R-local homotopy theory (original) (raw)
2019
Building on the seminal works of Quillen [12] and Sullivan [16], Bousfield and Guggenheim [3] developed a "homotopy theory" for commutative differential graded algebras (cdgas) in order to study the rational homotopy theory of topological spaces. This "homotopy theory" is a certain categorical framework, invented by Quillen, that provides a useful model for the non-abelian analogs of the derived categories used in classical homological algebra. In this masters thesis, we use K. Brown’s generalization [5] of Quillen’s formalism to present a homotopy theory for the category of semi-free, finite-type cdgas over a field k of characteristic 0. In this homotopy theory, the "weak homotopy equivalences" are a refinement of those used by Bousfield and Guggenheim. As an application, we show that the category of finite-dimensional Lie algebras over k faithfully embeds into our homotopy category of cdgas via the Chevalley-Eilenberg construction. Moreover, we prove ...
On the Space of Maps between R-Local CW Complexes
Asterisque- Societe Mathematique de France
The papers [A1,A2] infroduced and studied a differential graded Lie algebra (dgL) associated as a model to certain spaces. Building on that work, we construct in this note a simplicial skeleton for the space of pointed maps between two R-local simply-connected CW complexes (R ⊆ Q). The construction entails two steps. First is the construction, in the category of dgL's, of a cosimplicial resolution and an associated "function conplex" valid in a range of dinensions; and second is the connection with the topological mapping space via the above-mentioned models.
On the locally finite chain algebra of a proper homotopy type
Bulletin of The Belgian Mathematical Society-simon Stevin, 1996
In the classical paper (A-H) Adams-Hilton constructed a free chain algebra which is an important algebraic model of a simply connected homotopy type. We show that this chain algebra (endowed with an additional structure given by a \height function") yields actually an invariant of a proper homotopy type. For this we introduce the homotopy category of locally nite chain algebras
The homotopy lie algebra for finite complexes
Publications mathématiques de l'IHÉS, 1982
L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THE HOMOTOPY LIE ALGEBRA FOR FINITE COMPLEXES by YVES FfiLIX (1), STEPHEN HALPERIN (2) and JEAN-CLAUDE THOMAS (3) i. Introduction.-A generic question in topology asks how geometric restrictions on a topological space S are reflected in restrictions on^(S). A classical example is this: which discrete groups G admit a finite GW complex as classifying space? In this paper we shall deal with an analogous question for Lie algebras and simply connected spaces. Henceforth, and throughout this paper, we shall consider only those spaces which are simply connected and have the homotopy type ofGW complexes whose rational homology is finite dimensional in each degree. Such spaces will be called i-connected GW spaces of finite Qj-tyRe. For such spaces 7Tp(S)®Q^ is finite dimensional (each p), and the Whitehead product in ^(S), transferred to T^(OS) by the canonical isomorphism, makes T^(QS) ® Q, into a connected graded Lie algebra of finite type {i.e. finite dimensional in each degree): the rational homotopy Lie algebra of S. A striking result of Quillen [QJ asserts that every connected graded Lie algebra (over QJ of finite type arises in this way. The situation for finite complexes is very different, and the question referred to above, which forms the starting point of this paper, can be stated as the Problem.-What restrictions are imposed on the rational homotopy Lie algebra of a space S, if S is a finite, i-connected, GW complex? We shall establish serious restrictions, both on the integers dimTTp(S) ®Q,, and on the Lie structure. These restrictions, moreover, turn out to hold for the much larger class of those i-connected GW spaces of finite Q^-type whose rational Lusternik-Schnirelmann category is finite. Recall that the Lusternik Schnirelmann category of a space S, as normalized by Ganea [Ga], is the least integer m such that S can be covered by m + i open sets, each (1) Chercheur qualifie au F.N.R.S. (2) During this research the second named author enjoyed the hospitality of the Sonderforschungsbereich (40) Mathematik at the University of Bonn,
Towards an integral version of rational homotopy theory
In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod cup-one products and compatible binomial operations. Given such an R-dga, (A,d), defined over the ring R=Z or Z_p (for p a prime) and with H^1(A) a finitely generated, free R-module, we show that A admits a functorially defined 1-minimal model, unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, G(A), which only depends on the 1-quasi-isomorphism type of A. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic (torsion-free) cohomology that share the same rational 1-minimal model, yet whose integral 1-minimal models are not isomorphic. This is joint work with Richard Porter.
