A Peculiar Model Structure on Simplicial Sets?

Question

I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do homological algebra for commutative monoids, but first let me explain some background, my motivation, and articulate more precisely what I am after.

Background

A Quillen Model structure on a category has three classes of morphisms: fibrations, weak equivalences, and cofibrations. This structure allows one to do many advanced homotopical constructions mimicking the homotopy theory of (nice) topological spaces. There is a notion of Quillen equivalence between model categories which consists of a particular adjunction between the two model categories in question. This gives you "equivalent homotopy theories" for the two model categories in question.

The usual Model structure on simplicial sets has fibrations the Kan fibrations, the weak-equivalences are the maps which induces isomorphisms of homotopy groups, and the cofibrations are the (levelwise) inclusions. This is equivalent to the usual model category of topological spaces, and the Quillen equivalence is realized by the adjoint pair of functors: the geometric realization functor and the singular functor.

Quillen model categories are also useful for doing homological algebra, and particularly for working with derived categories. For reasonable abelian categories there are several nice (Quillen equivalent) model category structures on the category of (possibly bounded) chain complexes which one can use which reproduce the derived category. More precisely the homotopy category of the model category is the derived category and the "homotopical constructions" I mentioned above, in this case, correspond to the notion of (total) derived functor.

This story is further enriched by the Dold-Kan correspondence which is an equivalence between the categories of positively graded chain complexes of, say, abelian groups and the category of simplicial abelian groups, a.k.a. simplicial sets which are also abelian group objects. This in turn is Quillen equivalent to a model category of topological abelian groups.

Previous MathOverflow Question and Progress

Previously I asked a question on MO about doing homological algebra for commutative monoids. I got many fascinating and exciting answers. Some were more or less "here is something you might try", some were more like "here is a bit that people have done, but the full theory hasn't been studied". After hearing those answers I'm much more excited about the field with one element. But still, in the end none of the answers really had what I was after. Then Reid Barton asked his MO question.

This got me thinking about commutative monoids and simplicial commutative monoids again. I had some nice observations which have lead me to the question at hand.

1. The first is that while a simplicial abelian group is automatically a Kan simplicial set (i.e. if you forget the abelian group structure what have sitting in front of you is a Kan simplicial set) this is not the case for simplicial commutative monoids. A simplicial commutative monoid does not have to be a Kan simplicial set.

2. Next, I realized that the (normalized!) Dold-Kan correspondence still seems to work. You can go from a simplicial commutative monoid to a "complex" of monoids, ( and back again It is just an adjunction. Thanks, Reid, for pointing this out!).

3. If your simplicial commutative monoid is also a Kan complex, then under the Dold-Kan correspondence you get a complex where the bottom object is a commutative monoid, but all the rest are abelian groups (this was pointed out to me by Reid Barton in a conversation we had recently). Thus the theory of topological commutative monoids (which was one of the suggested answers to my previous question), which has a nice model category structure (see Clark Barwick's answer to an MO question), should be equivalent to a theory of chain complexes of this type. It shouldn't model arbitrary complexes of commutative monoids.

4. If you have a simplicial set which is not necessarily a Kan complex you can still define the naive simplicial homotopy "groups" $\pi_iX$. I put "goups" in quotes because for a general simplicial set these are just pointed sets. If your complex is Kan, these are automatically groups (for $i>0$). If your simplicial set is a simplicial abelian group, these are abelian groups (for all $i$) and they are precisely the homology groups of the chain complex you get under the Dold-Kan correspondence. If your simplicial set is a commutative monoid, but not Kan, then they are commutative monoids and are again the "homology" monoids of the chain complex you get under the Dold-Kan correspondence.

The Question

All this suggests that there should be a Quillen model structure on simplicial commutative monoids in which the weak equivalences are the $\pi_{\bullet} $-isomorphisms, where here $\pi_{\bullet} $ denotes the naive simplicial version, i.e. these are commutative monoids, not groups. I'm sure if such a thing was well known then it would have been mentioned as an answer to my previous question. I'd really like to see something that generalizes the usual theory of abelian groups. That way if we worked with simplicial abelian groups and construct derived functors we would just reproduce the old answers. As a stepping stone, there should be a companion model structure on simplicial sets which, I think, is more likely to be well known.

One of the properties that I think this hypothetical model structure on simplicial sets should have is that every simplicial set is fibrant, not just the Kan simplicial sets.

Question: Is there a model structure on simplicial sets in which every simplicial set is fibrant, and such that the weak equivalences between the Kan complexes are exactly the usual weak equivalences? Specifically can the weak equivalences be taken to be those maps which induce $\pi_*$-isomorphism, where these are the naive simplicial homotopy sets?

If this model structure exists, I'd like to know as much as possible about it. If you have any references to the literature, I'd appreciate those too, but the main question is as it stands.

Answer 1

So the short answer is that there is not such a model structure. The difficulty arises in trying to show that the class of weak equivalences has all of the necessary properties; in particular, even two-of-three does not hold for the naive definition. The first difficulty arises even before that: on ordinary simplicial sets we can arrange for a model of every set that is "minimal" on the $\pi_0$-level, meaning that every connected component has exactly one $0$-simplex. In simplicial commutative monoids we can no longer do this. However, we could assume that in order to be a weak equivalence we need to be a $\pi_*$-isomorphism when choosing any (coherently chosen) basepoints.

