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bisimplicial group

Idea

bisimplicial objects in Group.

Bisimplicial abelian groups

Proposition

Let $A,B:{\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\to \mathrm{Ab}$ be bisimplicial abelian groups. A morphism $f:A\to B$ which is degreewise in one argument a weak equivalence $f_{n,•}:A(n,•)\to B(n,•)$ 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 Goerss Rick Jardine, Simplicial Homotopy Theory (dvi)

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

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