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giving this its own entry (the concept used to appear in-line at geometric realization of simplicial topological spaces) for ease of hyperlinking. But just the bare definition, for the moment.
I am suspecting that the following is true, but I don’t have a proof yet. Possibly this is an easy consequence of statements that are in the literature.
Given
a morphism $X_\bullet \overset{p_\bullet}{\longrightarrow} B_\bullet$ of simplicial k-topological spaces which is a homotopy Kan fibration
a Čech nerve $C(U)_\bullet$ of a good open cover over a manifold, also regarded as a simplicial topological space,
and writing
then presumably
is again a homotopy Kan fibration?! (What I am really after is that its realization commutes with taking homotopy fibers.)
Presumably I’ll have to convince myself that $C(U)_\bullet$ is “homotopy cofibrant” in the suitable sense and that simplicial $\infty$-groupoids, regarded as a “model $\infty$-category” is “cartesian closed model”.
All this sounds like it ought to be true. Can this be cited from anywhere?
The Čech nerve C(U) of a good open cover over a manifold, regarded as a simplicial topological space, is weakly equivalent to a discrete space in every simplicial degree.
Such simplicial spaces are cofibrant, since they can be obtained as transfinite compositions of cobase changes of generating cofibrations in the model ∞-structure on simplicial spaces, namely, boundary inclusions ∂Δ^n→Δ^n.
This answers the question in the affirmative. I guess one could cite Mazel-Gee’s articles, Part I deals specifically with simplicial spaces.
But how about the property that homming a homotopy-cofibrant simplicial object into a homotopy Kan fibration yields a homotopy Kan fibration (or more generally the $\infty$-version of the pullback-power axiom). Is that proven anywhere?
Yes, this follows from Proposition 5.6 in Mazel-Gee II, which shows that a derived two-variable Quillen adjunction induces a closed symmetric monoidal structure on the localization.
Since the monoidal product of two terminal objects is the terminal object, this resulting monoidal structure is cartesian.
Thus, the hom from an ∞-cofibrant to an ∞-fibrant simplicial space computes the correct derived hom with respect to diagonal weak equivalences.
Thanks for the pointer! Had not looked at part II before.
I’ll need to think about how that Prop. 5.6 implies the claim. At face value it looks like claiming something slightly weaker, no?
But i see that the pushout-product axiom is in Def. 5.1, and it holds for simplicial spaces by Ex. 5.3. Now if that example 5.3 simply included the word “closed”, we’d immediately have the pullback-power axiom by the usual dualization.
Concerning Example 5.3: it is closed by Definition 4.3, and Section 4 does establish the pullback-power axiom.
Oh, I see, it’s built into Def. 4.3. Great, thanks again.
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