Justin R. Smith
This paper constructs model structures on the categories of coalgebras and pointed irreducible coalgebras over an operad. The underlying chain-complex is assumed to be unbounded and the results for bounded coalgebras over an operad are derived from the unbounded case.
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We give an example of a non-trivial linear operad that only admits trivial coalgebras and give sufficient conditions ensuring that the cofree coalgebra functor be faithful.
This paper proves that homology equivalences of cogenerating complexes induce homology equivalences of the cofree coalgebras in many interesting cases. We show that the underlying chain complex of any cofree coalgebra is naturally a direct summand of the underlying chain-complex of a cofree coalgebra over a free operad. This is combined with the previous result to prove the homology invariance of all cofree coalgebras.
We give a recursive formula to compute the cofree coalgebra P^\vee(C) over any colored operad P in Set, CGHaus or (dg)Vect. The construction is closed to that of Smith but different. We use a more conceptual approach to simplify the proofs that P^\vee is the cofree P-coalgebra functor and also the comonad generating P-coalgebras. In a second part, when P is a linear or dg-operad over a field, we generalize the notion of representative functions of Block & Leroux and prove that P^\vee(C) is simply the subobject of representative elements in the "completed P-algebra" P^\wedge(C). This says that …
In this article, we describe how coalgebraic structures on operads induce algebraic structures on their categories of algebras and coalgebras.
Operadic twisting as an adjunction
本文引入了弯曲运算子扭转余单子，并研究其与运算子分裂的关系，同时给出了特定运算子的余结构公式，恢复了规范群作用。
We compare two models for $\infty$-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for $\infty$-operads are equivalent - for instance, it follows that the homotopy theory of Lurie's $\infty$-operads is equivalent to that of dendroidal sets and that of simplicial operads.
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 …
Cellular approximations for nonsymmetric homotopy operads
本文为拓扑着色非对称操作子构造CW同伦操作子及严格CW操作子的同伦同构，并推广至拓扑代数与有限CW复形图，增强Quillen伴随并嵌入预层范畴。
We extend bar-cobar duality, defined for operads of chain complexes by Getzler and Jones, to operads of spectra in the sense of stable homotopy theory. Our main result is the existence of a Quillen equivalence between the category of reduced operads of spectra (with the projective model structure) and a new model for the homotopy theory of cooperads of spectra. The crucial construction is of a weak equivalence of operads between the Boardman-Vogt W-construction for an operad P, and the cobar-bar construction on P. This weak equivalence generalizes a theorem of Berger and Moerdijk that says the…
We study extensively the homotopy theory of coalgebras. By coalgebras, we mean the full theory of coalgebras: with counits and not necessarily locally conilpotent. For example $\mathcal E_\infty$-coalgebras, $\mathcal A_\infty$-coalgebras, $\mathcal L_\infty$-coalgebras etc. To do so, we define the category of complete curved algebras -- where the notion of quasi-isomorphims does not make sense -- and endow it with a model category structure, equivalent to that of the category of coalgebras.
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.
Over suitable monoidal model categories, we construct a Dwyer-Kan model category structure on the category of algebras over an augmented operadic collection. As examples we obtain Dwyer-Kan model category structure on the categories of enriched wheeled props, wheeled properads, and wheeled operads, among others. This result extends known model category structure on the categories of operads, properads, and props enriched in simplicial sets and other monoidal model categories. We also show that our Dwyer-Kan model category structure is well behaved with respect to simultaneous changes of the un…
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.
The category of simplicial R-coalgebras over a presheaf of commutative unital rings on a small Grothendieck site is endowed with a left proper, simplicial, cofibrantly generated model category structure where the weak equivalences are the local weak equivalences of the underlying simplicial presheaves. This model category is naturally linked to the R-local homotopy theory of simplicial presheaves and the homotopy theory of simplicial R-modules by Quillen adjunctions. We study the comparison with the R-local homotopy category of simplicial presheaves in the special case where R is a presheaf of…
The homotopy theory of infinity-operads is defined by extending Joyal's homotopy theory of infinity-categories to the category of dendroidal sets. We prove that the category of dendroidal sets is endowed with a model category structure whose fibrant objects are the infinity-operads (i.e. dendroidal inner Kan complexes). This extends the theory of infinity-categories in the sense that the Joyal model category structure on simplicial sets whose fibrant objects are the infinity-categories is recovered from the model category structure on dendroidal sets by simply slicing over the point.
