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There is actually not so direct a connection with the derived functor point of view, at least as far as I know, and for a pretty simple reason: the derived functor POV concerns "abelian" stuff, abelian groups or chain complexes being acted on by $G$. But extensions are nonabelian groups! So how are they supposed to appear?

Actually the connection is with the homotopy-theoretic point of view on group cohomology (if you like this has to do with "nonabelian derived functors"). From this point of view group cohomology is the cohomology of a space called the classifying space $BG$, also known as the Eilenberg-MacLane space $K(G, 1)$. This is a space with $\pi_1(BG) \cong G$ and all higher homotopy groups trivial, and group cohomology

$$H^n(G, A) \cong H^n(BG, A) \cong [BG, B^n A]$$

is the cohomology of this space, which can in turn be identified with homotopy classes of maps from $BG$ to spaces of the form $B^n A$, which can either be thought of as iterated classifying spaces or as Eilenberg-MacLane spaces $K(A, n)$, with $\pi_n(B^n A) \cong A$ and all other homotopy groups trivial.

So what does this have to do with group extensions? Given a homotopy class of maps $f : BG \to B^n A$ representing a class in group cohomology we can take its homotopy fiber, which is a "nonabelian derived kernel," if you like. This produces a space $X$ fitting into a fiber sequence (a homotopical version of a long exact sequence)

$$\dots \to B^{n-1} A \to X \to BG \xrightarrow{f} B^n A$$

and this space $X$ has the following properties, which can be shown using the long exact sequence in homotopy (which is the result of applying $\pi_0$ to the fiber sequence):

• If $n = 2$, then $\pi_1(X)$ is a group fitting into a short exact sequence $1 \to A \to \pi_1(X) \to G \to 1$ and all higher homotopy is trivial. This turns out to be the central extension classified by $f$.
• If $n \ge 3$, then $\pi_1(X) \cong G$ and $\pi_{n-1}(X) \cong A$, and all other homotopy groups are trivial. Furthermore every space with only these two nontrivial homotopy groups and trivial $\pi_1$ action on $\pi_{n-1}$ arises in this way. $f$ here is one of the simplest examples of a $k$-invariant and this becomes one of the simplest examples of a Postnikov tower.

For more on this you can check out, for example, John Baez's Lectures on $n$-categories and cohomology. This version of the story only covers central extensions and trivial $\pi_1$ action on $\pi_{n-1}$; non-central extensions and arbitrary action of $\pi_1$ on $\pi_{n-1}$ can also be classified this way but one needs to work with cohomology with local / twisted coefficients. Instead of working with homotopy types you can tell this story in terms of, equivalently, either homotopy types with a $G$-action or homotopy types equipped with a map to $BG$, which is like a "nonabelian derived category of $G$-modules."

There is actually not so direct a connection with the derived functor point of view, at least as far as I know, and for a pretty simple reason: the derived functor POV concerns "abelian" stuff, abelian groups or chain complexes being acted on by $G$. But extensions are nonabelian groups! So how are they supposed to appear?

Actually the connection is with the homotopy-theoretic point of view on group cohomology (if you like this has to do with "nonabelian derived functors"). This concerns the classifying space $BG$, also known as the Eilenberg-MacLane space $K(G, 1)$. This is a space with $\pi_1(BG) \cong G$ and all higher homotopy groups trivial, and group cohomology

$$H^n(G, A) \cong H^n(BG, A) \cong [BG, B^n A]$$

is the cohomology of this space, which can in turn be identified with homotopy classes of maps from $BG$ to spaces of the form $B^n A$, which can either be thought of as iterated classifying spaces or as Eilenberg-MacLane spaces $K(A, n)$, with $\pi_n(B^n A) \cong A$ and all other homotopy groups trivial.