The structure of homotopy Lie algebras
Commentarii Mathematici Helvetici, 2009
In this paper we consider a graded Lie algebra, L, of finite depth m, and study the interplay between the depth of L and the growth of the integers dim L i. A subspace W in a graded vector space V is called full if for some integers d , N , q, dim V k Ä d P kCq iDk dim W i , i N. We define an equivalence relation on the subspaces of V by U W if U and W are full in U C W. Two subspaces V , W in L are then called L-equivalent (V L W) if for all ideals K L, V \ K W \ K. Then our main result asserts that the set L of L-equivalence classes of ideals in L is a distributive lattice with at most 2 m elements. To establish this we show that for each ideal I there is a Lie subalgebra E L such that E \ I D 0, E˚I
Lie coalgebras and rational homotopy theory, I
2006
use of general theory when it is convenient, for example in placing model category structures on dga and dge (Theorem 4.16) and in our brief discussion of minimal models (Section 4.6). We expect this framework to be useful in a range of applications. In one sequel to this work, we will show that the functor E applied to the cochains of a space encodes generalized Hopf invariants. The Lie coalgebra point of view thus leads to a way to pass between cochain and homotopy data where the formalism, the combinatorics, and the geometry are unified. Indeed, it was an investigation of generalized Hopf invariants which led us to the framework of this paper. Another promising application is to use the functor E to understand classical Harrison homology. Our work throughout is over a field of characteristic zero. We emphasize that we are adding a finiteness hypothesis, namely that our algebras and coalgebras are finite-dimensional in each positive degree, for the sake of linear duality theorems. Under this hypothesis the category of chain complexes is canonically isomorphic to that of cochain complexes, and we will use this isomorphism without further comment, by abuse denoting both categories by dg. To clarify when possible, we have endeavored to use V to denote a chain complex and W to denote a cochain complex. Many of the facts we prove are true without the finiteness hypothesis, as we may indicate. We plan to remove this hypothesis, as well as the (simple-) connectivity hypotheses in the third paper in this series. Contents 1. Introduction 1 2. The Eil cooperad and its pairing with the Lie operad 2 3. The perfect pairing between free Lie algebras and cofree Eil coalgebras 4 3.1. Basic manipulations of cofree Eil coalgebras 4 3.2. Duality of free algebras and cofree coalgebras 5 3.3. Cofree Eil coalgebras as quotients of cotensor coalgebras 8 4. The functors E and A, a Quillen pair 9 4.1. The Quillen functors L and C 10 4.2. The functors E and A 10 4.3. Adjointness of E and A 12 4.4. The main diagram 13 4.5. Model structures and rational homotopy theory. 15 4.6. Minimal models 17 References 18 2. The Eil cooperad and its pairing with the Lie operad The combinatorial heart of our work is a pairing between rooted and unrooted trees, developed in [14]. Definition 2.1. (1) An n-tree is an isotopy class of acyclic graph whose vertices are either trivalent or univalent, with a distinguished univalent vertex called the root, embedded in the upper half-plane with the root at the origin. Univalent vertices, other than the root, are called leaves, and they are labeled by n = {1,. .. , n}. Trivalent vertices are also called internal vertices. (2) The height of a vertex in a n-tree is the number of edges between that vertex and the root. (3) Define the nadir of a path in a n-tree to be the vertex of lowest height which it traverses. (4) A n-graph is a connected oriented acyclic graph with vertices labeled by n.
An algebraic model for homotopy fibers
Homology Homotopy and Applications, 2002
Let F be the homotopy fiber of a continuous map f : X@ >>> Y , and let R be a commutative, unitary ring. Given a morphism of chain Hopf algebras that models (Ωf ) : C * (ΩX; R)@ >>> C * (ΩY ; R), we construct a cochain algebra that models C * (F ; R). We explain how to simplify the model for certain large classes of maps f and provide examples of the application of our model.
