Complete DG Lie Algebra Models

• Complete differential graded Lie algebra models are algebraic tools that encode the structure of a differential graded Lie algebra with an additional completeness property, representing rational homotopy types.
• They utilize explicit model and realization adjunctions, forming a Quillen pair that connects simplicial sets with complete DG Lie algebras.
• These models extend classical rational homotopy theory to non-simply-connected spaces by incorporating Maurer–Cartan elements and filtered completions for effective computations.

A complete differential graded Lie algebra (cdgLA or cDGL) is a central object in rational homotopy theory, encoding both the algebraic structure of a differential graded Lie algebra (dgLA) and a notion of completeness with respect to an appropriate filtration. These models serve as algebraic representatives for topological spaces, especially within the context of rational homotopy types—including non-simply-connected and non-nilpotent spaces—via Quillen’s and Sullivan’s paradigms. The development of explicit model and realization adjunctions, the transfer of homotopical structures, and compatibility with filtered completions make complete DG Lie algebra models essential in both theory and computations (Buijs et al., 2016, Buijs et al., 2015, Félix et al., 2017, Félix et al., 2022).

1. Definition and Structural Properties

A cdgLA over $\mathbb{Q}$ consists of a $\mathbb{Z}$-graded vector space $L = \bigoplus_{n} L_{n}$ equipped with a graded Lie bracket $[\,\cdot\,,\,\cdot\,]$ and a differential $d$ of degree $-1$ satisfying:

• Graded antisymmetry: $[x, y] = -(-1)^{|x||y|}[y,x]$,
• Jacobi identity, and
• Leibniz rule: $d[x, y] = [dx,y] + (-1)^{|x|}[x,dy]$, with $d^2 = 0$.

A complete filtration is a descending chain of dg-Lie subspaces:

$$L=F^{1}L \supset F^{2}L \supset F^{3}L \supset \cdots$$

with $d(F^{p}L) \subset F^{p}L$, $[F^{p}L, F^{q}L] \subset F^{p+q}L$, and completeness $L \cong \varprojlim_p L/F^{p}L$. Pronilpotence refers to the lower central series filtration $F^{p+1}L = [L,F^{p}L]$ being cofinal (Buijs et al., 2016, Félix et al., 2017, Félix et al., 2021).

The canonical example is the completed free Lie algebra on a graded space $V$:

$\widehat{L}(V) := \varprojlim_k L(V)/L^{>k}(V)$

where $L(V)$ is the free Lie algebra, and $L^{>k}(V)$ the ideal generated by brackets of length $>k$.

2. The Quillen Model/Realization Adjoint Pair

Given a simplicial set $X$, one constructs a Lie model in cDGL via cosimplicial methods. For each $n$, form the completed free Lie algebra on desuspended chains of the $n$-simplex:

$\mathfrak{L}_n := \widehat{L}(s^{-1}C_*(\Delta^n))$

with the linear part of the differential corresponding to the desuspended simplex boundary. The structure maps make $\{\mathfrak{L}_n\}_{n\ge0}$ a cosimplicial object (Buijs et al., 2015, Buijs et al., 2016, Buijs et al., 2017).

The model functor assigns to a simplicial set $K$ the colimit:

$\mathfrak{L}(K) := \colim_{\Delta^n \to K} \mathfrak{L}_n$

which, for finite $K$, equals the completed free Lie algebra on the desuspended normalized chains.

The realization functor assigns to a cDGL $L$ the simplicial set:

$$\langle L \rangle_n := \operatorname{Hom}_{\mathrm{cDGL}}(\mathfrak{L}_n, L)$$

yielding a Kan complex naturally isomorphic to the Deligne–Getzler–Hinich Maurer–Cartan simplicial set (Félix et al., 2022, Buijs et al., 2017).

Model and realization form an adjoint pair:

$$\text{sSet} \;\underset{\text{real}}{\stackrel{\text{model}}{\rightleftarrows}}\; \text{cDGL}$$

which is a Quillen pair with respect to appropriate model category structures inherited from the Kan–Quillen structure on simplicial sets (Buijs et al., 2016).

3. Homotopical and Model Category Structure

The transferred model category structure on cDGL is determined as follows (Buijs et al., 2016):

• Weak equivalences: $f:L\to M$ is a weak equivalence if: (i) The induced map on Maurer–Cartan sets $MC(L)\to MC(M)$ is a bijection, and (ii) for each basepoint $a$, the component map $f_{a}:L_{a}\to M_{f(a)}$ is a quasi-isomorphism.
• Fibrations: Maps that are surjective in degrees $\ge0$.
• Cofibrations: Maps having the left lifting property with respect to acyclic fibrations, and generated by maps associated to inclusions of $S^{n-1} \to D^n$.

This model structure realizes a Quillen adjunction, descending to an equivalence at the homotopy category level for finite-type spaces and finite-type, pronilpotent cDGLs. Every finite simplicial set admits both a cdgLA model $\mathfrak{L}_X$ and a Sullivan–de Rham commutative dg-algebra model, related via bar/cobar duality (Buijs et al., 2016, Félix et al., 2022).

