Joana Cirici, Anna Sopena
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
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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.
The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods of homotopical algebra, in the formalisms of stable infinity-categories, stable model categories, and pretriangulated, idempotent-complete dg categories. We characterize the filtered stable infinity-category Fil(C) of a stable infinity-category C as the left exact localization of sequences in C along the infinity-categorical version of completion (and prove a…
In this paper we consider $L_{\infty}$-algebras equipped with complete descending filtrations. We prove that, under some mild conditions, an $L_{\infty}$ quasi-isomorphism $U: L \to \tilde{L}$ induces a weak equivalence between the Deligne-Getzler-Hinich (DGH) $\infty$-groupoids corresponding to $L$ and $\tilde{L}$, respectively. This paper may be considered as a modest addition to foundational paper arXiv:math/0404003 by Ezra Getzler.
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.
In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its run-time optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more…
Tensor triangular geometry of filtered objects and sheaves
本文利用∞-范畴与无点方法，通过分配格和半格等工具计算了滤过或分次对象所成张量三角范畴中紧对象的 Balmer 谱，并应用于概形滤过导出范畴与滤过谱同伦范畴。
Filtered formal groups, Cartier duality, and derived algebraic geometry
本文在滤过背景下发展形式群的概念，描述其与一类滤过Hopf代数的对偶关系，并研究导出代数几何中法锥形变构造，应用于形式群的单位截面，得到等变退化及其坐标代数的滤过性质，进而恢复滤过圆上的滤过，并探讨形式群Hochschild同调的性质及其在谱代数几何中的提升。
Given a flat connection on a manifold with values in a filtered L-infinity-algebra, we construct a morphism of coalgebras that generalizes the holonomies of flat connections with values in Lie algebras. The construction is based on Gugenheim's A-infinity version of de Rham's theorem, which in turn is based on Chen's iterated integrals. Finally, we discuss examples related to the geometry of configuration spaces of points in Euclidean space, and to generalizations of the holonomy representations of braid groups.
We study the Koszul duality between augmented $E_n$-algebras and augmented $E_n$-coalgebras in a symmetric monoidal stable infinity $1$-category equipped with a filtration in a suitable sense. We obtain that the Koszul duality constructions restrict to an equivalence between augmented algebras and coalgebras which have some positivity and completeness with respect to the filtration. We also obtain that the Koszul duality construction is functorial between carefully constructed generalized Morita categories consisting of those algebras/coalgebras in each dimension.
We study the moduli of G-local systems on smooth but not necessarily proper complex algebraic varieties. We show that, when suitably considered as derived algebraic stacks, they carry natural Poisson structures, generalizing the well known case of curves. We also construct symplectic leaves of this Poisson structure by fixing local monodromies at infinity, and show that a new feature, called strictness, appears as soon as the divisor at infinity has non-trivial crossings.
We give a universal construction of a derived affine group scheme and its representation category from a symmetric monoidal infinity-category, which we shall call the tannnakization of a symmetric monoidal infinity-category. This can be viewed as infinity-categorical generalization of the work of Joyal-Street and Nori. We then apply it to the stable infinity-category of mixed motives equipped with the realization functor of a mixed Weil cohomology and obtain a derived motivic Galois group whose representation category has a universality, and which represents the automorphism group of the reali…
The notion of a derived A-infinity algebra, considered by Sagave, is a generalization of the classical notion of A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r-homotopy, for non-negative integers r, in such a way that r-homotopy equivalences underlie E_r-quasi-isomorphisms, defined via an associated spectral sequence. We study the special case of twisted complexes (also known as…
We develop novel methods for using persistent homology to infer the homology of an unknown Riemannian manifold $(M, g)$ from a point cloud sampled from an arbitrary smooth probability density function. Standard distance-based filtered complexes, such as the Čech complex, often have trouble distinguishing noise from features that are simply small. We address this problem by defining a family of "density-scaled filtered complexes" that includes a density-scaled Čech complex and a density-scaled Vietoris--Rips complex. We show that the density-scaled Čech complex is homotopy-equivalent to $M$ for…
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。
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…
A stratified space is a topological space together with a decomposition into strata corresponding to different types of singularities. Examples of such spaces appear everywhere in topology and geometry. The study of stratified spaces involves invariants such as intersection cohomology which are only invariants under stratification-preserving homotopies. In this thesis, we study the homotopy theory of stratified spaces with respect to those stratified homotopies. To do so, we construct model categories for stratified spaces and we introduce new invariants to characterize them, the filtered homo…
On one hand, together with Pelle Steffens, we recently characterized the infinity category of derived manifolds up to equivalence by a universal property. On the other hand, it is shown in recent work of Behrend-Liao-Xu that the category of differential graded manifolds admits a homotopy theory. In this paper, we prove that the associated infinity category of dg-manifolds is equivalent to that of derived manifolds. This allows the well developed theory of dg-geometry to live in symbiosis with infinity categorical approach to derived differential geometry. As a consequence, we prove that a dg-C…
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。
In this paper, we present an infinity-categorical version of the theory of monoidal categories. We show that the infinity category of spectra admits an essentially unique monoidal structure (such that the tensor product preserves colimits in each variable), and thereby recover the classical smash-product operation on spectra. We develop a general theory of algebras in a monoidal infinity category, which we use to (re)prove some basic results in the theory of associative ring spectra. We also develop an infinity-categorical theory of monads, and prove a version of the Barr-Beck theorem.
We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived al…
In paper arXiv:1406.1744, we constructed a symmetric monoidal category $LIE^{MC}$ whose objects are shifted (and filtered) L-infinity algebras. Here, we fix a cooperad $C$ and show that algebras over the operad $Cobar(C)$ naturally form a category enriched over $LIE^{MC}$. Following arXiv:1406.1744, we "integrate" this $LIE^{MC}$-enriched category to a simplicial category $HoAlg^Δ_C$ whose mapping spaces are Kan complexes. The simplicial category $HoAlg^Δ_C$ gives us a particularly nice model of an $(\infty,1)$-category of $Cobar(C)$-algebras. We show that the homotopy category of $HoAlg^Δ_C$ …
We define and study the degeneration property for BV-infinity algebras and show that it implies that the underlying L-infinity algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity Δ(e^x)=e^x(Δ(x)+[x,x]/2) which holds in all BV algebras. As an application we show that the higher Koszul brackets on the cohomology of a manifold supplied with a generalised Poisson structure all vanish.
Localization and flexibilization in symplectic geometry
本文提出了一种基于辛几何的P-柔化方法，通过局部化Weinstein无穷范畴，实现了对缠绕Fukaya范畴中P的可逆性控制，并展示了其在构造E-infinity交换代数中的应用。
We promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions a…
In this paper, we prove some foundational results on the deformation theory of E-infinity ring spectra.
Dirac geometry II: Coherent cohomology
本文将Dirac概形范畴嵌入Dirac叠的infinity-范畴，发展了带有相容t-结构的凝聚上同调理论，并应用复配边与对偶Steenrod代数定理识别了MU和F_p对应的Dirac叠。
Magnitude and magnitude homology of filtered set enriched categories
本文提出滤集富集范畴的 magnitude 与 magnitude homology 框架，统一有限范畴 Euler 示性数和度量空间 magnitude，并推广 Grothendieck 纤维化。
The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a topological, not a metric concept. That leads quickly to the notion of connectedness without any need to dwell on the definition of topology. In our approach line segments must be connected. Lines and planes are unified via the concept of separation: lines are separated into two components by each point, planes contain lines that separate them into two components as …
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…
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