MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)

share|cite|improve this question
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
up vote 9 down vote accepted

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

share|cite|improve this answer
Let this be the only answer I ever vote up that begins with "Harry is wrong!". – Harry Gindi Feb 23 '11 at 8:17
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.