Iterated Bar Complexes and the Poset of Pruned Trees Addendum to the Paper: “iterated Bar Complexes of E-infinity Algebras and Homology Theories”

The purpose of these notes is to explain the relationship between Batanin’s categories of pruned trees and iterated bar complexes. This article is an appendix of the article [4]. Our purpose is to explain the relationship between Batanin’s categories of pruned trees (see [1, 2]) and iterated bar complexes and to revisit some constructions [4] in this formalism. The reader can use this appendix as an informal introduction to the constructions of [4]. These notes is the appendix part of a preliminary version of [4], extracted without changes from this article except that we have removed the appendix mark from paragraph numberings. Thus the reader can easily retrieve references to former versions of [4] in this manuscript. 1. Level trees and sequences of non-decreasing surjections. Our first aim is to make explicit the expansion of iterated tensor coalgebras (T Σ)(M), for a connected Σ∗module M . For this purpose, we use the category Ord formed by the non-empty ordered sets r = {1, . . . , r}, r ∈ N∗, together with non-decreasing surjections as morphisms. The one-point set 1 forms a final object of Ord and we also use the notation ∗ to refer to this object. For any finite set e (not necessarily equipped with an ordering), we consider the set Π(e) defined by sequences of the form e ρ0 −→ r0 ρ1 −→ · · · ρn −→ rn = 1, where ρ0 is any surjective map and {r0 ρ1 −→ · · · ρn −→ rn} is an n-simplex of the category Ord such that rn = 1 is the final object. Equivalently, if we consider the category Surj of non-empty finite sets and surjections, then we have e ∈ Surj and we can identify Π(e) to a subset of the n-dimensional simplices in the nerve of the comma category e /Ord , where we consider the obvious forgetful functor from Ord to Surj . In the sequel, we also consider the subset Π n (e) ⊂ Π(e) formed by the simplices ρ ∈ Π(e) such that the initial arrow ρ0 : e → r0 is a bijection. An n-simplex ρ ∈ Π(e) defines the structure of a planar tree with inputs indexed by e and n + 1 levels indexed by 0, . . . , n. The ordered set ri gives the vertices of Date: 31 March 2010 (current version – preliminary version in October 2008). 2000 Mathematics Subject Classification. Primary: 57T30; Secondary: 55P48, 18G55, 55P35. Research supported in part by ANR grant JCJC06-0042 OBTH. 1