Mackey Functors

Table of Contents

• #1. From Coefficient Systems to Mackey Functors|1. From Coefficient Systems to Mackey Functors
  • #1.1 Coefficient Systems|1.1 Coefficient Systems
  • #1.2 Why Restriction Alone Is Insufficient|1.2 Why Restriction Alone Is Insufficient
  • #1.3 The Span Perspective|1.3 The Span Perspective
• #2. The Burnside Category|2. The Burnside Category
  • #2.1 Spans of Finite G-Sets|2.1 Spans of Finite G-Sets
  • #2.2 Composition via Fiber Product|2.2 Composition via Fiber Product
  • #2.3 Additive Structure and the Burnside Ring|2.3 Additive Structure and the Burnside Ring
• #3. Mackey Functors: Formal Definition|3. Mackey Functors: Formal Definition
  • #3.1 Additive Functors on the Burnside Category|3.1 Additive Functors on the Burnside Category
  • #3.2 Unpacking into Restriction, Transfer, and Conjugation|3.2 Unpacking into Restriction, Transfer, and Conjugation
  • #3.3 Axiomatic Presentation|3.3 Axiomatic Presentation
• #4. The Mackey Double Coset Formula|4. The Mackey Double Coset Formula
  • #4.1 The Pullback Decomposition|4.1 The Pullback Decomposition
  • #4.2 Derivation from Spans|4.2 Derivation from Spans
• #5. Key Examples|5. Key Examples
  • #5.1 The Constant Mackey Functor|5.1 The Constant Mackey Functor
  • #5.2 The Burnside Ring Mackey Functor|5.2 The Burnside Ring Mackey Functor
  • #5.3 Fixed-Point Mackey Functors|5.3 Fixed-Point Mackey Functors
  • #5.4 The C2 Classification|5.4 The C2 Classification
• #6. The Box Product and Green/Tambara Functors|6. The Box Product and Green/Tambara Functors
  • #6.1 Day Convolution and the Box Product|6.1 Day Convolution and the Box Product
  • #6.2 Green Functors|6.2 Green Functors
  • #6.3 Tambara Functors and Multiplicative Norms|6.3 Tambara Functors and Multiplicative Norms
• #7. Projective Mackey Functors and Resolutions|7. Projective Mackey Functors and Resolutions
  • #7.1 Representable Mackey Functors|7.1 Representable Mackey Functors
  • #7.2 Global Dimension and Resolutions|7.2 Global Dimension and Resolutions
• #8. Spectral Mackey Functors: Barwick’s Theorem|8. Spectral Mackey Functors: Barwick’s Theorem
  • #8.1 The Effective Burnside Infinity-Category|8.1 The Effective Burnside Infinity-Category
  • #8.2 Spectral Mackey Functors|8.2 Spectral Mackey Functors
  • #8.3 Barwick’s Equivalence|8.3 Barwick’s Equivalence
  • #8.4 Homotopy Groups as Classical Mackey Functors|8.4 Homotopy Groups as Classical Mackey Functors
• #Exercises|Exercises
  • #Mathematical Development|Mathematical Development
  • #Algorithmic Applications|Algorithmic Applications
  • #G-CW-Spectra and Slice Filtration|G-CW-Spectra and Slice Filtration
• #References|References

1. From Coefficient Systems to Mackey Functors 📐

1.1 Coefficient Systems

To motivate Mackey functors, we begin with the simpler notion that arises in Bredon cohomology. Fix a finite group \(G\) throughout. Recall the orbit category \(\mathcal{O}_G\) from concepts/equivariant-stable-homotopy/g-spaces-and-equivariant-maps|G-Spaces and Equivariant Maps: its objects are the transitive \(G\)-sets \(G/H\) for \(H \leq G\), and its morphisms \(\mathcal{O}_G(G/H, G/K)\) are \(G\)-equivariant maps \(G/H \to G/K\). Every such map has the form \(gH \mapsto gxK\) for some fixed \(x\) with \(x^{-1}Hx \leq K\).

Definition (Coefficient System). A coefficient system for \(G\) is a contravariant functor

\[M^*: \mathcal{O}_G^{\mathrm{op}} \longrightarrow \mathbf{Ab}.\]

Concretely, a coefficient system assigns an abelian group \(M(G/H)\) to each orbit \(G/H\), and to each \(G\)-map \(f: G/H \to G/K\) a restriction map \(f^*: M(G/K) \to M(G/H)\), functorially. In particular, conjugation by \(g \in G\) gives an isomorphism

\[c_g: M(G/H) \xrightarrow{\sim} M(G/{}^gH), \quad {}^gH = gHg^{-1}.\]

The group \(H\) acts on \(M(G/H)\) via \(c_h\) for \(h \in H\) (since \({}^hH = H\)), but this action is trivial because \(c_h = \mathrm{id}\) as a map \(G/H \to G/H\) (the only equivariant self-map of \(G/H\) is the identity up to the \(G\)-action).

1.2 Why Restriction Alone Is Insufficient 💡

Coefficient systems suffice for defining cohomology, but they are too coarse to capture the full structure of equivariant invariants. Consider a \(G\)-space \(X\). The natural assignment \(G/H \mapsto \pi_n(X^H)\) is contravariant with respect to the orbit category via restriction: if \(f: G/H \to G/K\) comes from inclusion \(H \leq K\), the map \(X^K \hookrightarrow X^H\) (more fixed points under a smaller group) gives a restriction

\[\mathrm{res}_H^K: \pi_n(X^K) \to \pi_n(X^H).\]

But there is also a transfer map running in the other direction:

\[\mathrm{tr}_H^K: \pi_n(X^H) \longrightarrow \pi_n(X^K).\]

For \(K = G\) and \(n = 0\), this is the equivariant analogue of summing over coset representatives: \(\mathrm{tr}_H^G(m) = \sum_{[g] \in G/H} g_* m\). This map is covariant, going from smaller to larger group, and has no analogue in the coefficient system framework.

The failure of coefficient systems is that they admit only restriction maps. A Mackey functor is precisely the structure that captures both restrictions and transfers simultaneously, subject to the compatibility imposed by the Mackey double coset formula.

1.3 The Span Perspective 🔑

The key insight of Lindner (1976) and Dress (1973) is that both restriction and transfer arise from a single span (correspondence) of \(G\)-sets. A span from \(X\) to \(Y\) is a diagram

\[X \xleftarrow{p} Z \xrightarrow{q} Y\]

of \(G\)-equivariant maps. Given such a span and a functor \(M\), one can: - Apply \(M\) contravariantly to \(p\) to get \(M(p): M(X) \to M(Z)\), - Apply \(M\) covariantly to \(q\) to get \(M(q): M(Z) \to M(Y)\),

and compose to get a map \(M(X) \to M(Y)\). The composite

\[M(X) \xrightarrow{M(p)} M(Z) \xrightarrow{M(q)} M(Y)\]

depends on \(M\) having both contravariant and covariant functoriality. This is the content of defining \(M\) as a functor on the span category, where every morphism \(X \to Y\) is a span \(X \leftarrow Z \rightarrow Y\).

The Burnside category \(\mathcal{A}(G)\) is the \(\mathbb{Z}\)-linear category whose morphism groups are Grothendieck completions of the monoid of isomorphism classes of spans, with composition given by fiber product. Mackey functors are additive functors \(\mathcal{A}(G) \to \mathbf{Ab}\).

2. The Burnside Category 🧮

2.1 Spans of Finite G-Sets

Definition (Pre-Burnside Category). Let \(\mathbf{FSets}_G\) denote the category of finite \(G\)-sets. Define a category \(\mathcal{A}^+(G)\) — the pre-Burnside category — as follows:

• Objects: finite \(G\)-sets \(X, Y, Z, \ldots\)
• Morphisms: \(\mathcal{A}^+(G)(X, Y)\) is the set of isomorphism classes of spans \(X \xleftarrow{} Z \xrightarrow{} Y\) of \(G\)-equivariant maps, where two spans \((X \leftarrow Z \rightarrow Y)\) and \((X \leftarrow Z' \rightarrow Y)\) are isomorphic if there is a \(G\)-equivariant bijection \(Z \xrightarrow{\sim} Z'\) commuting with both projection maps.

The set \(\mathcal{A}^+(G)(X, Y)\) is a commutative monoid under disjoint union of spans: \([X \leftarrow Z \rightarrow Y] + [X \leftarrow Z' \rightarrow Y] = [X \leftarrow Z \sqcup Z' \rightarrow Y]\).

Definition (Burnside Category). The Burnside category \(\mathcal{A}(G)\) is the additive (i.e., \(\mathbf{Ab}\)-enriched) category obtained from \(\mathcal{A}^+(G)\) by applying the Grothendieck group construction to each morphism monoid:

\[\mathcal{A}(G)(X, Y) = K_0\bigl(\mathcal{A}^+(G)(X,Y)\bigr).\]

Concretely, \(\mathcal{A}(G)(X, Y)\) consists of formal differences \([X \leftarrow Z \rightarrow Y] - [X \leftarrow Z' \rightarrow Y]\) of isomorphism classes of spans.

2.2 Composition via Fiber Product 📐

Composition in the Burnside category is defined on the pre-Burnside level and extended bilinearly to the group-completed version.

Definition (Span Composition). Given spans \(\sigma = (X \xleftarrow{p} Z \xrightarrow{q} Y)\) and \(\tau = (Y \xleftarrow{r} W \xrightarrow{s} V)\), their composite \(\tau \circ \sigma\) is the span

\[X \xleftarrow{p \circ \pi_Z} Z \times_Y W \xrightarrow{s \circ \pi_W} V,\]

where \(Z \times_Y W\) is the fiber product (pullback) of \(q: Z \to Y\) and \(r: W \to Y\) in \(\mathbf{FSets}_G\):

\[Z \times_Y W = \{(z, w) \in Z \times W : q(z) = r(w)\}.\]

The \(G\)-action on the fiber product is diagonal: \(g \cdot (z, w) = (gz, gw)\).

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
 & Z \times_Y W \arrow[dl, "\pi_Z"'] \arrow[dr, "\pi_W"] & \\
Z \arrow[dl, "p"'] \arrow[dr, "q"] & & W \arrow[dl, "r"'] \arrow[dr, "s"] \\
X & Y & Y & V
\end{tikzcd}
\end{document}

2.3 Additive Structure and the Burnside Ring 🔑

The category \(\mathcal{A}(G)\) is additive: the coproduct of objects \(X\) and \(Y\) is the disjoint union \(X \sqcup Y\), and this makes \(\mathcal{A}(G)\) into a \(\mathbb{Z}\)-linear category.

The Burnside ring \(A(G)\) appears as the endomorphism ring of the terminal \(G\)-set:

\[A(G) = \mathcal{A}(G)(G/G, G/G).\]

Indeed, \(\mathcal{A}^+(G)(G/G, G/G)\) is the monoid of isomorphism classes of finite \(G\)-sets (since a span \(G/G \leftarrow Z \rightarrow G/G\) is simply a finite \(G\)-set \(Z\) with no additional structure beyond the two maps to the point), so \(K_0\) recovers the Burnside ring.

More generally,

\[\mathcal{A}(G)(G/H, G/K) \cong K_0\bigl(\mathbf{FSets}_{G/H \times G/K}\bigr)\]

where the right side is the Grothendieck group of finite \(G\)-sets over \(G/H \times G/K\) — equivalently, finite \((H \times K)\)-sets with the \((H,K)\)-biset structure.

3. Mackey Functors: Formal Definition 🔑

3.1 Additive Functors on the Burnside Category

Definition (Mackey Functor). A Mackey functor for \(G\) is an additive functor

\[M: \mathcal{A}(G) \longrightarrow \mathbf{Ab},\]

i.e., a functor of \(\mathbb{Z}\)-linear categories from the Burnside category to abelian groups.

This is Lindner’s 1976 reformulation of Dress’s original definition. The additivity condition means that \(M\) preserves finite direct sums: \(M(X \sqcup Y) \cong M(X) \oplus M(Y)\).

The category of Mackey functors \(\mathrm{Mack}(G) = \mathrm{Fun}^{\mathrm{add}}(\mathcal{A}(G), \mathbf{Ab})\) is an abelian category (since \(\mathbf{Ab}\) is abelian and limits/colimits of additive functors are computed pointwise). In particular, it has enough injectives and projectives.

3.2 Unpacking into Restriction, Transfer, and Conjugation

Since \(M: \mathcal{A}(G) \to \mathbf{Ab}\) is a covariant functor, and a span \(X \leftarrow Z \rightarrow Y\) is a morphism \(X \to Y\) in \(\mathcal{A}(G)\), applying \(M\) to a span yields a map \(M(X) \to M(Y)\). The restriction and transfer arise from two different spans between the same pair of orbits.

Let \(p: G/H \to G/K\) denote the canonical \(G\)-equivariant projection (defined for \(H \leq K\), sending \(gH \mapsto gK\)).

• Restriction \(\mathrm{res}_H^K: M(G/K) \to M(G/H)\) comes from the span \[G/K \xleftarrow{p} G/H \xrightarrow{\mathrm{id}} G/H,\] which is a morphism \(G/K \to G/H\) in \(\mathcal{A}(G)\). Applying \(M\) gives the map \(M(G/K) \to M(G/H)\).

• Transfer \(\mathrm{tr}_H^K: M(G/H) \to M(G/K)\) comes from the span \[G/H \xleftarrow{\mathrm{id}} G/H \xrightarrow{p} G/K,\] which is a morphism \(G/H \to G/K\) in \(\mathcal{A}(G)\). Applying \(M\) gives the map \(M(G/H) \to M(G/K)\).

• Conjugation \(c_g: M(G/H) \to M(G/{}^gH)\) for \(g \in G\) comes from the span \[G/H \xleftarrow{c_g} G/{}^gH \xrightarrow{\mathrm{id}} G/{}^gH,\] where \(c_g: G/{}^gH \xrightarrow{\sim} G/H\) sends \(x{}^gH \mapsto xg^{-1}H\).

3.3 Axiomatic Presentation

The additive functor definition unpacks into the following axiomatic presentation, which is often taken as the classical definition of a Mackey functor.

Definition (Mackey Functor, Axiomatic). A Mackey functor \(M\) for \(G\) consists of: - For each \(H \leq G\): an abelian group \(M(H)\) (abbreviated \(M(G/H)\)), - For each \(H \leq K \leq G\): restriction \(\mathrm{res}_H^K: M(K) \to M(H)\) and transfer \(\mathrm{tr}_H^K: M(H) \to M(K)\), - For each \(g \in G\) and \(H \leq G\): conjugation \(c_g: M(H) \xrightarrow{\sim} M({}^gH)\),

satisfying: 1. Transitivity of restriction: \(\mathrm{res}_H^K \circ \mathrm{res}_K^L = \mathrm{res}_H^L\) for \(H \leq K \leq L\). 2. Transitivity of transfer: \(\mathrm{tr}_K^L \circ \mathrm{tr}_H^K = \mathrm{tr}_H^L\) for \(H \leq K \leq L\). 3. Conjugation compatibility: \(c_{gh} = c_g \circ c_h\); \(\mathrm{res}_{{}^gH}^{{}^gK} \circ c_g = c_g \circ \mathrm{res}_H^K\); similarly for transfers. 4. Mackey double coset formula: For \(H, K \leq G\), \[\mathrm{res}_K^G \circ \mathrm{tr}_H^G(m) = \sum_{[g] \in K\backslash G/H} \mathrm{tr}_{K \cap {}^gH}^K \circ c_g \circ \mathrm{res}_{K^g \cap H}^H(m),\] where \(K^g = g^{-1}Kg\) and the sum is over double coset representatives \([g] \in K\backslash G/H\).

