Peiffer elements in simplicial groups and algebras

Abstract

The main objective of this paper is to prove in full generality the following two facts:A. For an operad O in Ab, let A be a simplicial O-algebra such that A m is generated as an O-ideal by ( m−1 i=0 s i (A m−1 )), for m > 1, and let NA be the Moore complex of A. Theni∈I p ker d i   where the sum runs over those partitions of [m − 1], I = (I 1 , . . . , I p ), p ≥ 1, and γ is the action of O on A. B. Let G be a simplicial group with Moore complex NG in which G n is generated as a normal subgroup by the degenerate elements in dimension n > 1, then d(N n G) = I,J [ i∈I ker d i , j∈J ker d j ], for I, J ⊆ [n − 1] with I ∪ J = [n − 1]. In both cases, d i is the i-th face of the corresponding simplicial object. The former result completes and generalizes results from Akça and Arvasi [I. Akça, Z. Arvasi, Simplicial and crossed Lie algebras, Homology Homotopy Appl. 4 (1) (2002) 43-57], and Arvasi and Porter [Z. Arvasi, T. Porter, Higher dimensional Peiffer elements in simplicial commutative algebras, Theory Appl. Categ. 3 (1) (1997) 1-23]; the latter completes a result from Mutlu and Porter [A. Mutlu, T. Porter, Applications of Peiffer pairings in the Moore complex of a simplicial group, Theory Appl. Categ. 4 (7) (1998) 148-173].Our approach to the problem is different from that of the cited works. We have first succeeded with a proof for the case of algebras over an operad by introducing a different description of the inverse of the normalization functor N : Ab ∆ op → Ch ≥0 . For the case of simplicial groups, we have then adapted the construction for the inverse equivalence used for algebras to get a simplicial group NG Λ from the Moore complex NG of a simplicial group G. This construction could be of interest in itself.

3-types of simplicial groups and braided regular crossed modules

“…Mutlu and Porter [24] have also adapted their method to simplicial groups. Castiglioni and Ladra [15] generalized the results proved in [1], [4] and [24].…”

“…Castiglioni and Ladra [15] gave a general proof for the inclusions partially proved by Arvasi and Porter in [4], Arvasi and Akça in [1] and Mutlu and Porter in [24]. Their approach to the problem is different from that of the cited works.…”

Homotopy theory and generalized dimension subgroups

“…Cycles of this complex we denote by Z n G = Ker(d : N n G → N n−1 G), and boundaries by B n G = Im(d : N n+1 G → N n G). Then π n G = Z n G/B n G. The following theorem is Theorem B of [4].…”

Peiffer pairings in multisimplicial groups and crossed n-cubes and applications for bisimplicial groups

“…Castiglioni and Ladra [7] gave a general proof over operads for these inclusions given by Mutlu and Porter in [16].…”