arXiv subjects
Publications and source records attributed to Yves Félix.
We prove that the unit of the Quillen pair ${\mathfrak{L}}\colon {\bf sset}\rightleftarrows {\bf cdgl}\colon {\langle\,\cdot\,\rangle}$ given by the model and realization functor is, up to homotopy, the Bousfield-Kan ${\mathbb{Q}}$-completion.
This work develops a comprehensive algebraic model for rational stable parametrized homotopy theory over arbitrary base spaces. Building on the simplicial analogue of the foundational framework of May-Sigurdsson for parametrized spectra, and the homotopy theory of complete differential graded Lie algebras, we construct an explicit sequence of Quillen equivalences that translate the homotopy theory of rational spectra of retractive simplicial sets into the purely algebraic framework of complete differential graded modules over the completed universal enveloping algebra $\widehat{UL}$ of a Lie model $L$ of the base simplicial set $B$. Explicitly, there is a sequence of Quillen adjunctions $$ \mathbf{Sp}_B \leftrightarrows \mathbf{Sp}_L \leftrightarrows \mathbf{Sp}_{\widehat{UL}}^0 \leftrightarrows \mathbf{cdgm}_{\widehat{UL}} $$ which induces a natural, strong monoidal equivalence of categories $$ {\rm Ho}\,\mathbf{Sp}_B^{\Bbb Q}\cong {\rm Ho}\, \mathbf{cdgm}_{\widehat{UL}}. $$ This equivalence is highly effective in practice as it provides direct computational access to invariants of simplicial spectra by translating them into homotopy invariants of $\widehat{UL}$-modules. Here $\mathbf{Sp}_B$ denotes the stable model category of spectra of retractive simplicial sets over $B$, $\mathbf{Sp}_L$ denotes the stable model category of spectra of retractive complete differential graded Lie algebras over $L$, $\mathbf{Sp}_{\widehat{UL}}^0$ denotes the stable model category of connected $\widehat{UL}$-module spectra, and $\mathbf{cdgm}_{\widehat{UL}}$ denotes the category of complete differential graded $\widehat{UL}$-modules.
For any group $G$ of self homotopy equivalences of the finite nilpotent complex $X$, acting nilpotently on its homology, and for any nilpotent subcomplex $A$, we prove that the universal fibration $$ X \longrightarrow B(*,{\rm aut}^{A}_G(X),X)\longrightarrow B{\rm aut}^{A}_G(X), $$ which classifies $A$-fibrations for which the image of the $A$-holonomy action lies in $G$, has a Lie model of the form $$ L\longrightarrow L\widetilde\times {\cal D}er^ML\longrightarrow{\cal D}er^ML $$ in which: $M\hookrightarrow L$ is a Lie model of $A\hookrightarrow X$ and ${\cal D}er^ML$ is a connected complete differential graded Lie algebra of derivations of $L$ which vanish on $M$. The rational homotopy type of extended relative mapping fibrations is also similarly characterized.
We prove that for any reduced differential graded Lie algebra L, the classical Quillen geometrical realization $\langle L\rangle_Q$ is homotopy equivalent to the realization $\langle L\rangle= Hom_{\bf cdgl}(\mathfrak{L}_\bullet, L)$ constructed via the cosimplicial free complete differential graded Lie algebra $\mathfrak{L}_\bullet$. As the latter is a deformation retract of the Deligne-Getzler-Hinich realization MC${}_\bullet(L)$ we deduce that, up to homotopy, there is only one realization functor for complete differential graded Lie algebras. Immediate consequences include an elementary proof of the Baues-Lemaire conjecture and the description of the Quillen realization as a representable functor.
We construct Lie algebras of derivations (and identify their geometrical realization) whose Maurer-Cartan sets provide moduli spaces describing the classes of homotopy types of rational spaces sharing either the same homotopy Lie algebra, homology or cohomology.
The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of non-simply connected spaces. In particular, there is a realization functor, $\langle -\rangle$, of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, $L_{0}$, concentrated in degree 0 and proved that $\langle L_{0}\rangle$ is isomorphic to the usual bar construction on the Malcev group associated to $L_{0}$. Here we consider the case of a complete differential graded Lie algebra, $L=L_{0}\oplus L_{1}$, concentrated in degrees 0 and 1. We establish that the category of such two-stage Lie algebras is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Malcev groups. In particular, there is a crossed module $\mathcal{C}(L)$ associated to $L$. We prove that $\mathcal{C}(L)$ is isomorphic to the Whitehead crossed module associated to the simplicial pair $(\langle L\rangle, \langle L_{0}\rangle)$. Our main result is the identification of $\langle L\rangle$ with the classifying space of $\mathcal{C}(L)$.
