Is there a combinatorial way to factor a map of simplicial sets as a weak equivalence followed by a fibration?

Question

Background on why I want this:

I'd like to check that suspension in a simplicial model category is the same thing as suspension in the quasicategory obtained by composing Rezk's assignment of a complete Segal space to a simplicial model category with Joyal and Tierney's "first row functor" from complete Segal spaces to quasicategories. Rezk's functor first builds a bisimplicial set (I will say simplicial space) with a very nice description in terms of the model category, then takes a Reedy fibrant replacement in the category of simplicial spaces. Taking this fibrant replacement is the only part of the process which doesn't feel extremely concrete to me.

To take a Reedy fibrant replacement $X_*'$ for a simplicial space $X_*$ you can factor a sequence of maps of simplicial sets as weak equivalences followed by fibrations. The first such map is $X_0 \rightarrow *$. If I factor that as $X_0 \rightarrow X_0' \rightarrow *$ the next map to factor is $X_1 \rightarrow X_0' \times X_0'$ and factors as $X_1 \rightarrow X_1' \rightarrow X_0' \times X_0'$. (The maps are all from $X_n$ into the $n$th matching space of the replacement you're building, but these first two matching spaces are so easy to describe I wanted to write them down explicitly.)

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Background on the question:

The small object argument gives us a way to factor any map $X \rightarrow Y$ in a cofibrantly generated model category as a weak equivalence $X \rightarrow Z$ followed by a fibration $Z \rightarrow Y$. (In fact, as an acyclic cofibration followed by a fibration.) However, the object $Z$ this produces is in general hard to understand.

If the model category we're interested in is the usual one on simplicial sets (weak equivalences are weak homotopy equivalences on the geometric realizations, fibrations are Kan fibrations, and cofibrations are inclusions) we have some special ways of factoring maps $X \rightarrow *$ as a weak equivalence followed by a fibration. Probably the most familiar is using the singular chains on the geometric realization of $X$ to make $X \rightarrow S(\mid X\mid ) \rightarrow *$. This is again big and rather hard to understand.

Another method is Kan's Ex$^\infty$ functor, which I learned about in Goerss and Jardine's book. Ex is the right adjoint to the subdivision functor, which is defined first for simplices in terms of partially ordered sets, so the subdivision functor and Ex are both very combinatorial. There is a natural map $X \rightarrow$ Ex $X$, and Ex$^\infty X$ is the colimit of the sequence $X \rightarrow$ Ex $X \rightarrow$ Ex$^2 X \rightarrow$... . It turns out that $X \rightarrow $Ex$^\infty X \rightarrow *$ is a weak equivalence followed by a fibration.

I would like a way of factoring a map $X \rightarrow Y$ of simplicial sets as a weak equivalence followed by a fibration that is similar in flavor to the use of Kan's Ex$^\infty$ functor to find a fibrant replacement for a simplicial set $X$.

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Question:

In the standard model category structure on simplicial sets, is there a combinatorial way of factoring a map as a weak equivalence followed by a fibration?

Answer 1

I don't know if this does what you want, but it's one thing I know how to do.

If $X\to Y$ is a map between Kan complexes, then you can build a factorization using the path space construction. Thus, $P=Y^I\times_Y X$, and $P\to Y$ is given by evaluation, while $X\to P$ is produced using the constant path. Here $I=\Delta[1]$, and $Y^I$ is the mapping object in simplicial sets. Since $Y$ is a Kan complex, $P\to Y$ is a Kan fibration (using the fact that $Y^I\to Y^{\partial I}$ is a Kan fibration).

For a general map $f: X\to Y$, consider $Ex^\infty(f): Ex^\infty X\to Ex^\infty Y$. This is a map between Kan complexes, and therefore the path space construction on $Ex^\infty(f)$ gives a Kan fibration $P\to Ex^\infty Y$. Now pull back along $Y\to Ex^\infty Y$ to get a Kan fibration $Q\to Y$. Because pullbacks of fibrations in simplicial sets always are homotopy pullbacks, it should be possible to see that $X\to Q$ is a weak equivalence.

(I think this construction goes back to Quillen's book on model categories.)