nLab
bisimplicial group

Contents

Idea

bisimplicial objects in Group.

Bisimplicial abelian groups

Proposition

Let A,B:Δ op×Δ opAbA,B : \Delta^{op} \times \Delta^{op} \to Ab be bisimplicial abelian groups. A morphism f:ABf : A \to B which is degreewise in one argument a weak equivalence f n,:A(n,)B(n,)f_{n,\bullet} : A(n,\bullet) \to B(n,\bullet) induces a weak equivalence d(f):d(A)d(B)d(f) : d(A) \to d(B) of the associated diagonal complexes.

Proof

This is Lemma 2.7 in chapter 4 of (GoerssJardine)

References

  • Rick Jardine, Lecture 008 (2010) (pdf)

  • Samuel Issacson?, Excercises in homotopy colimits (pdf)

Created on August 9, 2011 01:02:51 by Urs Schreiber (194.81.173.175)