So what does this have to do with group extensions? Given a homotopy class of maps $f : BG \to B^n A$ representing a class in group cohomology we can take its homotopy fiber, which is a "nonabelian derived kernel," if you like. This produces a space $X$ fitting into a fiber sequence (a homotopical version of a long exact sequence)

$$\dots \to B^{n-1} A \to X \to BG \xrightarrow{f} B^n A$$

and this space $X$ has the following properties, which can be shown using the long exact sequence in homotopy (which is the result of applying $\pi_0$ to the fiber sequence):

• If $n = 2$, then $\pi_1(X)$ is a group fitting into a short exact sequence $1 \to A \to \pi_1(X) \to G \to 1$ and all higher homotopy is trivial. This turns out to be the central extension classified by $f$.
• If $n \ge 3$, then $\pi_1(X) \cong G$ and $\pi_{n-1}(X) \cong A$, and all other homotopy groups are trivial. Furthermore every space with only these two nontrivial homotopy groups and trivial $\pi_1$ action on $\pi_{n-1}$ arises in this way. $f$ here is one of the simplest examples of a $k$-invariant and this becomes one of the simplest examples of a Postnikov tower.

For more on this you can check out, for example, John Baez's Lectures on $n$-categories and cohomology. This version of the story only covers central extensions and trivial $\pi_1$ action on $\pi_{n-1}$; non-central extensions and arbitrary action of $\pi_1$ on $\pi_{n-1}$ can also be classified this way but one needs to work with cohomology with local / twisted coefficients. Instead of working with homotopy types you can tell this story in terms of, equivalently, either homotopy types with a $G$-action or homotopy types equipped with a map to $BG$, which is like a "nonabelian derived category of $G$-modules."

There is actually not so direct a connection with the derived functor point of view, at least as far as I know, and for a pretty simple reason: the derived functor POV concerns "abelian" stuff, abelian groups or chain complexes being acted on by $G$. But extensions are nonabelian groups! So how are they supposed to appear?

Actually the connection is with the homotopy-theoretic point of view on group cohomology (if you like this has to do with "nonabelian derived functors"). From this point of view group cohomology is the cohomology of a space called the classifying space $BG$, also known as the Eilenberg-MacLane space $K(G, 1)$. This is a space with $\pi_1(BG) \cong G$ and all higher homotopy groups trivial, and group cohomology

$$H^n(G, A) \cong H^n(BG, A) \cong [BG, B^n A]$$

is the cohomology of this space, which can in turn be identified with homotopy classes of maps from $BG$ to spaces of the form $B^n A$, which can either be thought of as iterated classifying spaces or as Eilenberg-MacLane spaces $K(A, n)$, with $\pi_n(B^n A) \cong A$ and all other homotopy groups trivial.

So what does this have to do with group extensions? Given a homotopy class of maps $f : BG \to B^n A$ representing a class in group cohomology we can take its homotopy fiber, which is a "nonabelian derived kernel," if you like. This produces a space $X$ fitting into a fiber sequence (a homotopical version of a long exact sequence)

$$\dots \to B^{n-1} A \to X \to BG \xrightarrow{f} B^n A$$

and this space $X$ has the following properties, which can be shown using the long exact sequence in homotopy (which is the result of applying $\pi_0$ to the fiber sequence):

• If $n = 2$, then $\pi_1(X)$ is a group fitting into a short exact sequence $1 \to A \to \pi_1(X) \to G \to 1$. This turns out to be the central extension classified by $f$.
• If $n \ge 3$, then $\pi_1(X) \cong G$ and $\pi_{n-1}(X) \cong A$. Furthermore every space with only these two nonzero homotopy groups and trivial $\pi_1$ action on $\pi_{n-1}$ arises in this way. $f$ here is one of the simplest examples of a $k$-invariant and this becomes one of the simplest examples of a Postnikov tower.

For more on this you can check out, for example, John Baez's Lectures on $n$-categories and cohomology. This version of the story only covers central extensions and trivial $\pi_1$ action on $\pi_{n-1}$; non-central extensions and arbitrary action of $\pi_1$ on $\pi_{n-1}$ can also be classified this way but one needs to work with cohomology with local / twisted coefficients. Instead of working with homotopy types you can tell this story in terms of, equivalently, either homotopy types with a $G$-action or homotopy types equipped with a map to $BG$, which is like a "nonabelian derived category of $G$-modules."