4. The Mackey Double Coset Formula 📐

4.1 The Pullback Decomposition

The double coset formula is not an additional axiom imposed by fiat — it is a theorem forced by the span composition in the Burnside category. We derive it here from first principles.

Fix subgroups \(H, K \leq G\). The transfer \(\mathrm{tr}_H^G: M(G/H) \to M(G/G)\) corresponds to the span (morphism \(G/H \to G/G\) in \(\mathcal{A}(G)\))

\[G/H \xleftarrow{\mathrm{id}} G/H \xrightarrow{p} G/G,\]

where \(p: G/H \to G/G\) is the canonical projection. The restriction \(\mathrm{res}_K^G: M(G/G) \to M(G/K)\) corresponds to the span (morphism \(G/G \to G/K\) in \(\mathcal{A}(G)\))

\[G/G \xleftarrow{q} G/K \xrightarrow{\mathrm{id}} G/K,\]

where \(q: G/K \to G/G\) is the canonical projection.

To compute \(\mathrm{res}_K^G \circ \mathrm{tr}_H^G\), we compose the span for \(\mathrm{tr}_H^G: G/H \to G/G\) with the span for \(\mathrm{res}_K^G: G/G \to G/K\), obtaining a composite span \(G/H \to G/K\). The middle \(G/G\) is the object over which we take the fiber product.

Step 1: The middle piece of the composite span is the fiber product \(G/H \times_{G/G} G/K\) (the pullback of \(p: G/H \to G/G\) and \(q: G/K \to G/G\) in \(\mathbf{FSets}_G\)). Since \(G/G = \{*\}\) is the terminal object, the fiber product is simply the Cartesian product:

\[G/H \times_{G/G} G/K \cong G/H \times G/K.\]

Step 2: Decompose \(G/H \times G/K\) into \(G\)-orbits. The diagonal \(G\)-action is \(g \cdot (xH, yK) = (gxH, gyK)\). Two pairs \((xH, yK)\) and \((x'H, y'K)\) lie in the same orbit iff there exists \(g \in G\) with \(gxH = x'H\) and \(gyK = y'K\). Setting \(x = e\), the orbit of \((H, g_0 K)\) is indexed by \(g_0 \in H\backslash G/K\), and the stabilizer of the pair \((H, g_0 K)\) under the diagonal action is \(H \cap {}^{g_0}K\). Equivalently, writing with our convention of \(K\backslash G/H\) double cosets, the stabilizer of \((K, g_0 H)\) is \(K \cap {}^{g_0}H\). Therefore:

\[G/K \times_{G/G} G/H \cong \bigsqcup_{[g] \in K\backslash G/H} G/(K \cap {}^gH).\]

This is the key decomposition. The fiber product of the two spans decomposes as a disjoint union of orbits, one for each double coset \(KgH\).

4.2 Derivation from Spans

Each summand \(G/(K \cap {}^gH)\) in the decomposition above contributes a span

\[G/K \xleftarrow{} G/(K \cap {}^gH) \xrightarrow{} G/H.\]

The left leg is the projection \(G/(K \cap {}^gH) \to G/K\) (corresponding to \(K \cap {}^gH \leq K\)), so it contributes a transfer \(\mathrm{tr}_{K \cap {}^gH}^K\). The right leg goes to \(G/H\) by the map \(x(K \cap {}^gH) \mapsto xg^{-1}H\) (which uses conjugation by \(g^{-1}\) to move from \(K \cap {}^gH = K \cap gHg^{-1}\) to \(g^{-1}Kg \cap H\)), contributing \(c_g \circ \mathrm{res}_{g^{-1}Kg \cap H}^H\).

Summing over all double coset representatives and applying \(M\):

\[\boxed{\mathrm{res}_K^G \circ \mathrm{tr}_H^G(m) = \sum_{[g] \in K\backslash G/H} \mathrm{tr}_{K \cap {}^gH}^K \circ c_g \circ \mathrm{res}_{K^g \cap H}^H(m)}\]

where \(K^g = g^{-1}Kg\).

This is the Mackey double coset formula, derived purely from the fiber product decomposition of spans in \(\mathbf{FSets}_G\).

5. Key Examples 🧮

5.1 The Constant Mackey Functor

Definition (Constant Mackey Functor). For an abelian group \(A\), the constant Mackey functor \(\underline{A}\) is defined by: - \(\underline{A}(G/H) = A\) for all \(H \leq G\), - \(\mathrm{res}_H^K = \mathrm{id}_A\) for all \(H \leq K\), - \(\mathrm{tr}_H^K = [K:H] \cdot \mathrm{id}_A\) (multiplication by the index \([K:H]\)), - \(c_g = \mathrm{id}_A\) for all \(g\).

Let us verify the Mackey formula. Take \(H, K \leq G\) and \(m \in \underline{A}(G/H) = A\). Then:

\[\mathrm{res}_K^G \circ \mathrm{tr}_H^G(m) = \mathrm{id}_A([G:H] \cdot m) = [G:H] \cdot m.\]

The right side of the Mackey formula gives:

\[\sum_{[g] \in K\backslash G/H} \mathrm{tr}_{K \cap {}^gH}^K \circ c_g \circ \mathrm{res}_{K^g \cap H}^H(m) = \sum_{[g] \in K\backslash G/H} [K : K \cap {}^gH] \cdot m.\]

Since \(\sum_{[g] \in K\backslash G/H} [K : K \cap {}^gH] = [G:H]\) (this is the standard double coset counting formula), the Mackey formula holds. ✓

5.2 The Burnside Ring Mackey Functor

The most fundamental Mackey functor is the Burnside ring Mackey functor \(\underline{A}\).

Definition (Burnside Ring Mackey Functor). Define \(\underline{A}\) by: - \(\underline{A}(G/H) = A(H)\), the Burnside ring of \(H\), - \(\mathrm{res}_H^K: A(K) \to A(H)\) is restriction of the \(K\)-action to \(H\): \([S] \mapsto [S|_H]\), - \(\mathrm{tr}_H^K: A(H) \to A(K)\) is induction: \([S] \mapsto [K \times_H S]\) where \(K \times_H S = K \times S / (kh, s) \sim (k, hs)\), - \(c_g: A(H) \to A({}^gH)\) sends \([S]\) to \([{}^gS]\) where \({}^gS\) has the action \(h \cdot s = g^{-1}hg \cdot s\).

The Burnside ring Mackey functor is the representable Mackey functor \(\mathcal{A}(G)(G/G, -)\) — it is represented by the terminal \(G\)-set. In terms of the Yoneda lemma for additive categories, \(\underline{A}(G/H) = \mathcal{A}(G)(G/G, G/H) = A(H)\).

5.3 Fixed-Point Mackey Functors

Definition (Fixed-Point Mackey Functor). For a \(G\)-space \(X \in G\mathbf{Top}\) (see concepts/equivariant-stable-homotopy/g-spaces-and-equivariant-maps|G-Spaces and Equivariant Maps) and \(n \geq 0\), define the homotopy Mackey functor \(\underline{\pi}_n(X)\) by:

\[\underline{\pi}_n(X)(G/H) = \pi_n(X^H),\]

where \(X^H = \{x \in X : hx = x \text{ for all } h \in H\}\) is the \(H\)-fixed-point subspace.

The restriction maps \(\mathrm{res}_H^K: \pi_n(X^K) \to \pi_n(X^H)\) are induced by the inclusion \(X^K \hookrightarrow X^H\) (for \(H \leq K\)). The transfer maps are more subtle and require the equivariant transfer map in homotopy theory — they are not simply functorial in the obvious sense.

For a genuine \(G\)-spectrum \(E\) (see concepts/equivariant-stable-homotopy/g-spectra|G-Spectra (no note yet)), the homotopy groups

\[\underline{\pi}_n(E)(G/H) = \pi_n(E^H)\]

form a Mackey functor for every \(n \in \mathbb{Z}\), and the collection \(\{\underline{\pi}_n(E)\}_{n \in \mathbb{Z}}\) is the primary algebraic invariant of \(E\).

5.4 The C2 Classification

For \(G = C_2 = \{e, \tau\}\), there is a simple classification of all \(C_2\)-Mackey functors. A \(C_2\)-Mackey functor consists of a diagram

\[M(C_2/C_2) \underset{\mathrm{tr}}{\overset{\mathrm{res}}{\rightleftharpoons}} M(C_2/e)\]

where \(\tau\) acts on \(M(C_2/e)\) (by \(c_\tau\)), subject to:

1. Mackey formula: \(\mathrm{res} \circ \mathrm{tr}(m) = m + \tau \cdot m\) for all \(m \in M(C_2/e)\).
2. Trace formula: \(\mathrm{tr} \circ \mathrm{res}(n) = \mathrm{tr}(\mathrm{res}(n))\) for \(n \in M(C_2/C_2)\) (automatic from transitivity).

(The conjugation axiom for \(M(C_2/C_2)\) is trivial since \(C_2/C_2\) has only one coset.)

The four most important \(C_2\)-Mackey functors are:

Let us verify the Mackey formula for \(\underline{\mathbb{Z}}\): \(\mathrm{res} \circ \mathrm{tr}(m) = \mathrm{id}(2m) = 2m = m + \tau \cdot m = m + m = 2m\) ✓ (since \(\tau\) acts by identity on \(M(C_2/e) = \mathbb{Z}\)).

6. The Box Product and Green/Tambara Functors 💡

6.1 Day Convolution and the Box Product

The category \(\mathrm{Mack}(G)\) carries a symmetric monoidal product, the box product \(\square\), which is the analogue of the tensor product of abelian groups.

Definition (Box Product). For Mackey functors \(M\) and \(N\), their box product \(M \square N\) is the Day convolution of \(M\) and \(N\) with respect to the symmetric monoidal structure on \(\mathcal{A}(G)\) given by Cartesian product of \(G\)-sets:

\[(M \square N)(G/H) = \int^{X, Y \in \mathcal{A}(G)} \mathcal{A}(G)(X \times Y, G/H) \otimes_{\mathbb{Z}} M(X) \otimes_{\mathbb{Z}} N(Y).\]

Here the coend \(\int^{X,Y}\) runs over pairs of objects of \(\mathcal{A}(G)\), and \(\otimes_\mathbb{Z}\) denotes the usual tensor product of abelian groups.

Concretely, \((M \square N)(G/H)\) is generated by symbols \(m \otimes_H n\) for \(m \in M(G/H)\), \(n \in N(G/H)\), subject to the bilinearity relations coming from the \(\mathcal{A}(G)\)-enrichment. The key relations imposed are:

\[\mathrm{tr}_K^H(m) \otimes_H n = \mathrm{tr}_K^H(m \otimes_K \mathrm{res}_K^H(n))\]

(the Frobenius reciprocity-type relation in the definition of the box product).

The unit for \(\square\) is the Burnside ring Mackey functor \(\underline{A}\): there are natural isomorphisms \(\underline{A} \square M \cong M \cong M \square \underline{A}\).

6.2 Green Functors

Definition (Green Functor). A Green functor for \(G\) is a commutative monoid in \((\mathrm{Mack}(G), \square, \underline{A})\). Explicitly, it is a Mackey functor \(R\) equipped with: - A unit \(\eta: \underline{A} \to R\), - A multiplication \(\mu: R \square R \to R\),

satisfying commutativity, associativity, and unitality in \(\mathrm{Mack}(G)\).

Unpacking the definition: a Green functor \(R\) is a Mackey functor such that each \(R(G/H)\) is a commutative ring with unit, the restriction maps \(\mathrm{res}_H^K: R(K) \to R(H)\) are ring homomorphisms, and the Frobenius reciprocity condition holds:

\[\mathrm{tr}_H^K(a) \cdot b = \mathrm{tr}_H^K\bigl(a \cdot \mathrm{res}_H^K(b)\bigr) \quad \text{for all } a \in R(G/H), b \in R(G/K).\]

6.3 Tambara Functors and Multiplicative Norms 🔑

Green functors capture additive ring structure with transfers. But genuine equivariant commutative ring spectra have multiplicative norms — maps \(N_H^G: M(G/H) \to M(G/G)\) that are multiplicative, not just additive. This extra structure is axiomatized by Tambara functors.

Definition (Tambara Functor). A Tambara functor for \(G\) is a commutative monoid in a suitable symmetric monoidal \(\infty\)-category of Mackey functors that also carries multiplicative norm maps \(N_H^K: M(G/H) \to M(G/K)\) for \(H \leq K\), satisfying:

1. \(N_H^K\) is a (not necessarily additive) multiplicative map: \(N_H^K(1) = 1\) and \(N_H^K(ab) = N_H^K(a) N_H^K(b)\),
2. Distributivity: \(\mathrm{tr}_H^K(a) \cdot b = \mathrm{tr}_H^K(a \cdot \mathrm{res}_H^K(b))\) (Frobenius) and \(N_H^K(\mathrm{res}_H^K(b) \cdot a) = b^{[K:H]} \cdot N_H^K(a)\) (norm-restriction compatibility),
3. Norm-transfer compatibility: \(N_H^K(\mathrm{tr}_{H'}^H(a)) = \mathrm{tr}_{K'}^K(N_{H'}^{K'}(a) \cdot c)\) for suitable \(H' \leq H\) and \(K' = \mathrm{ind}_H^K H'\) (this is the norm-transfer formula, which encodes the Tambara polynomial functor structure).

The key example: if \(E\) is a genuine equivariant commutative ring \(G\)-spectrum (an \(E_\infty\)-algebra in \(\mathrm{Sp}^G\)), then \(\underline{\pi}_0(E)\) is a Tambara functor. The norm maps \(N_H^G\) on \(\underline{\pi}_0(E)\) come from the multiplicative norm maps \(N_H^G: E^H \to E^G\) in spectra (the Hill-Hopkins-Ravenel norm).

7. Projective Mackey Functors and Resolutions 🧮

7.1 Representable Mackey Functors

The category \(\mathrm{Mack}(G)\) is an abelian category, and like any functor category, it has a natural supply of projective objects coming from the Yoneda lemma.

Definition (Representable Mackey Functor). For a subgroup \(H \leq G\), define the representable Mackey functor \(\mathbb{Z}[G/H, -] = \mathcal{A}(G)(G/H, -)\) by:

\[\mathbb{Z}[G/H, -](G/K) = \mathcal{A}(G)(G/H, G/K).\]

Since \(\mathcal{A}(G)\) is additive, \(\mathbb{Z}[G/H, -]\) is an additive (hence Mackey) functor. By the Yoneda lemma for additive categories, there is a natural isomorphism

\[\mathrm{Hom}_{\mathrm{Mack}(G)}(\mathbb{Z}[G/H, -], M) \cong M(G/H)\]

for any Mackey functor \(M\).

Proposition. The representable Mackey functor \(\mathbb{Z}[G/H, -]\) is projective in \(\mathrm{Mack}(G)\).

Proof sketch. The Yoneda isomorphism shows that \(\mathrm{Hom}(\mathbb{Z}[G/H, -], -)\) is naturally isomorphic to evaluation at \(G/H\), which is exact (since limits/colimits in \(\mathrm{Mack}(G)\) are computed pointwise). \(\square\)

The representable Mackey functor \(\mathbb{Z}[G/H, -]\) evaluates explicitly as:

\[\mathbb{Z}[G/H, -](G/K) = \mathcal{A}(G)(G/H, G/K) \cong \bigoplus_{[x] \in H\backslash G/K} \mathbb{Z} \cdot [KxH],\]

i.e., the free abelian group on the set of double cosets \(H\backslash G/K\).

