Justin R. Smith
This paper explicitely constructs cofree coalgebras over operads in the category of DG-modules. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). It is shown that the existence of an operad-action on a coalgebra implies a ``generalized coassociativity'' that facilitates the construction. Some areas of application are discussed.
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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 …
This paper considers a class of coalgebras over the Barratt-Eccles operad and shows that they classify Z-completions of pointed, reduced simplicial sets. As a consequence, they encapsulate the homotopy types of nilpotent simplicial sets. This result is a direct generalization of Quillen's result characterizing rational homotopy types via cocommutative coalgebras.
A recognition principle for iterated suspensions as coalgebras over the little cubes operad
本文提出了一种识别作为小立方体操作元的迭代悬垂的原理，通过构造余代数并证明其同伦等价性，揭示了与May结果的对偶关系。
The aim of this paper is to construct an $E_\infty$-operad inducing an $E_\infty$-coalgebra structure on chain complexes with coefficients in $\mathbb{Z}$, which is an alternative description to the $E_\infty$-coalgebra by the Barrat-Eccles operad.
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 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.
We study a special type of $E_\infty$-operads that govern strictly unital $E_\infty$-coalgebras (and algebras) over the ring of integers. Morphisms of coalgebras over such an operad are defined by using universal $E_\infty$-bimodules. Thus we obtain a category of $E_\infty$-coalgebras. It turns out that if the homology of an $E_\infty$-coalgebra have no torsion, then there is a natural way to define an $E_\infty$-coalgebra structure on the homology so that the resulting coalgebra be isomorphic to the initial $E_\infty$-coalgebra in our category. We also discuss some invariants of the $E_\infty…
We study the Eckmann-Hilton dual of the little disks algebra structure on iterated loop spaces: With the right definitions, every $n$-fold suspension is a coalgebra over the little $n$-disks operad. This structure induces non-trivial cooperations on the rational homotopy groups of an $n$-fold suspension. We describe the Eckmann-Hilton dual of the Browder bracket, which is a cooperation that forms an obstruction for an $n$-fold suspension to be an $(n+1)$-fold suspension, i.e. if this cooperation is non-zero then the space is not an $(n+1)$-fold suspension. We prove several results in equivaria…
We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second one we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson…
From the `cofree' cooperad $T'(A[-1])$ on a collection $A$ together with a differential, we construct an $L_\infty$-algebra structure on the total space $\bigoplus_nA(n)$ that descends to coinvariants. We use this construction to define an $L_\infty$-algebra controlling deformations of the operad $P$ under $Q$ from a cofibrant resolution for an operad $Q$, and an operad map $Q\longrightarrow P$. Starting from a diffent cofibrant resolution one obtains a quasi isomorohic $L_\infty$-algebra. This approach unifies Markl's cotangent cohomology of operads and the approaches to deformation of $Q$-al…
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。
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 construct, using finitely many generating cell and relations, props in the category of CW-complexes with the property that their associated operads are models for the $E_\infty$-operad. We use one of these to construct a cellular $E_\infty$-bialgebra structure on the interval and derive from it a natural cellular $E_\infty$-coalgebra structure on the geometric realization of a simplicial set which, passing to cellular chains, recovers up to signs the Barratt-Eccles and Surjection coalgebra structures introduced by Berger-Fresse and McClure-Smith. We use another prop, a quotient of the first…
The partition poset complex and the Goodwillie derivatives of the identity in spaces
该研究通过上单纯形对象配对构造空间恒等函子导数的典范同伦协调 operad 结构，证明其与 Ching 描述的结构一致，还揭示谱上交换余代数的导出本原元为该 operad 上的代数。
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 extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads O. To solve this problem, we introduce what we call thick maps of O-algebras and special thick maps that we call pseudo-derivations, which serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory. As an application, we derive explicit formulas for transferring Cobar(C)-algebra structures along contractions, where C is any …
In this paper we introduce the concept of L-algebras, which can be seen as a generalization of the structure determined by the Eilenberg-Mac lane transformation and Alexander-Whitney diagonal in chain complexes. In this sense, our main result states that L-algebras are endowed with an E-infinity coalgebra struture, like the one determined by the Barrat-Eccles operad in chain complexes. This results implies that the canonical L-algebra of spaces contains as much homotopy information as its usually associated E-infinity coalgebras, suggesting L-algebras as a tool for the study of homotopy types.
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 introduce a finitely presented prop $\mathcal{S} = \{\mathcal{S}(n,m)\}$ in the category of differential graded modules whose associated operad $U(\mathcal{S})=\{\mathcal{S}(1,m)\}$ is a model for the $E_\infty$-operad. This finite presentation allows us to describe a natural $E_\infty$-coalgebra structure on the chains of any simplicial set in terms of only three maps: the Alexander-Whitney diagonal, the augmentation map, and an algebraic version of the join of simplices. The first appendix connects our construction to the Surjection operad of McClure-Smith and Berger-Fresse. The second es…
We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which we construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this…
Curved operadic calculus
本文建立弯曲 operad 的演算与同伦理论，提出弯曲绝对 operad、完备 Bar–Cobar 伴随及模型结构，并据此得到弯曲泛包络代数、最小余纤维分解与弯曲代数的同伦转移定理。
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。
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 M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural ``fattening'' of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. Co-rings over operads are thus ``relative operads,'' parametrizing morphisms as operads parametrize (co)algebras. Let A denote the associative operad in the category of chain complexes. We de…
Working in the context of symmetric spectra, we consider any higher algebraic structures that can be described as algebras over an operad O. We prove that the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated comonad K, via topological Quillen homology (or TQ-homology), can be turned into an equivalence of homotopy theories by replacing O-algebras with the full subcategory of 0-connected O-algebras. This resolves in the affirmative the 0-connected case of a conjecture of Francis-Gaitsgory. This derived Koszul duality result can be thought of as the s…
We show that morphisms from n A_infinity-algebras to a single one are maps over an operad module with n+1 commuting actions of the operad A_infinity, whose algebras are conventional A_infinity-algebras. Similar statement holds for homotopy unital A_infinity-algebras. The operad A_infinity and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of A_infinity-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures. For this sake notions of lax Cat-span multicategories and multifunctors a…
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…
In previous works by the authors, a bifunctor was associated to any operadic twisting morphism, taking a coalgebra over a cooperad and an algebra over an operad, and giving back the space of (graded) linear maps between them endowed with a homotopy Lie algebra structure. We build on this result by using a more general notion of $\infty$-morphism between (co)algebras over a (co)operad associated to a twisting morphism, and show that this bifunctor can be extended to take such $\infty$-morphisms in either one of its two slots. We also provide a counterexample proving that it cannot be coherently…
We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of infinity-categories C. We classify such Goodwillie towers in terms of the derivatives of the identity functor of C. As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras with Tate diagonals. Our classification of Goodwillie …
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