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Added a note on cotorsion pairs
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Denis Nardin
Denis Nardin
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This is not quite the answer to your question as you pose it. I hope it will be useful anyway. By and large I am just expanding user337830 comments. Everything will use homological grading (what can I say, I am a homotopy theorist :)).

The relationship between derived functors in homological algebra and derived functors in homotopy theory is very close. In fact the notion of derived functor in homotopy theory is a (very successful) attempt to generalize the homological algebra notion.

Let me talk about right derived functors for one second. Let $A$ be a Grothendieck abelian category. Then it is known it has enough injectives. This allows us to define a model structure on the category $Ch(A)$ of chain complexes in $A$ such that

  • weak equivalences are quasi-isomorphisms;
  • cofibrations are chain maps $C_*\to D_*$ that are levelwise injective;
  • fibrations are chain maps $C_*\to D_*$ that are levelwise surjective and such that the kernel $K_*$ is a dg-injective complex (that is all $K_n$ are injective objects of $A$ and for all acyclic complexes $E_*$ every chain map $E_*\to K_*$ is nullhomotopic, bounded above complexes of injectives are an example of such).

This model structure is sometimes called the injective model structure on $Ch(A)$.

In particular if $M\in A$, an injective resolution of $M$ gives a fibrant replacement for the chain complex consisting of $M$ in degree 0. So it is easy to see that if $A,A'$ are two Grothendieck abelian categories and $F:A\to A'$ is a left-exact functor, the derived functor of $F$ is precisely the right-derived functor of $F_*(-):Ch(A)\to Ch(A')$.

Similarly if $A$ has enough projectives you can form a projective model structure that will allow you to describe left derived functors of right-exact functors.

Now, these two model structures in fact present the same homotopy theory (e.g. they have the same weak equivalences! More to the point, they are Quillen equivalent), so you can pass from one to the other with a minimum of fuss. Their common homotopy category is called the (unbounded) derived category of $A$. 

There are also other model structures on $Ch(A)$ with the same weak equivalences that are more convenient for other purposes, e.g. the flat model structure (where the fibrant objects are complexes of flat modules $C_*$ such that $C_*\otimes-$ preserves quasi-isomorphisms). In general there is a correspondence between algebraic model structures on $Ch(A)$ and cotorsion pairs on $A$. See this paper by Hovey for more details.

This is not quite the answer to your question as you pose it. I hope it will be useful anyway. By and large I am just expanding user337830 comments. Everything will use homological grading (what can I say, I am a homotopy theorist :)).

The relationship between derived functors in homological algebra and derived functors in homotopy theory is very close. In fact the notion of derived functor in homotopy theory is a (very successful) attempt to generalize the homological algebra notion.

Let me talk about right derived functors for one second. Let $A$ be a Grothendieck abelian category. Then it is known it has enough injectives. This allows us to define a model structure on the category $Ch(A)$ of chain complexes in $A$ such that

  • weak equivalences are quasi-isomorphisms;
  • cofibrations are chain maps $C_*\to D_*$ that are levelwise injective;
  • fibrations are chain maps $C_*\to D_*$ that are levelwise surjective and such that the kernel $K_*$ is a dg-injective complex (that is all $K_n$ are injective objects of $A$ and for all acyclic complexes $E_*$ every chain map $E_*\to K_*$ is nullhomotopic, bounded above complexes of injectives are an example of such).

This model structure is sometimes called the injective model structure on $Ch(A)$.

In particular if $M\in A$, an injective resolution of $M$ gives a fibrant replacement for the chain complex consisting of $M$ in degree 0. So it is easy to see that if $A,A'$ are two Grothendieck abelian categories and $F:A\to A'$ is a left-exact functor, the derived functor of $F$ is precisely the right-derived functor of $F_*(-):Ch(A)\to Ch(A')$.

Similarly if $A$ has enough projectives you can form a projective model structure that will allow you to describe left derived functors of right-exact functors.

Now, these two model structures in fact present the same homotopy theory (e.g. they have the same weak equivalences! More to the point, they are Quillen equivalent), so you can pass from one to the other with a minimum of fuss. Their common homotopy category is called the (unbounded) derived category of $A$. There are also other model structures on $Ch(A)$ with the same weak equivalences that are more convenient for other purposes, e.g. the flat model structure (where the fibrant objects are complexes of flat modules $C_*$ such that $C_*\otimes-$ preserves quasi-isomorphisms).

