Let $\mathcal{C}$ be a small category and let $\text{sPre}(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, i.e. $\text{sPre}(\mathcal{C})=\mathbf{sSet}^{\mathcal{C}^{\text{op}}}$. Let us denote by $U\mathcal{C}$ the model category obtained by endowing $\text{sPre}(\mathcal{C})$ with the projective model structure (so that weak equivalences or fibrations in $U\mathcal{C}$ are the natural transformations of simplicial presheaves which are sectionwise weak equivalences or fibrations for the Kan-Quillen model structure on $\mathbf{sSet}$).

Question 1: Can we provide an example of a small category $\mathcal{C}$ for which $U\mathcal{C}$ has the following property: there is a fibration $f\colon X\rightarrow Y$ and a trivial cofibration $i\colon A\rightarrow Y$ in $U\mathcal{C}$ such that the pulled back map $f^*(i)\colon X\times_{Y}A\rightarrow X$ is not a projective cofibration?

Question 2: Are there (interesting) sufficient conditions on $\mathcal{C}$ which automatically ensure that $U\mathcal{C}$ has the property of Question 1?

In general, these questions seem a bit tricky to approach since we do not have a very good descriptions of (trivial) cofibrations for the projective model structure, but maybe there are some easy/obvious/trivial cases I am missing.

Any comment/idea would be very much appreciated. Thanks in advance.

  • $\begingroup$ Take A=pt and choose i so that its image is outside of the image of f. Then X ×_Y A = ∅, so you're demanding that X is always cofibrant, which is only true if C=∅ or C=1. $\endgroup$ – Dmitri Pavlov Oct 6 '16 at 16:15
  • $\begingroup$ @DmitriPavlov Thanks for your answer. I can follow your argument, except for the fact that, in order for $i$ to be a trivial cofibration, I think $Y$ needs to be sectionwise contractible, right? If $f\colon X\rightarrow Y$ is now a projective fibration (I edited my question, as I want $f$ to be a fibration, not a random map) with $Y$ sectionwise contractible, does this make a difference in your argument somehow? $\endgroup$ – Marco Vergura Oct 12 '16 at 14:44
  • $\begingroup$ Your additional assumptions imply that f*i is an injective acyclic cofibration because simplicial sets and UC are right proper and monomorphisms are pullback-stable. However, you are still far from getting a projective cofibration. Take Y=A×Δ^1, X=B×Δ^1, f=(B→A)×Δ^1 for some fibration B→A. Then f*i: B→B×Δ^1 has to be a projective (acyclic) cofibration, which is hardly ever true for an arbitrary B (unless B itself is projectively cofibrant). $\endgroup$ – Dmitri Pavlov Oct 12 '16 at 18:45
  • $\begingroup$ In such cases it helps a lot if you also describe the original problem you're trying to solve in this way because alternative solutions may also be possible. This is in fact encouraged by the MO rules. $\endgroup$ – Dmitri Pavlov Oct 12 '16 at 18:51

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