# Universal model category as a $\text{sSet}$-enriched co-completion

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 $$\mathcal{V}=\text{sSet}$$?

• Naive comment: did you try to consider the category of presheaves valued in $\mathcal{V}$ ? – Philippe Gaucher May 16 at 19:19
• @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. – 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 $$$$ gives a left Quillen equivalence $$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 $$$$.

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).

• 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? – giuseppe May 16 at 20:03
• @IvanDiLiberti this the paper mentioned in the question we are talking about ? – Simon Henry May 16 at 20:18
• @IvanDiLiberti I linked to that paper in my question. – giuseppe May 16 at 20:18
• @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. – Simon Henry May 16 at 20:20
• @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) – giuseppe May 16 at 20:30