Chris: I’d appreciate it if someone could add an explanation of fiber products as limits of compsimplicial diagrams $A \times B, A \times C \times B, A \times C \times C \times A,...$, making the coface maps explicit.

Urs: I think it should work like this:

the first two coface maps are in full detail given by

and

With that it is clear that the fiber product $A \times_C B$ is the equalizer of these two maps, i.e. the limit over the diagram

I am guessing, but haven’t really checked in detail that the further coface maps continue this: the “inner” ones come from the diagonal $C \to C \times C$ and the outer ones from $f$ and $g$ as above. So for instance the next three would be

and

and

This way, unless I am making a mistake, the ordinary limit over the diagram

would still be $A \times_C B$, but the homotopy limit would pick up the right first degree correction for the homotopy fiber product.

Chris: Thanks, Urs. I’ll continue thinking about this. While I was at the coffee shop, I realized that to form $A \times_C B$ in a homotopically meaningful way, we should somehow resolve $B$. So if $\mathcal{D}$ is our category and $\mathcal{D}_C$ is the category of objects over $C \in \mathcal{D}$, then we have an adjoint pair of functors $R:= C\times? : \mathcal{D} \rightarrow \mathcal{D}_C$ and $L:=\mathcal{D}_C \rightarrow \mathcal{D}$ forgetting the map to $C$ . (Strange to me, but it seems that forgetting here is the left adjoint.) Then we can apply the comonad monad$RL$ to $B \rightarrow C$ to get a cobar resolution of $B$, take the product with $A$, and so get the cosimplicial object $A \times C \times B, A \times C \times C \times B,...$.

Perhaps this is the wrong section to discuss this, and when we figure it out, maybe we should make a section on fiber products of derived spaces.