Suppose I have two bisimplicial set K∙,∙K_{\bullet,\bullet} and L∙,∙L_{\bullet,\bullet} and a bisimplicial map

f∙,∙:K∙,∙⟶L∙,∙.f_{\bullet,\bullet}:K_{\bullet,\bullet}\longrightarrow L_{\bullet,\bullet}.

Now I can take the homotopy of the bisimplicial sets with respect to the first index, which I’ll denote π(1)\pi^{(1)}, or with respect to the second index, π(2)\pi^{(2)}. Assume fp,∙f_{p,\bullet} induces an isomorphism of π(2)\pi^{(2)} for all pp’s. What conditions would be sufficient for this to imply that f∙,qf_{\bullet,q} induces isomorphisms of π(1)\pi^{(1)}’s?

This situation makes me think of (some non-linear version of) spectral sequences. I am not an expert of that kind of methods, but I’m willing to give it a try if it can be applied.

Possible additional assumptions: In the specific case I’m looking at, we can further assume if needed

L=KL=K,

Kp,q=Kq,pK_{p,q} = K_{q,p}, but fp,q≠fq,pf_{p,q}\neq f_{q,p},

fp,qf_{p,q} and fq,pf_{q,p} probably have some compatibility or nice relations, and

in each slot, KK is a Kan complex.

Any ideas, help or references would be greatly appreciated.

=================

=================

=================