background on Mackey functors -- a brief mathematical intro to the theory of Mackey functors

Mackey functors were introduced by Andreas Dress [D71] as a way to encode a system of abelian groups indexed along conjugacy classes of a subgroup, with homomorphisms between them behaving analogously to restriction and induction of representations. They are ubiquitous in mathematics, appearing in representation theory, group cohomology, equivariant cohomology of $G$-spaces, algebraic $K$-theory of group rings, algebraic number theory (where they go by the name modulations), and many other contexts.

Assumption: In this package, we will work with cyclic groups of prime order. The reason for this reduction is the simple subgroup structure of $C_p$, which reduces our data structure dramatically. Furthermore, the theory of $C_p$-Mackey functors is already highly complicated - for example no classification result is known.

Definition: Let $p$ be a prime. A $C_p$-Mackey functor is the data of two abelian groups $M(C_p/e)$ (called the underlying) and $M(C_p/C_p)$ (called the fixed), together with three abelian group homomorphisms, called restriction, transfer, and conjugation, respectively:

\[\text{res} \colon M(C_p/C_p) \to M(C_p/e),\] \[\text{tr} \colon M(C_p/e) \to M(C_p/C_p),\] \[\text{conj} \colon M(C_p/e) \to M(C_p/e)\]

subject to the following axioms:

0. $\text{conj}$ is an automorphism of order dividing $p$ (encoding an action of the cyclic group $C_p$ on $M(C_p/e)$)
1. $\text{conj}\circ\text{res} = \text{res}$ and $\text{tr}\circ\text{conj} = \text{tr}$
2. $\text{res}(\text{tr}(x)) = \sum_{i=0}^{p-1}\text{conj}^{i}(x)$ for every $x\in M(C_p/e)$

Example: The easiest example is when all the abelian groups are the trivial group, and hence all maps are trivial. This is called the zero Mackey functor.

Example: A prototypical example has underlying module $\ZZ$ (i.e. the complex representation ring of the trivial group) and fixed module given by the complex representation ring of $C_p$, with transfer and restriction coming from restriction and transfer of $C_p$-representations. This is called the representation Mackey functor.

Note: For general examples and their constructors, see constructing examples of Mackey functors and for some common Mackey functors used over the group $C_p$, see list of common Mackey functors.

We frequently use $M$ as shorthand for all the data $(M(C_p/e), M(C_p/C_p), \text{res},\text{tr},\text{conj})$.

Definition: If $M$ and $N$ are both $C_p$-Mackey functors, a Mackey functor homomorphism $f \colon M \to N$ is the data of a morphism $f_{C_p/e} \colon M(C_p/e) \to N(C_p/e)$ and $f_{C_p/C_p} \colon M(C_p/C_p) \to N(C_p/C_p)$ which commute with transfer, restriction, and conjugation. Explicitly (if we decorate things like $\text{res}$ with a subscript to indicate which Mackey functor they are coming from) we mean that the following relations must hold:

0. $\text{res}_N\circ f_{C_p/C_p} = f_{C_p/e}\circ \text{res}_M$
1. $\text{tr}_N \circ f_{C_p/e} = f_{C_p/C_p} \circ \text{tr}_M$
2. $\text{conj}_N\circ f_{C_p/e} = f_{C_p/e} \circ \text{conj}_M$

References:

• [D71] A. Dress, Notes on the theory of representations of finite groups. Part I: The Burnside ring of a finite group and some AGN-applications. Bielefeld.
• [W00] P. Webb, A guide to mackey functors Handbook of Algebra, 2000.