Algebraic models of non-connected spaces and homotopy theory ofL∞algebras

Abstract

We develop a homotopy theory of L ∞ algebras based on the Lawrence-Sullivan construction, a complete differential graded Lie algebra which, as we show, satisfies the necessary properties to become the right cylinder in this category. As a result, we obtain a general procedure to algebraically model the rational homotopy type of nonconnected spaces.

Photophysical properties of betaxanthins: Vulgaxanthin I in aqueous and alcoholic solutions

“…The general claim, in the non-completed context, was made in [6, Theorem 6.4], but we have not been able to parse the proof in [6]. This example is simple enough to be worked out by hand, although the calculations are still nontrivial.…”

“…The operation of a disjoint product of two dglas was considered (without this name) in [6] and, in a more general, operadic context in [7]. The operation of a disjoint product of two dglas was considered (without this name) in [6] and, in a more general, operadic context in [7].…”

“…A non-complete version of the disjoint product was considered in [6]. One should view the construction g 0 as the Lie analogue of adjoining an isolated base point to a topological space.…”

The Lawrence–Sullivan construction is the right model forI+

“…That the spatial realization of this complete Lie algebra is in fact of the homotopy type of two contractible components is a particular instance of the more general context of homotopy theory in the category of (unbounded) differential graded Lie algebras, or more generally, L ∞ algebras [5].…”

Formality criteria in terms of higher Whitehead brackets

“…To achieve the mentioned result, we recall in Section 3.1 the necessary background. Then we recall in Section 3.2 that, under certain assumptions which we fix for the rest of the article, L ∞ algebras uniquely correspond to Sullivan algebras ( [10,7]). Finally, we achieve the main goal of this section by showing how some of our previous results in [5] generalize and/or complement the main result of [3].…”