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2 math error corrected: Dold-Kan is NOT an equiv. for simp. comm. monoids.

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, I think)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.

1

# A Peculiar Model Structure on Simplicial Sets?

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, I think).

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.