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I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral approach and the simplicial approach. There's these excellent answers in What is a simplicial commutative ring from the point of view of homotopy theory? . However I still have some doubts.

Let $R$ be a commutative ring. In page 46 of Lurie's thesis (http://dspace.mit.edu/handle/1721.1/30144), he says that there are functors $$R\text{-Alg}^{{\Delta}^{op}} \xrightarrow{\phi} \text{comm. DG} \ R\text{-Alg} \xrightarrow{\psi} E_{\infty} \ R\text{-Alg}$$ and that if $R$ is a $\mathbb{Q}$-algebra, then both arrows are an equivalence of categories when the codomain of $\psi$ is restricted to the connectives objects.

Furthermore, in the above cited question, Lurie says that, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic.

Where can I find a proof for these facts? Furthermore, if possible, could someone, please, describe explicitly these functors briefly? I believe the first functor $\phi$ is just the ordinary Dold-Kan correspondence however I'm not sure since I've never seen this correspondence for $R$-algebras when $R$ is not a field...

So, summarizing the question:

1)What's $\phi$ and $\psi$?

2)Why, when $R$ is a $\mathbb{Q}$-algebra and $\psi$ is restricted in the codomain to the connective algebras, $\phi$ and $\psi$ are equivalences?

3)Why, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic?

Thanks in advance.

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral approach and the simplicial approach. There's these excellent answers in What is a simplicial commutative ring from the point of view of homotopy theory? . However I still have some doubts.

Let $R$ be a commutative ring. In page 46 of Lurie's thesis (http://dspace.mit.edu/handle/1721.1/30144), he says that there are functors $$R\text{-Alg}^{{\Delta}^{op}} \xrightarrow{\phi} \text{comm. DG} \ R\text{-Alg} \xrightarrow{\psi} E_{\infty} \ R\text{-Alg}$$ and that if $R$ is a $\mathbb{Q}$-algebra, then both arrows are an equivalence of categories when the codomain of $\phi$ is restricted to the connectives objects.

Furthermore, in the above cited question, Lurie says that, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic.

Where can I find a proof for these facts? Furthermore, if possible, could someone, please, describe explicitly these functors briefly? I believe the first functor $\phi$ is just the ordinary Dold-Kan correspondence however I'm not sure since I've never seen this correspondence for $R$-algebras when $R$ is not a field...

So, summarizing the question:

1)What's $\phi$ and $\psi$?

2)Why, when $R$ is a $\mathbb{Q}$-algebra and $\psi$ is restricted in the codomain to the connective algebras, $\phi$ and $\psi$ are equivalences?

3)Why, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic?

Thanks in advance.

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral approach and the simplicial approach. There's these excellent answers in What is a simplicial commutative ring from the point of view of homotopy theory? . However I still have some doubts.

Let $R$ be a commutative ring. In page 46 of Lurie's thesis (http://dspace.mit.edu/handle/1721.1/30144), he says that there are functors $$R\text{-Alg}^{{\Delta}^{op}} \xrightarrow{\phi} \text{comm. DG} \ R\text{-Alg} \xrightarrow{\psi} E_{\infty} \ R\text{-Alg}$$ and that if $R$ is a $\mathbb{Q}$-algebra, then both arrows are an equivalence of categories when the codomain of $\psi$ is restricted to the connectives objects.

Furthermore, in the above cited question, Lurie says that, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic.

Where can I find a proof for these facts? Furthermore, if possible, could someone, please, describe explicitly these functors briefly? I believe the first functor $\phi$ is just the ordinary Dold-Kan correspondence however I'm not sure since I've never seen this correspondence for $R$-algebras when $R$ is not a field...

So, summarizing the question:

1)What's $\phi$ and $\psi$?

2)Why, when $R$ is a $\mathbb{Q}$-algebra and $\psi$ is restricted in the codomain to the connective algebras, $\phi$ and $\psi$ are equivalences?

3)Why, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic?

