4
$\begingroup$

In a model category $\mathcal{C}$, is the filtered colimit of fibrations, resp. trivial fibrations, a fibration, resp. trivial fibration?

Thm. 1.2.3.5 in Toen-Vezzosi's "Homotopical algebraic geometry, II" (https://arxiv.org/pdf/math/0404373.pdf) seems to give a criterion, but it points to the wrong reference, as nothing of the sort is in Hovey's book "Model Categories".

$\endgroup$
4
  • 3
    $\begingroup$ Proposition 3.3 of "On combinatorial model categories" (arxiv.org/abs/0708.2185) can be helpful. The class of (trivial) fibrations in a combinatorial model category is accessible and accessibly embedded. $\endgroup$ – Philippe Gaucher Oct 2 '17 at 10:37
  • $\begingroup$ First we have a cofinal functor from a directed category to a given filtered category. Then a directed category is a union of a $\lambda-$sequence of directed sub categories (whose cardinalities are strictly less than that of the directed category). A colimit of a $\lambda-$sequence of fibrations is a fibration by Hovey's proof. So we are reduced to prove that for each directed sub-category, the directed colimit of fibrations is a fibration. To prove this, we can use induction on the cardinality of the directed category. $\endgroup$ – user12580 Oct 13 '18 at 8:44
  • $\begingroup$ Also in Thm 1.2.3.5 (4) he uses the smallness which by the above argument is equivalent to compactness since the ordinal is $\omega$. See mathoverflow.net/questions/188714/… $\endgroup$ – user12580 Oct 13 '18 at 8:48
  • $\begingroup$ Note that the definition of $\omega-$smallness of Hovey might not be what we want. Let the cardinal of $\lambda$ be $\aleph_0$. Then $\lambda$ is not $\lambda-$filtered in the sense of Hovey. But if the set of maps contains all identities, one may extend a smaller sequence to a larger sequence obviously. $\endgroup$ – user12580 Oct 13 '18 at 9:12
5
$\begingroup$

In Lemma 7.4.1 of Hovey's book, he does prove that colimits of $\lambda$-sequences of cofibrations preserve fibrations, respectively trivial fibrations. However, upon inspecting the proof, the assumption on the transition maps being cofibrations is used only insofar domains and codomains of the generating cofibrations and trivial cofibrations are assumed to be finite relative to cofibrations. In Toen-Vezzosi, such domains and codomains are finite relative to the whole category, hence you can run the same proof as in Hovey, removing this assumption.

I don't know of a general criterion about preservation of fibrations/trivial fibrations under a class of colimits, but I don't expect such preservation to hold in great generality. If filtered colimits preserve fibrations and trivial cofibrations, then, upon endowing the diagram category with the induced model structure constructed in $\S$ 5 of Hovey's book, $\text{colim}$ being a left Quillen functor (and hence already preserving cofibrations and trivial cofibrations) would come to preserve weak equivalences.

Simplicial commutative rings with the projective model structure is a counterexample: coproducts clearly don't preserve weak equivalences (pick two ring maps with same source and that are not $\text{Tor}$-independent).

$\endgroup$
1
$\begingroup$

If you are willing to restrict yourself to combinatorial model categories, there is a proposition in the paper by G.Raptis and J. Rosicky https://arxiv.org/abs/1403.3042, proposition 4.1. It only requires that for the generating set of cofibrations the domains and codomains are $\kappa$-presentable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy