Tom Leinster
I exhibit a pair of non-symmetric operads that, although not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if `algebraic theory' is understood as meaning `monad', operads cannot be regarded as algebraic theories of a special kind.
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We show that Lurie's model for $\infty$-operads (or more precisely a "flagged" or "pinned" version thereof) is equivalent to the analytic monads previously studied by Gepner, Kock, and the author, with an $\infty$-operad $\mathcal{O}$ corresponding to the monad for $\mathcal{O}$-algebras in spaces. In particular, the $\infty$-operad $\mathcal{O}$ is completely determined by this monad. To prove this we study the Lawvere theories of analytic monads, and show that these are precisely pinned $\infty$-operads in a slight (equivalent) variant of Lurie's definition, where finite pointed sets are rep…
Generalizing a result of Dwyer and Kan for simplicial categories, we characterize the morphisms of multi-sorted simplicial algebraic theories and simplicial coloured operads which induce a Quillen equivalence between the corresponding categories of algebras.
Relative operads model $\infty$-operads
本文通过局部化证明相对 operad 与 ∞-operad 的同伦理论等价，并解决了 Lurie 的 operadic nerve 神经 functor 的等价性问题。
We give a framework for comparing on the one hand theories of n-categories that are weakly enriched operadically, and on the other hand n-categories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory of n-categories of the former kind and produce a globular operad whose algebras are the n-categories we started with. We first provide a generalisation of Trimble's original theory that allows for the…
Envelopes for Algebraic Patterns
本文提出了一种代数模式的包络构造，展示了其在不同代数模式下的无穷操作符的等价性，特别证明了G-无穷操作符与纤维模式的等价关系。
We generalize the operadic approach to algebraic quantum field theory [arXiv:1709.08657] to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful q…
In this article, we describe how coalgebraic structures on operads induce algebraic structures on their categories of algebras and coalgebras.
We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain …
Given an algebraic theory which can be described by a (possibly symmetric) operad $P$, we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for $P$-algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature. Using this definition, we generalize the result that every monoidal category is monoidally equivalent to a strict monoidal category, and show that the "strictification" functor has an i…
We relate the operad FMan controlling the algebraic structure on the tangent sheaf of an $F$-manifold (weak Frobenius manifold) defined by Hertling and Manin to the operad PreLie of pre-Lie algebras: for the filtration of PreLie by powers of the ideal generated by the Lie bracket, the associated graded object is FMan.
Kleisli bicategories are a natural environment in which the combinatorics involved in various notions of algebraic theory can be handled in a uniform way. The setting allows a clear account of comparisons between such notions. Algebraic theories, symmetric operads and nonsymmetric operads are treated as examples.
The notion of a coalgebra measuring, introduced by Sweedler, is a kind of generalized ring map between algebras. We begin by studying maps on Hochschild homology induced by coalgebra measurings. We then introduce a notion of coalgebra measuring between Lie algebras and use it to obtain maps on Lie algebra homology. Further, these measurings between Lie algebras satisfy nice adjoint like properties with respect to universal enveloping algebras. More generally, we introduce and undertake a detailed study of the notion of coalgebra measuring between algebras over any operad $\mathcal O$. In case …
In this paper, we study algebras over the little cubes operads introduced by Boardman and Vogt, using the formalism of higher category theory.
Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field) theory, we develop the involutive analogs of colored operads and their algebras, named colored $\ast$-operads and $\ast$-algebras. Central to the definition of colored $\ast$-operads is the involutive monoidal category of symmetric sequences, which we obtain from a general product-exponential $2$-adjunction whose right adjoint forms involutive functor categorie…
In this paper we study spaces of algebras over an operad (non-symmetric) in symmetric monoidal model categories. We first compute the homotopy fiber of the forgetful functor sending an algebra to its underlying object, extending a result of Rezk. We then apply this computation to the construction of geometric moduli stacks of algebras over an operad in a homotopical algebraic geometry context in the sense of Toën and Vezzosi. We show under mild hypotheses that the moduli stack of unital associative algebras is a Zariski open substack of the moduli stack of non-necessarily unital associative al…
I show that the theories of enrichment in a monoidal infinity-category defined by Hinich and by Gepner-Haugseng agree, and that the identification is unique. Among other things, this makes the Yoneda lemma available in the former model.
