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David White
David White
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Yes, there are many references that discuss this. The first is Quillen's Homotopical Algebra. Chapter II contains much of what you're asking about, especially II.4 and II.6

For a more modern version, see Schwede-Shipley Algebras and Modules in Monoidal Model Categories. Section 5 is all examples and contains precise references in Quillen's work above.

This covers your (2), (3), and (6). I agree with Zhen Lin that (4) is the wrong question to ask, since the structure as an abelian category is less important than the model structure if you are trying to study the homotopy category from (3). For (5), check out the Dold-Kan theorem. A nice write-up is in Goerss-Schemmerhorn Model Categories and Simplicial Methods 4.1. This also includes the model structure on simplicial R modules and what resolutions look like for simplicial R modules, answering your (4) in the correct homotopically meaningful sense. It is clear from 4.1 that $N$ is both an equivalence of categories (if by DG we mean non-negatively graded) and a Quillen equivalence, so the homotopy theories agree, and this answers your (5)

Yes, there are many references that discuss this. The first is Quillen's Homotopical Algebra. Chapter II contains much of what you're asking about, especially II.4 and II.6

For a more modern version, see Schwede-Shipley Algebras and Modules in Monoidal Model Categories. Section 5 is all examples and contains precise references in Quillen's work above.

This covers your (2), (3), and (6). I agree with Zhen Lin that (4) is the wrong question to ask, since the structure as an abelian category is less important than the model structure if you are trying to study the homotopy category from (3). For (5), check out the Dold-Kan theorem. A nice write-up is in Goerss-Schemmerhorn Model Categories and Simplicial Methods 4.1. This also includes the model structure on simplicial R modules and what resolutions look like for simplicial R modules, answering your (4) in the correct homotopically meaningful sense. It is clear from 4.1 that $N$ is both an equivalence of categories and a Quillen equivalence, so the homotopy theories agree, and this answers your (5)

Yes, there are many references that discuss this. The first is Quillen's Homotopical Algebra. Chapter II contains much of what you're asking about, especially II.4 and II.6

For a more modern version, see Schwede-Shipley Algebras and Modules in Monoidal Model Categories. Section 5 is all examples and contains precise references in Quillen's work above.

This covers your (2), (3), and (6). I agree with Zhen Lin that (4) is the wrong question to ask, since the structure as an abelian category is less important than the model structure if you are trying to study the homotopy category from (3). For (5), check out the Dold-Kan theorem. A nice write-up is in Goerss-Schemmerhorn Model Categories and Simplicial Methods 4.1. This also includes the model structure on simplicial R modules and what resolutions look like for simplicial R modules, answering your (4) in the correct homotopically meaningful sense. It is clear from 4.1 that $N$ is both an equivalence of categories (if by DG we mean non-negatively graded) and a Quillen equivalence, so the homotopy theories agree, and this answers your (5)

Source Link
David White
David White
  • 30.3k
  • 9
  • 153
  • 250

Yes, there are many references that discuss this. The first is Quillen's Homotopical Algebra. Chapter II contains much of what you're asking about, especially II.4 and II.6

For a more modern version, see Schwede-Shipley Algebras and Modules in Monoidal Model Categories. Section 5 is all examples and contains precise references in Quillen's work above.

This covers your (2), (3), and (6). I agree with Zhen Lin that (4) is the wrong question to ask, since the structure as an abelian category is less important than the model structure if you are trying to study the homotopy category from (3). For (5), check out the Dold-Kan theorem. A nice write-up is in Goerss-Schemmerhorn Model Categories and Simplicial Methods 4.1. This also includes the model structure on simplicial R modules and what resolutions look like for simplicial R modules, answering your (4) in the correct homotopically meaningful sense. It is clear from 4.1 that $N$ is both an equivalence of categories and a Quillen equivalence, so the homotopy theories agree, and this answers your (5)