I am reading this paper of Rezk's http://arxiv.org/abs/0901.3602 A cartesian presentation of weak n-categories, and as it is pointed out in the introduction, it contained a wrong statement (2.19 in the old version) and now there is a new proof for the main result, which is 6.6. Now, section 6.9 of the new proof refers to Prop. 6.4, which uses 2.19 towards the end.
Am I missing something? How is this apparent circularity resolved?
Thanks in advance!
It looks like I completely missed this.
Here's what I guess happens: although the original 2.19 was wrong, there is a weaker version that is true (I'll just state it for simplicial sets): If $X$ is a simplicial set, and $\mathcal{P}$ a finite collection of subobjects of $X$ which is closed under intersections, and the $\bigcup_{K\in\mathcal{P}} K=X$, then $\mathrm{hocolim}_{\mathcal{P}} K \approx X$. I think the original non-proof of 2.19 that I gave was really trying to prove this weaker version (I'll try to get that straight when I have some time).
Now the weak version of 2.19 applies to the poset $\mathcal{P}_K$. Almost: $\mathcal{P}_K$ isn't necessarily closed under intersection (finite limits), but only because it fails to contain an initial object, corresponding to $\varnothing \to K$. If you append this object to $\mathcal{P}_K$, the argument seems to go through (note that $V_\varnothing(A_1,\dots,A_m)$ is itself empty).
Working this out ...
Let $\{S_i\}_{i\in I}$ be an indexed collection of (distinct) subsets of a set $S$, implicitly making $I$ into a poset by set-inclusion, whence a functor $F\colon I\to Set_{/S}$. Suppose (*) every element of $S$ is contained in a minimal $S_i$, i.e., for each $x\in S$, there exists $i(x)\in I$ such that $x\in S_j$ iff $S_{i(x)}\subseteq S_j$. (This holds in particular if the indexed collection is closed under intersection, but is a weaker condition in general.)
Condition (*) implies that the functor $F$ is a coproduct of free functors: $F\approx \coprod_{x\in X} \mathrm{Hom}_I(i(x),-)$. In particular, $\mathrm{colim} F \approx S$.
Now suppose $\{S_i\}_{i\in I}$ is an indexed collection of subobjects of a simplicial set $S$, satisfying (*) in each degree. Using the above observation (applied inductively by degree to non-degenerate simplicies of $S$), we see that the functor $F\colon I\to sSet$ is cofibrant in the projective model structure. As a consequence, $\mathrm{hocolim} F \xrightarrow{\sim} \mathrm{colim} F\approx S$. We can extend the claim to replace $sSet$ with $Fun(C,sSet)$, since hocolims in functor categories are computed termwise.
This gives a version of 2.19 which is true, and which should apply to the $\mathcal{P}_K$ of 6.4. I think.
