A canonical enriched Adams–Hilton model for simplicial sets

Abstract

For any 1-reduced simplicial set K we define a canonical, coassociative coproduct on ΩC(K), the cobar construction applied to the normalized, integral chains on K, such that any canonical quasi-isomorphism of chain algebras from ΩC(K) to the normalized, integral chains on GK, the loop group of K, is a coalgebra map up to strong homotopy. Our proof relies on the operadic description of the category of chain coalgebras and of strongly homotopy coalgebra maps given in [HPS]. ≃ − → C * ΩX, such that θ X restricts to quasi-isomorphisms (T V ≤n , d) ≃ − → C * ΩX n+1 , where X n+1 denotes the (n + 1)-skeleton of X, T V denotes the free (tensor) algebra on a free, 1991 Mathematics Subject Classification. Primary: 55P35 Secondary: 16W30, 18D50, 18G35, 55U10, 55U35, 57T05, 57T30.

The Hochschild complex of a twisting cochain

“…In the case K = K , these Eilenberg-Zilber data give rise to the Alexander-Whitney coalgebra structure on C * K [14]. In order to prove Theorem 3.12, we consider another special case of the EilenbergMac Lane SDR.…”

“…We must therefore prove local finiteness of the associated F k 's, which follows from a technical result proved in [14] …”

“…Definition 3.4. (See [14].) A weak Alexander-Whitney coalgebra consists of a connected, coaugmented chain coalgebra C such that the comultiplication Δ : C → C ⊗ C is a DCSH map, together with a choice of chain algebra map ω : ΩC → Ω(C ⊗ C) that realizes the DCSH structure of Δ.…”

A chain coalgebra model for the James map

“…We now describe the coproduct structure on Ω(C(K)) for a 1-reduced simplicial set K. It was defined in [11] and shown there to be identical to the Baues coproduct. Recall that the Alexander-Whitney map f K,L is naturally a DCSH map.…”

“…The Alexander-Whitney diagonal on CL is comultiplicative up to strong homotopy or DCSH [8], and hence is the linear part in a morphism of chain algebras,Ω∆ : Ω(CK) → Ω(CL ⊗ CL). By [11], ψ = q •Ω∆, where q : Ω(CL ⊗ CL) → ΩCL ⊗ ΩCL is the Milgram equivalence [15]. 1 Furthermore, the Szczarba equivalence is a DCSH morphism.…”