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I have had a problem I've been thinking of recently but can't seem to make anything of it.

Let $X$ be a simplicial set. It is well known that one can construct a Postnikov tower $$P_nX \rightarrow P_{n-1}X \rightarrow \cdots \rightarrow P_1 X \rightarrow P_0 X$$
of X through , for example, letting $P_n$ be the $n$-th coskeleton of $X$. If $X$ is a (pointed) fibrant simplicial set, the above tower is actually a tower of fibrations.

When we have a cosimplicial space $Y$ which we assume to be pointed fibrant in the Reedy model structure, we also have the Tot-tower $$Tot_n Y \rightarrow Tot_{n-1} Y \rightarrow \cdots \rightarrow Tot_1 Y \rightarrow Tot_0 Y$$ which is a tower of fibrations as well.

The question I have been having is the following:

Given a fibrant simplicial set $X$ as above, can we recover the Postnikov tower of $X$ as some sort of totalization of some cosimplicial space? I would prefer if there were a somewhat canonical way to do this.

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    $\begingroup$ The Postnikov tower is not always a tower or principal fibrations. I have the feeling that the tot-tower is (because it is associated to simplicial skeleta, whose inclusions are principal cofibrations). Hence the answer to your question might be 'no'. But I'm just speculating with this. I didn't check anything. $\endgroup$ Sep 14, 2014 at 21:13
  • $\begingroup$ @FernandoMuro I believe that's why Tedar restricted X to being a fibrant simplicial set, so that he/she could compare the Tot-tower and the Postnikov tower. (Unless I misunderstood the question, of course.) $\endgroup$
    – user62675
    Dec 7, 2014 at 18:31
  • $\begingroup$ @RingSpectra my previous comment also applies to fibrant simplicial sets. $\endgroup$ Dec 8, 2014 at 15:35

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