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Let's say we are working with a fibrant simplicially enriched category $\mathbf{B}$ that has all limits and all homotopy limits, and let $\mathbf{A}$ be a full subcategory that is closed under weak equivalences (in the simplicially enriched sense).

  1. Is it the case that a homotopy limit can be computed as the strict limit of a homotopy equivalent diagram? (In the style of model categories, where we first choose a suitable replacement diagram and then take the strict limit of that, as shown in Chapter 20 of this book in the fullest generality).
  2. If I know that a certain diagram in $\mathbf{A}$ has a strict limit, is it true that an objectwise homotopy equivalent diagram also has a strict limit? (Remember that $\mathbf{B}$ has all of them by assumption, although the inclusion doesn't necessarily preserve them).
  3. Both the above questions in a particularly explicit example, such as $\mathcal{sSet}^{\circ}$, i.e. the simplicial category of all Kan complexes.

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  • $\begingroup$ Without additional assomption, that's probably false... are you willing to assume that for e.g. B has simplicially enriched limits (co-tensor)? That might help... $\endgroup$
    – Simon Henry
    Commented Aug 12, 2022 at 17:41
  • $\begingroup$ The point I'm getting at is that if homotopy limits in B can be described using a Bousfield-Kan formula, then at least in B homotopy limits are équivalent to strict limits of an equivalent diagram... $\endgroup$
    – Simon Henry
    Commented Aug 12, 2022 at 17:46
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    $\begingroup$ Yes I can assume tensors and cotensors. In fact, as a general guideline, I can have basically all sorts of nice assumptions on $\mathbf{B}$, but $\mathbf{A}$ should be as general as possible, except that I can close it under weak equivalences. I don't even want to assume completeness, or homotopy completeness on $\mathbf{A}$. $\endgroup$
    – Giulio Lo Monaco
    Commented Aug 12, 2022 at 21:12
  • $\begingroup$ I edited the question, asking if something can be said at least in the very special case of the category of fibrant objects in the classical model structure on simplicial sets. $\endgroup$
    – Giulio Lo Monaco
    Commented Aug 17, 2022 at 13:52

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