The transfer maps constructed in the previous part suggest that some constructions, such as homotopy orbits and fixed-points, have pretty interesting functoriality in the group $G$ itself, or more generally in (finite) $G$-sets. Namely they have the usual functoriality induced by $G$-equivariant functions, but they also have maps going “the wrong way”, which usually behave like some sort of transfer.

Perhaps we can package this data into a functor, not from the category $\mathbf{FinSet}_G$, but from a larger category which also has those “wrong way” maps.

Definition: Let $\mathcal{C}$ be a category. A span (aka correspondence aka generalized function) from $X$ to $Y$ in $\mathcal{C}$ is a diagram of the form

\[X \leftarrow A \rightarrow Y\]

In particular every “right way” map $r : X \to Y$ gives a span $X \xleftarrow{\mathrm{id}} X \xrightarrow{r} Y$ and every “wrong way” map $w : Y \to X$ gives a span $X \xleftarrow{w} Y \xrightarrow{\mathrm{id}} Y$. Spans can be composed by taking pullbacks of their intermediate objects:

Thus if $\mathcal{C}$ has all pullbacks, then one can always compose spans. In this case, they form the category of spans $\mathbf{Span}(\mathcal{C})$, which has the same objects as $\mathcal{C}$ but morphisms replaced by spans between those objects. If $\mathcal{C}$ has coproducts then spans (between fixed objects) actually form a monoid:

Under some mild assumptions on $\mathcal{C}$, this monoid structure becomes compatible with the composition of spans defined above, and so $\mathbf{Span}(\mathcal{C})$ becomes enriched over monoids. We let $\mathbf{Span}^+(\mathcal{C})$ be the group completion (aka Grothendieck construction) obtained by freely adding inverses to all homsets, thus making them into (abelian) groups. The resulting category is enriched over $\mathbf{Ab}$, perhaps it may be thought of as the “universal $\mathbf{Ab}$-enrichment” of $\mathcal{C}$ (I am not making any guarantees for a precise universal property, though I wouldn’t be surprised if one exists. Let me know if you happen to think about this!).

Definition: The Burnside category of $G$ is $\mathbf{B}_G := \mathbf{Span}^+(\mathbf{FinSet}_G)$.

Fun fact: The category $\mathbf{B}_G$ is actually equivalent to the full subcategory of spectra spanned by suspension spectra of the form $\Sigma^\infty T_+$ where $T \in \mathbf{FinSet}_G$. Proving this equivalence essentially amounts to constructing transfer maps between these spectra.

Now we can finally carry out our plan, of packaging up wrong-way information into a single functor:

Definition: A Mackey functor is an (additive) functor $\mathbf{B}_G^{op} \to \mathbf{Ab}$. A spectral Mackey functor is defined similarly, but valued in spectra instead of abelian groups. One may also discuss more general Mackey functors valued in other additive target categories. I’m not sure why, but it’s conventional to use here the opposite of the Burnside category, instead of the category itself. This is immaterial in practice because $\mathbf{B}_G \simeq \mathbf{B}_G^{op}$.

Notations: Mackey functors hold a fair bit of data, so here are some notational conventions to keep in mind. Some of these are standard, while others are my own:

  • I will often denote Mackey functors with an underline, e.g. $\underline{M}$. Underlined functors will always act on $G$-sets.
  • We will also define a non-underlined version, acting on subgroups: if $H\leq G$, set $M(H) := \underline{M}(G/H)$.
  • If $f : T \to T’$ is a map of $G$-sets, we will write $f_* : \underline{M}(T) \to \underline{M}(T’)$ for the map induced by the “right-way span” and $f^* : \underline{M}(T’) \to \underline{M}(T)$ for the map induced by the “wrong-way span”.
  • If $K\leq H\leq G$ are nested subgroups, then we get a map of $G$-sets $f : G/K \to G/H$. I will denote the two induced maps using arrow notation: $\uparrow_K^H := f_* : M(K) \to M(H)$ and $\downarrow^H_K := f^* : M(H) \to M(K)$.
  • It shouldn’t be hard to convince yourself that $\underline{M}$ is entirely determined by $M$. For example, additivity implies that $\underline{M}$ commutes with coprducts, and every finite $G$-set is a coproduct of those of the form $G/H$ (for varying $H$).

The category of Mackey functors is then simply the (enriched) functor category $[\mathbf{B}_G^{op},\mathbf{Ab}]$. This is clearly also functorial in the target category, so for instance the operation $\pi_0 : \mathbf{Sp} \to \mathbf{Ab}$ induces a functor from spectral Mackey functors to “plain” Mackey functors.

