Q:

Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice $\newcommand{\SSet}{\mathbf{SSet}}\SSet/X$.  Most obviously, one can take a homotopy pushout of $X \leftarrow Y \rightarrow X$, to obtain a non-fibrant model $\newcommand{\pre}{\mathrm{nf}} \Sigma^\pre_X Y \to X$, and then factorise this map as a trivial cofibration followed by a fibration to obtain a fibrant model $\Sigma_X Y \to X$.
However, in general, this construction will not be stable up to isomorphism under pullback in the base, only stable up to weak equivalence.  That is, given $p : Y \to X$ as before, and a map $f : X' \to X$, then $f^*(\Sigma_X Y)$ and $\Sigma_{X'} f^*Y$ will be weakly equivalent over $X'$, but not isomorphic.  The non-fibrant part $\Sigma^\pre$ can easily be constructed so that it is stable up to iso.  But the factorisation step isn’t so stable, at least using off-the-shelf constructions (a general choice of (TC,F) factorisations on the category can’t be pullback-stable in the codomain, for fairly obvious reasons).
Is there an alternative construction of the fibrant fibrewise suspension of a fibration, that is stable up to iso under pullback in the base?  Or is there some known obstacle to this?
I’d be most interested in answers for simplicial sets, or some class of model categories including it, or at least of a similar flavour.  My motivation is in modelling suspensions and related constructions in type theory (precisely, suspensions axiomatised as a higher inductive type).

A:

I'm thick as would be notorious, only you're all too polite to mention; is there something difficult about:
The Universal Example is the suspension of the tautological fibration $$ E \to \mathbf{B}(Aut(F))$$ which comes with a map $$\mathbf{B}(Aut(F)) \to \mathbf{B}(Aut(\Sigma F))$$ over which there is again the tautological fibration $$ E' \to \mathbf{B}Aut(\Sigma F).\tag{*}$$ So take pullbacks of $(\mathrm{*})$ along composites $X \to \mathbf{B}(Aut(F)) \to \mathbf{B}(Aut(\Sigma F))$ as the definition of fiberwise suspension.

A:

There is an answer to this question in the comments that works for simplicial sets, and more generally for projective model structures on simplicial presheaves.  Since 3 years later it hasn't been recorded as an "answer", I thought I would do the service, and provide the references.
For simplicial sets, it suffices to show that given Kan fibrations $A,B,C$ over a base $X$, and maps $f:A\to B$ and $g:A\to C$ one of which is a cofibration (a monomorphism), there is a pullback-stable fibrant replacement of the pushout $B+_A C$.  The fiberwise suspension $\Sigma_X Y$ can then be constructed as a pushout of $X+X \leftarrow Y+Y \to Y\times \Delta^1$, the latter map being a cofibration.  One can similarly construct general homotopy pushouts as double mapping cylinders.
To prove this, apply geometric realization to the whole diagram, landing in some convenient category of topological spaces (e.g. compactly generated).  As noted in the comments, Quillen proved that the geometric realization of a Kan fibration is a Serre fibration.  Moreover, since a Serre fibration between CW complexes is a Hurewicz fibration (h/t Peter May), in fact the geometric realization of a Kan fibration is a Hurewicz fibration.  However, Monica Clapp proved that Hurewicz fibrations are closed under pushouts of cofibrations.  Thus, $|B| +_{|A|} |C| \cong |B+_A C|$ is again a Hurewicz fibration over $|X|$.  Now apply the total singular complex, obtaining a Kan fibration $S|B+_A C| \to S|X|$, and pull back along the acyclic cofibration $X \to S|X|$ to obtain a fibrant model of $B+_A C$ over $X$.  Since geometric realization and total singular complex both preserve finite limits, the whole construction is stable under pullback (up to isomorphism).
Moreover, since all the functors involved are simplicially enriched, the construction can be applied pointwise in a category $[{\scr C}^{op},\mathrm{sSet}]$ of simplicial presheaves on any simplicially enriched small category $\scr C$ to yield a pointwise-fibrant, i.e. projectively fibrant, model for homotopy pushouts and suspensions that is stable under pullback.  Note that the cofibration also only needs to be pointwise, i.e. an injective cofibration rather than a projective one.
I believe this also solves the related problem of size preservation of fibers, since geometric realization and total singular complex both preserve the property of having small fibers.  Hence it implies that the universe of Kan fibrations in simplicial sets is closed under HIT pushouts.  (Unfortunately, the projective version doesn't help directly for modeling type theory, since projective model structures don't seem to have dependent products or univalent universes.)

