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Given a bisimplicial manifold (or set, topological space) there is the bar construction, which assigns to it a simplicial manifold, called the total simplicial manifold.

Now stepping back for a moment, you can put functions on each component of the bisimplicial space, valued in some abelian group, and then there are two cohomologies you can calculate: one is of the bisimplicial space, using both the horizontal and vertical differentials (each given by the pullback of the alternating sum of the face maps) induced by the horizontal and vertical simplicial spaces, and the other is by computing the cohomology of the total simplicial manifold (again using the differential given by the pullback of the alternating sum of the face maps).

I believe I have an argument that these are the same, but I'm interested in knowing if anyone has a reference for this fact.

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  • $\begingroup$ It appears that the Eilenberg–Zilber theorem is relevant for answering your question, but there are far too few details to say for sure. Perhaps you can provide a reference to the source that motivated your question? Right now it is not clear what the definition of a “total simplicial manifold” is (is it the Artin–Mazur codiagonal or something else?). $\endgroup$
    – Dmitri Pavlov
    Commented Mar 17, 2022 at 16:41

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