4. Maurer–Cartan Elements and Component Decomposition

A Maurer–Cartan (MC) element in a cDGL $L$ is $a\in L_{-1}$ solving $d a + \frac12[a,a]=0$. The gauge group $G = \exp(L_0)$ acts on $MC(L)$ by a BCH-style action, and the set of orbits $\widetilde{MC}(L) = MC(L)/G$ indexes the connected components in the realization (Buijs et al., 2016, Félix et al., 2021).

In the explicit models for finite simplicial complexes $K$, each $0$-simplex in $K$ yields an MC element in $L(K)$. The Deligne groupoid of $L(K)$ is in bijection with $\pi_0(K)$. Thus, for connected $K$, $MC(L(K))$ forms a single gauge orbit and $L(K)$ recovers the classical Quillen–Malcev model (Buijs et al., 2016). The homotopy groups of a component $(L)_a$ in the realization correspond to the homology $H_{n-1}(L_a)$, making the functor $L \mapsto \langle L \rangle$ a fully homotopy-theoretic construction (Buijs et al., 2016, Félix et al., 2022).

5. Uniqueness and Comparison of Realization Functors

All geometric realization functors for (reduced) cDGLs—Quillen's original realization, the cosimplicial free Lie realization, and the Deligne–Getzler–Hinich Maurer–Cartan realization—are equivalent up to homotopy (Félix et al., 2022, Buijs et al., 2017). The identification relies on producing natural surjective quasi-isomorphisms and deformation retracts at the level of underlying simplicial sets, allowing one to freely select whichever is most computationally advantageous (Félix et al., 2022).

This homotopical uniqueness underlies the categorical equivalence between rational homotopy types of nilpotent spaces and the homotopy category of pronilpotent, finite-type cDGLs (Félix et al., 2017).

6. Applications and Consequences in Rational Homotopy Theory

Complete DGLA models extend rational homotopy theory to non-nilpotent, non-simply-connected, and even non-connected spaces, by providing fully algebraic models in Lie-theoretic terms (Buijs et al., 2016, Buijs et al., 2015). In particular:

• For connected finite simplicial sets, the realization of their model is weakly equivalent to the Bousfield–Kan $\mathbb{Q}$-completion of the space plus a disjoint point (Buijs et al., 2016).
• For 1-connected finite-type complexes, the minimal Lie model recovers the rational homotopy groups, and the Sullivan–Quillen cdga is dual to the completed universal enveloping algebra.
• For two-stage cDGLs, there is a functorial equivalence with crossed modules of Malcev-type groups, and realizations classify their associated crossed modules, connecting Whitehead's model for 2-types to complete Lie algebraic data (Félix et al., 2021).
• The completeness (pronilpotence) criterion ensures that only when homology is a finite-type pronilpotent Lie algebra does the inclusion into the completed model induce a quasi-isomorphism; otherwise, essential information may be lost (Félix et al., 2017).

These models unify the Quillen and Sullivan approaches, admit explicit functorial constructions, and offer a framework for open problems concerning partial completions and the extension to spaces with infinite-type or more general completions (Buijs et al., 2016).

7. Limitations and Open Questions

Despite their strength, the theory of complete DGLA models faces several limitations:

• They require finite simplicial sets (or finite-type homology) in foundational theorems.
• Comparison between different types of completions (Malcev, Bousfield–Kan, fiberwise) remains subtle.
• A Lie-theoretic description of partial completions, such as fiberwise rationalizations, is an open problem.
• Dropping the finite-type assumption while retaining equivalences is unresolved in various geometric settings (Buijs et al., 2016).

Future research directions include extending these constructions to broader classes of spaces, further elucidating the connections between Lie algebraic and higher categorical models, and refining the algebraic models to encompass unstable or equivariant phenomena.

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Key references:

• (Buijs et al., 2016) Buijs–Félix–Murillo–Tanré, "Homotopy theory of complete Lie algebras and Lie models of simplicial sets"
• (Buijs et al., 2015) Buijs–Félix–Murillo–Tanré, "Lie models of simplicial sets and representability of the Quillen functor"
• (Buijs et al., 2016) Buijs–Félix–Murillo–Tanré, "Maurer-Cartan elements in the Lie models of finite simplicial complexes"
• (Félix et al., 2017) Félix–Moreno-Fernández–Tanré, "Lie models for nilpotent spaces"
• (Félix et al., 2021) Félix–Tanré, "Realization of Lie algebras and classifying spaces of crossed modules"
• (Félix et al., 2022) Félix–Fuentes–Murillo, "All known realizations of complete Lie algebras coincide"
• (Buijs et al., 2017) Buijs–Félix–Murillo–Tanré, "The infinity Quillen functor, Maurer-Cartan elements and DGL realizations"