Martin Markl
We give explicit formulas for transfers of $A_\infty$-structures and related maps and homotopies in the most easy situation in which these transfers exist. One half of our formulas was already known to Kontsevich-Soibelman and to Merkulov who derived them, without explicit signs, under slightly stronger assumptions than those made in this note.
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We show that the sum over planar trees formula of Kontsevich and Soibelman transfers C-infinity structures along a contraction. Applying this result to a cosimplicial commutative algebra A^* over a field of characteristic zero, we exhibit a canonical unital C-infinity structure on Tot(A^*), which is unital if A^* is; in particular, we obtain a canonical C-infinity structure on the cochain complex of a simplicial set.
Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative H-spaces in the sense that any map f:X->Y to a homotopy associative, homotopy commutative H-space extends to a uniquely determined H-map F: U(X)->Y. Developing a method for recognising certain universal spaces, we show the existence of the universal space F_2(n) of a certain three-cell complex L. Using this specific example, we derive some consequences for the calcula…
We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures.
Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting A-infinity algebras have an interpretation as totally ramified extensions of discrete valuation rings. All A-infinity algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest.
Deformations and homotopy theory of Nijenhuis associative algebras
本文引入同伦Nijenhuis结合代数概念，证明其控制算子为极小模型，构造控制结合积与Nijenhuis算子同时形变的 $L_\infty$-代数，并导出形变复形与上同调理论。
The Anick spaces play a key role in an unstable filtration of the stable homotopy of V(0) to produce secondary EHP sequences. This work establishes that the Anick spaces are homotopy associative and homotopy commutative H-spaces, and that they have a universal mapping property for maps into a homotopy commutative and homotopy associative H-space with a (necessary) condition on the growth of the p torsion. This is utilized to establish an unstable composition theory. The techniques involve generalized Whitehead products based on co-H spaces, calculations in a congruence category that lies betwe…
Let $P^{2n+1}$ be a two-cell complex which is formed by attaching a $(2n+1)$--cell to a $2m$--sphere by a suspension map. We construct a universal space $U$ for $P^{2n+1}$ in the category of homotopy associative, homotopy commutative $H$--spaces. By universal we mean that $U$ is homotopy associative, homotopy commutative, and has the property that any map $f\colon P^{2n+1}\lra Y$ to a homotopy associative, homotopy commutative $H$--space $Y$ extends to a uniquely determined $H$--map $\bar{f}\colon U\lra Y$. We then prove upper and lower bounds of the $H$--homotopy exponent of $U$. In the case …
A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that - similar to spaces - the derived category obtained by inverting the quasi-isomorphisms is naturally left triangulated. Also, homology theories on rings are studied. These must be homotopy invariant in the algebraic sense, meet the Mayer-Vietoris property and plus some minor natural axioms. To …
We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff-Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a cup-1 product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense of Munkholm.
The aim of this brief note is mainly to advocate our approach to homotopy algebras based on the minimal model of an operad. Our exposition is motivated by two examples which we discuss very explicitly - the example of strongly homotopy associative algebras and the example of strongly homotopy Lie algebras. We then indicate what must be proved in order to show that these homotopy algebraic structures are really `stable under a homotopy.' The paper is based on a talk given by the author on June 16, 1998, at University of Osnabrueck, Germany.
We prove that the homotopy theory of $N_\infty$ operads is equivalent to a homotopy theory of discrete operads, and we construct free and associative operadic realizations of every indexing system. This resolves a conjecture of Blumberg and Hill in the affirmative.
Lie, associative and commutative quasi-isomorphism
本文证明特征零下交换 dg 代数的交换与结合拟同构等价，并给出同伦完备 dg 李代数的 Koszul 对偶结果，从而确定有理同伦型并推出幂零李代数由泛包络代数识别。
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。
In an application of the notion of twisting structures introduced by Hess and Lack, we define twisted composition products of symmetric sequences of chain complexes that are degreewise projective and finitely generated. Let Q be a cooperad and let BP be the bar construction on the operad P. To each morphism of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes the two-sided Koszul and bar constructions. When the co-unit from K(g) to P is a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic to the category of P-algebras and of their morphisms up…
The symmetric spectra introduced by Hovey, Shipley and Smith are a convenient model for the stable homotopy category with a nice associative and commutative smash product on the point set level and a compatible Quillen closed model structure. About the only disadvantage of this model is that the stable equivalences cannot be defined by inverting those morphisms which induce isomorphisms on homotopy groups, because this would leave too many homotopy types. In this sense the naively defined homotopy groups are often `wrong`, and then their precise relationship to the `true` homotopy groups (i.e.…
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…
We prove that the Real Johnson-Wilson theories ER(n) are homotopy associative and commutative ring spectra up to phantom maps. We further show that ER(n) represents an associatively and commutatively multiplicative cohomology theory on the category of (possibly non-compact) spaces.
