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Take a topologically enriched small category $\mathcal{P}$ and the category of enriched diagrams of spaces $[\mathcal{P},\mathrm{Top}]_0$. We work with the category of $\Delta$-generated spaces equipped with the mixed model structure. Suppose that the injective model structure exists (the paper http://dx.doi.org/10.4310/HHA.2019.v21.n2.a15 gives some sufficient conditions).

Is there an explicit description of a fibrant replacement somewhere ?

I can only understand that the injective fibrant diagrams are some kind of cofree enriched diagrams.

EDIT: by explicit, I mean which enables us to make some calculations.

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Section 8 of my paper All (∞,1)-toposes have strict univalent universes shows that under fairly general conditions, injective fibrant replacements can be given by cobar constructions (e.g. the dual of Corollary 8.16). I think this will apply to your situation if the hom-objects of $\mathcal{P}$ are cofibrant and the inclusions of identity morphisms are cofibrations. I don't know what sort of calculations you want to do, but cobar constructions come with a filtration that sometimes gives rise to calculational tools like spectral sequences.

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  • $\begingroup$ If you mean by cofibrant: m-cofibrant. Yes they are. The category $\mathcal{P}$ has for objects the strictly positive real numbers and $\mathcal{P}(\ell,\ell')$ is the space of surjective nondecreasing maps from $[0,\ell]$ to $[0,\ell']$, space which is contractible. And your second condition holds as well. As for the calculation, I want to lift some model structure along a right adjoint and I want to apply the Quillen Path Object argument. Maybe I will ask another question if I can't use your result. $\endgroup$
    – Philippe Gaucher
    Commented May 2, 2022 at 18:01

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