7.2 Global Dimension and Resolutions 🔑

Theorem (Projectivity and Generators). The representable Mackey functors \(\{\mathbb{Z}[G/H, -] : H \leq G\}\) (one for each conjugacy class of subgroups) form a generating set of projectives for \(\mathrm{Mack}(G)\). Every Mackey functor \(M\) admits a projective resolution:

\[0 \longleftarrow M \longleftarrow P_0 \longleftarrow P_1 \longleftarrow \cdots\]

where each \(P_i\) is a direct sum of representable Mackey functors.

Theorem (Global Dimension). The global dimension of \(\mathrm{Mack}(G)\) is finite. Specifically, if \(G\) has virtual cohomological dimension \(\mathrm{vcd}(G)\), then every Mackey functor has a projective resolution of length at most \(\mathrm{vcd}(G) + 1\).

For a finite group \(G\) of order \(n\), the global dimension is bounded: projective resolutions exist of length \(\leq 2\) after inverting \(|G|\) (since \(|G|\)-torsion is the obstruction). Over \(\mathbb{Z}[1/|G|]\), the category \(\mathrm{Mack}(G)[\frac{1}{|G|}]\) is semisimple.

The derived functors of \(\mathrm{Hom}_{\mathrm{Mack}(G)}\) give Ext groups for Mackey functors:

\[\mathrm{Ext}^n_{\mathrm{Mack}(G)}(M, N) = H^n(\mathrm{Hom}_{\mathrm{Mack}(G)}(P_\bullet, N))\]

and similarly for \(\mathrm{Tor}_n^{\mathrm{Mack}(G)}(M, N)\) using the box product. These Ext groups appear as the \(E_2\)-page of the slice spectral sequence for a genuine \(G\)-spectrum \(E\):

\[E_2^{s,t} \cong \mathrm{Ext}^s_{\mathrm{Mack}(G)}(\underline{\pi}_t(E), \underline{A}) \Rightarrow \pi_{t-s}(E^G).\]

8. Spectral Mackey Functors: Barwick’s Theorem 💡

8.1 The Effective Burnside Infinity-Category

The classical Burnside category \(\mathcal{A}(G)\) is a 1-category. To relate Mackey functors to genuine \(G\)-spectra, we need the \(\infty\)-categorical lift due to Barwick (2014).

Definition (Effective Burnside \(\infty\)-Category). Let \(\mathcal{F}_G\) denote the \(\infty\)-category of finite \(G\)-sets (regarded as a 1-category, hence an \(\infty\)-category with only homotopy-discrete morphism spaces). The effective Burnside \(\infty\)-category \(\mathcal{A}^{\mathrm{eff}}_\infty(G)\) is the \(\infty\)-category of spans in \(\mathcal{F}_G\): its objects are finite \(G\)-sets, and its morphisms are spans \(X \leftarrow Z \rightarrow Y\) where composition is via homotopy pullback (which for sets is the ordinary pullback, but the \(\infty\)-categorical framework tracks all higher coherences).

More precisely, \(\mathcal{A}^{\mathrm{eff}}_\infty(G)\) is constructed as the effective Burnside \(\infty\)-category of the disjunctive triple \((\mathcal{F}_G, \mathcal{F}_G, \mathcal{F}_G)\) in Barwick’s sense: a triple \((\mathcal{C}, \mathcal{C}^\dagger, \mathcal{C}_\dagger)\) where both the “ingressive” maps \(\mathcal{C}^\dagger\) and “egressive” maps \(\mathcal{C}_\dagger\) are taken to be all maps (since all maps of finite \(G\)-sets may appear as either leg of a span).

The key property is that \(\mathcal{A}^{\mathrm{eff}}_\infty(G)\) is the homotopy-coherent version of span composition: the associativity of fiber products is handled up to coherent homotopy, not just up to isomorphism.

8.2 Spectral Mackey Functors

Definition (Spectral Mackey Functor). A spectral Mackey functor for \(G\) is an additive (equivalently, finite-coproduct-preserving) functor

\[\mathcal{M}: \mathcal{A}^{\mathrm{eff}}_\infty(G) \longrightarrow \mathbf{Sp}\]

where \(\mathbf{Sp}\) is the \(\infty\)-category of spectra. Additivity means \(\mathcal{M}(X \sqcup Y) \simeq \mathcal{M}(X) \vee \mathcal{M}(Y)\) (wedge of spectra).

The \(\infty\)-category of spectral Mackey functors is

\[\mathrm{SpMack}(G) = \mathrm{Fun}^{\oplus}(\mathcal{A}^{\mathrm{eff}}_\infty(G), \mathbf{Sp}).\]

This is a presentable stable \(\infty\)-category and carries a symmetric monoidal structure (the spectral Day convolution).

8.3 Barwick’s Equivalence

The central theorem of Barwick’s 2014 paper is:

Theorem (Barwick 2014). There is an equivalence of \(\infty\)-categories

\[\mathrm{Fun}^{\oplus}(\mathcal{A}^{\mathrm{eff}}_\infty(G), \mathbf{Sp}) \simeq \mathrm{Sp}^G,\]

where \(\mathrm{Sp}^G\) is the \(\infty\)-category of genuine \(G\)-spectra (orthogonal \(G\)-spectra localized at genuine equivalences).

Proof sketch. The forward direction sends a spectral Mackey functor \(\mathcal{M}\) to the genuine \(G\)-spectrum \(\mathcal{M}(G/G)\) together with the action of the Burnside category on the collection \(\{\mathcal{M}(G/H)\}\). The reverse direction sends a genuine \(G\)-spectrum \(E\) to the functor \(G/H \mapsto E^H\) (categorical fixed-point spectrum), with the restriction and transfer maps encoded by the span functoriality.

The key technical input is that the span functoriality on \(\{E^H\}\) — specifically the existence of coherent transfer maps — is exactly what genuine \(G\)-spectra have and naive \(G\)-spectra lack.

Corollary. Eilenberg-MacLane spectral Mackey functors \(H\underline{M}\) for classical Mackey functors \(\underline{M} \in \mathrm{Mack}(G)\) correspond under Barwick’s equivalence to the genuine Eilenberg-MacLane \(G\)-spectra of equivariant stable homotopy theory. These are the coefficient objects for RO(G)-graded Bredon cohomology.

8.4 Homotopy Groups as Classical Mackey Functors 🔑

Under Barwick’s equivalence, the relationship between spectral and classical Mackey functors is mediated by the truncation functors on spectra.

Proposition. Let \(\mathcal{M}: \mathcal{A}^{\mathrm{eff}}_\infty(G) \to \mathbf{Sp}\) be a spectral Mackey functor, and let \(E = \mathcal{M}(G/G)\) be the corresponding genuine \(G\)-spectrum. Then for each \(n \in \mathbb{Z}\), the classical Mackey functor

\[\underline{\pi}_n(\mathcal{M}): G/H \longmapsto \pi_n(\mathcal{M}(G/H))\]

is the \(n\)-th homotopy Mackey functor of \(E\).

Proof sketch. Since \(\mathcal{M}\) sends spans to maps of spectra, and restriction/transfer in the \(G\)-spectrum \(E\) are encoded by specific spans in \(\mathcal{A}^{\mathrm{eff}}_\infty(G)\), applying \(\pi_n\) levelwise gives a classical Mackey functor with the correct restrictions and transfers. The Mackey formula holds because it holds at the level of \(\pi_n\) applied to the span decomposition of pullbacks. \(\square\)

The homotopy groups of a genuine \(G\)-spectrum \(E\) are not merely abelian groups — they are Mackey functors. This is the fundamental reason Mackey functors appear as the coefficient objects for equivariant cohomology: they encode the simultaneous data of all fixed-point homotopy groups together with the restriction and transfer maps between them.

Exercises 📝

Mathematical Development

Problem 1: Morphism Groups of the Burnside Category for C2

This problem makes the Burnside category concrete by computing all four morphism groups \(\mathcal{A}(C_2)(X, Y)\) explicitly, verifying the note’s claim that spans between the two orbits \(C_2/e\) and \(C_2/C_2\) generate free abelian groups of specified ranks.

Prerequisites: cf. #2.1 Spans of Finite G-Sets|§2.1 — Spans of Finite G-Sets; cf. #2.3 Additive Structure and the Burnside Ring|§2.3 — Additive Structure and the Burnside Ring

Let \(G = C_2 = \{e, \tau\}\). Write \(* = C_2/C_2\) and \(C_2 = C_2/e\) for the two isomorphism classes of transitive \(G\)-sets.

1. List all isomorphism classes of spans \(* \leftarrow Z \rightarrow *\). Show that \(\mathcal{A}^+(C_2)(*, *) \cong \mathbb{N} \times \mathbb{N}\) as a commutative monoid, with generators \([* \leftarrow * \rightarrow *]\) and \([* \leftarrow C_2 \rightarrow *]\). Conclude \(\mathcal{A}(C_2)(*, *) \cong \mathbb{Z}^2\).

2. List all isomorphism classes of spans \(C_2 \leftarrow Z \rightarrow *\). Show that \(\mathcal{A}(C_2)(C_2, *) \cong \mathbb{Z}\), generated by the span \(C_2 \xleftarrow{\mathrm{id}} C_2 \xrightarrow{p} *\) where \(p\) is the unique map to the point.

3. List all isomorphism classes of spans \(* \leftarrow Z \rightarrow C_2\). Show that \(\mathcal{A}(C_2)(*, C_2) \cong \mathbb{Z}\), generated by the span \(* \xleftarrow{p} C_2 \xrightarrow{\mathrm{id}} C_2\).

4. Show that \(\mathcal{A}(C_2)(C_2, C_2) \cong \mathbb{Z}^2\) by classifying all spans \(C_2 \leftarrow Z \rightarrow C_2\) up to isomorphism and identifying the two generators.

[!TIP]- Solution to Exercise 1 Key insight: Every \(C_2\)-set decomposes uniquely as a disjoint union of copies of \(*\) and \(C_2\), so the middle piece \(Z\) of any span is determined by two non-negative integers \((a, b)\) counting the two orbit types.

Sketch:

A span \(* \leftarrow Z \rightarrow *\) is determined by the isomorphism class of \(Z\) (both maps to \(*\) are forced). Every \(C_2\)-set \(Z \cong (*^{\sqcup a}) \sqcup (C_2^{\sqcup b})\), giving a monoid \(\mathbb{N} \times \mathbb{N}\) with generators \([*]\) and \([C_2]\); group completion yields \(\mathbb{Z}^2\).

For \(\mathcal{A}(C_2)(C_2, *)\): a span \(C_2 \leftarrow Z \rightarrow *\) requires a \(G\)-equivariant map \(Z \to C_2\). The equivariant maps \(Z \to C_2\) exist only when \(Z\) has a free orbit; the only connected such \(Z\) is \(Z = C_2\) with the identity. Thus the monoid is \(\mathbb{N}\), giving \(\mathbb{Z}\).

For \(\mathcal{A}(C_2)(C_2, C_2)\): the middle piece \(Z\) admits equivariant maps to \(C_2\) on both sides. The connected possibilities are \(Z = C_2\) (identity span) or \(Z = *\) (folding span). The two generators are \([\mathrm{id}_{C_2}]\) and \([* \leftarrow * \rightarrow C_2]\); group completion gives \(\mathbb{Z}^2\).

Problem 2: Morphism Groups of the Burnside Category for C3

This problem extends the \(C_2\) computation to \(C_3\), where the orbit structure of the middle piece \(Z\) is richer, producing a rank-2 morphism group for \(\mathcal{A}(C_3)(C_3, C_3)\) from different generators.

Prerequisites: cf. #2.1 Spans of Finite G-Sets|§2.1 — Spans of Finite G-Sets; requires Problem 1

Let \(G = C_3\). Write \(* = C_3/C_3\) and \(C_3 = C_3/e\).

1. List all isomorphism classes of \(C_3\)-sets of order \(\leq 6\).

2. Compute the monoid \(\mathcal{A}^+(C_3)(*, *)\) and show \(\mathcal{A}(C_3)(*, *) \cong \mathbb{Z}^2\).

3. Show that the morphism group \(\mathcal{A}(C_3)(C_3, *) \cong \mathbb{Z}\), with the sole generator being the span \(C_3 \xleftarrow{\mathrm{id}} C_3 \xrightarrow{p} *\).

4. Classify all spans \(C_3 \leftarrow Z \rightarrow C_3\) by listing all possible middle pieces \(Z\). Show that \(\mathcal{A}(C_3)(C_3, C_3) \cong \mathbb{Z}^2\), and identify the two generators.

[!TIP]- Solution to Exercise 2 Key insight: The argument is identical to the \(C_2\) case — the orbit classification forces every middle piece to be a disjoint union of \(*\) and \(C_3\), and one counts equivariant maps to each endpoint to enumerate span types.

Sketch:

Every finite \(C_3\)-set is \(*^{\sqcup a} \sqcup C_3^{\sqcup b}\). The monoid \(\mathcal{A}^+(C_3)(*, *) \cong \mathbb{N} \times \mathbb{N}\) (generators \([*]\) and \([C_3]\)), so \(\mathcal{A}(C_3)(*,*) \cong \mathbb{Z}^2\).

For \(\mathcal{A}(C_3)(C_3, *)\): the only equivariant map \(Z \to C_3\) from a transitive \(C_3\)-set is \(Z = C_3\). The monoid is \(\mathbb{N}\), giving \(\mathbb{Z}\).

For \(\mathcal{A}(C_3)(C_3, C_3)\): connected middle pieces admitting equivariant maps to \(C_3\) on both sides: (i) \(Z = C_3\) giving the identity span and “diagonal” span \(C_3 \xleftarrow{g \mapsto g} C_3 \xrightarrow{g \mapsto g\sigma} C_3\); (ii) \(Z = *\) with constant maps (a “folding” span). Group completion of the two-generator monoid gives \(\mathbb{Z}^2\).

Problem 3: The Burnside Ring as an Endomorphism Ring

This problem proves the identification \(A(G) = \mathcal{A}(G)(G/G, G/G)\) rigorously, by showing that spans \(G/G \leftarrow Z \rightarrow G/G\) are simply finite \(G\)-sets and that composition becomes Cartesian product.

Prerequisites: cf. #2.3 Additive Structure and the Burnside Ring|§2.3 — Additive Structure and the Burnside Ring

1. Show that a span \(G/G \leftarrow Z \rightarrow G/G\) is determined up to isomorphism by the \(G\)-set \(Z\) alone. Conclude that \(\mathcal{A}^+(G)(G/G, G/G)\) is the commutative monoid of isomorphism classes of finite \(G\)-sets under disjoint union.

2. Show that the composition of spans \(* \leftarrow Z \rightarrow *\) and \(* \leftarrow W \rightarrow *\) in \(\mathcal{A}(G)\) corresponds to Cartesian product \(Z \times W\) of finite \(G\)-sets. Conclude that the ring structure on \(K_0(\mathbf{FSets}_G)\) induced by span composition is the Burnside ring product.

3. For \(G = C_2\): write out the multiplication table of \(\mathcal{A}(C_2)(*, *)\) in the basis \(\{[*], [C_2]\}\) using part (b). Verify that \([C_2]^2 = 2[C_2]\).

[!TIP]- Solution to Exercise 3 Key insight: Since \(G/G = *\) is the terminal object, both legs of any span \(* \leftarrow Z \rightarrow *\) are the unique map to \(*\), so the span is completely determined by the \(G\)-set \(Z\) itself; composition of spans then becomes Cartesian product.

Sketch:

For any \(G\)-set \(Z\), the unique maps \(Z \to *\) provide the legs; two spans are isomorphic iff \(Z \cong Z'\) as \(G\)-sets. So \(\mathcal{A}^+(G)(*,*) \cong (\mathbf{FSets}_G)^{\cong}\), and \(\mathcal{A}(G)(*,*) = K_0(\mathbf{FSets}_G) = A(G)\).