There is actually not so direct a connection with the derived functor point of view, at least as far as I know, and for a pretty simple reason: the derived functor POV concerns "abelian" stuff, abelian groups or chain complexes being acted on by $G$. But extensions are nonabelian groups! So how are they supposed to appear?

Actually the connection is with the homotopy-theoretic point of view on group cohomology (if you like this has to do with "nonabelian derived functors"). From this point of view group cohomology is the cohomology of a space called the classifying space $BG$, also known as the Eilenberg-MacLane space $K(G, 1)$. This is a space with $\pi_1(BG) \cong G$ and all higher homotopy groups trivial, and group cohomology

$$H^n(G, A) \cong H^n(BG, A) \cong [BG, B^n A]$$

is the cohomology of this space, which can in turn be identified with homotopy classes of maps from $BG$ to spaces of the form $B^n A$, which can either be thought of as iterated classifying spaces or as Eilenberg-MacLane spaces $K(A, n)$, with $\pi_n(B^n A) \cong A$ and all other homotopy groups trivial.

So what does this have to do with group extensions? Given a homotopy class of maps $f : BG \to B^n A$ representing a class in group cohomology we can take its homotopy fiber, which is a "nonabelian derived kernel," if you like. This produces a space $X$ fitting into a fiber sequence (a homotopical version of a long exact sequence)

$$\dots \to B^{n-1} A \to X \to BG \xrightarrow{f} B^n A$$

and this space $X$ has the following properties, which can be shown using the long exact sequence in homotopy (which is the result of applying $\pi_0$ to the fiber sequence):

• If $n = 2$, then $\pi_1(X)$ is a group fitting into a short exact sequence $1 \to A \to \pi_1(X) \to G \to 1$ and all higher homotopy is trivial. This turns out to be the central extension classified by $f$.
• If $n \ge 3$, then $\pi_1(X) \cong G$ and $\pi_{n-1}(X) \cong A$, and all other homotopy groups are trivial. Furthermore every space with only these two nontrivial homotopy groups and trivial $\pi_1$ action on $\pi_{n-1}$ arises in this way. $f$ here is one of the simplest examples of a $k$-invariant and this becomes one of the simplest examples of a Postnikov tower.

For more on this you can check out, for example, John Baez's Lectures on $n$-categories and cohomology. This version of the story only covers central extensions and trivial $\pi_1$ action on $\pi_{n-1}$; non-central extensions and arbitrary action of $\pi_1$ on $\pi_{n-1}$ can also be classified this way but one needs to work with cohomology with local / twisted coefficients. Instead of working with homotopy types you can tell this story in terms of, equivalently, either homotopy types with a $G$-action or homotopy types equipped with a map to $BG$, which is like a "nonabelian derived category of $G$-modules."

There is actually not so direct a connection with the derived functor point of view, at least as far as I know, and for a pretty simple reason: the derived functor POV concerns "abelian" stuff, abelian groups or chain complexes being acted on by $G$. But central extensions are nonabelian groups! So how are they supposed to appear?

Actually the connection is with the homotopy-theoretic point of view on group cohomology (if you like this has to do with "nonabelian derived functors"). From this point of view group cohomology is the cohomology of a space called the classifying space $BG$, also known as the Eilenberg-MacLane space $K(G, 1)$. This is a space with $\pi_1(BG) \cong G$ and all higher homotopy groups trivial, and group cohomology

$$H^n(G, A) \cong H^n(BG, A) \cong [BG, B^n A]$$

is the cohomology of this space, which can in turn be identified with homotopy classes of maps from $BG$ to spaces of the form $B^n A$, which can either be thought of as iterated classifying spaces or as Eilenberg-MacLane spaces $K(A, n)$, with $\pi_n(B^n A) \cong A$ and all other homotopy groups trivial.

