Homotopy theory of complete Lie algebras and Lie models of simplicial sets

Abstract

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,cDGL 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 LX coincides with Q∞X + , the disjoint union of the Bousfield-Kan completion of X with an external point. We also define a model category structure on cDGL making L and − 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 cDGL rational homotopy theory for all spaces. Contents

Gauge equivalence for complete $L_\infty$-algebras

“…We start by recalling basic facts on L ∞ -and A ∞ -algebras, Maurer-Cartan elements, Sullivan homotopy, and gauge equivalence for the differential graded case. All definitions given in this section are standard and agree with those commonly found in the literature, except for the notion of completeness: in particular, the definition given here agrees with [LM15] but not with [BFMT18].…”

“…The results in this section should be considered Koszul dual to the approach taken in the rest of the paper, where we choose to work in the setting of cdgas in order for results to immediately generalise to the setting of L ∞ -algebras. There are also close parallels between the approach in Section 3 and the recent papers [BM13b,BFMT18], in which it is shown that gauge equivalence coincides with left homotopy for a larger class of dglas with a different model structure. However, their result only holds in the generality of dglas, and the method used does not seem to easily generalise to L ∞ -algebras.…”

“…In this section, we show that gauge equivalence for complete dglas coincides with left homotopy between complete dgla morphisms, with respect to the model category structure of [LM15]. An analogous result is proved in [BM13b,BFMT18] for a different model structure.…”

Lie models of homotopy automorphism monoids and classifying fibrations

“…Assume the assertion holds if each y i ∈ L >p and assume y i ∈ L p for each i. By formula (9) there are elements z i ∈ Γ p so that y i − z i ∈ Γ p+1 . Hence,…”

“…In this section we recall the basics for complete differential graded Lie algebras, from the homotopy point of view. For it, original references are [7,8,9,10] whose main results are developed in the complete and detailed monograph [11]. Sometimes we will also use classical facts from the Sullivan commutative approach to rational homotopy theory.…”