# 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 $\mathcal{V}$ ? May 16, 2020 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. May 16, 2020 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).

• 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? May 16, 2020 at 20:03
• @IvanDiLiberti this the paper mentioned in the question we are talking about ? May 16, 2020 at 20:18
• @IvanDiLiberti I linked to that paper in my question. May 16, 2020 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. May 16, 2020 at 20:20
• @giuseppe I think the fact you can’t recover the result straight from the idea of the free cocompletion is just an indication of the depth of the result. The purely simplicial theorem would have been considerably less significant a result. May 16, 2020 at 22:00