Question

The Artin-Mazur codiagonal $\nabla:ssSet \to sSet$ is right adjoint to the total decalage functor $Dec:sSet \to ssSet$. The total decalage functor is defined to be precomposition with the ordinal sum functor $+:\Delta\times \Delta \to \Delta$, $+([n],[m]) = [n+m+1]$.

If we agree to call a bisimplicial set $X= ([n]\times [m] \mapsto X_{nm})$ a bi-Kan complex if each simplicial set $[k] \mapsto X_{km}$ and $[k] \mapsto X_{nk}$ is a Kan complex (for all $n$ and $m$), then I'm wondering if it is known in the literature whether $\nabla X$ is a Kan complex. Perhaps in Artin-Mazur's monograph?

Note that a bi-Kan complex is not necessarily the same as an internal Kan complex in the category of Kan complexes (I haven't checked this - just a warning)

Answer 1

Harry is wrong! Antonio CEGARRA, B. A. HEREDIA, and J. REMEDIOS: see http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.3820v1.pdf, page 9. Fact 2.8. give the answer in a recent preprint. They know that stuff much better than I do!