So what does this have to do with group extensions? Given a homotopy class of maps $f : BG \to B^n A$ representing a class in group cohomology we can take its homotopy fiber, which is a "nonabelian derived kernel," if you like. This produces a space $X$ fitting into a fiber sequence (a homotopical version of a long exact sequence)

$$\dots \to B^{n-1} A \to X \to BG \xrightarrow{f} B^n A$$

and this space $X$ has the following properties, which can be shown using the long exact sequence in homotopy (which is the result of applying $\pi_0$ to the fiber sequence):

• If $n = 2$, then $\pi_1(X)$ is a group fitting into a short exact sequence $1 \to A \to \pi_1(X) \to G \to 1$. This is the central extension classified by $f$.
• If $n \ge 3$, then $\pi_1(X) \cong G$ and $\pi_{n-1}(X) \cong A$. Furthermore every space with only these two nonzero homotopy groups and trivial $\pi_1$ action on $\pi_{n-1}$ arises in this way. $f$ here is one of the simplest examples of a $k$-invariant and this becomes one of the simplest examples of a Postnikov tower.

For more on this you can check out, for example, John Baez's Lectures on $n$-categories and cohomology. This version of the story only covers central extensions and trivial $\pi_1$ action on $\pi_{n-1}$; non-central extensions and arbitrary action of $\pi_1$ on $\pi_{n-1}$ can also be classified this way but one needs to work with cohomology with local / twisted coefficients.

There is actually not so direct a connection with the derived functor point of view, at least as far as I know, and for a pretty simple reason: the derived functor POV concerns "abelian" stuff, abelian groups or chain complexes being acted on by $G$. But extensions are nonabelian groups! So how are they supposed to appear?

Actually the connection is with the homotopy-theoretic point of view on group cohomology (if you like this has to do with "nonabelian derived functors"). From this point of view group cohomology is the cohomology of a space called the classifying space $BG$, also known as the Eilenberg-MacLane space $K(G, 1)$. This is a space with $\pi_1(BG) \cong G$ and all higher homotopy groups trivial, and group cohomology

$$H^n(G, A) \cong H^n(BG, A) \cong [BG, B^n A]$$

is the cohomology of this space, which can in turn be identified with homotopy classes of maps from $BG$ to spaces of the form $B^n A$, which can either be thought of as iterated classifying spaces or as Eilenberg-MacLane spaces $K(A, n)$, with $\pi_n(B^n A) \cong A$ and all other homotopy groups trivial.

So what does this have to do with group extensions? Given a homotopy class of maps $f : BG \to B^n A$ representing a class in group cohomology we can take its homotopy fiber, which is a "nonabelian derived kernel," if you like. This produces a space $X$ fitting into a fiber sequence (a homotopical version of a long exact sequence)

$$\dots \to B^{n-1} A \to X \to BG \xrightarrow{f} B^n A$$

and this space $X$ has the following properties, which can be shown using the long exact sequence in homotopy (which is the result of applying $\pi_0$ to the fiber sequence):

• If $n = 2$, then $\pi_1(X)$ is a group fitting into a short exact sequence $1 \to A \to \pi_1(X) \to G \to 1$. This turns out to be the central extension classified by $f$.
• If $n \ge 3$, then $\pi_1(X) \cong G$ and $\pi_{n-1}(X) \cong A$. Furthermore every space with only these two nonzero homotopy groups and trivial $\pi_1$ action on $\pi_{n-1}$ arises in this way. $f$ here is one of the simplest examples of a $k$-invariant and this becomes one of the simplest examples of a Postnikov tower.

For more on this you can check out, for example, John Baez's Lectures on $n$-categories and cohomology. This version of the story only covers central extensions and trivial $\pi_1$ action on $\pi_{n-1}$; non-central extensions and arbitrary action of $\pi_1$ on $\pi_{n-1}$ can also be classified this way but one needs to work with cohomology with local / twisted coefficients. Instead of working with homotopy types you can tell this story in terms of, equivalently, either homotopy types with a $G$-action or homotopy types equipped with a map to $BG$, which is like a "nonabelian derived category of $G$-modules."