范畴以及整理条件等不同方式刻画了这种双弱双范畴。
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…
An $(\infty,n)$-categorical pasting theorem
本文证明了一个 (∞,n)-范畴粘贴定理，识别出严格 n-范畴中被包含函子保持的一大类推出，适用于无挠复形，为 Henry 猜想提供部分进展，并可用于构造 (∞,n)-范畴新模型和 Gray 张量积。
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.
Computads with invertible generators for weak ω-categories
扩展弱ω-范畴的computad概念以允许标记可逆生成元，并归纳描述它们生成的自由ω-范畴，从而给出行走等价和分类可逆胞腔的ω-范畴的简单有限描述。
Quantaloid-enriched categories: Factorization, weak classifiers, and symmetry
本文研究了quantaloid-enriched范畴的基础性质，证明了其左伴随分配子范畴存在(epi, extremal mono)-分解系统，并在稳定性条件下满足弱子对象分类子公理，进而将结构结果推广至quantale值集。
$ω$-equifibrations between strict and weak $ω$-categories
本文研究Batanin-Leinster意义下弱ω-范畴间的ω-等纤维化，通过严格ω-函子集合J的右提升性质刻画该概念，证明相关构造与已有结论一致，且严格ω-范畴间的该类纤维化与folk模型结构纤维化重合。
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.
Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful "fibre functors" to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras with weak Hopf algebras and strong monoidal functors with separable Frobenius monoidal functors; second, we replace the category of vector spaces with an arbitrary braided monoidal category. To accomplish thi…
There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened -- these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category $Δ$ is replaced by a certain `fat' delta of `coloured ordinals', where the degeneracy maps are only up to homotopy. The first part of this exposition is aimed at a broad mathematical readership and contains also a brief introduction to simplicial viewpoints on higher categori…
Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke or D. Dugger. We are contributing to this endeavour by proving that weak equivalences in a combinatorial model category form an accessible category. We also present some new results about weak equivalences and cofibrations in combinatorial model categories.
We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith's result to left Bousfield localizations of categories of monoids in a suitable monoidal model category.
Weakly invertible cells in a weak $ω$-category
本文采用余归纳定义研究Batanin-Leinster弱$ω$-范畴中的弱可逆胞腔，证明其在球状粘贴下封闭，并将严格$ω$-范畴中的基本性质推广至弱$ω$-范畴，进而证明每个弱$ω$-范畴都存在最大的弱$ω$-子群胚。
We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to the cartesian monoidal structure. As these are a form of tricategory it follows that doubly-degenerate ones are braided monoidal categories. We show that this form of semi-strict tricategory is weak enough to produce all braided monoidal categories. That is, given any braided monoidal category $B$ there is a doubly-degenerate ``vertically weak'' semi-strict tricategory whose associated braided m…
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 how to "interleave" the monad for operads and the monad for contractions on the category \coll of collections, to construct the monad for the operads-with-contraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit mutual stability conditions that enable us to alternate the dimension-by-dimension free functors. Hence we give an explicit construction of a left adjoint for the forgetful functor $\owc \lra \coll$, fr…
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…
In this note, we study the connection between Gray tensor product and suspension. We derive a characterization of weak equivalences as fully faithful and essentially surjective functors. We construct the $co$ duality, a weak involution that reverses the direction of even dimensional cells. We conclude by studying Grothendieck fibrations of complicial sets.
Model categories have long been a useful tool in homotopy theory, allowing many generalizations of results in topological spaces to other categories. Giving a localization of a model category provides an additional model category structure on the same base category, which alters what objects are being considered equivalent by increasing the class of weak equivalences. In some situations, a model category where the class of weak equivalences is restricted from the original one could be more desirable. In this situation we need the notion of a delocalization. In this paper, right Bousfield deloc…
Computads for weak $ω$-categories as an inductive type
本文通过自由词的显式归纳定义给出弱 $\omega$-范畴的 computads 新描述，避免球状操作技术，并证明新定义与 Batanin-Leinster 及 Garner 的结果一致。
We prove that a weak factorization system on a locally presentable category is accessible if and only if it is small generated in the sense of R. Garner. Moreover, we discuss an analogy of Smith's theorem for accessible model categories.
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.
Weakly globular double categories and weak units
本文建立弱球状双范畴与 fair 2-范畴的直接比较，证明二者在关于 2-等价局部化后等价，并揭示其严格结合复合与弱单位结构。
We show that the strict 1-category $\square$ of cubes -- defined to be the full subcategory of strict $ω$-categories whose objects are the Gray tensor powers of the arrow category -- are dense in the $(\infty,1)$-category $\mathsf{Cat}_ω$ of weak $(\infty,\infty)$-categories, in both Rezk-complete and incomplete variants. More precisely, we show that Joyal's category $Θ$ is contained in the idempotent completion of $\square$, and in fact that the idempotent completion of $\square$ is closed under suspensions and wedge sums. This result extends a theorem of Campbell and Maehara in dimension 2. …
An enriched small object argument over a cofibrantly generated base
本文提出一种在余纤维生成弱分解系统上富化的小对象论证变体，统一了范畴及多种高阶范畴中的小对象论证方法。
Homotopy theories of $(\infty, \infty)$-categories as universal fixed points with respect to enrichment
本文证明了具归纳与余归纳等价的$(\infty, \infty)$-范畴满足弱富化泛性质，分别构成局部可表示不动点子范畴的始对象与富化不动点范畴的终对象。
Let $\mathscr{C}$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. We introduce the notions of left Frobenius pairs, left ($n$-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to $ξ$ in $\mathscr{C}$. We show how to construct left cotorsion pais from left $n$-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. We also study the relation between a certain class of cotorsion pairs and that of $n$-cotorsion pairs. These work generalize Ma-Zhao-Huang's results…
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…
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 develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R \otimes R.
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 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.
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.
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 develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and classify module categories over the fusion category of $\hat{sl}(2)$ at a positive integer level where we meet once again ADE classification pattern.
We develop the theory of weak Fraisse categories, where the crucial concept is the weak amalgamation property, discovered relatively recently in model theory. We show that, in a suitable framework, every weak Fraisse category has its unique limit, a special object in a bigger category, characterized by certain variant of injectivity. This significantly extends the present theory of Fraisse limits.
Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg-Moore category of the induced monad. Making appropriate ass…
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…
We consider four categories: the category of diagrams of small categories indexed by a given small category O, the (comma) category of small categories over O, the category of diagrams of simplicial sets indexed by O, and the category of simplicial sets over the nerve of O. Fritsch and Golasinski claimed that these four categories have equivalent homotopy categories but, in fact, their proof contains an error and the homotopy categories are not equivalent with the weak equivalences they use in the comma categories. We show here that the correct weak equivalences are the ``weak fibre homotopy e…
Incidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner's bigraphs and crossed modules are considered.
This thesis is devoted to the proof of a theorem showing the existence of a closed model category structure for weakly enriched categories. It requires first of all the definitions of weakly enriched categories and equivalences of weakly enriched categories such that these definitions recover some existing notions of higher order weak categories, for example Segal categories, Tamsamani n-categories and strict n-categories. In order to prove our theorem, we elaborate a theory of plans for cell addition following the approach of the small object argument a la Quillen. We conclude this work with …
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 …
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