This paper introduces a new category, Edgl, of enriched differential graded Lie algebras (edgl), directly related to the topology of all connected CW complexes and simplicial sets. It is equipped with a homotopy theory analogous to that developed by Sullivan for commutative differential graded algebras. Each connected space has a unique minimal edgl model, and an algebraic process connects this to the minimal Sullivan model. Minimal edgl models naturally represent cofibrations and, in particular cell attachments, and the interplay between edgl and Sullivan models permits the extension to all path connected spaces of results previously established only for simply connected spaces. This, in particular, provides applications and interesting examples of the classical Sullivan rationalization $X\to X_{\mathbb Q}$ of a path connected space.
Given $X$ a finite nilpotent simplicial set, consider the classifying fibrations $$ X\to Baut_G^*(X)\to Baut_G(X),\qquad X\to Z\to Baut_π^*(X), $$ where $G$ and $π$ denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed self homotopy equivalences of $X$ which act nilpotently on $H_*(X)$ and $π_*(X)$. We give algebraic models, in terms of complete differential graded Lie algebras (cdgl's), of the rational homotopy type of these fibrations. Explicitly, if $L$ is a cdgl model of $X$, there are connected sub cdgl's $Der^G L$ and $Der^π L$ of the Lie algebra $Der L$ of derivations of $L$ such that the geometrical realization of the sequences of cdgl morphisms $$ L\stackrel{ad}{\to} Der^G L\to Der^G L\widetilde\times sL,\qquad L\to L\widetilde\times Der^π L\to Der^π L $$ have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev $Q$-completion of $G$ and $π$ together with the rational homotopy type of the classifying spaces $BG $ and $Bπ$.
We show that the associated Lie algebra of the Malcev $\mathbb{Q}$-completion of the lamplighter group is the pronilpotent completion of the lamplighter Lie algebra. We also prove that the homology of this completed Lie algebra is of uncountable dimension on each degree.
We prove that the second homology group of the completion of an infinite dimensional free Lie algebra is uncountable.
We prove that the spatial realization of a rational complete Lie algebra $L$, concentrated in degree 0, is isomorphic to the simplicial bar construction on the group, obtained from the Baker-Campbell-Hausdorff product on $L$.
R. Lawrence and D. Sullivan have constructed a Lie model for an interval from the geometrical idea of flat connections and flows of gauge transformations. Their model supports an action of the symmetric group $Σ_2$ reflecting the geometrical symmetry of the interval. In this work, we present a Lie model of the triangle with an action of the symmetric group $Σ_3$ compatible with the geometrical symmetries of the triangle. We also prove that the model of a graph consisting of a circuit with $k$ vertices admits a Maurer-Cartan element stable by the automorphisms of the graph.
In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, 'model' and 'realization', between the categories of simplicial sets and complete differential graded Lie algebras. This paper is a follow up of this work. We show that when X is a finite connected simplicial set, then the realization of the model of X is the disjoint union of the Bousfield-Kan completion of X with an external point. We also define a model category structure on the category of complete differential graded algebras making the two previous functors a Quillen pair, and we construct an explicit cylinder. In particular, these functors preserve homotopies and weak equivalences and therefore, this gives the basis for developing a Lie rational homotopy theory for all spaces.
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 $\text{MC}(\mathscr{A}_\bullet\widehat{\otimes} L)$.
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.
This survey contains the main results in rational homotopy, from the beginning to the most recent ones. It makes the status of the art, gives a short presentation of some areas where rational homotopy has been used, and contains a lot of important open problems
Extending the model of the interval, we explicitly define for each $n\ge 0$ a free complete differential graded Lie algebra $\mathfrak{L}_n$ generated by the simplices of $Δ^n$, with desuspended degrees, in which the vertices are Maurer-Cartan elements and the differential extends the simplicial chain complex of the standard $n$-simplex. The family $\{\mathfrak{L}_\bullet\}_{n\ge 0}$ is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct a pair of adjoint functors between the categories of simplicial sets and complete differential graded Lie algebras given by $\langle L\rangle_\bullet=\text{ DGL} (\mathfrak{L}_\bullet,L)$ and $ \mathfrak{L}(K)=\varinjlim_K\mathfrak{L}_{\bullet} $. This new tools let us extend Quillen rational homotopy theory approach to any simplicial set $K$ whose path components are non necessarily simply connected. We prove that $\mathfrak{L} (K)$ contains a model of each component of $K$. When $K$ is a $1$-connected finite simplicial complex, the Quillen model of $K$ can be extracted from $\mathfrak{L} (K)$. When $K$ is connected then, for a perturbed differential $\partial_a$, $H_0(\mathfrak{L} (K),\partial_a)$ is the Malcev Lie completion of $π_1(K)$. Analogous results are obtained for the realization $\langle L\rangle$ of any complete $\text{DGL}$.
In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.