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 $E_\infty$ topologically equivalent $E_\infty$ DGAs. Also, we show using these obstruction theories that for co-connective $E_\infty$ DGAs, $E_\infty$ topological equivalences and quasi-isomorphisms agree. For $E_\infty$ $\mathbb{F}_p$-DGAs with trivial first homology, we show that an $E_\infty$ topological equivalence induces an isomorphism in homology that preserves the Dyer-Lashof operations and therefore induces an $H_\infty $ $\mathbb{F}_p$-equivalence.
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…
We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg-Mac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasi-isomorphic dgas are topologically equivalent, but we produce explicit counter-examples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting.
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.
We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential …
We describe a general method for algorithmic construction of G-equivariant chain homotopy equivalences from non-equivariant ones. As a consequence, we obtain an algorithm for computing equivariant (co)homology of Eilenberg-MacLane spaces K(pi,n), where pi is a finitely generated ZG-module. This result will be used in a forthcoming paper to construct equivariant Postnikov towers of simply connected spaces with free actions of a finite group G and further to compute stable equivariant homotopy classes of maps between such spaces. The methods of this paper work for modules over any non-negatively…
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 $\text{MC}(\mathscr{A}_\bullet\widehat{\otimes} L)$.
It is well known that the differential graded operad of A_infinity-algebras is a cofibrant replacement (a dg-resolution) of the operad of associative differential graded algebras without units. In this article we find a cofibrant replacement of the operad of associative differential graded algebras with units. Algebras over it are called homotopy unital A_infinity-algebras. We prove that the operad bimodule of A_infinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras without units. Similarly we show that the operad bimodule of homotopy unital A_infinity…
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…
In this paper we introduce a functor, called the simplicial nerve of an A-infinity category, defined on the category of (small) A-infinity categories with values in simplicial sets. We prove that the simplicial nerve of any A-infinity category is an infinity category. This construction extends functorially the nerve construction for differential graded categories proposed by J.Lurie in Higher Algebra. We prove that if a differential graded category is pretriangulated in the sense of A.I.Bondal-M.Kapranov, then its nerve is a stable infinity category in the sense of J.Lurie.
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 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…
We compare two approaches to the homotopy theory of infinity-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and infinity-categories which replaces simplices by trees. The other is based on a certain homotopy theory of marked simplicial sets over the nerve of Segal's category Gamma. In this paper we prove that for operads without constants these two theories are equivalent, in the precise sense of the existence of a zig-zag of Quillen equivalences between the respective model categories.
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.
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.
Derived A-infinity algebras were developed recently by Sagave. Their advantage over classical A-infinity algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A-infinity algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad dAs encoding bidgas. This generalises the established result describing the operad A-infinity as a resolution of the operad As encoding associative algebras. We further show Sagave's definition of morphisms agrees…
We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential graded and simplicial categories. In an intermediate step we generalize Shipley-Schwede's work on connective DG algebras by extending the Dold-Kan correspondence to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial k-modules. As an application we obtain a conceptual explanation of Simpson's homotopy fiber construction.
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.
This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a certain way. We prove an equivalence between the homotopy category of schematic homotopy types and a homotopy category of closed dg-categories. We give a description of homotopy invariants of spaces in terms of minimal models. The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence be…
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 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 develop a notion of an algebra over an infinity-operad with values in infinity-categories which is completely intrinsic to the formalism of dendroidal sets. Its definition involves the notion of a coCartesian fibration of dendroidal sets and extends Lurie's definition of a coCartesian fibration of simplicial sets. We show how, for a dendroidal set X, the coCartesian fibrations over X fit together to form an infinity-category coCart(X). Using a generalization of the Grothendieck construction, we prove that coCart(X) is equivalent to the infinity-category of algebras in infinity-categories ov…
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the world of C-infinity rings. The opposite of the category of differential graded C-infinity algebras contains the category of differential graded manifolds as a full subcategory. More generally, this notion of differential graded algebra makes sense for algebras over any (super) Ferm…
We introduce a notion of left homotopy for Maurer--Cartan elements in $L_{\infty}$-algebras and $A_{\infty}$-algebras, and show that it corresponds to gauge equivalence in the differential graded case. From this we deduce a short formula for gauge equivalence, and provide an entirely homotopical proof to Schlessinger--Stasheff's theorem. As an application, we answer a question of T. Voronov, proving a non-abelian Poincaré lemma for differential forms taking values in an $L_{\infty}$-algebra.
We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to the homotopy theory of Eilenberg-Mac Lane spectral categories. Finally, we show that over the rationals two dg categories are topological equivalent if and only if they are quasi-equivalent.
In this brief postscript to our paper "Integral transforms and Drinfeld centers in derived algebraic geometry", we describe a Morita equivalence for derived, categorified matrix algebras implied by theory developed since its appearance. We work in the setting of perfect stacks X and their stable infinity-categories Q(X) of quasicoherent sheaves. Perfect stacks include all varieties and common stacks in characteristic zero, and their stable infinity-categories of sheaves are well behaved refinements of their quasicoherent derived categories, satisfying natural analogues of common properties of …
