It's a standard fact that given a small category $\mathcal{C},$ the category of pre-sheaves $\text{Psh}(\mathcal{C})$ is the free co-completion of it.

I'm sure this can be done not only for $\text{Set}$-enriched categories but for general $\mathcal{V}$-enriched categories, with the appropriate notions of $\mathcal{V}$-enriched colimit, and functor preserving the enrichment, and I just found it in section 4.4. of Kelly's Basic Concepts of Enriched Category Theory.

So now the question is: can one prove proposition 2.3 in this paper about simplicial pre-sheaves on $\mathcal{C}$ being the universal model category on $\mathcal{C}$ by just doing the $\mathcal{V}$-enriched co-completion with $\mathcal{V}=\text{sSet}$?

  • $\begingroup$ Naive comment: did you try to consider the category of presheaves valued in $\mathcal{V}$ ? $\endgroup$ – Philippe Gaucher May 16 at 19:19
  • $\begingroup$ @PhilippeGaucher Yes, of course it will be that. But there are details to be added to show the universal property in the general case, and I did not want to rediscover the wheel. $\endgroup$ – giuseppe May 16 at 19:23

Given $C$ a small category (eventually, a small simplicial category) I denote by $UC$ the projective model structure on the category of simplicial presheaves on $C$ as in the paper. Using the kind of argument you have in mind we obtain the following theorem:

Theorem: If $M$ is a simplicial model category, then there is an equivalence of categories between:

  • (Simplicial) Functors $C \to M$ taking values in the full subcategory of cofibrant objects.
  • Simplicial left Quillen functor $UC \to M$.

In one direction, the equivalence is simply given by restricting to the Yoneda embedding $ C \to UC$ as representable are cofibrant in the projective model structure, this forces the composite functor $C \to UC \to M$ to take values in cofibrant objects. In the converse direction, one takes the unique simplicial left adjoint functor $UC \to M$ and check, using the axiom of simplicial model category for $M$ that this is a left Quillen functor.

However, this is not what the paper you mention proves.

There, they start from a model category $M$ that is not assumed to be a simplicial model category, and a functor $C \to M$ not assumed to takes values in cofibrant objects. And construct a left Quillen functor $UC \to M$ by considering (and choosing) a cofibrant simplicial resolution of the functor $C \to M$ they started from. In particular, the "uniqueness" of the left Quillen functor obtained this way, is only up to homotopy (to be more precise, up to a contractible space of choices).

One abstract way to understand the relation between the two is as follows:

Given $M$ a combinatorial left proper model category, there is a Quillen equivalent simplicial model structure on the category $sM$ on the category of simplicial objects of $M$, (this is explained in the paper "Replacing model categories with simplicial one" by Dugger)

The evaluation at $[0]$ gives a left Quillen equivalence $sM \to M$

One way to understand the non-simplicial theorem is that if you start from $C \to M$, you can see it as a functor $C \to sM$ taking values in constant simplicial objects, then take a levelwise cofibrant replacement to obtain a functor $C \to sM$ taking value in cofibrant object, apply the "simplicial theorem" to get a Quillen functor $UC \to sM$ and finally, post compose with Quillen functor $sM \to M$ that evaluate at $[0]$.

Now for the model structure on $sM$ to exist we need $M$ to be combinatorial and left proper, if you are willing to work with a left semi-model structure instead it is enough to assume that $M$ is an accessible model category (no properness assumption).

But in some sense the central observation of the paper you quote, is that, even if the model structure on $sM$ cannot be constructed, the overall construction make sense with no assumption $M$ (other than being a model category, I guess they also need functorial factorization, I do not remember).

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  • $\begingroup$ Thanks for the explanation. I am wondering if from the theorem about simplicial model categories one can then derive also the result in the paper about general model categories? $\endgroup$ – giuseppe May 16 at 20:03
  • $\begingroup$ @IvanDiLiberti this the paper mentioned in the question we are talking about ? $\endgroup$ – Simon Henry May 16 at 20:18
  • $\begingroup$ @IvanDiLiberti I linked to that paper in my question. $\endgroup$ – giuseppe May 16 at 20:18
  • $\begingroup$ @giuseppe : I've added a comment on the relation between the two at the end. Does it helps ? At some point, even if it would be not true that you cannot use the result you mention to prove the proposition. This is really the best one can do: To apply the theorem you have in mind you need a simplicial model category, and I do not know any other way to make one appear, than this construction. $\endgroup$ – Simon Henry May 16 at 20:20
  • $\begingroup$ @SimonHenry Thanks very much. Yes this new edit is helpful and it's the kind of argument I was looking for. It's still disappointing to me that it can't be done to recover the full result of the paper though. (Yes they also need functorial factorization) $\endgroup$ – giuseppe May 16 at 20:30

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