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##Previous MathOverflow Question and Progress##

##The Question##

Previous MathOverflow Question and Progress

The Question

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.

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.

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.

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.

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.

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.