Consider the following category $\mathbf{GroupMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ is a pair $(\varphi,f)$, where $\varphi:G \to H$ is a group homomorphism and $f:\varphi^* N \to M$ is a $\Bbb Z[G]$-module homomorphism, where $\varphi^*$ denotes restriction of scalars along $\varphi$. The motivation for considering this category is that for each $n$, group cohomology defines a contravariant functor $H^n:\mathbf{GroupMod}^* \to \mathbf{Ab}$ and it is this functoriality that gives rise to all the usual ways of changing the group (restriction, corestriction, inflation, conjugation).

This is "correct" category for group cohomology if one wants to vary the group.

There's a very similar category for group homology called $\mathbf{GroupMod}$, the definition is basically the same, but in the definition of a morphism, one has a map $f:M \to \varphi^*N$ instead of $\varphi^*N \to M$. This is the "correct" category for group homology if one wants to vary the group.

So the question arises, what properties do these categories have? I think they are not abelian. Are they semiabelian? If not, homological? Protomodular?

That would allow one to apply some version of non-abelian homological algebra (such as in the paper "Homology and homotopy in semi-abelian categories" by Van der Linden). The idea would be to take in some suitable form the "derived functor" of the invariants functor $\mathbf{GroupMod}^* \to \mathbf{Ab}$, (so we're taking group cohomology without fixing the group, in some sense), maybe this would yield something interesting about group cohomology.

Edit I think there's a very natural description of the category $\mathbf{GroupMod}$ that is maybe helpful. Consider the contravariant pseudofunctor $F:\mathrm{Grp}^\text{op} \to \mathbf{Cat}$ sending $G$ to the category of left $\Bbb Z[G]$-modules and sending morphisms of groups to restriction of scalars, then $\mathbf{GroupMod}$ is the Grothendieck construction $\int F$.

Consider the following category $\mathbf{GrpMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ is a pair $(\varphi,f)$, where $\varphi:G \to H$ is a group homomorphism and $f:\varphi^* N \to M$ is a $\Bbb Z[G]$-module homomorphism, where $\varphi^*$ denotes restriction of scalars along $\varphi$. The motivation for considering this category is that for each $n$, group cohomology defines a contravariant functor $H^n:\mathbf{GrpMod}^* \to \mathbf{Ab}$ and it is this functoriality that gives rise to all the usual ways of changing the group (restriction, corestriction, inflation, conjugation).

This is "correct" category for group cohomology if one wants to vary the group.

There's a very similar category for group homology called $\mathbf{GrpMod}$, the definition is basically the same, but in the definition of a morphism, one has a map $f:M \to \varphi^*N$ instead of $\varphi^*N \to M$. This is the "correct" category for group homology if one wants to vary the group.

So the question arises, what properties do these categories have? I think they are not abelian. Are they semiabelian? If not, homological? Protomodular?

That would allow one to apply some version of non-abelian homological algebra (such as in the paper "Homology and homotopy in semi-abelian categories" by Van der Linden). The idea would be to take in some suitable form the "derived functor" of the invariants functor $\mathbf{GrpMod}^* \to \mathbf{Ab}$, (so we're taking group cohomology without fixing the group, in some sense), maybe this would yield something interesting about group cohomology.

Edit I think there's a very natural description of the category $\mathbf{GrpMod}$ that is maybe helpful. Consider the contravariant pseudofunctor $F:\mathrm{Grp}^\text{op} \to \mathbf{Cat}$ sending $G$ to the category of left $\Bbb Z[G]$-modules and sending morphisms of groups to restriction of scalars, then $\mathbf{GrpMod}$ is the Grothendieck construction $\int F$.

Consider the following category $\mathbf{GroupMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ is a pair $(\varphi,f)$, where $\varphi:G \to H$ is a group homomorphism and $f:\varphi^* N \to M$ is a $\Bbb Z[G]$-module homomorphism, where $\varphi^*$ denotes restriction of scalars along $\varphi$. The motivation for considering this category is that for each $n$, group cohomology defines a contravariant functor $H^n:\mathbf{GroupMod}^* \to \mathbf{Ab}$ and it is this functoriality that gives rise to all the usual ways of changing the group (restriction, corestriction, inflation, conjugation).

This is "correct" category for group cohomology if one wants to vary the group.

There's a very similar category for group homology called $\mathbf{GroupMod}$, the definition is basically the same, but in the definition of a morphism, one has a map $f:M \to \varphi^*N$ instead of $\varphi^*N \to M$. This is the "correct" category for group homology if one wants to vary the group.

So the question arises, what properties do these categories have? I think they are not abelian. Are they semiabelian? If not, homological? Protomodular?

That would allow one to apply some version of non-abelian homological algebra (such as in this paper by Van der Linden). The idea would be to take in some suitable form the "derived functor" of the invariants functor $\mathbf{GroupMod}^* \to \mathbf{Ab}$, (so we're taking group cohomology without fixing the group, in some sense), maybe this would yield something interesting about group cohomology.

