Johannes Huebschmann
We relate a construction of Kadeishvili's establishing an A-infinity-structure on the homology of a differential graded algebra or more generally of an A-infinity algebra with certain constructions of Chen and Gugenheim. Thereafter we establish the links of these constructions with subsequent developments.
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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 differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and show that any dga A over an arbitrary commutative ground ring k is equivalent to a minimal derived A-infinity algebra. Such a minimal derived A-infinity algebra model for A is a k-projective resolution of the homology algebra of A together with a family of maps satisfying appropriate relations. As in the case of A-infinity algebras, it is possible to recover…
We study in this article a possible further structure of homotopic nature on multiplicative spectral sequences. More precisely, since Kadeishvili's theorem asserts that, given a dg (or A-infinity-)algebra, its cohomology has also a structure of A-infinity-algebra such that both become quasi-isomorphic, and in a multiplicative spectral sequence one considers the cohomology of dg algebras when moving from a term to the next one, a natural problem that arises is to study how this two possible structures intertwine. We give such a homotopic structure proposal, called A-infinity-enhancement of mult…
Building on Kadeishvili's original theorem inducing $A_\infty$-algebra structures on the homology of dg-algebras, several directions of algorithmic research in $A_\infty$-algebras have been pursued. In this paper we will survey work done on calculating explicit $A_\infty$-algebra structures from homotopy retractions; in group cohomology; and in persistent homology.
Kadeishvili's proof of the minimality theorem induces an algorithm for the inductive computation of an $A_\infty$-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete $A_\infty$-algebra structure after a finite amount of computational work.
Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in a finite dimensional vector space induces a multiplicative filtration (which would not be a so harsh hypothesis in our setting) on the dg algebra given by the complex of simplicial cochains, we may use a result by T. Kadeishvili to get a unique (up to noncanonical equivalence) A_infinity-algebra structure on the complete persistent cohomology of the filtered simplicial (or topological) set. We th…
We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goa…
Hochschild cohomology and deformation theory of stable infinity-categories with t-structures
本文研究了配备左完备 t-结构的稳定可呈现无穷范畴的形变理论，证明其 Hochschild 上同调控制了该范畴关于左完备 t-结构的形变。
We show that algebraic K-theory KGL, the motivic Adams summand ML and their connective covers acquire unique E-infinity structures refining naive multiplicative structures in the motivic stable homotopy category. The proofs combine Gamma-homology computations and work due to Robinson giving rise to motivic obstruction theory. As an application we employ a motivic to simplicial delooping argument to show a uniqueness result for E-infinity structures on the K-theory Nisnevich presheaf of spectra.
We prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory
We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of multiplicative algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne'…
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。
We give a source of examples of H_infinity ring structures that do not lift to E_infinity ring structures, based on Mandell's equivalence between certain cochain algebras and spaces.
We study several homotopical and geometric properties of Maurer-Cartan spaces for L-infinity algebras which are not nilpotent, but only filtered in a suitable way. Such algebras play a key role especially in the deformation theory of algebraic structures. In particular, we prove that the Maurer-Cartan simplicial set preserves fibrations and quasi-isomorphisms. Then we present an algebraic geometry viewpoint on Maurer-Cartan moduli sets, and we compute the tangent complex of the associated algebraic stack.
We study the moduli space of A-infinity structures on a topological space as well as the moduli space of A-infinity-ring structures on a fixed module spectrum. In each case we show that the moduli space sits in a homotopy fiber sequence in which the other terms are representing spaces for Hochschild cohomology.
In this paper, we provide a conceptual new construction of the algebraic structure on the pair of the Hochschild cohomology spectrum (cochain complex) and Hochschild homology spectrum, which is analogous to the structure of calculus on a manifold. This algebraic structure is encoded by a two-colored operad introduced by Kontsevich and Soibelman. We prove that for a stable idempotent-complete infinity-category, the pair of its Hochschild cohomology and homology spectra naturally admits the structure of algebra over the operad. Moreover, we prove a generalization to the equivariant context.
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 show how and when it is possible to detect and recover higher Massey products on the cohomology $H$ of a differential graded algebra $A$ with higher multiplications on quasi-isomorphic $A_\infty$ structures on $H$.
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 describe dual notions of tangent bundle for an infinity-topos, each underlying a tangent infinity-category in the sense of Bauer, Burke and the author. One of those notions is Lurie's tangent bundle functor for presentable infinity-categories, and the other is its adjoint. We calculate that adjoint for injective infinity-toposes, where it is given by applying Lurie's tangent bundle on infinity-categories of points.
We provide an example of a spectrum over S^0 with an H_\infty structure which does not rigidify to an E_3 structure. It follows that in the category of spectra over S^0 not every H_\infty ring spectrum comes from an underlying E_\infty ring spectrum. After comparing definitions, we obtain this example by applying Σ^\infty_+ to the counterexample to the transfer conjecture constructed by Kraines and Lada.
The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of E-infinity algebras. We prove that the bar complex of any E-infinity algebra can be equipped with the structure of an E-infinity algebra so that the bar construction defines a functor from E-infinity algebras to E-infinity algebras. We prove the homotopy uniqueness of such natural E-infinity structures on the bar construction. We apply our construction to cochain complexes of topol…
The space of E-infinity structures on an simplicial operad C is the limit of a tower of fibrations, so its homotopy is the abutment of a Bousfield-Kan fringed spectral sequence. The spectral sequence begins (under mild restrictions) with the stable cohomotopy of the graded right Gamma-module formed by the homotopy groups of C ; the fringe contains an obstruction theory for the existence of E-infinity structures on C. This formulation is very flexible: applications extend beyond structures on classical ring spectra to examples (in references) in motivic homotopy theory.
We show that an A-infinity algebra structure can be transferred to a projective resolution of the complex underlying any A-infinity algebra. Under certain connectedness assumptions, this transferred structure is unique up to homotopy. In contrast to the classical results on transfer of A-infinity structures along homotopy equivalences, our result is of interest when the ground ring is not a field. We prove an analog for A-infinity module structures, and both transfer results preserve strict units.
Transfers of $A_\infty$- and other homotopy structures as Grothendieck bifibrations
本文证明 $A_\infty$ 态射到其底层链映射的函子为离散 Grothendieck 双纤维化，并推广到特征零域上任意二次 Koszul 操作子的 $P_\infty$ 结构，从而建立严格函子化的同伦结构转移框架与一般转移定理。
Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also …
Let A be a connected graded algebra and let E denote its Ext-algebra. There is a natural A-infinity algebra structure on E, and we prove that this structure is mainly determined by the relations of A. In particular, the coefficients of the A-infinity products m_n restricted to the tensor powers of Ext^1 give the coefficients of the relations of A. We also relate the m_n's to Massey products.
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.
Let p be an odd prime. When n>2, we show that each tensor factor of form E \otimes Γin H(Z,n;Z_p) is an A-infinity bialgebra with non-trivial structure. We give explicit formulas for the structure maps and the quadratic relations among them. Thus E \otimes Γis a naturally occurring example of an A-infinity bialgebra whose internal structure is well-understood.
An A-infinity bialgebra of type (m,n) is a Hopf algebra H equipped with a "compatible" operation ω: H^{\otimes m} \to H^{\otimes n} of positive degree. We determine the structure relations for A-infinity bialgebras of type (m,n) and construct a purely algebraic example for each m \geq 2 and m+n \geq 4.
The present work re-enacts the classical theory of t-structures reducing the classical definition given in *Faisceaux Pervers* to a rather primitive categorical gadget: suitable reflective factorization systems. This translation is only possible due to the virtues of the infinity-categorical setting. Once the basic theory of t-structure has been developed, a natural direction of research is to develop a comprehensive treatment of t-structure in the "torsio-centric" perspective: we apply the technology of factorization systems to the theory of Postnikov towers, to the proof of the abelianity of…
Homotopy transfer for L-infinity structures and the BV-formalism
本文利用BV形式体系和积分微扰展开，给出了广义和幺模L-无穷代数结构极小模型的显式构造，证明了其作为同伦转移定理的实例，并揭示了有根树公式与费曼图的关系。
We show how the model structure on the category of simplicially-enriched (colored) props induces a model structure on the category of simplicially-enriched (colored) properads. A similar result holds for dioperads.