Composition: \((* \leftarrow Z \rightarrow *) \circ (* \leftarrow W \rightarrow *) = (* \leftarrow Z \times_* W \rightarrow *)\) and \(Z \times_* W = Z \times W\). This matches the Burnside ring product \([Z] \cdot [W] = [Z \times W]\).

For \(C_2\): \([C_2]^2 = [C_2 \times C_2]\). The diagonal \(C_2\)-action on \(C_2 \times C_2\) has orbits \(\{(e,e),(\tau,\tau)\}\) and \(\{(e,\tau),(\tau,e)\}\), each a copy of \(C_2\). So \([C_2]^2 = 2[C_2]\) in \(A(C_2)\). ✓

Problem 4: Span Composition and the Mackey Formula for C4

This problem uses the fiber product formula to compute \(\mathrm{res}_{C_2} \circ \mathrm{tr}_{C_2}^{C_4}\) for the unique index-2 subgroup \(C_2 \leq C_4\), verifying that the answer is \(1 + \tau_*\) where \(\tau \in C_2\) is the generator.

Prerequisites: cf. #4.1 The Pullback Decomposition|§4.1 — The Pullback Decomposition; cf. #4.2 Derivation from Spans|§4.2 — Derivation from Spans

Let \(G = C_4 = \langle \sigma \rangle\) and \(H = K = C_2 = \langle \sigma^2 \rangle \leq C_4\).

1. List the double cosets in \(K\backslash G/H = C_2\backslash C_4/C_2\) and find a set of representatives.

2. For each double coset representative \([g]\), compute the intersection \(K \cap {}^gH\), the conjugation map \(c_g\), and the contribution \(\mathrm{tr}_{K \cap {}^gH}^K \circ c_g \circ \mathrm{res}_{K^g \cap H}^H\) to the Mackey formula.

3. Conclude that \(\mathrm{res}_{C_2}^{C_4} \circ \mathrm{tr}_{C_2}^{C_4}(m) = m + \tau_*(m)\) for any Mackey functor \(M\) and \(m \in M(C_4/C_2)\), where \(\tau = \sigma^2\).

4. Verify this formula on the constant Mackey functor \(\underline{\mathbb{Z}}\): confirm that \(\mathrm{res} \circ \mathrm{tr}(m) = 2m\) and that \(m + \tau_*(m) = 2m\).

[!TIP]- Solution to Exercise 4 Key insight: The double coset space \(C_2\backslash C_4/C_2\) has exactly two classes — \([e]\) and \([\sigma]\) — because \(C_4/C_2\) has two elements; the corresponding Mackey formula contributions are \(\mathrm{id}\) and \(\tau_*\).

Sketch:

\(C_2\backslash C_4/C_2\): the orbits of \(C_2 = \{e, \sigma^2\}\) acting on \(C_4/C_2 = \{eC_2, \sigma C_2\}\) by left multiplication are each singletons. So \(K\backslash C_4/H = \{[e], [\sigma]\}\).

For \([g] = [e]\): \(K \cap {}^eH = C_2 \cap C_2 = C_2\), so the contribution is \(\mathrm{tr}_{C_2}^{C_2} \circ c_e \circ \mathrm{res}_{C_2}^{C_2} = \mathrm{id}\).

For \([g] = [\sigma]\): \(K \cap {}^\sigma H = C_2 \cap \sigma C_2 \sigma^{-1} = C_2\) (since \(C_4\) is abelian). The contribution is \(\mathrm{tr}_{C_2}^{C_2} \circ c_\sigma \circ \mathrm{res}_{C_2}^{C_2} = c_{\sigma^2} = \tau_*\).

Summing: \(\mathrm{res}_{C_2} \circ \mathrm{tr}_{C_2}^{C_4} = \mathrm{id} + \tau_*\). For \(\underline{\mathbb{Z}}\): \(\tau = \mathrm{id}\), so \(\mathrm{id} + \tau_* = 2\cdot\mathrm{id}\), which matches \(\mathrm{res}(\mathrm{tr}(m)) = 2m\). ✓

Problem 5: The Mackey Formula for S3

This problem applies the double coset formula to a non-abelian group, computing \(\mathrm{res}_K^{S_3} \circ \mathrm{tr}_H^{S_3}\) for two non-conjugate subgroups \(H = \langle (12) \rangle\) and \(K = \langle (13) \rangle\) of order 2, a case where the double coset decomposition is non-trivial.

Prerequisites: cf. #4. The Mackey Double Coset Formula|§4 — The Mackey Double Coset Formula

Let \(G = S_3\), \(H = \langle (12) \rangle \cong C_2\), and \(K = \langle (13) \rangle \cong C_2\).

1. Compute the double coset decomposition \(K\backslash G/H\). List representative elements and the corresponding double cosets \(KgH\) explicitly as subsets of \(S_3\).

2. For each double coset representative \([g]\), compute \(K \cap {}^gH\) and the conjugate \({}^gH = g H g^{-1}\).

3. Write out the full Mackey formula: \[\mathrm{res}_K^{S_3} \circ \mathrm{tr}_H^{S_3}(m) = \sum_{[g] \in K\backslash S_3/H} \mathrm{tr}_{K \cap {}^gH}^K \circ c_g \circ \mathrm{res}_{K^g \cap H}^H(m).\]

4. Specialize to the constant Mackey functor \(\underline{\mathbb{Z}}\) and verify numerically that both sides give the same value.

[!TIP]- Solution to Exercise 5 Key insight: The two subgroups \(H = \langle(12)\rangle\) and \(K = \langle(13)\rangle\) are non-conjugate in \(S_3\) but both of order 2; the double coset decomposition \(K\backslash S_3/H\) has two elements, one with trivial intersection and one where \(K \cap {}^gH = K\).

Sketch:

Partitioning \(S_3\) into double cosets \(KgH\): by the double coset size formula \(|KgH| = |K||H|/|K \cap {}^gH|\): - \([g] = e\): \(K \cap H = \{e\}\), so \(|KeH| = 4\). - \([g] = [(132)]\): \((132)(12)(123) = (13)\), so \({}^{(132)}H = K\). Then \(K \cap K = K \cong C_2\), so \(|K(132)H| = 2\). Total: \(4 + 2 = 6 = |S_3|\). ✓ Two double cosets.

Mackey formula: \[\mathrm{res}_K \circ \mathrm{tr}_H = (c_e \circ \mathrm{res}_{\{e\}}^H) + (\mathrm{tr}_K^K \circ c_{(132)} \circ \mathrm{res}_{K^{(132)} \cap H}^H).\] Here \(K^{(132)} = \langle(12)\rangle = H\), so \(K^{(132)} \cap H = H\), and the second term is \(c_{(132)} \circ \mathrm{id}\).

For \(\underline{\mathbb{Z}}\): first term gives \(\mathrm{id}\) (restriction from \(H\) to \(e\) then covariant), contributing \(m\) at each of the \([K:e]\) indices… full count gives \(3m = [S_3:H] \cdot m\), consistent with \(\mathrm{res}_K(\mathrm{tr}_H^{S_3}(m)) = 3m\).

Problem 6: The Norm Map for Cp

This problem establishes the general formula \(\mathrm{res}_e^{C_p} \circ \mathrm{tr}_e^{C_p} = \sum_{g \in C_p} g_*\) for any prime \(p\), which is the fundamental example of the Mackey formula for a cyclic group of prime order.

Prerequisites: cf. #4.2 Derivation from Spans|§4.2 — Derivation from Spans

Let \(G = C_p\) for a prime \(p\), \(H = K = \{e\}\).

1. List all double cosets in \(\{e\}\backslash C_p/\{e\}\). Show there are exactly \(p\) double cosets, one for each \(g \in C_p\).

2. For each double coset representative \(g \in C_p\), compute \(K \cap {}^gH = \{e\}\) and show each contributes the map \(c_g: M(C_p/e) \to M(C_p/e)\).

3. Conclude that for any \(C_p\)-Mackey functor \(M\): \(\mathrm{res}_e^{C_p} \circ \mathrm{tr}_e^{C_p}(m) = \sum_{g \in C_p} c_g(m) = \mathrm{Nm}(m)\), the norm map.

4. For the constant Mackey functor \(\underline{\mathbb{Z}}\), verify this gives \(pm\). For the fixed-point Mackey functor of the regular representation \(M(C_p/e) = \mathbb{Z}[C_p]\), compute the norm \(\sum_{g \in C_p} g \cdot m\) explicitly for \(m = e\).

[!TIP]- Solution to Exercise 6 Key insight: When both \(H\) and \(K\) are trivial, every element of \(C_p\) is its own double coset, so the Mackey formula is a sum of \(p\) conjugation maps — by definition the norm map.

Sketch:

\(\{e\}\backslash C_p/\{e\} = C_p\) (each element \(g\) is the singleton double coset \(\{g\}\)). There are exactly \(p\) double cosets.

For each \(g \in C_p\): \(K \cap {}^gH = \{e\}\), so \(\mathrm{tr}_{\{e\}}^{\{e\}} = \mathrm{id}\) and \(\mathrm{res}_{\{e\}}^{\{e\}} = \mathrm{id}\). The contribution is \(c_g\).

Summing: \(\mathrm{res}_e^{C_p} \circ \mathrm{tr}_e^{C_p}(m) = \sum_{g \in C_p} c_g(m) = \mathrm{Nm}(m)\).

For \(\underline{\mathbb{Z}}\): all \(c_g = \mathrm{id}\), so \(\mathrm{Nm}(m) = pm\). Also \(\mathrm{tr}_e^{C_p}(m) = pm\) and \(\mathrm{res}_e^{C_p}(pm) = pm\). ✓

For \(\mathbb{Z}[C_p]\): \(c_g(e) = g\), so \(\mathrm{Nm}(e) = \sum_{g \in C_p} g = N\) (the norm element of the group ring).

Problem 7: The C2 Mackey Functor Classification

This problem classifies all \(C_2\)-Mackey functors with \(M(C_2/e) = \mathbb{Z}\) by determining which pairs (res, tr) are compatible with the Mackey formula.

Prerequisites: cf. #5.4 The C2 Classification|§5.4 — The C2 Classification

Let \(G = C_2\) and suppose \(M\) is a \(C_2\)-Mackey functor with \(M(C_2/e) = \mathbb{Z}\).

1. Write \(A = M(C_2/C_2)\), \(r = \mathrm{res}: A \to \mathbb{Z}\), \(t = \mathrm{tr}: \mathbb{Z} \to A\), and \(\tau = c_\tau: \mathbb{Z} \to \mathbb{Z}\). Show \(\tau\) must be an involution, so either \(\tau = \mathrm{id}\) or \(\tau = -\mathrm{id}\).

2. In the case \(\tau = \mathrm{id}\): show the Mackey formula forces \(r \circ t = 2 \cdot \mathrm{id}_\mathbb{Z}\). Show that \(A = \mathbb{Z}\), \(r = \mathrm{id}\), \(t = \times 2\) gives the constant Mackey functor \(\underline{\mathbb{Z}}\).

3. In the case \(\tau = -\mathrm{id}\): show the Mackey formula forces \(r \circ t = 0\). Show the choice \(A = \mathbb{Z}\), \(r = 0\), \(t = 0\), \(\tau = -1\) gives \(\underline{\mathbb{Z}}^-\).

4. Are there Mackey functors with \(M(C_2/e) = \mathbb{Z}\) and \(A = M(C_2/C_2) = \mathbb{Z}/2\)? Identify any example or prove none exist.

[!TIP]- Solution to Exercise 7 Key insight: The Mackey formula \(\mathrm{res} \circ \mathrm{tr} = \mathrm{id} + \tau\) is the single constraint; with \(\tau \in \{\pm\mathrm{id}\}\), it forces \(r \circ t = 2\) or \(r \circ t = 0\), each case determining a one-parameter family of extensions.

Sketch:

Since \(c_\tau: \mathbb{Z} \to \mathbb{Z}\) must satisfy \(c_\tau^2 = \mathrm{id}\) and be a group automorphism of \(\mathbb{Z}\), we get \(c_\tau \in \{\mathrm{id}, -\mathrm{id}\}\).

Case \(\tau = \mathrm{id}\): Mackey gives \(r \circ t = 2\). The pair \((r, t)\) is characterized by \(m = t(1) \in A\) with \(r(m) = 2\). The minimal choice \(A = \mathbb{Z}\), \(t(n) = 2n\), \(r = \mathrm{id}\) gives \(\underline{\mathbb{Z}}\).

Case \(\tau = -\mathrm{id}\): Mackey gives \(r \circ t = \mathrm{id} + (-\mathrm{id}) = 0\). The choice \(A = \mathbb{Z}\), \(r = 0\), \(t = 0\) gives \(\underline{\mathbb{Z}}^-\).

For \(A = \mathbb{Z}/2\): \(\mathrm{Hom}(\mathbb{Z}/2, \mathbb{Z}) = 0\) (since \(\mathbb{Z}\) is torsion-free), so \(r = 0\). The Mackey formula in the case \(\tau = -\mathrm{id}\) gives \(0 = 0\), which is consistent. So there do exist Mackey functors with \(A = \mathbb{Z}/2\), \(r = 0\), \(t: \mathbb{Z} \to \mathbb{Z}/2\) any homomorphism, and \(\tau = -\mathrm{id}\).

Problem 8: The Constant Mackey Functor and Bredon Cohomology

This problem computes \(H^0_{C_2}(C_2; \underline{\mathbb{Z}})\) using the Lewis diagram of \(\underline{\mathbb{Z}}\), providing a concrete first example of Bredon cohomology computed directly from the Mackey functor structure.

Prerequisites: cf. #5.1 The Constant Mackey Functor|§5.1 — The Constant Mackey Functor; cf. #5.4 The C2 Classification|§5.4 — The C2 Classification

For \(G = C_2\), the constant Mackey functor \(\underline{\mathbb{Z}}\) has Lewis diagram \(\mathbb{Z} \underset{2}{\overset{1}{\rightleftharpoons}} \mathbb{Z}\).

1. Verify the Mackey formula for \(\underline{\mathbb{Z}}\): confirm that \(\mathrm{res} \circ \mathrm{tr}(m) = 2m\) agrees with \(m + \tau \cdot m = 2m\).

2. Recall that \(H^0_{C_2}(C_2; M)\) is computed from the complex \(0 \to M(C_2/C_2) \xrightarrow{\mathrm{res}} M(C_2/e) \to 0\). Compute \(H^0_{C_2}(C_2; \underline{\mathbb{Z}}) = \ker(1 - \tau)/\mathrm{im}(\mathrm{res})\).

3. Since \(\tau = \mathrm{id}\) on \(M(C_2/e) = \mathbb{Z}\), the map \(1 - \tau = 0\), so \(\ker(1-\tau) = \mathbb{Z}\). Compute \(\mathrm{im}(\mathrm{res})\) and conclude. Explain geometrically why the 0th Bredon cohomology of a free orbit should vanish.

[!TIP]- Solution to Exercise 8 Key insight: For the free orbit \(C_2/e\), the fixed-point space is empty at the group level; the \(C_2\)-equivariant Bredon cohomology in degree 0 sees only \(C_2\)-invariants, which for a free orbit are trivial.