Edit I think there's a very natural description of the category $\mathbf{GroupMod}$ that is maybe helpful. Consider the contravariant pseudofunctor $F:\mathrm{Grp}^{op} \to \mathbf{Cat}$ sending $G$ to the category of left $\Bbb Z[G]$-modules and sending morphisms of groups to restriction of scalars, then $\mathbf{GroupMod}$ is the Grothendieck construction $\int F$.

Consider the following category $\mathbf{GroupMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ is a pair $(\varphi,f)$, where $\varphi:G \to H$ is a group homomorphism and $f:\varphi^* N \to M$ is a $\Bbb Z[G]$-module homomorphism, where $\varphi^*$ denotes restriction of scalars along $\varphi$. The motivation for considering this category is that for each $n$, group cohomology defines a contravariant functor $H^n:\mathbf{GroupMod}^* \to \mathbf{Ab}$ and it is this functoriality that gives rise to all the usual ways of changing the group (restriction, corestriction, inflation, conjugation).

This is "correct" category for group cohomology if one wants to vary the group.

There's a very similar category for group homology called $\mathbf{GroupMod}$, the definition is basically the same, but in the definition of a morphism, one has a map $f:M \to \varphi^*N$ instead of $\varphi^*N \to M$. This is the "correct" category for group homology if one wants to vary the group.

So the question arises, what properties do these categories have? I think they are not abelian. Are they semiabelian? If not, homological? Protomodular?

That would allow one to apply some version of non-abelian homological algebra (such as in the paper "Homology and homotopy in semi-abelian categories" by Van der Linden). The idea would be to take in some suitable form the "derived functor" of the invariants functor $\mathbf{GroupMod}^* \to \mathbf{Ab}$, (so we're taking group cohomology without fixing the group, in some sense), maybe this would yield something interesting about group cohomology.

Edit I think there's a very natural description of the category $\mathbf{GroupMod}$ that is maybe helpful. Consider the contravariant pseudofunctor $F:\mathrm{Grp}^\text{op} \to \mathbf{Cat}$ sending $G$ to the category of left $\Bbb Z[G]$-modules and sending morphisms of groups to restriction of scalars, then $\mathbf{GroupMod}$ is the Grothendieck construction $\int F$.

Consider the following category $\mathbf{GroupMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ is a pair $(\varphi,f)$, where $\varphi:G \to H$ is a group homomorphism and $f:\varphi^* N \to M$ is a $\Bbb Z[G]$-module homomorphism, where $\varphi^*$ denotes restriction of scalars along $\varphi$. The motivation for considering this category is that for each $n$, group cohomology defines a contravariant functor $H^n:\mathbf{GroupMod}^* \to \mathbf{Ab}$ and it is this functoriality that gives rise to all the usual ways of changing the group (restriction, corestriction, inflation, conjugation).

This is "correct" category for group cohomology if one wants to vary the group.

There's a very similar category for group homology called $\mathbf{GroupMod}$, the definition is basically the same, but in the definition of a morphism, one has a map $f:M \to \varphi^*N$ instead of $\varphi^*N \to M$. This is the "correct" category for group homology if one wants to vary the group.

So the question arises, what properties do these categories have? I think they are not abelian. Are they semiabelian? If not, homological? Protomodular?

That would allow one to apply some version of non-abelian homological algebra (such as in this paper by Van der Linden). The idea would be to take in some suitable form the "derived functor" of the invariants functor $\mathbf{GroupMod}^* \to \mathbf{Ab}$, (so we're taking group cohomology without fixing the group, in some sense), maybe this would yield something interesting about group cohomology.

Consider the following category $\mathbf{GroupMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ is a pair $(\varphi,f)$, where $\varphi:G \to H$ is a group homomorphism and $f:\varphi^* N \to M$ is a $\Bbb Z[G]$-module homomorphism, where $\varphi^*$ denotes restriction of scalars along $\varphi$. The motivation for considering this category is that for each $n$, group cohomology defines a contravariant functor $H^n:\mathbf{GroupMod}^* \to \mathbf{Ab}$ and it is this functoriality that gives rise to all the usual ways of changing the group (restriction, corestriction, inflation, conjugation).

This is "correct" category for group cohomology if one wants to vary the group.

There's a very similar category for group homology called $\mathbf{GroupMod}$, the definition is basically the same, but in the definition of a morphism, one has a map $f:M \to \varphi^*N$ instead of $\varphi^*N \to M$. This is the "correct" category for group homology if one wants to vary the group.

So the question arises, what properties do these categories have? I think they are not abelian. Are they semiabelian? If not, homological? Protomodular?

That would allow one to apply some version of non-abelian homological algebra (such as in this paper by Van der Linden). The idea would be to take in some suitable form the "derived functor" of the invariants functor $\mathbf{GroupMod}^* \to \mathbf{Ab}$, (so we're taking group cohomology without fixing the group, in some sense), maybe this would yield something interesting about group cohomology.

Edit I think there's a very natural description of the category $\mathbf{GroupMod}$ that is maybe helpful. Consider the contravariant pseudofunctor $F:\mathrm{Grp}^{op} \to \mathbf{Cat}$ sending $G$ to the category of left $\Bbb Z[G]$-modules and sending morphisms of groups to restriction of scalars, then $\mathbf{GroupMod}$ is the Grothendieck construction $\int F$.