Thanks in advance.

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral approach and the simplicial approach. There's these excellent answers in What is a simplicial commutative ring from the point of view of homotopy theory? . However I still have some doubts.

Let $R$ be a commutative ring. In page 46 of Lurie's thesis (http://dspace.mit.edu/handle/1721.1/30144), he says that there are functors $$R\text{-Alg}^{{\Delta}^{op}} \xrightarrow{\phi} \text{comm. DG} \ R\text{-Alg} \xrightarrow{\psi} E_{\infty} \ R\text{-Alg}$$ and that if $R$ is a $\mathbb{Q}$-algebra, then both arrows are an equivalence of categories when the codomain of $\psi$ is restricted to the connectives objects.

Furthermore, in the above cited question, Lurie says that, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic.

Where can I find a proof for these facts? Furthermore, if possible, could someone, please, describe explicitly these functors briefly? I believe the first functor $\phi$ is just the ordinary Dold-Kan correspondence however I'm not sure since I've never seen this correspondence for $R$-algebras when $R$ is not a field...

So, summarizing the question:

1)What's $\phi$ and $\psi$?

2)Why, when $R$ is a $\mathbb{Q}$-algebra and $\psi$ is restricted in the codomain to the connective algebras, $\phi$ and $\psi$ are equivalences?

3)Why, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic?

Thanks in advance.

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral approach and the simplicial approach. There's these excellent answers in What is a simplicial commutative ring from the point of view of homotopy theory? . However I still have some doubts.

Let $R$ be a commutative ring. In page 46 of Lurie's thesis (http://dspace.mit.edu/handle/1721.1/30144), he says that there are functors $$R\text{-Alg}^{{\Delta}^{op}} \xrightarrow{\phi} \text{comm. DG} \ R\text{-Alg} \xrightarrow{\psi} E_{\infty} \ R\text{-Alg}$$ and that if $R$ is a $\mathbb{Q}$-algebra, then both arrows are an equivalence of categories when the codomain of $\psi$ is restricted to the connectives objects.

Furthermore in the above cited question Lurie says that, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic.

Where can I find a proof for these facts? Furthermore, if possible, could someone, please, describe explicitly these functors briefly? I believe the first functor $\phi$ is just the ordinary Dold-Kan correspondence however i'm not sure since I've never seen this correspondence for $R$-algebras when $R$ is not a field...

So, summarizing the question:

1)What's $\phi$ and $\psi$?

2)Why, when $R$ is a $\mathbb{Q}$-algebra $\phi$ and $\psi$ are equivalences when $\psi$ is restricted to the connective algebras?

3)Why, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic?

Thanks in advance.

I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral approach and the simplicial approach. There's these excellent answers in What is a simplicial commutative ring from the point of view of homotopy theory? . However I still have some doubts.

Let $R$ be a commutative ring. In page 46 of Lurie's thesis (http://dspace.mit.edu/handle/1721.1/30144), he says that there are functors $$R\text{-Alg}^{{\Delta}^{op}} \xrightarrow{\phi} \text{comm. DG} \ R\text{-Alg} \xrightarrow{\psi} E_{\infty} \ R\text{-Alg}$$ and that if $R$ is a $\mathbb{Q}$-algebra, then both arrows are an equivalence of categories when the codomain of $\psi$ is restricted to the connectives objects.

Furthermore, in the above cited question, Lurie says that, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic.

Where can I find a proof for these facts? Furthermore, if possible, could someone, please, describe explicitly these functors briefly? I believe the first functor $\phi$ is just the ordinary Dold-Kan correspondence however I'm not sure since I've never seen this correspondence for $R$-algebras when $R$ is not a field...

So, summarizing the question:

1)What's $\phi$ and $\psi$?

2)Why, when $R$ is a $\mathbb{Q}$-algebra and $\psi$ is restricted in the codomain to the connective algebras, $\phi$ and $\psi$ are equivalences?

3)Why, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic?

Thanks in advance.