What are the advantages of simplicial model categories over non-simplicial ones?

Question

Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a fan of quasicategories, for example). What do I miss out on?

I think the idea is supposed to be: the mapping spaces in a simplicial model category are functorial on the nose, rather than something which must be treated in an ad-hoc manner. But I'd like to better understand how this plays out.

Questions:

1. What constructions / results are only available for simplicial model categories, and not for non-simplicial model categories?

2. What constructions / results were originally proven for simplicial model categories and only later adapted to the non-simplicial case, or else are easier in the simplicial case?

3. When adapting results to the non-simplicial case, are the issues mainly set-theoretical, or are there other issues?

For example, there are general results on existence of Bousfield localization which require enrichment of the model structure; in order to get by without enrichment one needs to assume combinatorialness -- but this is basically a set-theoretical restriction.

Answer 1

(1) The main benefit of a simplicial model category structure is explicit formulas for (co)simplicial resolutions. Remark 5.2.10 in Hovey's book identifies the functor $A\mapsto \tilde{A}^m = A\otimes \Delta[m]$ as a cosimplicial resolution (over all $m$, obviously), if $A$ is cofibrant. I needed this in my thesis, specifically Corollary 6.7 here, and was never able to figure out how to get away from the simplicial assumption in that proof. There's also an analogous formula for simplicial resolutions of fibrant objects.

(3) I'm not aware of any Bousfield localization results that need $M$ to be a simplicial model category. They usually need simplicial mapping spaces, but those are easy to get from a framing, which exists whenever $M$ is cofibrantly generated (see Chapter 5 of Hovey's book) and not necessarily combinatorial. When adapting to the non-simplicial case, usually just a little bit more care is needed to deal with the framings. A good example is how early work of Rezk was always set in simplicial model categories, and later work was not (these are good examples for your question (2)). Authors usually leave out the details of making the simplicial framings work, and that's because it's really quite easy if all you need are simplicial mapping spaces and their connection to (co)fibrant objects and weak equivalences.

(2) Here are some explicit examples. Dugger's universal homotopy theory paper (and other papers of his from this era), Gutierrez-Roitzheim work with framings and localization, Barnes-Roitzheim framings and Bousfield localization, Rezk's thesis is written for simplicial model categories but works much more generally, and part of that general story is in my paper with Gutierrez.

Answer 2

I believe the main reasons enriched model category are simpler boils down to:

Tensoring and co-tensoring by $\Delta[1]$ gives very well behaved path objects and cylinder objects adjoint to each other

Slightly more generally, (co)tensoring by more general simplicial set gives an explicit construction of the (co)tensorization by spaces, and this also allows to recover the enrichment in spaces using the simplicial hom between cofibrant and fibrant object. But cylinder object plays a very special role in this story.

Having such well behaved cylinders makes the construction of model structure considerably easier, this is very apparent in M.Olschok works (see his theorem 3.16).

In fact I claim the following:

Theorem: Let $C$ be a presentable simplicially enriched category. Let $I$ and $J$ be sets of morphism in $C$ such that $J \subset I-Cof$ (or more generally, $J$ are cofibration in the sense below). I'm also assuming* that the domain of every arrow in $I$ and $J$ is cofibrant in the sense below.

Then there is a simplicial combinatorial left semi-model structure on $C$ such that:

• The cofibrations are generated by $I$ as a simplicially enriched class of cofibration, i.e. they are generated in the naive sense by the $(\partial \Delta[n] \hookrightarrow \Delta[n]) \widehat{\otimes} I $.

• The fibrations between fibrant objects are characterized by the right lifting property against the $(\Lambda^k[n] \hookrightarrow \Delta[n] )\widehat{\otimes} I$ and the $(\partial \Delta[n] \hookrightarrow \Delta[n]) \widehat{\otimes} J$.

• and Obviously, The trivial fibrations are characterized by the right lifting property against the $(\partial \Delta[n] \hookrightarrow \Delta[n]) \widehat{\otimes} I $.

where $\otimes$ denotes the tensoring by simplicial sets, and $\widehat{\otimes}$ denotes the corresponding corner-product/pushout-product.

This type of theorem makes it incredibly easy to construct simplicial model categories. For example, knowing this it is immediate the left bousfield localization exists, or that left transfer along simplicial adjunction essentially always exists.

The proof of this theorem is not available yet (in preparation, it should be available before in a few month). Though some very similar result are already available: if you add the assumption that every object in $C$ is cofibrant, then this should follows from Olschok's theorem 3.16 quoted above applied to the cylinder functor given by tensoring by $\Delta[1]$ (maybe with a little bit of work to check that the definition of the class of fibration and cofibration match those I have given). And If you are ok with getting something weaker than a left semi-model structure, this follows from my theorem 3.2.1 here (which produce what I call a "weak model structure").

(Regarding the assumption (*) it can actually be removed, but some modification needs to be made to the statement to keep it correct, and they are a bit to complicated to be explained here... also I don't understand them very well so far)

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Another concrete example of interest, which I think could also be somehow attributed to the theorem above, is the construction by Ching & Riehl of a model structure on the category of algebraically cofibrant objects of a Simplicial model category $C$, Quillen equivalent to $C$. Somehow a dual of the well known Nikolaus construction.

This construction uses in an essentially way that the category is simplicial, and more precisely that the the "cofibrant replacement comonad" used to define the category of algebraically cofibrant object is a simplicially enriched monad. This makes the category of cofibrant objects a simplicial category, and makes it easy to put a model structure on it.

In fact, I claim that this result fails without simplicial enrichment. More precisely, if you take $C$ to be the category of simplicial sets, and consider the (non-simplicial) algebraic weak factorization system generated by the $\partial \Delta[n] \hookrightarrow \Delta[n]$ then this category of algebraically cofibrant object has very few model structure, and they are all essentially trivial (their $\infty$-categorical localization are actually posets).

Unfortunately the proof of this last claim rellies on a lot of so far unpublished things so I'm not sure I can explain it on MO.

(This being said Ching & Riehl result might actually be true for any combinatorial model category as soon as one only wants a right semi-model structure on the category of algebraically cofibrant objects... but I don't know a proof of this)

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A final remark: I don't think these things are specifically about "simplicial model category", it is more about the difference between enriched model categories on non-enriched ones. You can do similar construction for essentially any kind of enrichment.