In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven by to compatible with the concepts of homotopy theory: $\infty$-morphisms and the Homotopy Transfer Theorem. We give …
An affine action of an associative algebra $A$ on a vector space $V$ is an algebra morphism $A \to V \rtimes {\rm End}(V)$, where $V$ is a vector space and $V \rtimes {\rm End}(V)$ is the algebra of affine transformations of $V$. The one dimensional version of the Swiss-Cheese operad, denoted ${\mathrm{\bf{sc}}}_1$, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by $({\mathrm{\bf{sc}}}_1)_\infty$. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an $A_\infty$-morphism $A \to V \rtim…
Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological p…
Over a field of characteristic zero, we show that the forgetful functor from the homotopy category of commutative dg algebras to the homotopy category of dg associative algebras is faithful. In fact, the induced map of derived mapping spaces gives an injection on all homotopy groups at any basepoint. We prove similar results both for unital and non-unital algebras, and also Koszul dually for the universal enveloping algebra functor from dg Lie algebras to dg associative algebras. An important ingredient is a natural model for these derived mapping spaces as Maurer-Cartan spaces of complete fil…
Starting from a biased definition of a properad, we describe explicitly algebras over the cobar construction of a properad. Equivalent description in terms of solutions of generalized master equations, which can be interpreted as homological differential operators, are explained from the properadic point of view. This is parallel to the Barannikov's theory for modular operads. In addition to well known IBL-homotopy algebras, the examples include their associative analogues, which we call $IBA$-homotopy algebras, and a combination of the above two.
For dimensions $n\geq 3$ and $k\in\{2, \cdots, n\}$, we show that the space of metrics of $k$-positive Ricci curvature on the sphere $S^{n}$ has the structure of an $H$-space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May that the path component of this space containing the round metric is weakly homotopy equivalent to an $n$-fold loop space.
Let X be a (not-necessarily homotopy-associative) H-space. We show that TC_{n+1}(X) = cat(X^n), for n >= 1, where TC_{n+1}(-) denotes the so-called higher topological complexity introduced by Rudyak, and cat(-) denotes the Lusternik-Schnirelmann category. We also generalize this equality to an inequality, which gives an upper bound for TC_{n+1}(X), in the setting of a space Y acting on X.
We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality t…
We give a proof of the Homotopy Transfer Theorem following Kadeishvili's original strategy. Although Kadeishvili originally restricted himself to transferring a dg algebra structure to an $A_\infty$-structure on homology, we will see that a small modification of his argument proves the general case of transferring any kind of $\infty$-algebra structure along a quasi-isomorphism, under weaker hypotheses than existing proofs of this result.
A modern perspective on rational homotopy theory
本文通过现代方法扩展了Quillen的模型结构，将理性同调等价和理性空间的概念推广至更广泛的空间范畴。
We provide explicit quasi-isomorphisms between the following three algebraic structures associated to the unit interval: i) the commutative dg algebra of differential forms, ii) the non-commutative dg algebra of simplicial cochains and iii) the Whitney forms, equipped with a homotopy commutative and homotopy associative, i.e. $C_\infty$, algebra structure. Our main interest lies in a natural `discretization' $C_\infty$ quasi-isomorphism $φ$ from differential forms to Whitney forms. We establish a uniqueness result that implies that $φ$ coincides with the morphism from homotopy transfer, and ob…
The goal of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the $B_\infty$-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a $B_\infty$-algebra, another, called a hom…
The classical Hopf invariant is an invariant of homotopy classes of maps from $S^{4n-1} $ to $S^{2n}$, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for $E_n$-operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space $X$ and the homotopy groups of $X$. In this paper we will give a generalization of the classical Hopf invariant by defining a pairing between the coho…
We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching $\infty$-category $\mathcal{V}$ is presentably symmetric monoidal then $\mathrm{Cat}^\mathcal{V}_\infty$ is as well. These features render the theory useful even when an $\infty$-category of enriched $\infty$-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and $(\infty,n)$-categories). This…
An $A_\infty$-version of the Eilenberg-Moore theorem
本文在双侧 bar 构造上建立 $A_\infty$-结构，将 Eilenberg-Moore 定理的乘法上同调同构提升为 $A_\infty$-代数拟同构，并证明所得微分挠积乘积与经典定义一致。
For a Poincare duality space X and a map X -> B, consider the homotopy fiber product X x^B X. If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X x^B X has the structure of a graded associative algebra. When X -> B is the diagonal map of a manifold X, one recovers a result of Chas and Sullivan about the homology of the free loop space LX.
This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer--Cartan element. On the way, we settle the integration theory of complete pre-Lie algebras in order to describe this twisting procedure in terms of gauge group action. We give a criterion on quadratic operads for the existence of a meaningful twisting procedure of their associated categories of (homotopy) algebras. We also give a new presentation of the twisting procedure f…
We construct an $(\infty,2)$-version of the (lax) Gray tensor product. On the 1-categorical level, this is a binary (or more generally an $n$-ary) functor on the category of $Θ_2$-sets, and it is shown to be left Quillen with respect to Ara's model structure. Moreover we prove that this tensor product forms part of a "homotopical" (biclosed) monoidal structure, or more precisely a normal lax monoidal structure that is associative up to homotopy.
We describe two constructions giving rise to curved $A_{\infty}$-algebras. The first consists of deforming $A_{\infty}$-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a vector bundle on a polyhedron $X$, we exhibit a curved $A_{\infty}$-structure on the complex of matrix-valued cochains of sufficiently fine triangulations of $X$. We use this structure as a motivation to develop a homotopy associative version of Chern-Weil theory.
First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyc…
Derived Smooth and Banach Higher Groupoids: Representability and Descent
本文研究特定几何中高阶群胚的同伦理论，通过区分可表示性与实现，建立了光滑与Banach情形的Brown结构，并推广至导出模型，涵盖有限与超完备情形。
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 …
We propose an operadic framework suitable for describing algebraic structures with operations being multilinear differential operators of varying orders or, more generally, formal series of such operators. The framework is built upon the notion of a multifiltration of a linear operad generalizing the concept of a filtration of an associative algebra. We describe a particular way of constructing and analyzing multifiltrations based on a presentation of a linear operad in terms of generators and relations. In particular, that allows us to observe a special role played in this context by Lie, Lie…
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