M. A. Batanin
We introduce a category of locally constant $n$-operads which can be considered as the category of higher braided operads. For $n=1,2,\infty$ the homotopy category of locally constant $n$-operads is equivalent to the homotopy category of classical nonsymmetric, braided and symmetric operads correspondingly.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
This short note reports on joint work with Michael Batanin towards a general machine for proving Baez-Dolan Stabilization Theorems for various models of higher categories, based on substitudes, Bousfield localization, and homotopical Beck-Chevalley squares. I provide a road map to our recent papers, and include new results proving Baez-Dolan Stabilization Theorems for Tamsamani weak $n$-categories, higher Segal categories, Ara's $n$-quasi-categories, and cartesian models of Segal and complete Segal objects due to Bergner and Rezk. I also attempt to clarify the connection to higher braided oper…
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.
A general notion of operad is given, which includes as instances, the operads originally conceived to study loop spaces, as well as the higher operads that arise in the globular approach to higher dimensional algebra. In the framework of this paper, one can also describe symmetric and braided analogues of higher operads, likely to be important to the study of weakly symmetric, higher dimensional monoidal structures.
The structure of a $k$-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schwänzl and Vogt can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of ($n$-fold) operads in a $k$-fold monoidal category which generalizes the definition of operads in a braided category. Furthermore, the inheritance of structure by the category of operads is actually an inheritance of iterated monoidal structure, decremented by at least two iterations. We prove that the category of $n$-fold oper…
Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce a large family of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras …
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. This is the first book on the subject and lays its foundations. Many examples are given throughout. There is also an introductory chapter motivating the subject for topologists.
We propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we proof a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type $F_\infty$. This unifies and extends existing proofs that certain Thompson-like groups are of type $F_\infty$.
We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for $\mathtt{W}=\mathtt{W}_{\infty}$ the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebra…
One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product [12] of Gray categories. Moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations, and we illustrate this by giving an expedited account of many aspects of Cheng's analysis [11] of Trimb…
The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its $Hom$-set is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is `the same' as a braided monoidal category. In this p…
We extend Thomason's homotopy colimit construction in the category of permutative categories to categories of algebras over an arbitrary $\Cat$ operad and analyze its properties. We then use this homotopy colimit to prove that the classifying space functor induces an equivalence between the category of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces after formally inverting certain classes of weak equivalences, where $\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an equivalence of the categories of $n$-fold monoidal categories and the categor…
A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and $n$-operad correspond to unital moment…
The higher algebra and geometry of monoidal bicategories
该研究通过分析小立方操作符的几何，揭示了辫状、硅勒克和对称单调双范畴作为$\mathsf{E}_k$-单子的结构特性。
We study bicolored configurations of points in the Euclidean $n$-space that are constrained to remain either inside or outside a fixed Euclidean $m$-subspace, with $n - m \ge 2$. We define a higher-codimensional variant of the Swiss-Cheese operad, called the complementarily constrained disks operad $\mathsf{CD}_{mn}$, associated to such configurations. The operad $\mathsf{CD}_{mn}$ is weakly equivalent to the operad of locally constant factorization algebras on the stratified space $\{\mathbb{R}^{m} \subset \mathbb{R}^{n}\}$. We prove that this operad is formal over $\mathbb{R}$.
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. Although it can be treated purely as an algebraic subject, it is inherently topological in nature: the higher-dimensional diagrams one draws to represent these structures can be taken quite literally as pieces of topology. Examples of this are the braids in a braided monoidal category, and the pentagon which appears in the definitions of both monoidal category and A_infinity space. I will try to give a Friday-afternoonish description of some of the dreams peo…
A braided monoidal $(\infty,2)$-category of Soergel bimodules
本文首次将 Hecke 辫子幺半范畴范畴化为 $\mathbb{E}_2$-幺半 $(\infty,2)$-范畴，基于 Soergel 双模链复形，旨在控制同伦相干链环同调及 4/5 维 TQFT，并发展了分解系统工具。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。