We construct a new spectrum of units for a commutative symmetric ring spectrum that detects the difference between a periodic ring spectrum and its connective cover. It is augmented over the sphere spectrum. The homotopy cofiber of its augmentation map is a non-connected delooping of the usual spectrum of units whose bottom homotopy group detects periodicity.
Our approach builds on the graded variant of E-infinity spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded E-infinity spaces and use it to exhibit our spectrum of units functor as right adjoint on the level of homotopy categories. The resulting group completion functor is an essential tool for studying ring spectra with graded logarithmic structures.
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.
Two DGAs are called topologically equivalent if the corresponding Eilenberg-Mac Lane ring spectra are weakly equivalent as ring spectra. Quasi-isomorphic DGAs are topologically equivalent but the converse is not necessarily true. As a counter-example, Dugger and Shipley showed that there are DGAs that are non-trivially topologically equivalent, i.e. topologically equivalent but not quasi-isomorphic. In this work, we define $E_\infty$ topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of non-trivially $E_\infty$ topo…
We show an alternative construction of the cosimplicial free complete diferential graded Lie algebra $\mathfrak{L}_\bullet=\widehat{\mathbb{L}}(s^{-1}Δ^\bullet)$ based on a new Lie bracket formulae for Lie polynomials on a general tensor algebra. Based on it,we prove that for any complete differential graded Lie algebra $L$, its geometrical realization $\langle L\rangle=\text{Hom}_{\text{cdgl}}(\mathfrak{L}_\bullet,L)$ is isomorphic to its nerve $γ_\bullet(L)$, a deformation retract of the Getzler-Hinich realization $\text{MC}(\mathscr{A}_\bullet\widehat{\otimes} L)$.
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…
Let $(L,d)$ be a differential graded Lie algebra, where $L=L(V)$ is free as graded Lie algebra and $V=V_{\geq 0}$ is a finite type graded vector space. We prove that the injection of $(L,d)$ into its completion $(\widehat{L},d)$ is a quasi-isomorphism if and only if $H(L,d)$ is a finite type pronilpotent graded Lie algebra. As a consequence, we obtain an equivalence between graded Lie models for nilpotent spaces in rational homotopy theory.
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 $H$.
在本文中，我们引入一个函子，称为单纯神经的A-无限范畴，定义在范畴的（小）A-无限范畴的值在单纯集。证明了任意A-无穷范畴的单纯神经是无穷范畴。这种构造在函数上推广了J.Lurie在《高等代数》中提出的微分分次范畴的神经构造。证明了如果微分分次范畴在A. I. Bondal-M.Kapranov意义下是预三角化的，则它的神经在J. Lurie意义下是稳定无穷范畴.
We establish a correspondence between infinity-enhanced Leibniz algebras, recently introduced in order to encode tensor hierarchies, and differential graded Lie algebras, which have been already used in this context. We explain how any Leibniz algebra gives rise to a differential graded Lie algebra with a corresponding infinity-enhanced Leibniz algebra. Moreover, by a theorem of Getzler, this differential graded Lie algebra canonically induces an $L_\infty$-algebra structure on the suspension of the underlying chain complex. We explicitly give the brackets to all orders and show that they agre…
A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose cohomologies have finitely many generators in each degree. Stacks represented by dg schemes are shown to be derived schemes under this correspondence.
Given an $\infty$-category with a set of weak equivalences which is stable under pullback, we show that the mapping spaces of the corresponding localization can be described as group completions of $\infty$-categories of spans. Furthermore, we show how these $\infty$-categories of spans are the mapping objects of an $(\infty, 2)$-category, which yields a Segal space model for the localization after a Kan fibrant replacement.
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.
本文的目的是比较，在一个精确的方式，两个概念的操作同伦出现在文献中。也就是说，我们构造了一个从严格酉同伦有色算子范畴到无穷算子范畴的函子。前一个概念，我们使精确，是操作推广的概念的A-无限范畴和后一个概念是由Moerdijk-Weiss定义的，以推广单纯概念的无限范畴的Joyal-Lurie。这个函子在两个方向上扩展了A-无穷范畴的Faonte-Lurie的单纯神经和微分分次算子的Moerdijk-Weiss的同伦相干神经;它也被证明等价于微分分次算子的大神经à la Lurie。我们证明了它满足一些同伦性质的弱等价和纤维化，例如，它是一个右Quillen函子。
Using the BV-formalism of mathematical physics an explicit construction for the minimal model of a quantum L-infinity-algebra is given as a formal super integral. The approach taken herein to these formal integrals is axiomatic; they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix. Additionally, there exists a canonical differential graded Lie algebra morphism mapping formal functions on homology to formal functions on the whole space. An L-infinity-algebra morphism inverse to this differential graded Lie algebra morphism on the level of ho…
We introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded algebras and show that the category of flat connections on such modules forms a Tannakian category. As such this category can be realised as the category of representations of an affine group scheme $G$, which in the classical case is (the pro-algebraic completion of) the usual fundamental group. This motivates us to define $G$ to be the fundamental group of the n…
