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I'm reading Illusie's book "complexe de cotangent et Deformations I". And I'm puzzled on the definition of cotangent complex.

I formulate my question as follows:

Suppose $C$ and $D$ are abelian categories, and $F:C \to D$ is a functor. I would like to consider its left-derived functor $LF$. There are two ways:

(1) $C$ embedded into $K(C)$ ,the homotopy category of complexes in $C$, and denote the left-derived functor of F as $L^1F$, so you calculate $L^1F$ by resolution of chain complexes.

(2) $C$ embedded into $sC$, the category of simplicial objects in C, and denote the left-derived functor of F as $L^2F$, so you calculate $L^2F$ by resolution of simplicial objects.

then, my question is:

1. for any object $X$ in $C$, what is the relation ship with $L^1F(X)$ and $L^2F(X)$? Is there any relation like Dold-Kan?

2. then what is the crucial reason for the defition of cotangent complex to use simplicial resolution? Is it simply because there is no left-derived functor for the Kahler differential functor if I embed the category $C$ into $K(C)$?

Appreciate very much!

1

resolution by simplicial objects versus resolution by chain complex

I'm reading Illusie's book "complexe de cotangent et Deformations I". And I'm puzzled on the definition of cotangent complex.

I formulate my question as follows:

Suppose $C$ and $D$ are abelian categories, and $F:C \to D$ is a functor. I would like to consider its left-derived functor $LF$. There are two ways:

(1) $C$ embedded into $K(C)$ ,the homotopy category of complexes in $C$, and denote the left-derived functor of F as $L^1F$, so you calculate $L^1F$ by resolution of chain complexes.

(2) $C$ embedded into $sC$, the category of simplicial objects in C, and denote the left-derived functor of F as $L^2F$, so you calculate $L^2F$ by resolution of simplicial objects.

then, my question is:

1. for any object $X$ in $C$, what is the relation ship with $L^1F(X)$ and $L^2F(X)$? Is there any relation like Dold-Kan?

2. then what is the crucial reason for the defition of cotangent complex to use simplicial resolution?

Appreciate very much!