Explicit calculations of small homotopy limits of CDGAs I would like to carry out explicit calculations of homotopy limits of certain simple diagrams of CDGAS. My set-up is the following : I have a finite graded poset $R$ with minimal element $0$ and a diagram of rational CDGAs
$$s:R^\mathrm{op}\to\mathrm{CDGA}$$
These diagrams shouldn't be Reedy fibrant.
The two simplest examples of such shapes $R^\mathrm{op}$ (represented by their Hasse diagram) are
        * -----> 0 <----- *

and 
                 *
               /   \
      * ----  *     *  ---- *
       \       \   /       /
        *  ----  0  ----  *
       /       /   \       \
      * ----  *     *  ---- *
               \   /
                 *                                                               

(all arrows pointing inwards)

The first diagram shape calculates the homotopy pullback, and there are explicit formulas for expressing it : according to the nLab, since all objects in CDGA are fibrant, the homotopy pullback of 
$$
\begin{array}{ccc}
&&A^*\\
&&\downarrow\\
B^*&\rightarrow & C^*
\end{array}
$$
can be calculated as the standard pullback of the associated diagram
$$
\begin{array}{ccc}
&&(C^*)^I\\
&&\downarrow\\
A^*\times B^*&\rightarrow & C^*\times C^*
\end{array}
$$
where $(C^*)^I$ is a path object for $C^*$, which apparently can be taken to be $\mathbb{Q}\oplus\big(\overline{C^*}\otimes S(t,dt)\big)$.

My question is three-fold :


*

*Is it possible to explicitely calculate the homotopy limit of a diagram of CDGAs of the second shape? (By which I mean obtain generators, relations and the value of the differential on those generators.)

*Can the calculation be split up into subcalculations? I.e. can I calculate homotopy limits of subpieces of the diagram and break the calculation into small steps?

*I have a conjectured description of this homotopy limit, how can I verify that it is correct?


All CDGAs that appear in these diagrams are finite dimensional and their differentials are zero (they are CGAs, actually cohomology algebras of certain manifolds); furthermore they all have explicit presentations in terms of generators and relations, and the presentations are compatible with the maps of the diagram. Also, the outmost layer of those diagrams is populated by copies of $\mathbb{Q}$.
I wrote "explicit calculation" in the bounty note; by that I mean, for instance, a simple associated diagram (finite if possible) whose ordinary limit can be identified with the homotopy limit I'm after.
 A: (3) is the easiest to answer. Once you have a conjectured description, just compute the honest limit of it and check that it's weakly equivalent to the limit of your original diagram. You can also check that your description is weakly equivalent in the Reedy model structure (i.e. entrywise weakly equivalent) to the original diagram. To check that your conjectured description is Reedy fibrant, use the dual of Proposition 10.6 from Dwyer-Spalinski, i.e. check that the maps in the diagram are fibrations and the pullback corner maps as well.
(1) is asking for quite a lot. For the diagram in your picture, I would guess that one Reedy fibrant replacement would involve replacing all the maps pointing at 0 by fibrations (this can be done via factorization into trivial cofibration followed by fibration) and then replacing the remaining objects by products of themselves with pullbacks of the new diagrams (i.e. the ones with fibrations). So, for example, if the top part of your diagram was $A\to 0 \gets B$ and $A\gets D\to B$, then we'd replace $A$ and $B$ by $A'$ and $B'$ so that $A'\to 0\gets B'$ are both fibrations, and then replace $D$ by $D\times P$ where $P$ is the pullback of the diagram $A'\to 0 \gets B'$. The map $D\to A'$ is the composition $D\to A\to A'$, and same for $B$. The maps from $P$ are the pullback maps.
This is complicated, but it's not impossible. In fact, you can work your way up the poset one vertex at a time, simply replacing the next vertex by the pullback of itself with everything new you've created so far. The dual story is presented beautifully in this paper by Luis Pereira. See Lemma 4.8. His $Q_{T_e}$ is the colimit of everything up to the new vertex $e$. This inductive process also answers your (2).
In the bounty note you mention being interested in simplicial methods. A great reference is Model Categories and Simplicial Methods by Goerss and Schemmerhorn, section 4. It turns out to be easy to make a simplicial resolution (Reedy fibrant replacement) using the Dold-Kan equivalence, i.e. normalized chains functor and it's inverse $\Gamma$. A detailed study of this equivalence and its relationship to simplicial mapping spaces is given in Schwede-Shipley Equivalences of monoidal model categories. A lot of this story goes back to lecture notes of Curtis from the 70s, and they remain very readable and relevant. Good luck!