Point-set models for homotopy coherent coalgebras
本文通过归纳法证明，当操作子为cofibrant时，通过局部化微分格子代数得到的∞-范畴与基于富集∞-操作子的余代数∞-范畴等价，从而为E_n和E_∞余代数提供显式点集模型。
We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of $(\infty,1)$-categories and $\infty$-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structu…
Koszul duality for algebras over infinity-operads
本文提出了一种新的∞-操作符代数与余代数的科兹尔对偶理论，并基于树的范畴对偶定义了同调，扩展了经典操作符的对偶结果。
We describe the category of homotopy coalgebras, concentrating on properties of relatively cofree homotopy coalgebras, morphisms and coderivations from an ordinary coalgebra to a relatively cofree homotopy coalgebra, morphisms and coderivations between coalgebras of latter type. Cobar- and bar-constructions between counit-complemented curved coalgebras, unit-complemented curved algebras and curved homotopy coalgebras are described. Using twisting cochains an adjunction between cobar- and bar-constructions is derived under additional assumptions.
We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by …
We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.
We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched non-symmetric monoidal model categories.
We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted $\mathbf{U}$, plays a similar role for modular operads that the dendroidal category $Ω$ plays for operads. We carefully study properties of $\mathbf{U}$, including the existence of certain factorization systems. Related structures, such as cyclic operads and stab…
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 correct a mistake in the construction of push-outs along free morphisms of algebras over a nonsymmetric operad in arXiv:1101.1634 [math.AT], and we fix the affected results in arXiv:1101.1634 [math.AT] and arXiv:1304.6641 [math.AT].
We prove that given $\mathcal{C}$ a presentably symmetric monoidal $\infty$-category, and any essentially small $\infty$-operad $\mathcal{O}$, the $\infty$-category of $\mathcal{O}$-algebras in $\mathcal{C}$ is enriched, tensored and cotensored over the presentably symmetric monoidal $\infty$-category of $\mathcal{O}$-coalgebras in $\mathcal{C}$. We provide a higher categorical analogue of the universal measuring coalgebra. For categories in the usual sense, the result was proved by Hyland, López Franco, and Vasilakopoulou.
We apply the Acyclicity Theorem of Hess, Kerdziorek, Riehl, and Shipley (recently corrected by Garner, Kedziorek, and Riehl) to establishing the existence of model category structure on categories of coalgebras over comonads arising from simplicial adjunctions, under mild conditions on the adjunction and the associated comonad. We study three concrete examples of such adjunctions where the left adjoint is comonadic and show that in each case the component of the derived counit of the comparison adjunction at any fibrant object is an isomorphism, while the component of the derived unit at any 1…
We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy P-algebra in M is, provided only that some of the morphisms in M have been marked out as `homotopy equivalences'. The bulk of the paper consists of examp…
The goal of the present paper is to compare, in a precise way, two notions of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk--Weiss in order to generalize the simplicial notion of infinity-category of Joyal--Lurie. This functor extends in two directions the simplicial nerve of Faonte--Lurie for A-infinity-…
We extend the theory of d-categories, by providing an explicit description of the right mapping spaces of the d-homotopy category of an $\infty$-category. Using this description, we deduce an invariant $\infty$-categorical characterization of the d-homotopy category. We then proceed to develop an analogous theory of d-operads, which model $\infty$-operads with (d -1)-truncated multi-mapping spaces, and prove analogous results for them.
We show how to induce products in sheaf cohomology for a wide variety of coefficients: sheaves of dg commutative and Lie algebras, symmetric Omega-spectra, filtered dg algebras, operads and operad algebras.
Generalizing the approach to pseudo monoidal DG-categories as certain colored non-symmetric DG-operads, we introduce a certain relaxed notion of a category enriched in DG-categories. We construct model structures on the category of colored non-symmetric DG-operads and on the category of DGCat-enriched categories with a fixed set of objects. This allows us to talk about strong homotopy maps in both settings. We discuss the notion of a strong homotopy monad in a DG-category and a notion of strong homotopy adjunction data for two DG-functors.
We compare two approaches to the homotopy theory of infinity-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and infinity-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Gamma. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories.
