the abelian category of Mackey functors -- background on the category of Mackey functors and how to code within it

The category of Mackey functors: Fixing a finite group $G$ (always cyclic of prime order in this package), we obtain a category $\text{Mack}_G$, whose objects are Mackey functors and whose morphisms are Mackey functor homomorphisms. Given any Mackey functor $M$ we can take its identity homomorphism as follows:

i1 : M = makeRandomCpMackeyFunctor(3,GenBound=>2);

i2 : id_M

o2 =                       fix : | 1 |
     cokernel | 6 0 0 | <--------------- cokernel | 6 0 0 |
             ^ |                                 ^ | 
             | |                                 | | 
             | v                                 | v 
     cokernel | 126 0 | <--------------- cokernel | 126 0 |
             ^ |           und : | 1 |           ^ | 
             └-┘                                 └-┘ 

o2 : MackeyFunctorHomomorphism

Composition of Mackey functor homomorphisms is done exactly the same as for composition of maps. If $f\colon M \to N$ and $g \colon N \to Q$ are two Mackey functor homomorphisms, their composite $g\circ f \colon M \to Q$ can be accessed as g*f.

The abelian category structure: Given any two Mackey functors $M$ and $N$, we can form the set $\text{Hom}_{\text{Mack}_G}(M,N)$ of homomorphisms from $M$ to $N$. Since we can add any two homomorphisms, we have that $\text{Hom}_{\text{Mack}_G}(M,N)$ is really an abelian group of homomorphisms. Composition is moreover $\ZZ$-linear, hence we can say that the category $\text{Mack}_G$ is pre-additive.

i3 : Hom(makeRandomCpMackeyFunctor(2,GenBound=>2),makeRandomCpMackeyFunctor(2,GenBound=>2))

o3 = 0

o3 : ZZ-module

i4 : prune oo

o4 = 0

o4 : ZZ-module

Given any two Mackey functors, we can take their direct sum and obtain another Mackey functor. This is a finite biproduct on the category $\text{Mack}_G$, in other words it is both a product and a coproduct. There is a zero object for this biproduct, namely the zero Mackey functor. This gives $\text{Mack}_G$ what is often called a semiadditive structure.

Moreover, any Mackey functor homomorphism admits a kernel and a cokernel. All this data we've collected so far is often called a pre-abelian structure. Finally, one can prove that images and coimages agree, hence $\text{Mack}_G$ is an abelian category.

Enough projectives: The category $\text{Mack}_{G}$ has exactly two projective objects, which we denote by $\underline{A}$ and $\underline{B}$. These are the Burnside Mackey functor and the underlying free Mackey functor, respectively. The category $\text{Mack}_G$ has enough projectives, meaning in particular that we can take projective resolutions in order to compute ext and tor groups.

Free resolutions: Given any Mackey functor we can take its free resolution. The projective dimension of an object of $\text{Mack}_G$ is generally infinite, so we cannot provide all of the data of a resolution -- nevertheless we can provide a computation that holds in a range. This is accomplished using the freeResolution (aka freeResolution) method, which inputs a Mackey functor $M$ and an integer $n$, and outputs a free resolution of the form \[M \leftarrow F_0 \leftarrow F_1 \leftarrow \cdots \leftarrow F_n\]

i5 : M = prune makeRandomCpMackeyFunctor(3,GenBound=>2);

i6 : class res(M,2)

o6 = List

o6 : Type

i7 : length resolution(M,2)

o7 = 3

In the notation above, the element res(M,2) will be a list of differentials $(M \leftarrow F_0, F_0 \leftarrow F_1, F_1 \leftarrow F_2)$, where each entry in the list is a Mackey functor homomorphism.

When a resolution is computed, this is cached in the Mackey functor itself, and can be accessed as M.ProjRes. This makes computations less costly, and moreover the cache helps us expand resolutions. For instance if we have already computed res(M,10) and want to compute res(M,20), it will access the cached 10-term resolution to shorten the computation for the 20-term resolution.

The monoidal structure: The category $\text{Mack}_G$ furthermore has a symmetric monoidal structure, given by the box product of two Mackey functors. Taking the box product admits a right adjoint, in other words there is an internal hom, which is a functor

\[\underline{\text{Hom}} \colon \text{Mack}_G^{\text{op}} \times \text{Mack}_G \to \text{Mack}_G,\]

whose fixed point module is the abelian group of homomorphisms from $M$ to $N$. The category $\text{Mack}_G$ is closed symmetric monoidal, meaning the internal hom and box product fit together into a natural isomorphism of abelian groups, natural in any three Mackey functors $M$, $N$, and $P$: \[\text{Hom}_{\text{Mack}_G}(M \square N, P) \cong \text{Hom}_{\text{Mack}_G}(M, \underline{\text{Hom}}(N,P)).\]

Ext and Tor: With free resolutions in mind, we can define internal ext and tor groups as the derived functors of the internal hom and box product, respectively:

\[\underline{\text{Ext}}^i(-,M):= \mathbf{R}^i \underline{\text{Hom}}_{\text{Mack}_G}(-,M),\]

\[\underline{\text{Tor}}^i(-,M) := \mathbf{L}_i (- \square M).\]

These are internal incarnations of $\text{Ext}$ and $\text{Tor}$. We can recover the abelian groups $\text{Ext}^i(M,N)$ and $\text{Tor}_i(M,N)$ as the fixed point module of the internal Ext and Tor.