Fun fact: I would be remiss if I went through a whole blogpost on equivariant homotopy theory without mentioning genuine equivariant spectra. Briefly, here is the idea behind them: ordinary spectra are obtained from the category of (pointed) spaces by formally inverting the suspension functor. Recall that the suspensions of a space are simply its smash product with spheres. Hence we may think of spectra as the “localization” of spaces at the spheres, with respect to their smash product monoidal structure

spectra $\approx$ spaces[1/spheres]

So far I’ve been working with the so-called “Borel $G$-spectra”, also known as “spectra with $G$-action” – these are just functors from $BG$, as we’ve used earlier. There is another way to obtain this category, more in line with this new slogan:

naïve $G$-spectra $\approx$ $G$-spaces[1/spheres]

In this “slogan”, the spheres are regarded as $G$-spaces by giving them the trivial $G$-action. But as the name suggests, this construction is too naïve for many practical applications. A better idea is as follows: if $V$ is a vector space (finite dimensional over $\mathbb{R}$), we write $S^V$ for its one-point compactification. Clearly an $n$-dimensional vector space gives the $n$-sphere. But now if $V$ is a nontrivial representation of $G$, then the sphere $S^V$ will have a more interesting $G$-action. Spaces obtained in this manner are called representation spheres, and using those, we can suspend not only an integer-amount of times, but actually a representation-amount of times if you will. The more accepted definition of equivariant spectra nowadays is:

genuine $G$-spectra $\approx$ $G$-spaces[1/representation spheres]

Why am I telling you all of this? Here’s the fun part: you can define equivariant homotopy groups of genuine spectra, and those will be indexed not only by the ordinary spheres, but by all representation spheres. These equivariant homotopy groups turn out to be functorial in the group $G$ itself, but they also possess transfer maps similar to what we have defined, and in fact they assemble into Mackey functors. The magic theorem is then that there’s an equivalence of categories:

genuine $G$-spectra $\approx$ spectral Mackey functors. Huzzah!

I’m not going to need this theorem in the rest of the post, so if you’re confused don’t worry. All I wanted was to showcase some cool aspects of Mackey functors for you. Here are some more neat examples of Mackey functors:

Example: The Burnside Mackey functor is the functor $\underline{A}_G : \mathbf{B}_G \to \mathbf{Ab}$ which sends the transitive $G$-set $G/H$ to the group completion of the semigroup of finite $H$-sets under products. This functor is easily seen to be represented by the singleton $G$-set (every $G$-set is an $H$-set). Its wrong-way maps are given by restricting the group action, and its right-way maps are given by induction. The group completion in question is in fact a ring (under direct product of $H$-sets), known as the Burnside ring of $H$.

Example: The representation Mackey functor is similar to the previous example, except instead of the group completion of finite $G$-sets, it gives the group completion of finite dimensional representations of $G$. Again, the wrong-way maps are given by restrictions of representations, while the right-way maps are given by induced representations.

Example: For the trivial group $G = e$, a finite $G$-set is just a finite set, which is determined up to isomorphism by its cardinality, and so the Burnside ring is $\mathbb{Z}$. By “forgetting the $G$-action” we get a degree map from every Burnside ring to $G$, and in fact we get a Mackey functor called the constant Mackey functor $\underline{\mathbb{Z}}$. This sends every transitive $G$-set to $\mathbb{Z}$, and extends to arbitrary $G$-sets by preserving coproducts. By studying the effect of restriction and induction, we find that the wrong-way maps of $\underline{\mathbb{Z}}$ are all identities, while the right-way maps $\uparrow_K^H : \mathbb{Z} \to \mathbb{Z}$ are multiplication by the index $[H:K]$. The degree map induces a degree morphism of Mackey functors $\varepsilon : \underline{A}_G \to \underline{\mathbb{Z}}$.

Even more examples:

  • Homotopy orbits & fixed points.
  • Equivariant homotopy groups of (genuine) equivariant spectra.
  • Group cohomology and homology.
  • Iwasawa modules.
  • Ideal class groups, idèle class groups.
  • Mordell-Weil groups.
  • Shafarevich-Tate groups.
  • and many more…!

The definition I gave above is quite abstract, which is very nice for proving general properties, but not so nice for working with examples and performing computations. Historically, Mackey functors had a much more explicit definition, so I’d like to go through some concrete consequences of all the abstract nonsense.

A basic consequence is the pullback axiom: given a pullback square of $G$-sets

One has $r^*\circ w_* = {w’}_* \circ {r’}^*$. This is a direct consequence of functoriality, applied to the composite of the right-way span $T \xleftarrow{\mathrm{id}} T \xrightarrow{r} T’$ and the wrong-way span $T’ \xleftarrow{w} T’’ \xrightarrow{\mathrm{id}} T’’$.