This is not quite the answer to your question as you pose it. I hope it will be useful anyway. By and large I am just expanding user337830 comments. Everything will use homological grading (what can I say, I am a homotopy theorist :)).

The relationship between derived functors in homological algebra and derived functors in homotopy theory is very close. In fact the notion of derived functor in homotopy theory is a (very successful) attempt to generalize the homological algebra notion.

Let me talk about right derived functors for one second. Let $A$ be a Grothendieck abelian category. Then it is known it has enough injectives. This allows us to define a model structure on the category $Ch(A)$ of chain complexes in $A$ such that

  • weak equivalences are quasi-isomorphisms;
  • cofibrations are chain maps $C_*\to D_*$ that are levelwise injective;
  • fibrations are chain maps $C_*\to D_*$ that are levelwise surjective and such that the kernel $K_*$ is a dg-injective complex (that is all $K_n$ are injective objects of $A$ and for all acyclic complexes $E_*$ every chain map $E_*\to K_*$ is nullhomotopic, bounded above complexes of injectives are an example of such).

This model structure is sometimes called the injective model structure on $Ch(A)$.

In particular if $M\in A$, an injective resolution of $M$ gives a fibrant replacement for the chain complex consisting of $M$ in degree 0. So it is easy to see that if $A,A'$ are two Grothendieck abelian categories and $F:A\to A'$ is a left-exact functor, the derived functor of $F$ is precisely the right-derived functor of $F_*(-):Ch(A)\to Ch(A')$.

Similarly if $A$ has enough projectives you can form a projective model structure that will allow you to describe left derived functors of right-exact functors.

Now, these two model structures in fact present the same homotopy theory (e.g. they have the same weak equivalences! More to the point, they are Quillen equivalent), so you can pass from one to the other with a minimum of fuss. Their common homotopy category is called the (unbounded) derived category of $A$. 

There are also other model structures on $Ch(A)$ with the same weak equivalences that are more convenient for other purposes, e.g. the flat model structure (where the fibrant objects are complexes of flat modules $C_*$ such that $C_*\otimes-$ preserves quasi-isomorphisms). In general there is a correspondence between algebraic model structures on $Ch(A)$ and cotorsion pairs on $A$. See this paper by Hovey for more details.

Source Link
Denis Nardin
Denis Nardin
  • 16.5k
  • 2
  • 69
  • 103

This is not quite the answer to your question as you pose it. I hope it will be useful anyway. By and large I am just expanding user337830 comments. Everything will use homological grading (what can I say, I am a homotopy theorist :)).

The relationship between derived functors in homological algebra and derived functors in homotopy theory is very close. In fact the notion of derived functor in homotopy theory is a (very successful) attempt to generalize the homological algebra notion.

Let me talk about right derived functors for one second. Let $A$ be a Grothendieck abelian category. Then it is known it has enough injectives. This allows us to define a model structure on the category $Ch(A)$ of chain complexes in $A$ such that

  • weak equivalences are quasi-isomorphisms;
  • cofibrations are chain maps $C_*\to D_*$ that are levelwise injective;
  • fibrations are chain maps $C_*\to D_*$ that are levelwise surjective and such that the kernel $K_*$ is a dg-injective complex (that is all $K_n$ are injective objects of $A$ and for all acyclic complexes $E_*$ every chain map $E_*\to K_*$ is nullhomotopic, bounded above complexes of injectives are an example of such).

This model structure is sometimes called the injective model structure on $Ch(A)$.

In particular if $M\in A$, an injective resolution of $M$ gives a fibrant replacement for the chain complex consisting of $M$ in degree 0. So it is easy to see that if $A,A'$ are two Grothendieck abelian categories and $F:A\to A'$ is a left-exact functor, the derived functor of $F$ is precisely the right-derived functor of $F_*(-):Ch(A)\to Ch(A')$.

Similarly if $A$ has enough projectives you can form a projective model structure that will allow you to describe left derived functors of right-exact functors.

Now, these two model structures in fact present the same homotopy theory (e.g. they have the same weak equivalences! More to the point, they are Quillen equivalent), so you can pass from one to the other with a minimum of fuss. Their common homotopy category is called the (unbounded) derived category of $A$. There are also other model structures on $Ch(A)$ with the same weak equivalences that are more convenient for other purposes, e.g. the flat model structure (where the fibrant objects are complexes of flat modules $C_*$ such that $C_*\otimes-$ preserves quasi-isomorphisms).