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Let $\mathcal{M}$ be a

locally finitely presentable model category, cofibrantly generated by two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial cofibrations with presentable domain and codomain.

I know that weak equivalences and fibrations are stable by filtered colimits.

  1. What can be said about cofibrations and trivial cofibrations?

  2. Is there a class of good examples in which this is known to be true?

  3. Are there additional axioms that can be imposed that ensure this?

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    $\begingroup$ The best account I am aware of is in Rosicky's "On combinatorial model categories". $\endgroup$ Nov 8, 2022 at 12:36
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    $\begingroup$ @IvanDiLiberti I have seen that paper, but I haven't found anything I could use about my question. I'll try to take another look. Thanks! $\endgroup$ Nov 8, 2022 at 12:49
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    $\begingroup$ Do you mean to assume $\mathcal{I}$ and $\mathcal{J}$ have finitely presentable domain and codomain? $\endgroup$
    – Zhen Lin
    Nov 8, 2022 at 22:59
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    $\begingroup$ @ZhenLin Yes. I mean to assume that hom(A, -) preserves with filtered colimits $\endgroup$ Nov 9, 2022 at 0:04

1 Answer 1

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If both cofibrations and weak equivalences are stable under filtered colimits, then so are trivial cofibrations. This happens for instance if $\mathcal{M}$ is a presheaf category on an elegant Reedy category (such as $\Delta$) with cofibrations the monomorphisms, whatever the weak equivalences are (see Cor. 3.4.41 & (the proof of) Prop. 8.2.9 here). This is also true for cochain complexes in a Grothendieck abelian category (with cofibrations as monomorphisms and quasi-isomorphisms as weak equivalences). Similarly, if you consider simplicial sheaves on a site. More generally, if you can define cofibrations and weak equivalences using functors which commute with filtered colimits (e.g. suitable kernels to detect monomorphisms in a topos, cohomology sheaves or sheaves of homotopy groups to detect weak equivalences), you have a good chance of survival. This kind of properties is easily seen to be preserved under left Bousfield localizations.

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  • $\begingroup$ so it seems that this property is quite common in examples arising in nature. thank you for the overview. $\endgroup$ Nov 9, 2022 at 15:38

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