Transfer of A-infinity structures to projective resolutions
本文研究交换环上沿拟同构转移或提升 A_infinity 代数结构，证明了若干相关结果。
We extend the Bousfield-Kan spectral sequence for the computation of the homotopy groups of the space of minimal A-infinity algebra structures on a graded projective module. We use the new part to define obstructions to the extension of truncated minimal A-infinity algebra structures. We also consider the Bousfield-Kan spectral sequence for the moduli space of A-infinity algebras. We compute up to the second page, terms and differentials, of these spectral sequences in terms of Hochschild cohomology.
We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative categories, not only with respect to their Reedy structures but also their Rezk structure, which turns them into models for the theories of (infinity, 1)- and (infinity, k)-categories, respectively.
We refine our earlier work on the existence and uniqueness of E-infinity structures on K-theoretic spectra to show that at each prime p, the connective Adams summand has an essentially unique structure as a commutative S-algebra. For the p-completion we show that the McClure-Staffeldt model for it is equivalent as an E-infinity ring spectrum to the connective cover of the periodic Adams summand. We establish Bousfield equivalence between the connective cover, c(E_n), of the Lubin-Tate spectrum E_n and BP<n> and propose c(E_n) as an E-infinity approximation to the latter.
In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered sutured Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure, allowing us to combine contact structures not just by gluing, but also by higher-order operations. In this paper we investigate such A-infinity structures and higher order operations on contact structures. We give explicit constructions of such A-infinity structures, and establish some of their pro…
We investigate Gamma-cohomology of some commutative cooperation algebras E_*E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Gamma-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E infinity structures. As a consequence we obtain an E infinity structure for the connective Adams summand. For the Johnson-Wilson spectrum E(n) with n > 0 we establish the existence of a unique E infinit…
Model Structures on Infinity-Categories of Filtrations
本文通过研究稳定∞-范畴的形变序列，推广Gugenheim-May关于矩阵Massey积生成元的结论，刻画滤过谱所诱导谱序列中存活至E_k页元素的生成机制。
The mod p homology of E-infinity spaces is a classical topic in algebraic topology traditionally approached in terms of Dyer--Lashof operations. In this paper, we offer a new perspective on the subject by providing a detailed investigation of an alternative family of homology operations equivalent to, but distinct from, the Dyer--Lashof operations. Among other things, we will relate these operations to the Dyer--Lashof operations, describe the algebra generated by them, and use them to describe the homology of free E-infinity spaces. We will also investigate the relationship between the operat…
We show that the complex $C_\bullet X$ of rational simplicial chains on a compact and triangulated Poincaré duality space $X$ of dimension $d$ is an A$_\infty$ coalgebra with $\infty$ duality. This is the structure required for an A$_\infty$ version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohomology $HH^{\bullet+d} (C^\bullet X, C_\bullet X)$ of the cochain algebra $C^\bullet X$ with values in $C_\bullet X$ has a BV structure. This implies, if $X$ is moreover simply connected, that the shifted homology $H_{\bullet+d}LX$ of the free loop space admits a BV …
Let R=S/I be a monomial ring whose minimal free resolution F is rooted. We describe an A-infinity algebra structure on F. Using this structure, we show that R is Golod if and only if the product on Tor^S(R,k) vanishes. Furthermore, we give a necessary and sufficient combinatorial condition for R to be Golod.
Using the methods of Ando-Hopkins-Rezk, we describe the characteristic series arising from E-infinity genera valued in topological modular forms with level structure. We give examples of such series for tmf_0(N) and show that the Ochanine genus comes from an E-infinity ring map. We also show that, away from 6, certain tmf orientations of String cobordism descend to orientations of Spin cobordism.
We introduce a strong homotopy notion of a cyclic symmetric inner product of an A-infinity algebra and prove a characterization theorem in the formalism of the infinity inner products by Tradler. We also show that it is equivalent to the notion of a non-constant symplectic structure on the corresponding formal non-commutative supermanifold. We show that (open Gromov-Witten type) potential for a cyclic filtered A-infinity algebra is invariant under the cyclic filtered A-infinity homomorphism up to reparametrization, cyclization and a constant addition, generalizing the work of Kajiura.
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.
We describe a comparison between pretriangulated differential graded categories and certain stable infinity categories. Specifically, we use a model category structure on differential graded categories over k (a field of characteristic 0) where the weak equivalences are the Morita equivalences, and where the fibrant objects are in particular pretriangulated differential graded categories. We show the underlying infinity category of this model category is equivalent to the infinity category of k-linear stable infinity categories.
For A a dg (or A-infinity) algebra and M a module over A, we study the image of the characteristic morphism $χ_M: HH^*(A, A) \to Ext_A(M, M)$ and its interaction with the higher structure on the Yoneda algebra $Ext_A(M, M)$. To this end, we introduce and study a notion of A-infinity centre for minimal A-infinity algebras, agreeing with the usual centre in the case that there is no higher structure. We show that the image of $χ_M$ lands in the A-infinity centre of $Ext_A(M, M)$. When A is augmented over k, we show (under mild connectedness assumptions) that the morphism $χ_k: HH^*(A, A) \to Ext…
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.
We introduce and study A-infinity persistence of a given homology filtration of topological spaces. This is a family, one for each n > 0, of homological invariants which provide information not readily available by the (persistent) Betti numbers of the given filtration. This may help to detect noise, not just in the simplicial structure of the filtration but in further geometrical properties in which the higher codiagonals of the A-infinity structure are translated. Based in the classification of zigzag modules, a characterization of the A-infinity persistence in terms of its associated barcod…
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 …
Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I
本文基于分次向量空间范畴的形式概形概念，提出研究A-infinity代数与范畴的几何方法，厘清相关结构问题，推广多个经典猜想与概念，并给出对应二维拓扑场论的结构等价性结论。
Models for cyclic infinity operads
本文在循环树状集与空间上构建了循环拟算畴与完备循环树状 Segal 空间的模型结构，证明其与单纯循环算畴 Quillen 等价，并推广至平面情形以支持 Walde 纲领。
Čech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction
本文提出一种基于无穷李理论的构造，将微分示性类实现为无穷群胚间的态射，以研究其同伦纤维并描述微分阻碍问题，尤其适用于扭曲微分弦结构和五膜结构。
We define the notion of a relative matrad and realize the free relative matrad as a free H_\infty-bimodule structure on cellular chains of bimultiplihedra JJ={JJ_{n,m} = JJ_{m,n}}. We define a morphism G:A => B of A_\infty-bialgebras as a bimodule over H_\infty and prove that the homology of every A_\infty-bialgebra over a commutative ring with unity admits an induced A_\infty-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of A_\infty-bialgebras and identify the A_\infty-bialgebra structure of H_*(ΩΣX; Q). For each n>1, we construct a space X_n and identify an i…
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…
The d.g. operad C of cellular chains on the operad of spineless cacti is isomorphic to the Gerstenhaber-Voronov operad codifying the cup product and brace operations on the Hochschild cochains of an associative algebra, and to the suboperad F_2X of the surjection operad. Its homology is the Gerstenhaber operad G. We construct an operad map psi from A-infinity to C such that psi(m_2) is commutative and the homology of psi is the canonical map A \to Com \to G. This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. …
Algebraic quantum field theory and prefactorization algebra are two mathematical approaches to quantum field theory. In this monograph, using a new coend definition of the Boardman-Vogt construction of a colored operad, we define homotopy algebraic quantum field theories and homotopy prefactorization algebras and investigate their homotopy coherent structures. Homotopy coherent diagrams, homotopy inverses, A-infinity-algebras, E-infinity-algebras, and E-infinity-modules arise naturally in this context. In particular, each homotopy algebraic quantum field theory has the structure of a homotopy …
We study the application of IBL-infinity-algebras to string topology and explicitly compute the case of spheres. This involves finding a Green kernel and computing integrals associated to trivalent ribbon graphs, which are similar to integrals from perturbative Chern-Simons theory. We generalize the computation and show that the twisted IBL-infinity-structure in question is in many cases trivial.
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