nLab bisimplicial group

Idea

bisimplicial objects in Group.

Bisimplicial abelian groups

Proposition

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

Proof

This is Lemma 2.7 in chapter 4 of (GoerssJardine)

References

• Paul GoerssRick Jardine, Simplicial Homotopy Theory (dvi)

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

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