Sketch:

Mackey formula check: \(\mathrm{res}(\mathrm{tr}(m)) = \mathrm{id}(2m) = 2m\) and \(m + \tau m = m + m = 2m\) (since \(\tau = \mathrm{id}\) on \(M(C_2/e) = \mathbb{Z}\)). ✓

For Bredon \(H^0\): \(H^0 = \ker(1 - \tau \mid_{M(C_2/e)}) / \mathrm{im}(\mathrm{res})\). With \(\tau = \mathrm{id}\): \(1 - \tau = 0\) so \(\ker = \mathbb{Z}\); \(\mathrm{im}(\mathrm{res}) = \mathrm{im}(\mathrm{id}: \mathbb{Z} \to \mathbb{Z}) = \mathbb{Z}\).

\(H^0_{C_2}(C_2; \underline{\mathbb{Z}}) = \mathbb{Z}/\mathbb{Z} = 0\).

Geometric interpretation: Bredon \(H^0_{C_2}(C_2; \underline{\mathbb{Z}})\) computes connected components of the orbit space \(C_2/C_2 = *\), weighted by the Mackey structure. Since \(C_2\) acts freely on itself, there are no genuine fixed points to contribute, and the single orbit \(*\) has \(H^0 = 0\) (vanishing due to the free action).

Problem 9: The Burnside Ring Mackey Functor for C2

This problem computes \(\underline{A}(C_2)\) explicitly as a Lewis diagram, verifies the Mackey formula for the Burnside ring Mackey functor, and confirms that \(A(C_2) \cong \mathbb{Z}^2\).

Prerequisites: cf. #5.2 The Burnside Ring Mackey Functor|§5.2 — The Burnside Ring Mackey Functor

Let \(G = C_2\). Recall \(\underline{A}(G/H) = A(H)\) (the Burnside ring of \(H\)).

1. Compute \(\underline{A}(C_2/C_2) = A(C_2)\) and \(\underline{A}(C_2/e) = A(e) \cong \mathbb{Z}\). Write the Lewis diagram with explicit groups.

2. Compute the restriction map \(\mathrm{res}_e^{C_2}: A(C_2) \to A(e) = \mathbb{Z}\): for each generator \([*]\) and \([C_2]\) of \(A(C_2) \cong \mathbb{Z}^2\), restrict the \(C_2\)-action to \(e\). Express \(\mathrm{res}\) as a matrix.

3. Compute the transfer map \(\mathrm{tr}_e^{C_2}: A(e) \to A(C_2)\): the transfer of an \(e\)-set \(S\) is \(C_2 \times_e S \cong C_2 \times S\). Compute \(\mathrm{tr}([*]) = [C_2]\) and write the Lewis diagram.

4. Verify the Mackey formula: \(\mathrm{res} \circ \mathrm{tr}(n) = 2n\) and \(n + \tau_*(n) = 2n\).

[!TIP]- Solution to Exercise 9 Key insight: The Burnside ring of \(C_2\) is \(A(C_2) = \mathbb{Z}\{[*],[C_2]\} \cong \mathbb{Z}^2\), and the restriction/transfer are precisely the “forget action” and “induce” operations on \(0\)-dimensional \(C_2\)-sets.

Sketch:

\(\underline{A}(C_2/C_2) = A(C_2) \cong \mathbb{Z}^2\) (generators \([*]\) and \([C_2]\)). \(\underline{A}(C_2/e) = A(e) \cong \mathbb{Z}\).

Restriction \(\mathrm{res}_e^{C_2}: A(C_2) \to A(e)\): forget the \(C_2\)-action. \(\mathrm{res}([*]) = 1\) and \(\mathrm{res}([C_2]) = 2\) (two isolated points). So \(\mathrm{res}(a, b) = a + 2b\) as a map \(\mathbb{Z}^2 \to \mathbb{Z}\).

Transfer \(\mathrm{tr}_e^{C_2}: A(e) \to A(C_2)\): \(\mathrm{tr}(1) = [C_2]\). So \(\mathrm{tr}(n) = n[C_2]\), i.e., \(\mathrm{tr} = (0, 1)^T: \mathbb{Z} \to \mathbb{Z}^2\).

Lewis diagram: \(\mathbb{Z}^2 \underset{(0,1)^T}{\overset{(1,2)}{\rightleftharpoons}} \mathbb{Z}\).

Mackey check: \(\mathrm{res} \circ \mathrm{tr}(n) = (1,2)(0,n)^T = 2n\) and \(n + \tau_*(n) = n + n = 2n\). ✓

Problem 10: The Mackey Formula from Fiber Products

This problem re-derives the Mackey double coset formula from the explicit fiber product decomposition in \(\mathbf{FSets}_G\), following the derivation in the note but working out all intermediate steps for \(G = C_4\), \(H = e\), \(K = C_2\).

Prerequisites: cf. #4.1 The Pullback Decomposition|§4.1 — The Pullback Decomposition; cf. #4.2 Derivation from Spans|§4.2 — Derivation from Spans

Let \(G = C_4 = \langle \sigma \rangle\), \(H = \{e\}\), \(K = C_2 = \langle \sigma^2 \rangle\).

1. Write the two spans corresponding to \(\mathrm{tr}_e^{C_4}\) and \(\mathrm{res}_{C_2}^{C_4}\).

2. Compute the fiber product \(G/e \times_{G/G} G/K = C_4 \times_* (C_4/C_2)\) as a \(C_4\)-set. Decompose into orbits under the diagonal \(C_4\)-action.

3. Identify each orbit in the decomposition with a coset space \(G/(K \cap {}^gH)\) for appropriate \(g\). Write the full orbit decomposition and read off the Mackey formula.

4. Verify the result matches the double coset formula: \(\mathrm{res}_{C_2}^{C_4} \circ \mathrm{tr}_e^{C_4}(m) = \sum_{g \in C_2\backslash C_4/\{e\}} c_g(m)\). How many terms appear?

[!TIP]- Solution to Exercise 10 Key insight: The fiber product \(C_4 \times_* (C_4/C_2)\) is just the Cartesian product (since the base is a point), and the diagonal \(C_4\)-orbits are indexed by double cosets in \(C_2\backslash C_4/\{e\} = C_2\backslash C_4\).

Sketch:

The span for \(\mathrm{tr}_e^{C_4}\) is \(C_4 \xleftarrow{\mathrm{id}} C_4 \xrightarrow{p} *\); for \(\mathrm{res}_{C_2}^{C_4}\) it is \(* \xleftarrow{q} C_4/C_2 \xrightarrow{\mathrm{id}} C_4/C_2\).

Fiber product: \(C_4 \times_* (C_4/C_2) = C_4 \times C_4/C_2\) with diagonal \(C_4\)-action.

Orbits: for each \(\sigma^i C_2 \in C_4/C_2\), the orbit of \((e, \sigma^i C_2)\) has stabilizer \(\{e\} \cap \sigma^i C_2 \sigma^{-i} = \{e\} \cap C_2 = \{e\}\). So each orbit is \(\cong C_4/\{e\} = C_4\).

There are \(|C_4/C_2| = 2\) orbits, each \(\cong C_4\). Reading off: \(\mathrm{res}_{C_2}^{C_4} \circ \mathrm{tr}_e^{C_4}(m) = c_e(m) + c_\sigma(m)\), a sum of 2 terms. ✓

Problem 11: The Yoneda Lemma for Mackey Functors

This problem proves the Yoneda isomorphism \(\mathrm{Hom}_{\mathrm{Mack}(G)}(\mathbb{Z}[G/H,-], M) \cong M(G/H)\) in detail, establishing that representable Mackey functors are the fundamental projective objects.

Prerequisites: cf. #7.1 Representable Mackey Functors|§7.1 — Representable Mackey Functors

Recall the representable Mackey functor \(\mathbb{Z}[G/H,-] = \mathcal{A}(G)(G/H,-)\).

1. For a natural transformation \(\phi: \mathbb{Z}[G/H,-] \to M\), let \(m = \phi_{G/H}(\mathrm{id}_{G/H}) \in M(G/H)\). Show that \(\phi_{G/K}(\alpha) = M(\alpha)(m)\) for every \(\alpha \in \mathcal{A}(G)(G/H, G/K)\) and every orbit \(G/K\).

2. Deduce that the assignment \(\phi \mapsto \phi_{G/H}(\mathrm{id}_{G/H})\) defines a bijection \(\mathrm{Hom}_{\mathrm{Mack}(G)}(\mathbb{Z}[G/H,-], M) \xrightarrow{\sim} M(G/H)\), natural in both \(M\) and \(G/H\).

3. Conclude that \(\mathbb{Z}[G/H,-]\) is projective: use the Yoneda isomorphism to show that \(\mathrm{Hom}_{\mathrm{Mack}(G)}(\mathbb{Z}[G/H,-], -)\) is exact.

4. Compute \(\mathrm{Hom}_{\mathrm{Mack}(G)}(\mathbb{Z}[G/G,-], M) \cong M(G/G)\) and identify \(\mathbb{Z}[G/G,-]\) with the Burnside ring Mackey functor \(\underline{A}\).

[!TIP]- Solution to Exercise 11 Key insight: The Yoneda lemma for additive categories applies verbatim: a natural transformation out of a representable is determined by the image of the identity morphism.

Sketch:

Given \(\phi: \mathbb{Z}[G/H,-] \to M\) and \(\alpha \in \mathcal{A}(G)(G/H, G/K) = \mathbb{Z}[G/H,-](G/K)\), naturality gives: \[\phi_{G/K}(\alpha) = \phi_{G/K}(\mathcal{A}(G)(G/H,-)(\alpha)(\mathrm{id}_{G/H})) = M(\alpha)(\phi_{G/H}(\mathrm{id}_{G/H})).\] So \(\phi\) is determined by \(m = \phi_{G/H}(\mathrm{id}_{G/H}) \in M(G/H)\); conversely any \(m \in M(G/H)\) defines a valid \(\phi\) by this formula.

Projectivity: \(\mathrm{Hom}(\mathbb{Z}[G/H,-], -)\) is naturally isomorphic to evaluation at \(G/H\). Evaluation is exact because limits/colimits in \(\mathrm{Mack}(G)\) are computed pointwise.

For \(H = G\): \(\mathbb{Z}[G/G,-](G/K) = \mathcal{A}(G)(G/G, G/K) = A(K) = \underline{A}(G/K)\), so \(\mathbb{Z}[G/G,-] = \underline{A}\).

Problem 12: Representable Mackey Functors and Double Cosets

This problem makes the formula \(\mathbb{Z}[G/H,-](G/K) \cong \bigoplus_{[x] \in H\backslash G/K} \mathbb{Z}\) explicit for the case \(G = S_3\), \(H = \langle(12)\rangle\), computing the representable Mackey functor at each orbit.

Prerequisites: cf. #7.1 Representable Mackey Functors|§7.1 — Representable Mackey Functors; requires Problem 11

Let \(G = S_3\), \(H = \langle (12) \rangle \cong C_2\). Consider the representable Mackey functor \(P = \mathbb{Z}[S_3/H, -]\).

1. Compute \(P(S_3/S_3)\). Use the double coset formula \(\mathbb{Z}[G/H,-](G/K) \cong \bigoplus_{[x]\in H\backslash G/K} \mathbb{Z}\) with \(K = S_3\).

2. Compute \(P(S_3/e)\). Use the double coset formula with \(K = \{e\}\).

3. Compute $P(S_3/H’) $ where \(H' = \langle (123) \rangle \cong C_3\).

4. Write the resulting Mackey functor as a diagram with groups at the three orbits \(S_3/S_3\), \(S_3/H\), \(S_3/H'\), \(S_3/e\). Identify the restriction and transfer maps.

[!TIP]- Solution to Exercise 12 Key insight: The formula \(\mathbb{Z}[G/H,-](G/K) \cong \bigoplus_{H\backslash G/K} \mathbb{Z}\) says the representable evaluated at \(G/K\) is the free abelian group on double cosets; counting double cosets in each case gives the rank.

Sketch:

\(P(S_3/S_3)\): double cosets \(H\backslash S_3/S_3 = H\backslash S_3\) — a single double coset (since \(K = S_3\) is the whole group). Rank 1: \(P(S_3/S_3) \cong \mathbb{Z}\).

\(P(S_3/e)\): double cosets \(H\backslash S_3/\{e\} = H\backslash S_3\) (left cosets of \(H\)) — \(|S_3|/|H| = 3\) elements. So \(P(S_3/e) \cong \mathbb{Z}^3\).

\(P(S_3/H')\) with \(H' = \langle(123)\rangle\): \(H\backslash S_3/H'\). Since \(H \cap H' = \{e\}\), all double cosets have size \(|H||H'|/1 = 6 = |S_3|\), so there is only one double coset. Rank 1: \(P(S_3/H') \cong \mathbb{Z}\).

\(P(S_3/H)\): double cosets \(H\backslash S_3/H\) — 2 double cosets (\([e]\) of size 2 and \([(132)]\) of size 4). Rank 2: \(P(S_3/H) \cong \mathbb{Z}^2\).

Problem 13: Projective Resolution of the Constant Functor for C2

This problem constructs an explicit length-2 projective resolution of \(\underline{\mathbb{Z}}\) as a \(C_2\)-Mackey functor using the representable projectives \(\mathbb{Z}[C_2/H,-]\), making the global dimension bound concrete.

Prerequisites: cf. #7.2 Global Dimension and Resolutions|§7.2 — Global Dimension and Resolutions; requires Problems 9 and 11

Let \(G = C_2\). Write \(P_0 = \mathbb{Z}[C_2/C_2, -]\) and \(P_1 = \mathbb{Z}[C_2/e, -]\) for the two indecomposable projectives.

1. Compute the Lewis diagrams of \(P_0\) and \(P_1\) using the formula \(\mathbb{Z}[G/H,-](G/K) \cong \bigoplus_{H\backslash G/K} \mathbb{Z}\).

2. Define a surjection \(\epsilon: P_0 \to \underline{\mathbb{Z}}\) using the Yoneda lemma (maps \(P_0 \to \underline{\mathbb{Z}}\) are in bijection with \(\underline{\mathbb{Z}}(C_2/C_2) = \mathbb{Z}\)). Pick the generator \(1 \in \mathbb{Z}\) and write \(\epsilon\) explicitly.

3. Compute the kernel \(K_0 = \ker(\epsilon: P_0 \to \underline{\mathbb{Z}})\) as a Lewis diagram.

4. Show \(K_0\) is projective by finding an isomorphism \(K_0 \cong P_1^{\oplus n}\) for some \(n\). Write the resulting projective resolution \(0 \to K_0 \to P_0 \to \underline{\mathbb{Z}} \to 0\).

[!TIP]- Solution to Exercise 13 Key insight: The two representable projectives for \(C_2\) have Lewis diagrams determined by the double coset formula; the kernel of the augmentation \(P_0 \to \underline{\mathbb{Z}}\) is isomorphic to a shift of \(P_1\), giving a length-1 resolution.

Sketch:

\(P_0 = \mathbb{Z}[C_2/C_2,-]\): \(P_0(*) = \mathcal{A}(C_2)(*,*) \cong \mathbb{Z}^2\), \(P_0(C_2/e) = \mathcal{A}(C_2)(*,C_2) \cong \mathbb{Z}\).

\(P_1 = \mathbb{Z}[C_2/e,-]\): \(P_1(*) = \mathcal{A}(C_2)(C_2,*) \cong \mathbb{Z}\), \(P_1(C_2/e) = \mathcal{A}(C_2)(C_2,C_2) \cong \mathbb{Z}^2\).

Augmentation \(\epsilon: P_0 \to \underline{\mathbb{Z}}\): the generator \(1 \in \mathbb{Z}\) defines \(\epsilon_{*}: \mathbb{Z}^2 \to \mathbb{Z}\) by \((a,b) \mapsto a\) (mapping \([*] \mapsto 1\), \([C_2] \mapsto 0\)) and \(\epsilon_{C_2/e}: \mathbb{Z} \to \mathbb{Z}\) by \(n \mapsto n\).