In this paper, we use the language of operads to study open dynamical systems. More specifically, we study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones. The syntactic architecture of such interconnections is encoded using the visual language of wiring diagrams. We define the symmetric monoidal category W, from which we may construct an operad O(W), whose objects are black boxes with input and output ports, and whose morphisms are wiring diagrams, thus prescribing the algebraic rules for interconnection. We then define two W-algebras, G an…
We survey Lawvere theories at the level of infinity categories, as an alternative framework for higher algebra (rather than infinity operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They are also better-suited than operads for equivariant homotopy theory and its relatives. Our main result establishes a universal property for the infinity category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its infinity category of models. Many familiar properties of Lawvere theories follow directly. …
Day convolution for algebraic patterns
本文通过扩展Day卷积到更广泛的结构，研究了广泛存在于∞-范畴代数中的可指数化对象，提出了适用于如等价∞-操作子和虚拟双∞-范畴的必要且充分条件。
Plus constructions, plethysm, and unique factorization categories with applications to graphs and operad-like theories
本文在范畴论框架下推广Baez-Dolan型Plus构造，证明其核心性质，并引入唯一分解范畴，解释operads等结构的幺半群表述存在性，为图论与operad-like理论提供统一视角。
The $ω$-categorification of Algebraic Theories
本文通过将Leinster的球状operad扩展为球状PRO并引入收缩结构，构造了一个monad，其代数为任意等式代数理论的弱$ω$-范畴化，其中初始弱化给出完全弱化的$ω$-范畴化。
Generalized operads, also called generalized multicategories and $T$-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study functoriality of the Kleisli construction, and correspondingly that of generalized operads. Motivated by this problem we develop a lax version of the formal theory of monads, and study its connection to …
Homotopical operadic calculus in positive characteristic
本文针对代数 operad 理论长期局限于特征零的问题，引入 quasi-planar cooperad 作为构建基石，将其同伦演算的主要结果与方法推广到正特征。
Higher structures - infinity algebras and other objects up to homotopy, categorified algebras, `oidified' concepts, operads, higher categories, higher Lie theory, higher gauge theory... - are currently intensively investigated in Mathematics and Physics. The present thesis deals with abstractions and flexibilizations of algebraic structures, and more precisely, with algebraic operads, homotopification and categorification.
It is well known since Stasheff's work that 1-fold loop spaces can be described in terms of the existence of higher homotopies for associativity (coherence conditions) or equivalently as algebras of contractible non-symmetric operads. The combinatorics of these higher homotopies is well understood and is extremely useful. For $n \ge 2$ the theory of symmetric operads encapsulated the corresponding higher homotopies, yet hid the combinatorics and it has remain a mystery for almost 40 years. However, the recent developments in many fields ranging from algebraic topology and algebraic geometry to…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this paper, we develop a unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category,…
This paper describes a higher-categorical version of the theory of colored operads, giving applications to the study of commutative ring spectra.
Enriched homotopy-coherent structures
本文基于“代数模式”框架，为Segal条件描述的同伦相干代数结构引入一般性富化概念，恢复了已知的富化∞-范畴等实例，并提出了富化模∞-代数操作等新实例。
Operads and the Markov Property on the square lattice
本文利用算子代数语言为任意维度的格点马尔可夫过程提供了完整的代数描述，提出了边界条件的新参数化方法，并利用Kolmogorov扩张构造了平移不变的无限体积Gibbs测度。
We unravel the algebraic structure which controls the various ways of computing the word ((xy)(zt)) and its siblings. We show that it gives rise to a new type of operads, that we call permutads. It turns out that this notion is equivalent to the notion of "shuffle algebra" introduced by the second author. It is also very close to the notion of "shuffle operad" introduced by V. Dotsenko and A. Khoroshkin. It can be seen as a noncommutative version of the notion of nonsymmetric operads. We show that the role of the associahedron in the theory of operads is played by the permutohedron in the theo…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as symmetric operads, non-symmetric operads, generalised operads, and monads. These variants of universal algebra are called notions of algebraic theory. In the first part of this thesis, we develop a unified framework for notions of algebraic theory which includes all of the above examples. Our key observation is that each notion of al…
Since its introduction by Loday in 1995 with motivation from algebraic K-theory, dendriform dialgebras have been studied quite extensively with connections to several areas in mathematics and physics. A few more similar structures have been found recently, such as the tri-, quadri-, ennea- and octo-algebras, with increasing complexity in their constructions and properties. We consider these constructions as operads and their products and duals, in terms of generators and relations, with the goal to clarify and simplify the process of obtaining new algebra structures from known structures and f…
Noetherian forms of free non-symmetric operads
本文研究带标签有限有根有序树范畴，通过正合性性质刻画其同构类，并证明附加严格初始对象后具有诺特形式。
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