For the purposes of our discussion we are going to assume that $\pi_*$-equivalences use the model $S^n = \Delta^n/\partial\Delta^n$. (This is the model that most closely mimicks the boundary maps in the Dold-Kan correspondence.) Now let $X = S^2$, and let $Y$ be $S^2$ with an extra $0$-simplex connected by a $1$-simplex to the original basepoint. (So it looks like a balloon on a string.) We define a map $X\rightarrow Y$ to be the inclusion of $S^2$ in the obvious manner, and a map $Y\rightarrow X$ to be collapsing the extra $1$-simplex back down. Then the composition of these two maps is the identity on $X$, so obviously a weak equivalence. The map $X\rightarrow Y$ is also a weak equivalence, because adding the "string" can't add any new homotopy groups to $X$. However, the map $Y\rightarrow X$ is not a weak equivalence, as $\pi_2Y$ based at the extra point is a one-point set but $\pi_2X$ at its image is a two-point set.

The problem arose because in order to show that $\pi_*$ was invariant of basepoint in the usual Kan complex model we needed to be able to "pull back" simplices along paths in the simplicial set, which used the Kan condition. The new model does not have such a condition, and thus we can't necessarily pull things back.

Another observation along these lines. Take any connected simplicial set $X$, and let $Y$ be $X$ with a "string" added to it at any basepoint. Then $*\rightarrow Y$ (including into the new point) is a weak equivalence, and $X\rightarrow Y$ (including into itself) is a weak equivalence. Thus in the homotopy category, $X$ is isomorphic to a point (and thus the homotopy category is just the category of sets) ... which is presumably not desired.

-- The Bourbon seminar

Answer 2

There is, of course, at least one model structure on any category in which every object is both fibrant and cofibrant, but it's not that interesting: the weak equivalences are just the isomorphisms and everything is a fibration and a cofibration. (-:

However, I'm pretty sure there is not a model structure on simplicial sets in which the cofibrations are the monomorphisms (as in the Kan and the Joyal model structures) and in which everything is fibrant. In fact, quite generally, for any class of cofibrations in a suitably well-behaved category there is a smallest class of weak equivalences forming a model structure (corresponding to a largest class of fibrations), called the "left-determined model structure." Of course, any model structure with the same cofibrations will then be a left Bousfield localization of the left-determined one. We can then ask what the fibrant objects are in the left-determined model structure.

Left-determined model structures were first studied by Rosicky and Tholen in "Left-determined model categories and universal homotopy theories," and the construction was given a more explicit description by Olschok in "Left-determined model categories for locally presentable categories." According to Olschok, in the model structure on simplicial sets that is left-determined by the monomorphisms, the fibrant objects can be described as those for which $X\to 1$ has the RLP relative to the class of maps of the form

$$ i_n \;\square\; \gamma_i \;\square\; \gamma \;\square\; \dots \;\square\; \gamma $$

where $\square$ denotes a pushout product, $i_n: \partial\Delta^n \hookrightarrow\Delta^n$ is the inclusion of a boundary, $J$ is the nerve of the walking isomorphism, $\gamma: 2\to J$ is the inclusion of the two vertices, and $\gamma_0,\gamma_1:1\to J$ are the two inclusions of single vertices. (This follows from his description because $2\to J \to 1$ is a (mono, trivial fibration) factorization, see his Remark 4.17.) Evidently not every simplicial set has this property.

The best intuition I have so far for an object $X$ like this is that (1) if we look at the "equivalences" in $X$, i.e. simplices equipped with specified inverses, then these equivalences can be "composed" and thus form a sort (cubical-ish) of $\infty$-groupoid (this corresponds to lifting for the above maps with $n=0$), while (2) the non-equivalence simplices can't necessarily be composed, but can be "transported" along equivalences on their boundaries (this is $n\gt 0$).

Answer 3

I've gotten a bit lost with the revisions and subquestions, but let me try to summarize some of the discussions of the Bourbon Seminar about this question.

First, note that the usual argument that abelian group objects are automatically fibrant falls apart entirely in the context of commutative monoids. More specifically, the "least fibrant" simplicial sets out there admit a commutative monoid structure.

Example. Consider the "principal spine" $P:=\Delta^1\sqcup^{\Delta^0}\Delta^1\sqcup^{\Delta^0}\cdots\sqcup^{\Delta^0}\Delta^1\subset\Delta^n$. The set $P_0=\{0,1,\dots,n\}$ of $0$-simplices admit a commutative monoid structure given by taking the maximum. This extends in a canonical fashion to the simplicial set $P$ itself. Observe that this kind of example satisfies no inner horn-filling condition.

Second, to enlarge slightly on Reid's comment above, note that, unlike the situation with abelian group objects, the homotopy groups of a commutative monoid based at the identity cannot characterize the weak equivalences, even between fibrant objects. That is, the functor $\pi_{\ast}:s\mathbf{Comm}\to\mathbf{Ab}^{\mathrm{gr}}$ is not conservative in the sense that a map $X\to Y$ between Kan-fibrant commutative monoids is a weak equivalence if and only if $\pi_{\ast}(X)\to\pi_{\ast}(Y)$ is an isomorphism. (Of course it would be if we restricted to the class of connected objects.) This leads to the following, which should be at the heart of any Dold-Kan correspondence.

Challenge. Find an "algebraic" category $\mathbf{A}$ and a conservative functor $\Pi:s\mathbf{Comm}\to\mathbf{A}$.

Of course there are plenty of pairs $(\mathbf{A},\Pi)$ of this kind. (After all, the initial one is the homotopy category.) But it is not obvious how algebraic these can be made.

Answer 4

Joyal's model category for quasicategories might be a step in the direction that you're going, though it does not answer your question in itself. In particular, it has more fibrant objects than the usual model structure does, but the same weak equivalences between Kan complexes. (In fact the usual structure is a Bousfield localization of Joyal's structure with Kan complexes the fibrant objects.)