The Lawrence–Sullivan construction is the right model forI⁺

Abstract

We prove that the universal enveloping algebra of the LawrenceSullivan construction is a particular perturbation of the complete Baues-Lemaire cylinder of S 0 . Together with other evidences we present, this exhibits the Lawrence-Sullivan construction as the right model of I + . From this, we also deduce a generalized Euler formula on Bernoulli numbers.

A Tale of Three Homotopies

“…Iterations of less than N decompositions of those would produce a tensor product where at least one factor belongs to Ω ∨ (2) , hence will be annihilated at the final step when we apply the map θ everywhere (this map vanishes on Ω ∨ (2) by construction). This guarantees that the computation in the spirit [8,9], even though either transfers higher structures from a space which is not a coalgebra or uses the contracting homotopy with the wrong codomain, nevertheless produces an honest A ∞ -coalgebra. Remark 6.…”

Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes

“…Here the B n are the well known Bernoulli numbers. This model has been described in detail in [9], [4]. In a cDGL (L, d), two Maurer-Cartan elements u 1 and u 2 are equivalent if they are in the same orbit for the gauge action.…”

The gauge action, DG Lie algebras and identities for Bernoulli numbers

“…Given a differential graded Lie algebra, the gauge action on its Maurer-Cartan set, which may be understood as an algebraic abstraction of the behavior of gauge infinitesimal transformations in classical gauge theory, can be encoded via the Lawrence-Sullivan construction L, see [2,Prop.3.1] or [3, 4.6]. Then, we show that Theorem 0.1 above is equivalent to Theorem 1 of [11], that is, L is indeed a differential graded Lie algebra.…”

“…Then, we show that Theorem 0.1 above is equivalent to Theorem 1 of [11], that is, L is indeed a differential graded Lie algebra. This is done by transporting L to the category of differential graded algebras via the universal enveloping functor and forcing it to be a cylinder in the corresponding homotopy category [2,Thm. 3.3].…”