Titlebook: Lie Models in Topology; Urtzi Buijs,Yves Félix,Daniel Tanré Book 2020 Springer Nature Switzerland AG 2020 rational homotopy theory.Lie mod

切割

The Quillen Functors L, C and their Duals A , E,Also, as a key tool in some forthcoming fundamental results, we need functors that behave as “duals” of L and C. Straight dualization defines a functor from ., but the dual of a dgl is not a dglc unless finite type assumptions are imposed.

LIMIT

Complete Differential Graded Lie Algebras,In this chapter we introduce and carefully study the category . of ., which is the main algebraic category in this text.

Mirage

,Maurer–Cartan Elements and the Deligne Groupoid,Whenever the dgl is complete, the set of these elements is endowed with a particular equivalence relation given by the ., a particular action of L. in the set of Maurer–Cartan elements.

NIB

CLAMP

,The Cosimplicial cdgl L•,In this chapter we further develop the ideas and machinery developed in Chapter 5, specially those involving the construction of the LS interval and the “model of the triangle” in Section 5.4, to find compatible models of the simplicial sets Δ., for any n ≥ 0.

Detonate

The Model and Realization Functors,The first thing to notice is that this pair provides the precise Eckmann–Hilton dual of the classical Sullivan approach to rational homotopy theory.

巩固

A Model Category for cdgl,As we recalled in Section 1.3.1, practically all categories of chain complexes enriched with some additional structure, in particular cdgl, also possess a model category structure in which fibrations and weak equivalences are surjections and quasi-isomorphisms, respectively.

吹牛者

Debility

Extracting the Sullivan, Quillen and Neisendorfer Models from the Global Model,On the other hand, this weak equivalence also constitutes the starting point for the construction of a bridge linking Sullivan models with Lie models. In fact, we provide explicit algorithms to obtain a Sullivan model from the minimal Lie model of a given connected simplicial set, and vice versa.

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