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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!

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    $\begingroup$ I will point out that the author has on his website Correction to “A cartesian presentation of weak n-categories”. There you can also find the version which was published - dx.doi.org/10.2140/gt.2010.14.521 - and compare it to the arxiv version if you want to. The correction was published in the same journal: dx.doi.org/10.2140/gt.2010.14.2301 (I guess this does not help with the circularity issue, but still mentioning the published versions seemed relevant enough to add a comment.) $\endgroup$ – Martin Sleziak May 31 '16 at 4:34
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    $\begingroup$ As you said, the (possibly apparent) circularity still remains, as I am referring to the updated version which incorporates the new proof. Also, if you read the new proof separately, you can see it still refers to Prop 6.4. $\endgroup$ – Edoardo Lanari May 31 '16 at 6:54
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    $\begingroup$ Did you try to contact the author? That may be a more efficient way around it. $\endgroup$ – Vladimir Dotsenko May 31 '16 at 14:35
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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.

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