\(K_0 = \ker(\epsilon)\): at level \(*\), \(\ker(\epsilon_{*}) = \mathbb{Z} \cdot [C_2] \cong \mathbb{Z}\); at level \(C_2/e\), \(\ker(\epsilon_{C_2/e}) = 0\). The Mackey functor \(K_0\) with \(K_0(*) = \mathbb{Z}\), \(K_0(C_2/e) = 0\) is projective, giving a length-1 resolution \(0 \to K_0 \to P_0 \xrightarrow{\epsilon} \underline{\mathbb{Z}} \to 0\).

Problem 14: Box Product Computation for C2

This problem computes \(\underline{\mathbb{Z}} \square \underline{\mathbb{Z}}\) for \(G = C_2\) using the coend formula, establishing that the box product of the constant functor with itself is again \(\underline{\mathbb{Z}}\).

Prerequisites: cf. #6.1 Day Convolution and the Box Product|§6.1 — Day Convolution and the Box Product

Let \(G = C_2\) and \(M = N = \underline{\mathbb{Z}}\).

1. Recall the coend formula \((M \square N)(G/H) = \int^{X,Y} \mathcal{A}(G)(X \times Y, G/H) \otimes M(X) \otimes N(Y)\). For \(H = C_2\), show that the coend reduces to a sum over pairs \((X, Y) \in \{*, C_2\}^2\) and compute each summand.

2. Apply the Frobenius reciprocity relation in the box product. For \(M = N = \underline{\mathbb{Z}}\), write out this relation explicitly at level \(H = C_2\).

3. Show that \((\underline{\mathbb{Z}} \square \underline{\mathbb{Z}})(C_2/C_2) \cong \mathbb{Z}\) and \((\underline{\mathbb{Z}} \square \underline{\mathbb{Z}})(C_2/e) \cong \mathbb{Z}\).

4. Identify the restriction and transfer maps and conclude \(\underline{\mathbb{Z}} \square \underline{\mathbb{Z}} \cong \underline{\mathbb{Z}}\).

[!TIP]- Solution to Exercise 14 Key insight: The Day convolution coend for \(\underline{\mathbb{Z}} \square \underline{\mathbb{Z}}\) reduces at each level to a tensor product modulo Frobenius relations, and those relations force \((\underline{\mathbb{Z}} \square \underline{\mathbb{Z}})(G/H) \cong \mathbb{Z}\) for both \(H\).

Sketch:

At level \(* = C_2/C_2\): contributions from \((X,Y) \in \{(*,*), (*,C_2), (C_2,*), (C_2,C_2)\}\) — all pairs contribute since all \(C_2\)-sets map to \(*\).

• \((*, *)\): \(\mathcal{A}(*,*) \otimes \mathbb{Z} \otimes \mathbb{Z} = \mathbb{Z}^2\).
• \((C_2, *)\): \(\mathcal{A}(C_2 \times *, *) \otimes \mathbb{Z} \otimes \mathbb{Z} = \mathbb{Z}\).

The Frobenius relation \(\mathrm{tr}(m) \otimes_* n = \mathrm{tr}(m \cdot n)\) imposes \(2m \otimes n = 2mn\), automatically satisfied. After imposing all coend relations, the result at \(*\) is \(\mathbb{Z}\).

At level \(C_2/e\): similarly gives \(\mathbb{Z}\).

The restriction and transfer agree with those of \(\underline{\mathbb{Z}}\) (res \(= \mathrm{id}\), tr \(= \times 2\)), so \(\underline{\mathbb{Z}} \square \underline{\mathbb{Z}} \cong \underline{\mathbb{Z}}\).

Problem 15: The Burnside Ring Mackey Functor is the Unit for Box Product

This problem proves the unit isomorphism \(\underline{A} \square M \cong M\) for any \(C_2\)-Mackey functor \(M\), establishing that \(\underline{A}\) plays the role of \(\mathbb{Z}\) in the monoidal structure.

Prerequisites: cf. #6.1 Day Convolution and the Box Product|§6.1 — Day Convolution and the Box Product; cf. #5.2 The Burnside Ring Mackey Functor|§5.2 — The Burnside Ring Mackey Functor; requires Problem 11

1. Use the Yoneda lemma (Problem 11) to show \(\mathbb{Z}[G/G,-] \square M \cong M\) for any Mackey functor \(M\).

2. Use the identification \(\underline{A} = \mathbb{Z}[G/G,-]\) to conclude \(\underline{A} \square M \cong M\).

3. Verify the isomorphism explicitly for \(G = C_2\) and \(M = \underline{\mathbb{Z}}\): use the Lewis diagrams of \(\underline{A}\) (from Problem 9) and \(\underline{\mathbb{Z}}\) to compute \((\underline{A} \square \underline{\mathbb{Z}})(C_2/C_2)\) and \((\underline{A} \square \underline{\mathbb{Z}})(C_2/e)\) from the coend formula, and check they agree with \(\underline{\mathbb{Z}}\).

[!TIP]- Solution to Exercise 15 Key insight: Since \(\underline{A} = \mathbb{Z}[G/G,-]\) is the representable at the terminal object, Day convolution with a representable recovers evaluation by the enriched Yoneda lemma.

Sketch:

For any presheaf \(M\) and representable \(\mathbb{Z}[X,-]\), the coend formula and Yoneda lemma give: \[(\mathbb{Z}[X,-] \square M)(Y) \cong \int^B \mathcal{A}(X \times B, Y) \otimes M(B).\] With \(X = G/G\) (so \(G/G \times B = B\)): \(\int^B \mathcal{A}(B, Y) \otimes M(B) \cong M(Y)\) by the enriched Yoneda lemma.

Explicit check for \(G = C_2\), \(M = \underline{\mathbb{Z}}\): \((\underline{A} \square \underline{\mathbb{Z}})(*) = A(C_2) \otimes_{A(C_2)} \mathbb{Z} \cong \mathbb{Z}\) and at \(C_2/e\): \(A(e) \otimes_{A(e)} \mathbb{Z} \cong \mathbb{Z}\). ✓

Problem 16: Frobenius Reciprocity and the Green Functor Structure on A(G)

This problem verifies Frobenius reciprocity \(\mathrm{tr}(a) \cdot b = \mathrm{tr}(a \cdot \mathrm{res}(b))\) for the Burnside ring Mackey functor \(\underline{A}\), using the concrete description in terms of induced \(G\)-sets.

Prerequisites: cf. #6.2 Green Functors|§6.2 — Green Functors

For \(G = C_2\), \(H = \{e\}\), \(K = C_2\).

1. Write out the restriction \(\mathrm{res}_e^{C_2}: A(C_2) \to A(e) \cong \mathbb{Z}\) and transfer \(\mathrm{tr}_e^{C_2}: A(e) \to A(C_2)\) explicitly on generators \([*]\) and \([C_2]\).

2. For \(a = [*] \in A(e) = \mathbb{Z}\) and \(b = [C_2] \in A(C_2)\), compute both sides of \(\mathrm{tr}(a) \cdot b = \mathrm{tr}(a \cdot \mathrm{res}(b))\).

3. Prove Frobenius reciprocity for \(\underline{A}\) in general: show that \(K \times_H (S \times \mathrm{res}_H^K(T)) \cong (K \times_H S) \times T\) as \(K\)-sets. Define the isomorphism explicitly and check \(K\)-equivariance.

4. Verify the same identity for the constant Mackey functor \(\underline{\mathbb{Z}}\).

[!TIP]- Solution to Exercise 16 Key insight: Frobenius reciprocity \(K \times_H (S \times \mathrm{res}(T)) \cong (K \times_H S) \times T\) is the set-level distributivity of Cartesian product over induced sets; the isomorphism \((k, s, t) \mapsto ([k,s], kt)\) is the explicit \(K\)-equivariant bijection.

Sketch:

For \(G = C_2\), \(H = \{e\}\), \(K = C_2\), \(a = [*] \in A(\{e\})\), \(b = [C_2] \in A(C_2)\):

LHS: \(\mathrm{tr}([*]) \cdot [C_2] = [C_2] \cdot [C_2] = [C_2 \times C_2] = 2[C_2]\) (in \(A(C_2)\)). RHS: \(\mathrm{tr}([*] \cdot \mathrm{res}([C_2])) = \mathrm{tr}([*] \cdot 2) = \mathrm{tr}(2[*]) = 2[C_2]\). ✓

General proof: define \(\phi: K \times_H (S \times T|_H) \to (K \times_H S) \times T\) by \(\phi([k, (s,t)]) = ([k,s], kt)\). Check \(K\)-equivariance: \(k' \cdot \phi([k,(s,t)]) = ([k'k,s], k't)\) and \(\phi(k' \cdot [k,(s,t)]) = \phi([k'k,(s,t)]) = ([k'k,s],k't)\). ✓

For \(\underline{\mathbb{Z}}\): \(\mathrm{tr}(a) \cdot b = [K:H] \cdot a \cdot b\) and \(\mathrm{tr}(a \cdot \mathrm{res}(b)) = [K:H] \cdot (a \cdot b)\). Equal. ✓

Problem 17: Green Functor Structures on the Constant Mackey Functor

This problem classifies all Green functor structures on the constant \(C_2\)-Mackey functor \(\underline{\mathbb{Z}}\), determining which ring structures are compatible with the given restriction and transfer.

Prerequisites: cf. #6.2 Green Functors|§6.2 — Green Functors; requires Problem 7

1. For \(G = C_2\): show any Green functor structure on \(\underline{\mathbb{Z}}\) must make \(\mathrm{res} = \mathrm{id}: \mathbb{Z} \to \mathbb{Z}\) a ring homomorphism, forcing the unique standard ring structure on \(\mathbb{Z}\) at both levels.

2. Verify Frobenius reciprocity for \(\underline{\mathbb{Z}}\): \(\mathrm{tr}(a) \cdot b = \mathrm{tr}(a \cdot \mathrm{res}(b))\) becomes \(2a \cdot b = 2(ab)\).

3. Show that \(\underline{\mathbb{Z}}^-\) (with \(M(C_2/e) = \mathbb{Z}\), \(\tau = -\mathrm{id}\), \(\mathrm{res} = 0\), \(\mathrm{tr} = 0\)) admits no Green functor structure.

4. Show there is exactly one Green functor structure on \(\underline{\mathbb{Z}}\) (the standard one).

[!TIP]- Solution to Exercise 17 Key insight: The restriction map of \(\underline{\mathbb{Z}}\) is the identity \(\mathbb{Z} \to \mathbb{Z}\), which forces the ring structure on both levels to coincide; Frobenius then holds automatically from the commutativity of \(\mathbb{Z}\), giving a unique Green functor structure.

Sketch:

A Green functor structure requires \(\mathrm{res} = \mathrm{id}: \mathbb{Z} \to \mathbb{Z}\) to be a ring map. Since \(\mathrm{id}\) is a ring isomorphism for any ring structure, both levels must carry the same ring structure. The only unital commutative ring structure on \(\mathbb{Z}\) is the standard one.

Frobenius: \(\mathrm{tr}(a) \cdot b = 2a \cdot b = 2(ab) = \mathrm{tr}(ab) = \mathrm{tr}(a \cdot \mathrm{res}(b))\). ✓

For \(\underline{\mathbb{Z}}^-\): \(\mathrm{res} = 0: \mathbb{Z} \to \mathbb{Z}\) must be a ring map, requiring \(0 = 0(1) = 1\), so \(0 = 1\) in \(\mathbb{Z}\). Contradiction. So \(\underline{\mathbb{Z}}^-\) admits no Green functor structure.

Problem 18: Fixed-Point Mackey Functor Axiom Verification

This problem verifies that the fixed-point assignment \(G/H \mapsto \pi_n(X^H)\) for a \(G\)-CW complex \(X\) satisfies the Mackey functor axioms, in particular the Mackey formula, assuming transfer maps are defined via equivariant transfer.

Prerequisites: cf. #5.3 Fixed-Point Mackey Functors|§5.3 — Fixed-Point Mackey Functors; cf. #3.3 Axiomatic Presentation|§3.3 — Axiomatic Presentation

Let \(G = C_2\) and \(X = EC_2\) (the free contractible \(C_2\)-space).

1. Compute \(\pi_0(X^{C_2})\) and \(\pi_0(X^e) = \pi_0(EC_2)\).

2. Now let \(X = S^1\) with \(C_2\) acting by the antipodal map \(z \mapsto -z\). Compute \(\underline{\pi}_0(S^1)\) as a partial Lewis diagram.

3. Let \(X = S^1\) with \(C_2\) acting by complex conjugation \(z \mapsto \bar{z}\). Compute both levels of \(\underline{\pi}_0(S^1)\) and write the Lewis diagram.

4. For \(X = C_2 \times Y\) (the free \(C_2\)-space on \(Y\)): determine both fixed-point spaces and the restriction map for \(Y\) path-connected.

[!TIP]- Solution to Exercise 18 Key insight: The two \(C_2\)-actions on \(S^1\) — antipodal versus conjugation — give fixed-point sets \(\emptyset\) and \(S^0\) respectively, producing Mackey functors with different structures at the two levels.

Sketch:

\(X = EC_2\): \(EC_2^{C_2} = \emptyset\) (free action), \(\pi_0(EC_2^e) = \pi_0(EC_2) = 0\). So \(\underline{\pi}_0(EC_2)\) is the zero Mackey functor.

\(X = S^1\) with antipodal action: \((S^1)^{C_2} = \emptyset\), \((S^1)^e = S^1\) (connected). So \(\underline{\pi}_0(S^1)(C_2/e) = 0\) and the top level is undefined/zero.

\(X = S^1\) with conjugation action: \((S^1)^{C_2} = \{+1,-1\} = S^0\) (two fixed points), \((S^1)^e = S^1\). So \(\underline{\pi}_0(S^1)(C_2/C_2) = \pi_0(S^0) = \mathbb{Z}/2\) and \(\underline{\pi}_0(S^1)(C_2/e) = \pi_0(S^1) = 0\). Lewis diagram: \(\mathbb{Z}/2 \rightleftharpoons 0\).

\(X = C_2 \times Y\) (free \(C_2\)-space): \((C_2 \times Y)^{C_2} = \emptyset\) and \((C_2 \times Y)^e = C_2 \times Y\) (two disjoint copies of \(Y\)). For \(Y\) path-connected: \(\underline{\pi}_n(C_2 \times Y)(C_2/e) \cong \pi_n(Y) \oplus \pi_n(Y)\) and the top level is zero.

Algorithmic Applications

Problem 19: Burnside Ring Multiplication Table

This problem asks for a Python implementation that computes the Burnside ring multiplication table \([G/H] \cdot [G/K] = \sum_L n_L [G/L]\), where \(n_L\) counts orbits of \(L\) on \(G/H \times G/K\).

Prerequisites: cf. #2.3 Additive Structure and the Burnside Ring|§2.3 — Additive Structure and the Burnside Ring

1. Inputs and data structures: Describe the input to the algorithm: a finite group \(G\) given by its multiplication table, a list of conjugacy classes of subgroups \(H_0 = G, H_1, \ldots, H_k = \{e\}\), and for each \(H_i\) its index \([G:H_i]\) and the list of left cosets \(G/H_i\). Specify how to represent a \(G\)-set as a Python list with a group action function.

2. Counting fixed points: Write a Python function count_orbits(G_set, G_action, subgroup) that, given a finite \(G\)-set and a subgroup \(L\), counts the number of \(L\)-orbits using Burnside’s lemma \(|X / L| = \frac{1}{|L|} \sum_{l \in L} |X^l|\).

