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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)

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For some reason, I think that the only people who can answer this are Tim Porter or Phil Ehlers. –  Harry Gindi Feb 22 '11 at 23:37

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up vote 9 down vote accepted

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!

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Let this be the only answer I ever vote up that begins with "Harry is wrong!". –  Harry Gindi Feb 23 '11 at 8:17
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Not that I have a problem with being wrong, just that I have a healthy respect for irony. –  Harry Gindi Feb 23 '11 at 8:18

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