Subgroup inclusion $K \leq H \leq G$ gives a map of $G$-sets $G/H \to G/K$, which then induces two maps that I will denote $\downarrow_K^H : M(H) \to M(K)$ and $\uparrow_K^H : M(K) \to M(H)$. I opt to use this arrow notation rather than more traditional names such as “restriction” and “transfer” due to the mismatch between homology-style and cohomology-style functors:

Mackey functor $\downarrow$ $\uparrow$
cohomology $(-)^{hG}$ restriction transfer
homology $(-)_{hG}$ transfer restriction

Some other easy consequences of the definition are functoriality of $M_*$ and $M^*$, and compatibility on isomorphisms: if $c : T \to T’$ is an isomorphism of finite $G$-sets then $M_*c$ and $M^*c$ are obviously also isomorphisms, and they are inverses of one another (exercise: prove this using the pullback axiom). In particular if $H\leq G$ is a subgroup and $z\in G$ any element, then conjugation by $z$ defines an isomorphism $c_z : G/H \to G/zHz^{-1}$ and hence an isomorphism $M(H) \to M(zHz^{-1})$, which by crude abuse of notation we also denote

\[c_z := M_*c_z = (M^*c_z)^{-1} = M^*(c_z^{-1})\]

But by far, the most interesting property of Mackey functors is what they do on arbitrary pullback squares. To understand this more deeply, let’s think about transitive $G$-sets: suppose $H,K \leq G$ are any two subgroups. How can we describe explicitly the product (special case of pullback) $G/H \times G/K$?

Remark: Since products distribute over coproducts, this would allow us to “compute” the product of arbitrary finite $G$-sets; just multiply separately each pair of orbits.

Let’s analyze the orbit structure of the $G$-set $G/H\times G/K$.

Instructive example: If $H,K = e$ are both the trivial subgroup, then the orbit of $(x,y)$ is uniquely determined by the “ratio” $xy^{-1}$. Indeed, if there exists $g$ which satisfies $(gx_1,gy_1) = (x_2,y_2)$ then it must be $x_2x_1^{-1} = g = y_2y_1^{-1}$ which is equivalent to $x_1^{-1}y_1 = x_2^{-1}y_2$.

For general $H,K$, we can still talk about “ratios”, except they’re not determined as elements of $g$ but rather as elements of the double coset space $H\backslash G/K$. Explicitly:

  • the orbit of $(xH,yK)$ corresponds to the double coset $Hx^{-1}yK$, and
  • the double coset $HzK$ corresponds to the orbit of, for instance, $(H,zK)$.

So we know how many orbits there are in $G/H\times G/K$. We can also figure out what each orbit looks like, by considering the stabilizer of any representative. For the orbit corresponding to $HzK$, let’s think about the stabilizer of $(H,zK)$. Certainly this would be the intersection of the stabilizers of $H$ and of $zK$, which is just $H \cap zKz^{-1}$. We end up with the following $G$-equivariant isomorphism, known as the Mackey decomposition formula:

\[G/H \times G/K \cong \coprod_{z \in H\backslash G/K}{G/(H\cap zKz^{-1})}\]

An analogous proof gives the slightly more general identity for arbitrary pullbacks (not just products): if $H,K$ are both contained in some intermediate subgroup $J\subseteq G$ then

\[G/H \times_{G/J} G/K \cong \coprod_{z \in H\backslash J/K}{J/(H\cap zKz^{-1})}\]

Finally, let’s see what happens when we apply our Mackey functor on this pullback. Allow me to be overly careful and go through this in detail. We have in our hands an explicit isomorphism of $G$-sets

\[\varphi : \coprod_{z\in H\backslash J/K}{G/(H\cap zKz^{-1})} \xrightarrow{\qquad\qquad} G/H \times_{G/J} G/K\]

On the summand indexed by $HzK$, it is given by

\[g(H\cap zKz^{-1}) \mapsto (gH, c_z^{-1}(gzKz^{-1}))\]

Hence if we apply our pullback axiom on the following pullback square

then we get the following formula, known as the double coset formula:

\[\boxed{\downarrow^J_K \circ \uparrow_H^J = \sum_{x\in H\backslash G/K}{\uparrow_{zHz^{-1}\cap K}^K \circ c_z \circ \downarrow^H_{H\cap zKz^{-1}}}}\]

Remark: There’s a few different ways to write this formula, by changing whether you write $c_z$ before the composite, in the middle of the composite (as I did), or after the composite. My convention is the most standard one, but overall it hardly matters too much – you should view the involvement of conjugation more as a technicality than anything.

Cards on the table, all of this is completely irrelevant to the question I originally posed. Indeed, what I care about is actually the composite $\uparrow_e^G \circ \downarrow^G_e$, so the double coset formula won’t be so helpful for us after all. But anyway this whole post is just an excuse to showcase fun math! It’s also meant to demonstrate that although Mackey functors seem quite abstract, they actually have very concrete consequences towards their algebraic & representation-theoretic origins.

Next time I will study interactions between Mackey functors, in the form of forming “modules” over each other. From this we will get a formula more helpful for the compsite of interest.