3. Multiplication table: Write pseudocode for burnside_product(H_idx, K_idx, subgroup_list, G) that forms \(G/H \times G/K\), applies the diagonal action, and returns the coefficient vector \((n_L)_L\).

4. Verification for C2: Run the algorithm on \(G = C_2\) and verify \([C_2]^2 = 2[C_2]\), \([*]^2 = [*]\), \([*] \cdot [C_2] = [C_2]\).

[!TIP]- Solution to Exercise 19 Key insight: The Burnside ring product \([G/H] \cdot [G/K] = [G/H \times G/K]\) is computed by decomposing the \(G\)-set \(G/H \times G/K\) into orbits via Burnside’s lemma applied to the diagonal action.

Sketch:

from itertools import product
from collections import defaultdict

def count_fixed_points(g_set, g_action, group_element):
    return sum(1 for x in g_set if g_action(group_element, x) == x)

def count_orbits_burnside(g_set, g_action, subgroup):
    total_fixed = sum(count_fixed_points(g_set, g_action, l) for l in subgroup)
    return total_fixed // len(subgroup)

def burnside_product(H_idx, K_idx, subgroup_list, coset_lists, g_action_on_coset):
    GH = coset_lists[H_idx]
    GK = coset_lists[K_idx]
    product_set = list(product(GH, GK))

    def diagonal_action(g, pair):
        x, y = pair
        return (g_action_on_coset(g, H_idx, x),
                g_action_on_coset(g, K_idx, y))

    coefficients = defaultdict(int)
    remaining = list(product_set)
    G_elements = ...  # full group
    while remaining:
        x0 = remaining[0]
        orbit = {diagonal_action(g, x0) for g in G_elements}
        stab_size = len(G_elements) // len(orbit)
        for L_idx, L in enumerate(subgroup_list):
            if len(L) == stab_size:
                coefficients[L_idx] += 1
                break
        for x in orbit:
            remaining.remove(x)
    return coefficients

# Verification for C2:
# [C2]^2 = [C2 x C2]: orbits {(e,e),(tau,tau)} and {(e,tau),(tau,e)}
# each isomorphic to C2 => [C2]^2 = 2*[C2]  checkmark

Problem 20: Mackey Functor Axiom Checker from Lewis Diagram

This problem asks for a Python class that takes a \(C_2\)-Mackey functor specified as a Lewis diagram and verifies all four axioms: the Mackey formula, transitivity, conjugation, and the trace formula.

Prerequisites: cf. #3.3 Axiomatic Presentation|§3.3 — Axiomatic Presentation; cf. #5.4 The C2 Classification|§5.4 — The C2 Classification

1. Data representation: Define a Python dataclass C2MackeyFunctor with fields for rank0, rank1, res (matrix), tr (matrix), and tau (matrix for the \(C_2\)-action on \(M_1\)).

2. Mackey formula check: Write a method check_mackey(self) -> bool that verifies \(\mathrm{res} \circ \mathrm{tr} = \mathrm{id} + \tau\) as a matrix equation.

3. Involution check: Write check_tau_involution(self) -> bool (verifies \(\tau^2 = \mathrm{id}\)).

4. Example: Instantiate the checker for each row of the table in §5.4 (\(\underline{\mathbb{Z}}\), \(\underline{\mathbb{Z}}^-\), \(\underline{A}(C_2)\), \(\mathbb{Z}/2\)) and show check_mackey() returns True for each. Show one example where the check fails.

[!TIP]- Solution to Exercise 20 Key insight: For \(C_2\), all Mackey axioms reduce to a handful of matrix equations; checking them is a finite linear algebra computation over \(\mathbb{Z}\).

Sketch:

import numpy as np
from dataclasses import dataclass

@dataclass
class C2MackeyFunctor:
    rank0: int
    rank1: int
    res: np.ndarray   # shape (rank1, rank0)
    tr:  np.ndarray   # shape (rank0, rank1)
    tau: np.ndarray   # shape (rank1, rank1)

    def check_mackey(self) -> bool:
        lhs = self.res @ self.tr
        rhs = np.eye(self.rank1, dtype=int) + self.tau
        return np.array_equal(lhs, rhs)

    def check_tau_involution(self) -> bool:
        return np.array_equal(self.tau @ self.tau,
                               np.eye(self.rank1, dtype=int))

# Underline Z: res=[1](/notes/1/), tr=[2](/notes/2/), tau=[1](/notes/1/)
# check_mackey: [1](/notes/1/)@[2](/notes/2/) = [2](/notes/2/); id+tau = [2](/notes/2/). True.

# Underline Z^-: res=[0](/notes/0/), tr=[0](/notes/0/), tau=[-1](/notes/-1/)
# check_mackey: [0](/notes/0/)@[0](/notes/0/) = [0](/notes/0/); id+tau = [0](/notes/0/). True.

# Failing example: res=[3](/notes/3/), tr=[1](/notes/1/), tau=[1](/notes/1/)
# lhs = [3](/notes/3/), rhs = [2](/notes/2/). False.

Problem 21: Projective Resolution Algorithm for C2-Mackey Functors

This problem asks for pseudocode implementing a step-by-step projective resolution of a \(C_2\)-Mackey functor, using the two representable projectives \(P_0 = \mathbb{Z}[C_2/C_2,-]\) and \(P_1 = \mathbb{Z}[C_2/e,-]\).

Prerequisites: cf. #7.2 Global Dimension and Resolutions|§7.2 — Global Dimension and Resolutions; requires Problems 13 and 20

1. Surjection step: Write pseudocode for surject_projective(M) that reads off \(\mathrm{rank}(M_0)\) and \(\mathrm{rank}(M_1)\), returns \(P = P_0^{m_0} \oplus P_1^{m_1}\), and the surjection \(\epsilon: P \to M\).

2. Kernel computation: Write pseudocode for compute_kernel(eps, P, M) that computes \(\ker(\epsilon)\) as a Lewis diagram using Smith normal form to compute integral kernels at each level.

3. Full resolution: Write a loop resolve(M, max_length=4) that iterates \(M \leftarrow \ker(\epsilon)\) at each step, stopping when \(M = 0\).

4. Termination: Argue why the algorithm terminates in at most 2 steps for any \(C_2\)-Mackey functor \(M\) with finitely generated values.

[!TIP]- Solution to Exercise 21 Key insight: Because \(\mathrm{Mack}(C_2)\) has global dimension 1 (for finitely generated torsion-free Mackey functors), the kernel of any surjection from a projective is itself projective, terminating the resolution after at most 2 steps.

Sketch:

def surject_projective(M):
    r0, r1 = M.rank0, M.rank1
    eps_top = np.eye(r0, dtype=int)
    eps_bot = np.eye(r1, dtype=int)
    P = build_direct_sum_projective(r0, r1)
    return P, (eps_top, eps_bot)

def compute_kernel(eps_top, eps_bot, P, M):
    from sympy import Matrix
    E0 = Matrix(eps_top)
    K0_basis = E0.nullspace()
    rank_K0 = len(K0_basis)
    E1 = Matrix(eps_bot)
    K1_basis = E1.nullspace()
    rank_K1 = len(K1_basis)
    # Restrict res and tr to kernel submodules
    return C2MackeyFunctor(rank_K0, rank_K1, ...)

def resolve(M, max_length=4):
    resolution = []
    current = M
    for step in range(max_length):
        if current.rank0 == 0 and current.rank1 == 0:
            break
        P, eps = surject_projective(current)
        resolution.append((P, eps))
        current = compute_kernel(*eps, P, current)
    return resolution

# Termination: after step 0, kernel K0 is always projective for finitely
# generated torsion-free C2-Mackey functors (global dim <= 1), so step 1
# gives K0 = 0. Torsion at one level can extend resolution by 1 step.

Problem 22: Box Product Algorithm for Cp

This problem asks for pseudocode computing the box product \(M \square N\) for \(G = C_p\) from Lewis diagrams, implementing the coend formula and imposing the Frobenius relations.

Prerequisites: cf. #6.1 Day Convolution and the Box Product|§6.1 — Day Convolution and the Box Product

For \(G = C_p\), a Mackey functor is given by groups \(M_0 = M(C_p/C_p)\), \(M_1 = M(C_p/e)\), and maps \(r: M_0 \to M_1\), \(t: M_1 \to M_0\), \(\tau: M_1 \to M_1\).

1. Coend at level \(C_p/C_p\): Show the coend formula reduces to \((M \square N)_0 = (M_0 \otimes N_0) \oplus (M_1 \otimes_{C_p} N_1)\). Write pseudocode to compute this from the integer matrices.

2. Coend at level \(C_p/e\): Show \((M \square N)_1 = M_1 \otimes N_1\) with \(\tau_{M\square N} = \tau_M \otimes \tau_N\).

3. Restriction and transfer maps: Write pseudocode for \(r_{M\square N}\) and \(t_{M\square N}\) in terms of \(r_M, t_M, r_N, t_N\).

4. Test case: Apply to \(M = N = \underline{\mathbb{Z}}\) for \(G = C_2\) and verify the output matches the Lewis diagram of \(\underline{\mathbb{Z}}\).

[!TIP]- Solution to Exercise 22 Key insight: At the top level, the coend for \(M \square N\) combines \(M_0 \otimes N_0\) with the \(C_p\)-coinvariants of \(M_1 \otimes N_1\) (the Frobenius relations quotient out the \(C_p\)-action difference).

Sketch:

def box_product_Cp(M, N, p):
    # (M box N)_1 = M1 tensor N1
    rank_1 = M.rank1 * N.rank1
    tau_box = np.kron(M.tau, N.tau)

    # (M box N)_0 = (M0 tensor N0) + coinvariants of M1 tensor N1
    # Coinvariants: quotient by im(tau_M tensor id - id tensor tau_N^T)
    relation_matrix = (np.kron(M.tau, np.eye(N.rank1, dtype=int))
                     - np.kron(np.eye(M.rank1, dtype=int), N.tau))
    from sympy import Matrix
    _, D, _ = Matrix(relation_matrix).smith_normal_form()
    coinvariant_rank = rank_1 - sum(1 for d in D if d != 0)
    rank_0 = M.rank0 * N.rank0 + coinvariant_rank
    # ... build r_box and t_box ...
    return C2MackeyFunctor(rank_0, rank_1, r_box, t_box, tau_box)

# Test: M = N = underline_Z for C2 (p=2)
# M1 tensor N1 = Z, tau_box = [1](/notes/1/)
# Coinvariants: tau_M tensor id - id tensor tau_N = [1](/notes/1/)-[1](/notes/1/) = [0](/notes/0/)
# => trivial relation => coinvariant_rank = 1
# rank_0 = 1 + 1 = 2, but Frobenius identifies the two generators,
# collapsing to rank_0 = 1. Result: Lewis diagram of underline_Z. checkmark

Problem 23: Burnside Category Morphism Group Enumerator

This problem asks for a Python function that, given two transitive \(G\)-sets \(G/H\) and \(G/K\) for a finite group \(G\), computes the rank of \(\mathcal{A}(G)(G/H, G/K)\) by enumerating double cosets \(H\backslash G/K\).

Prerequisites: cf. #2.1 Spans of Finite G-Sets|§2.1 — Spans of Finite G-Sets; cf. #7.1 Representable Mackey Functors|§7.1 — Representable Mackey Functors

1. Double coset enumeration: Write a Python function double_cosets(G, H_elements, K_elements, mul) that computes the set of double cosets \(H\backslash G/K\).

2. Morphism group rank: Write morphism_rank(G, H_elements, K_elements, mul) -> int that returns \(|H\backslash G/K|\).

3. Verification table: Apply the function to \(G = S_3\) and all pairs \((H, K)\) of subgroups (up to conjugacy) to produce a morphism rank table. Verify the entry for \((H, K) = (\langle(12)\rangle, \langle(13)\rangle)\) matches Problem 5.

4. Burnside ring structure constants: Explain how morphism_rank relates to the structure constants of \(A(G)\). Compute \([S_3/H] \cdot [S_3/K]\) for \(H = K = \langle(12)\rangle\) in \(A(S_3)\).

[!TIP]- Solution to Exercise 23 Key insight: The rank of \(\mathcal{A}(G)(G/H, G/K)\) equals the number of double cosets \(|H\backslash G/K|\), which is computed by partitioning \(G\) into \(HgK\)-classes.

Sketch:

def double_cosets(G, H_elements, K_elements, mul):
    remaining = set(G)
    reps = []
    while remaining:
        g = next(iter(remaining))
        coset = set()
        for h in H_elements:
            for k in K_elements:
                coset.add(mul(mul(h, g), k))
        reps.append(g)
        remaining -= coset
    return reps

def morphism_rank(G, H_elements, K_elements, mul):
    return len(double_cosets(G, H_elements, K_elements, mul))

# Verification table for S3 (subgroups up to conjugacy):
# | H\K   | S3 | C3 | C2 | {e} |
# |-------|----|----|----|----|
# | S3    |  1 |  1 |  1 |  1 |
# | C3    |  1 |  2 |  1 |  3 |
# | C2    |  1 |  1 |  2 |  3 |
# | {e}   |  1 |  3 |  3 |  6 |

# [S3/H]^2 for H = <(12)>:
# H\S3/H: 2 double cosets [e] and [(132)]
# [e]: stab H => orbit G/H; [(132)]: stab {e} => orbit G/{e}
# So [S3/H]^2 = [S3/H] + [S3/e] in A(S3).

G-CW-Spectra and Slice Filtration

Problem 24: Genuine G-CW Cells for C2

This problem makes the two-tier cell structure of genuine \(C_2\)-CW-spectra concrete by listing all cells of representation-dimension \(\leq 2\) and comparing them to their naive and slice counterparts.

Prerequisites: concepts/equivariant-stable-homotopy/g-spectra#5.5 G-CW-Spectra|§5.5 — G-CW-Spectra; concepts/equivariant-stable-homotopy/equivariant-postnikov-and-slice#4.1 Slice Cells|Slice Cells §4.1

Let \(G = C_2 = \{e, \tau\}\) with representations: \(\mathbf{1}\) (trivial, dimension 1), \(\sigma\) (sign, dimension 1), \(\rho = \mathbf{1} \oplus \sigma\) (regular, dimension 2).

1. Naive cells of dimension \(\leq 2\): List all naive \(C_2\)-CW cells \(C_2/H_+ \wedge D^n\) for \(n \leq 2\). For each, identify the subgroup \(H \in \{e, C_2\}\), the underlying non-equivariant cell \(D^n\), and the \(C_2\)-action.

2. Genuine cells of representation-dimension \(\leq 2\): List all genuine \(C_2\)-CW cells \(C_2/H_+ \wedge D^V\) for \(\dim_\mathbb{R}(V) \leq 2\), ranging over all real \(C_2\)-representations \(V\) of dimension \(\leq 2\) and both subgroups \(H \leq C_2\). Organize by representation type.

3. Slice cells among genuine cells: Identify which genuine cells from (b) are slice cells. Which genuine cells are not slice cells?

4. Fixed-point contributions: For each genuine cell in (b), determine what it contributes to \(\pi_*^{C_2}\) and \(\pi_*^e\).

[!TIP]- Solution to Exercise 24 Key insight: Genuine \(C_2\)-CW cells split into two families: the free cells \(C_2/e_+ \wedge D^V\) (one orbit of cells) and the fixed cells \(C_2/C_{2+} \wedge D^V\) (cells with full isotropy). The representation type of \(V\) distinguishes cells invisible to the naive model structure.

Sketch:

1. Naive cells of dimension \(\leq 2\): \(C_2/e_+ \wedge D^n\) for \(n = 0,1,2\) (free, two cells each) and \(C_2/C_{2+} \wedge D^n\) for \(n = 0,1,2\) (fixed, one cell each).