An explicit construction of the Quillen homotopical category of dg Lie algebras
2007
Let g1\g_1g1 and g2\g_2g2 be two dg Lie algebras, then it is well-known that the LinftyL_\inftyLinfty morphisms from g1\g_1g1 to g2\g_2g2 are in 1-1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra Bbbk(g1,g2)\Bbbk(\g_1,\g_2)Bbbk(g1,g2). Then the gauge action by exponents of the zero degree component Bbbk(g1,g2)0\Bbbk(\g_1,\g_2)^0Bbbk(g1,g2)0 on MCsubsetBbbk(g1,g2)1MC\subset\Bbbk(\g_1,\g_2)^1MCsubsetBbbk(g1,g2)1 gives an explicit "homotopy relation" between two LinftyL_\inftyLinfty morphisms. We prove that the quotient category by this relation (that is, the category whose objects are LinftyL_\inftyLinfty algebras and morphisms are LinftyL_\inftyLinfty morphisms modulo the gauge relation) is well-defined, and is a localization of the category of dg Lie algebras and dg Lie maps by quasi-isomorphisms. As localization is unique up to an equivalence, it is equivalent to the Quillen-Hinich homotopical category of dg Lie algebras [Q1,2], [H1,2]. Moreover, we prove that the Quillen's concept of a homotopy coincides with ours. The last result was conjectured by V.Do...
Rational homotopy theory for non-simply connected spaces
Transactions of the American Mathematical Society, 2000
We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.
Enriched Differential Lie Algebras in Topology
arXiv (Cornell University), 2022
The context of this article is the Sullivan rationalization X → X Q of path connected topological spaces. This is constructed by Sullivan via his Sullivan algebras, which are commutative differential graded algebras (cdga's), (∧V, d), in which: (i) V = V ≥1 , (ii) ∧V is the free graded commutative algebra generated by V , and (iii) d satisfies a certain "Sullivan condition". As Sullivan realized from the outset, the component, d 1 : V → V ∧ V of d, defines by duality a Lie bracket in L V := s −1 Hom(V ; Q); (s −1 is inverse suspension). Moreover, L V is a graded Lie algebra, the homotopy Lie algebra of (∧V, d). We recall now that each connected space admits a quasi-isomorphism (∧V, d)
Rational homotopy calculus of functors
Arxiv preprint math/0603336, 2006
We construct a homotopy calculus of functors in the sense of Goodwillie for the categories of ratio-nal homotopy theory. More precisely, given a homotopy functor between any of the categories of differential graded vector spaces (DG), reduced differential graded vector spaces, ...
A Cartan–Eilenberg approach to homotopical algebra
Journal of Pure and Applied Algebra, 2010
In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.
Cyclic theory for commutative differential graded algebras and s-cohomology
Eprint Arxiv 0905 1489, 2009
In this paper one considers three homotopy functors on the category of manifolds , hH * , cH * , sH * , and parallel them with other three homotopy functors on the category of connected commutative differential graded algebras, HH * , CH * , SH *. If P is a smooth 1-connected manifold and the algebra is the de-Rham algebra of P the two pairs of functors agree but in general do not. The functors HH * and CH * can be also derived as Hochschild resp. cyclic homology of commutative differential graded algebra, but this is not the way they are introduced here. The third SH * , although inspired from negative cyclic homology, can not be identified with any sort of cyclic homology of any algebra. The functor sH * might play some role in topology. Important tools in the construction of the functors HH * , CH * and SH * , in addition to the linear algebra suggested by cyclic theory, are Sullivan minimal model theorem and the "free loop" construction described in this paper. (dedicated to A. Connes for his 60-th birthday) Contents 1. Introduction 2 2. Mixed complexes, a formalism inspired from Connes' cyclic theory 3 3. Mixed commutative differential graded algebras 8 4. De-Rham Theory in the presence of a smooth vector field 11 5. The free loop space and s-cohomology 15 6. The free loop construction for CDGA 18 7. Minimal models and the proof of Theorem 3 21 References 23
Drinfel'd algebra deformations, homotopy comodules and the associahedra
1994
The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel d algebra A and thus finish the program which began in , . The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of a map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition 3.3 below) on the bar resolution of a general Drinfel d algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in .