What properties do the categories $\mathbf{GrpMod}$ and $\mathbf{GrpMod}^*$ of compatible pairs have? Can we do homological algebra with them?

Question

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 $\int F$.

Answer 1

@Z.M pointed out in the comments that a useful framework to look into is animation. If we want a $1$-category to to have an animation, the natural question to ask is if is an algebraic category, which here means being cocomplete and generated under sifted colimits by a set of compact projective objects.

As I have written in the OP, I realized after asking the question that $\mathbf{GrpMod}$, or to be precise the fibered category $\mathbf{GrpMod} \to \mathbf{Grp}$ is actually the Grothendieck construction of the pseudofunctor $G \mapsto \Bbb Z[G]\textrm{-}\mathbf{Mod}$.

Thus to answer the question, I will provide a sufficient criterion for the Grothendieck construction of a pseudofunctor to be algebraic which implies that $\mathbf{GrpMod}$ is algebraic.

So let $C$ be a category and let $F:C^{op} \to \mathbf{Cat}$ a pseudo-functor. Suppose that:

1. $C$ is algebraic
2. For all $c \in C$, $F(c)$ is algebraic.
3. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ has a left adjoint $f_! \dashv f^*$.
4. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ preserves sifted colimits.
5. For all $c \xrightarrow{f} c'$ in $C$, the functor $f^*:F(c') \to F(c)$ is conservative.

Then $\int F$ is algebraic.

Condition $3.$ implies that $\int F \to C$ is in fact a Grothendieck bifibration. Because it is an opfibration and the base category and the pointwise images are cocomplete, the total category is cocomplete and colimits may be computed as follows: for a diagram $I \to \int F, i \mapsto (c_i,a_i)$, first take the colimit $\varinjlim c_i$ in $C$, denote the structure morphisms of that as $\iota_i$ and then take the colimit in $F(\varinjlim c_i)$ of $(\iota_i)_!a_i$, then the pair $(\varinjlim c_i, \varinjlim (\iota_i)_! a_i) \in \int F$ is the desired colimit. Now we can choose for $C$ a set of compact projectives $P_C$ that generates $C$ under sifted colimits. And then for each $p \in P_C$, we may choose a set of compact projectives $P_{F(p)}$ in $F(p)$ that generate it under sifted colimits, then the claim is that the set $\{(p,q_p) \mid p \in P_C, q_p \in P_{F(p)}\}$ is a set of compact projectives for $\int F$ that generates it under sifted colimits.

Fix such a pair $(p,q_p)$. Let $I$ be a sifted category and consider a diagram $I \to \int F, i \mapsto (c_i,a_i)$. As described above, we have the colimit of the diagram given by $(\varinjlim c_i, \varinjlim (\iota_i)_!a_i)$. We have:

$$ \mathrm{Hom}_{\int F}((p,q_p),(\varinjlim c_i, \varinjlim (\iota_i)_!a_i))=\{(f,\alpha) \mid f \in \mathrm{Hom}_C(p,\varinjlim c_i),\alpha \in \mathrm{Hom}_{F(p)}(q_p,f^*(\varinjlim (\iota_i)_!a_i))\}$$ Now we can use condition 4. and that $q_p$ is compact projective, so we get that this is $$\{(f,\alpha) \mid f \in \mathrm{Hom}_C(p,\varinjlim c_i), \varinjlim \mathrm{Hom}_{F(p)}(q_p, f^*((\iota_i)_!a_i))\}$$ $p$ is also compact projective, so this is (using that $I$ is sifted) $$\varinjlim \{(f,\alpha) \mid f \in \mathrm{Hom}_C(p,c_i),\alpha \in \mathrm{Hom}_{F(p)}(q_p,f^*a_i)\}=\varinjlim \mathrm{Hom}_{\int F}((p,q_p),(c_i,a_i))$$

Conditions 4. and 5. imply that for each $p \xrightarrow{f} c$ the set $\{f_!q_p\mid q_p \in P_{F(p)}\}$ generates $F(c)$ under sifted colimits. Now take $(c,a) \in \int F$. Then we can find a filtered $I$ category and a diagram $I \to C, i \mapsto p_i$ such that $c=\varinjlim\limits_{i \in I} p_i$ with structure morpisms $\iota_i:p_i \to c$. Then for all $i \in I$, we can find a filtered category $J_i$ and a diagram $J_i \to F(c), j_i \mapsto (\iota_i)_!q_{p_i,j_i}$ such that $\varinjlim\limits_{j_i \in J_i}(\iota_i)_!q_{p_i,j_i}=a$. Now we get that $\varinjlim\limits_{i \in I} \varinjlim\limits_{j_i \in J_i}(p_i,q_{p_i,j_i})=(c,a)$

Therefore we get the animation $\mathbf{Ani}(\mathbf{GrpMod})$. The coinvariants functor $(-)_G:\mathbf{GrpMod} \to \mathbf{Ab}$ is cocontinuous and hence induces a functor $H_{\mathrm{Ani}}:\mathbf{Ani}(\mathbf{GrpMod}) \to \mathbf{Ani}(\mathbf{Ab})$ which deserves to be called animated group homology. One can recover the usual $n$-th homology functor as the composition $$\mathbf{GrpMod} \to \mathbf{Ani}(\mathbf{GrpMod}) \xrightarrow{H_{\mathrm{Ani}}} \mathbf{Ani}(\mathbf{Ab})=\mathbf{D}^-_{\geq 0}(\mathbf{Ab})\xrightarrow{\pi_n} \mathbf{Ab}$$

This answers the question on whether one can do a form of homological algebra with $\mathbf{GrpMod}$: through animation we have injected it with homotopical/derived information and obtained a form of a derived category.

Intuitively, I would expect that, like $\mathbf{GrpMod}$ is the collection of the categories of $\Bbb Z[G]\textrm{-}\mathbf{Mod}$ for all groups $G$ "glued together" along all the morphisms in the category $\mathbf{Grp}$, the $\infty$-category $\mathbf{Ani}(\mathbf{GrpMod})$ is the collection of all derived $\infty$-categories $\mathbf{D}^-_{\geq 0}(\Bbb Z[G]\textrm{-}\mathbf{Mod})$ glued together over $\mathbf{Ani}(\mathbf{Grp})$.

To maybe make this precise one can note that $\mathbf{GrpMod} \to \mathbf{Grp}$ is cocontinuous, so we obtain an animated functor $\mathbf{Ani}(\mathbf{GrpMod}) \to \mathbf{Ani}(\mathbf{Grp})$. I'm not sure if it's a reasonable conjecture, but one could certainly raise the question whether this is a Cartesian (op?) fibration, and whether the fiber over an object $G$ in the image of $\mathbf{Grp} \to \mathbf{Ani}(\mathbf{Grp})$ is of the form $\mathbf{Ani}(\Bbb Z[G]\textrm{-}\mathbf{Mod})$. If that holds, the other fibers should then be the analogue of the derived category of $\Bbb Z[G]$-modules for animated groups.