Dwyer--Kan homotopy theory for cyclic operads
该研究提出对称幺半基范畴上着色循环 operad 的通用定义，借助其到着色 operad 的遗忘函子的左右伴随显式公式，将单纯集富集着色 operad 范畴的 Cisinski--Moerdijk 模型结构提升到对应循环 operad 范畴。
Profinite completions of topological operads
本文将拓扑算畴的特定有限完备纳入左 Quillen 函子框架，构建了有限同伦算畴的模型范畴，刻画了其底层 $\infty$-范畴与弱等价，并在 Quillen 等价背景下获得了类似结果。
The purpose of this foundational paper is to introduce various notions and constructions in order to develop the homotopy theory for differential graded operads over any ring. The main new idea is to consider the action of the symmetric groups as part of the defining structure of an operad and not as the underlying category. We introduce a new dual category of higher cooperads, a new higher bar-cobar adjunction with the category of operads, and a new higher notion of homotopy operads, for which we establish the relevant homotopy properties. For instance, the higher bar-cobar construction provi…
We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads, respectively. Algebras for these models are homotopy versions of the corresponding structures.
We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.
A universal characterization of the curved homotopy Lie and associative operads
该研究刻画了带权分次与特定微分的非对称 dg operad 范畴，证明其初始对象为弯曲 A-无穷 operad，伴随函子对应弯曲 A-无穷代数的 Maurer-Cartan 扭转，对称情形则对应弯曲 L-无穷 operad 及扭转过程。
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.
Enriched coalgebras are sometimes comonadic
本文引入了在对称单子V-范畴C中定义的余代数概念，当C是半笛卡尔时，构造了与P相关的V-函子，并在特定条件下证明其为V-余单子，从而计算V-余代数的范畴。
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。
Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold `up to homotopy' in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis of some examples. It is a much-abbreviated version of the paper `Homotopy Algebras for Operads' (math.QA/0002180), and does not assume any knowledge of operads.
This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher homotopies. We try to show how universal this theory is by giving many applications in Algebra, Geometry, Topology, and Mathematical Physics. (This text is accessible to any student knowing what tensor products, chain complexes, and categories are.)
We define the $m$th Veronese power of a weight graded operad $\mathcal{P}$ to be its suboperad $\mathcal{P}^{[m]}$ generated by operations of weight $m$. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads $\mathcal{P}^{[m]}$ are related by Koszul duality to operads describing strongly homotopy …
This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion of morphisms called infinity-morphisms. The method consists in using the operadic calculus to endow the category of coalgebras over the Koszul dual cooperad or the bar construction with a new type of model category structure, Quillen equivalent to that of algebras. We provide an explicit homotopy equivalence for infinity-morphisms, which gives a simple description of the homotopy category, and we endo…
Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for infinity-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and infinity-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a b…
Let K be a comonad on a model category M. We provide conditions under which the associated category of K-coalgebras admits a model category structure such that the forgetful functor to M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf-Galois extensions. These examples are specific instances of the following categories of comodules over a coring. For any semihereditary commutative ring R, let A be a dg R-algebra that is homologically simply connected. Let V be an A-coring that …
The notion of 2--monoidal category used here was introduced by B.~Vallette in 2007 for applications in the operadic context. The starting point for this article was a remark by Yu. Manin that in the category of quadratic algebras (that is, "quantum linear spaces") one can also define 2--monoidal structure(s) with rather unusual properties. Here we give a detailed exposition of these constructions, together with their generalisations to the case of quadratic operads. Their parallel exposition was motivated by the following remark. Several important operads/cooperads such as genus zero quantum c…
Open Dynamical Systems as Coalgebras for Polynomial Functors, with Application to Predictive Processing
将具有一般时间演化的开放动力系统范畴化为多项式接口反索引的余代数范畴，扩展余代数框架与operad Org以构建分层开放系统，并以拉普拉斯学说为例展示其在主动推断等科学应用中的动力学语义。
In our recent papers [Sh1,2], we introduced a {\it twisted tensor product} of dg categories, and provided, in terms of it, {\it a contractible 2-operad $\mathcal{O}$}, acting on the category of small dg categories, in which the "natural transformations" are derived. We made use of some homotopy theory developed in [To] to prove the contractibility of the 2-operad $\mathcal{O}$. The contractibility is an important issue, in vein of the theory of Batanin [Ba1,2], according to which an action of a contractible $n$-operad on $C$ makes $C$ a weak $n$-category. In this short note, we provide a new e…
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