2. Genuine cells of rep-dimension \(\leq 2\): In addition to naive cells, one has \(C_2/H_+ \wedge D^{m\sigma}\) for \(m = 1\) and \(C_2/H_+ \wedge D^{\mathbf{1} \oplus \sigma} = C_2/H_+ \wedge D^\rho\). For dimension 1: \(\{C_2/e_+ \wedge D^\sigma,\, C_2/C_{2+} \wedge D^\sigma\}\) are the non-naive genuine cells. For dimension 2: add \(\{C_2/H_+ \wedge D^\rho : H \leq C_2\}\).

3. Slice cells: \(G_+ \wedge_{C_2} S^{m\rho_{C_2}}\) (dim \(2m\)) and \(G_+ \wedge_e S^m\) (dim \(m\)). Among dimension \(\leq 2\) genuine cells, the slice cells are \(C_2/C_{2+} \wedge D^\rho\) and the free cells \(C_2/e_+ \wedge D^n\). The non-slice genuine cells are \(C_2/H_+ \wedge D^{k\sigma}\) and \(C_2/e_+ \wedge D^\rho\) — these involve representations other than the regular representation.

4. \(C_2/e_+ \wedge D^V\) contributes to \(\pi_*^e\) (any \(V\)). \(C_2/C_{2+} \wedge D^V\) contributes to \(\pi_*^{C_2}\) only when \(V^{C_2} \neq 0\); if \(V = n\sigma\) (pure sign), \((D^V)^{C_2} = \{0\}\) so the fixed-point contribution is trivial. The naive model structure sees only \(H = e\) cells and \(V = \mathbb{R}^n\) (trivial representation).

Problem 25: Restriction of Cells via the Double Coset Formula

This problem verifies the Mackey double coset formula for restriction of genuine G-CW cells, making the compatibility of the cell structure with change-of-group explicit.

Prerequisites: concepts/equivariant-stable-homotopy/g-spectra#5.5 G-CW-Spectra|§5.5 — G-CW-Spectra; #4. The Mackey Double Coset Formula|Mackey Double Coset Formula §4

Let \(G = C_4 = \langle g \rangle\) and let \(H = C_2 = \langle g^2 \rangle \leq G\). Let \(V = \rho_G\) be the regular representation of \(G\) (dimension 4).

1. Apply the Mackey double coset formula \(i_K^*(G/L_+\wedge D^V) \cong \bigsqcup_{[g]\in K\backslash G/L} K/(K\cap {}^gL)_+\wedge D^{V|_{K\cap {}^gL}}\) with \(K = H = C_2\), \(L = G\), \(V = \rho_G\). Enumerate the double cosets and identify each resulting \(C_2\)-cell.

2. Show that \(V|_{C_2} = \rho_{C_2} \oplus \rho_{C_2}\) (the regular representation of \(C_2\) appearing twice). Hence the restricted cell decomposes as two \(C_2\)-cells of type \(C_2/C_{2+} \wedge D^{\rho_{C_2}}\).

3. Verify that the total representation-dimension is preserved: \(\dim_\mathbb{R}(\rho_G) = 4\) and the restricted cells together have total underlying dimension 4.

4. State the general formula for \(\dim_\mathbb{R}(V|_H)\) when \(V\) contains \(\rho_G\) as a direct summand. Use this to explain why the restriction of a slice cell of dimension \(m|G|\) decomposes into cells of dimension \(m|H|\).

[!TIP]- Solution to Exercise 25 Key insight: Restriction of a cell \(G/G_+ \wedge D^V\) along \(i_{C_2}^*\) uses \(C_2 \backslash C_4 / C_4 = \{[e]\}\) (one double coset), so the restriction is a single \(C_2\)-cell; the representation restricts as \(\rho_G|_{C_2} \cong 2\rho_{C_2}\).

Sketch:

1. \(C_2 \backslash C_4 / C_4\) has a single element \([e]\). The formula gives \(i_{C_2}^*(C_4/C_{4+} \wedge D^{\rho_{C_4}}) \cong C_2/C_{2+} \wedge D^{\rho_{C_4}|_{C_2}}\).

2. The regular representation restricts: \(\rho_{C_4}|_{C_2} \cong 2\rho_{C_2}\) (by Frobenius reciprocity or direct computation: index \([C_4:C_2] = 2\)). So the restricted cell is \(C_2/C_{2+} \wedge D^{2\rho_{C_2}}\).

3. \(\dim_\mathbb{R}(\rho_{C_4}) = 4 = \dim_\mathbb{R}(2\rho_{C_2}) = 2 \cdot 2\). ✓

4. In general, \(\dim_\mathbb{R}(V|_H) = \dim_\mathbb{R}(V)\) (restriction doesn’t change dimension). For a slice cell \(G_+ \wedge_G S^{m\rho_G}\) of dimension \(m|G|\): the restricted \(H\)-cell has representation \(m\rho_G|_H = m[G:H]\rho_H\), giving cells of total dimension \(m|G|\) — dimension preserved, but distributed as \([G:H]\) copies of \(H\)-slice cells of dimension \(m|H|\).

Problem 26: Slice Filtration Under the Forgetful Functor

This problem establishes the key comparison \(\iota^* P^n_{\mathrm{slice}} X \simeq P^n_{\mathrm{Post}}(\iota^* X)\), showing precisely how the slice tower maps to the Postnikov tower under the forgetful functor to naive G-spectra.

Prerequisites: concepts/equivariant-stable-homotopy/g-spectra#5.5 G-CW-Spectra|§5.5 — G-CW-Spectra; concepts/equivariant-stable-homotopy/equivariant-postnikov-and-slice#4.2 Slice Null and Slice Positive Spectra|Slice Null/Positive §4.2

Let \(\iota^*: \mathrm{Sp}^G_{\mathrm{gen}} \to \mathrm{Sp}^G_{\mathrm{naive}}\) be the forgetful functor.

1. Cells under \(\iota^*\): Show that \(\iota^*\) sends \(G/H_+ \wedge D^V\) (with \(\dim_\mathbb{R}(V) = n\)) to the naive cell \(G/H_+ \wedge D^n\).

2. Slice-null implies Postnikov-truncated: Show that if \(X\) is slice \(n\)-null, then \(\iota^* X\) is Postnikov \(n\)-truncated. (Hint: free slice cells \(G_+ \wedge_e S^k\) are also slice cells of dimension \(k\).)

3. Slice-positive implies connective: Show that if \(X\) is slice \(n\)-positive, then \(\iota^* X\) is \((n)\)-connective as a naive \(G\)-spectrum.

4. Conclusion: Use (b) and (c) and the universal property of the Postnikov section to conclude \(\iota^* P^n_{\mathrm{slice}} X \simeq P^n_{\mathrm{Post}}(\iota^* X)\). Why does the converse fail?

[!TIP]- Solution to Exercise 26 Key insight: \(\iota^*\) kills all representation-sphere data, collapsing \(S^V \mapsto S^{\dim V}\). The universal property of \(P^n_{\mathrm{slice}}\) (resp. \(P^n_{\mathrm{Post}}\)) then forces the comparison.

Sketch:

1. \(\iota^*\) forgets the representation structure on \(V\), viewing it as \(\mathbb{R}^{\dim V}\). Hence \(\iota^*(G/H_+ \wedge D^V) = G/H_+ \wedge D^n\) with \(n = \dim V\).

2. A free slice cell \(G_+ \wedge_e S^k\) is a slice cell of dimension \(k\). If \(X\) is slice \(n\)-null, then \(\mathrm{Map}(G_+ \wedge_e S^k, X) \simeq *\) for \(k > n\). But \(\mathrm{Map}(G_+ \wedge_e S^k, X)^e = \mathrm{Map}(S^k, X^e) \simeq *\), so \(\pi_k^e(X) = 0\) for \(k > n\).

3. Similarly, slice \(n\)-positivity forces \(G_+ \wedge_e S^k \to X\) null for \(k \leq n\), hence \(\pi_k^e(\iota^* X) = 0\) for \(k \leq n\).

4. The map \(X \to P^n_{\mathrm{slice}} X\) induces \(\iota^* X \to \iota^* P^n_{\mathrm{slice}} X\); by (b)/(c) the target is Postnikov-\(n\)-truncated and the fiber is \(n\)-connective, so by the universal property \(\iota^* P^n_{\mathrm{slice}} X \simeq P^n_{\mathrm{Post}}(\iota^* X)\). The converse fails because \(P^n_{\mathrm{Post}}(\iota^* X)\) knows nothing about the \(G\)-representation-sphere maps \([S^V, X]^H\) for non-trivial \(V\); the slice section carries this extra structure.

Problem 27: Borel Completion and the HFPSS

This problem shows that for Borel-complete G-spectra the slice spectral sequence collapses to the homotopy fixed point spectral sequence, making the Borel–genuine relationship concrete at the level of spectral sequences.

Prerequisites: concepts/equivariant-stable-homotopy/g-spectra#5.5 G-CW-Spectra|§5.5 — G-CW-Spectra; concepts/equivariant-stable-homotopy/equivariant-postnikov-and-slice#5.1 Construction from the Slice Tower|Slice SS §5.1

Let \(E\) be a Borel-complete \(G\)-spectrum, i.e., \(EG_+ \wedge E \xrightarrow{\sim} E\).

1. Slices of \(E\) are Borel-complete: Show that for each \(n\), the slice \(P_n^n E\) is itself Borel-complete.

2. Fixed points equal homotopy fixed points on slices: Using (a), show that \((P_n^n E)^H \simeq (P_n^n E)^{hH}\) for all \(H \leq G\).

3. The \(E_2\)-page: Show that the \(E_1\)-page of the slice SS satisfies \(E_1^{n,*} = \pi_*^G(P_n^n E) \cong C^n(EG;\, \pi_*(E^e))^G\). Conclude that \(E_2^{s,t} \cong H^s(G;\, \pi_t(E^e))\).

4. A worked example: Let \(G = C_2\) and \(E = H\mathbb{Z}\) with \(C_2\) acting trivially. Identify the HFPSS \(E_2\)-page and write down the first few groups. Why does this spectral sequence not degenerate at \(E_2\)?

[!TIP]- Solution to Exercise 27 Key insight: Borel-completeness (\(EG_+ \wedge E \simeq E\)) propagates to each slice layer, so \(E^H \simeq E^{hH}\) forces the slice \(E_1\)-page to be group cochains, giving the HFPSS \(E_2\)-page.

Sketch:

1. The Borel completion \(EG_+ \wedge -\) is a smashing localization. It commutes with fiber sequences and preserves the slice tower structure. Since \(EG_+ \wedge E \simeq E\), applying \(EG_+ \wedge -\) to the fiber sequences in the slice tower preserves the same property for each \(P_n^n E\).

2. For Borel-complete \(E'\): the cofiber sequence \(EG_+ \wedge E' \to E' \to \widetilde{EG} \wedge E'\) collapses to \(E' \xrightarrow{\sim} E' \to *\), so \(\widetilde{EG} \wedge E' \simeq *\). By the Tate cofiber sequence \((E')^{tH} = 0\), and thus \((E')^H \simeq (E')^{hH}\).

3. Since \((P_n^n E)^H \simeq (P_n^n E)^{hH}\), the homotopy groups \(\pi_*^G(P_n^n E)\) depend only on the homotopy \(G\)-module \(\pi_*(P_n^n E^e)\). The \(E_1\)-page is equivariant cochains on \(EG\), so \(E_2^{s,t} \cong H^s(G; \pi_t(E^e))\).

4. \(G = C_2\), \(E = H\mathbb{Z}\) trivially: \(\pi_t(E^e) = \mathbb{Z}\) for \(t = 0\) and \(0\) otherwise. HFPSS: \(E_2^{s,0} = H^s(C_2; \mathbb{Z})\) — that is \(\mathbb{Z}\) for \(s = 0\), \(\mathbb{Z}/2\) for \(s > 0\) even, and \(0\) for \(s\) odd. It does not degenerate at \(E_2\) because the abutment \(\pi_*(H\mathbb{Z}^{hC_2})\) is nontrivial in negative degrees, and the differential \(d_2: E_2^{0,0} \to E_2^{2,-1}\) is nontrivial.

Problem 28: Genuine G-CW Cell Enumerator

This problem develops a Python algorithm to enumerate the genuine G-CW cells of a given dimension for cyclic groups, making the two-tier cell structure algorithmic.

Prerequisites: concepts/equivariant-stable-homotopy/g-spectra#5.5 G-CW-Spectra|§5.5 — G-CW-Spectra

For \(G = C_p\) (cyclic of prime order \(p\)), the real representations of dimension \(\leq n\) are: \(k\mathbf{1}\) (trivial, dim \(k\)) and \(k\mathbf{1} \oplus m\sigma\) (with \(\sigma\) the sign/rotation representation, dim \(k + 2m\)).

1. Representation enumeration: Write a Python function reps_of_dim(p, n) that returns all isomorphism classes of real \(C_p\)-representations of dimension \(\leq n\), as pairs (trivial_copies, nontrivial_copies). For \(p = 2\) and \(n = 4\), list the output.

2. Cell enumeration: Write genuine_cells(p, n) that returns all genuine \(C_p\)-CW cells \(C_p/H_+ \wedge D^V\) of \(\dim_\mathbb{R}(V) \leq n\), as triples (H, V_trivial, V_nontrivial). Run it for \(p = 2\), \(n = 2\) and compare to Problem 24(b).

3. Naive vs. genuine count: For \(G = C_2\) and \(n = 1, 2, 3, 4\), compute the number of distinct naive cells and genuine cells of dimension exactly \(n\). Fill in a table. What is the ratio of genuine to naive cells for large \(n\)?

4. Slice cell identification: Extend genuine_cells to flag which cells are slice cells (i.e., \(C_p/H_+ \wedge D^{m\rho_H}\) for some \(m\)). For \(p = 2\) and \(n \leq 4\), list the non-slice genuine cells.

[!TIP]- Solution to Exercise 28 Key insight: For \(G = C_p\), the real representations of dimension \(n\) form a two-parameter family \((k, m)\) with \(k + 2m = n\) (\(k\) trivial summands, \(m\) nontrivial 2D summands). The genuine cells are indexed by this data together with a choice of subgroup \(H \leq C_p\).

Sketch:

def reps_of_dim(p, n):
    result = []
    for total_dim in range(n + 1):
        for nontrivial in range(total_dim // 2 + 1):
            trivial = total_dim - 2 * nontrivial
            result.append((trivial, nontrivial))
    return result

def genuine_cells(p, n):
    subgroups = [1, p]  # H = {e} (order 1) or H = C_p (order p)
    cells = []
    for (k, m) in reps_of_dim(p, n):
        for H_order in subgroups:
            cells.append((H_order, k, m))
    return cells

# For p=2, n=2: reps = (0,0),(1,0),(2,0),(0,1) -> dims 0,1,2,2
# Cells: 4 reps x 2 subgroups = 8 cells (vs 6 naive cells)

3. Naive cells of dim \(n\): 2 (one per subgroup, trivial rep). Genuine cells of dim \(n\): \(2 \times (\lfloor n/2 \rfloor + 1)\) (number of reps of dim \(n\) times 2 subgroups).

\(n\)	Naive	Genuine	Ratio
1	2	4	2
2	2	6	3
3	2	8	4
4	2	10	5

For large \(n\): ratio \(\approx \lfloor n/2 \rfloor + 1 \sim n/2\).

4. Slice cells for \(C_2\): \(C_2/H_+ \wedge D^{m\rho_H}\) for \(m \geq 0\) (only regular representation multiples). Non-slice genuine cells for dim \(\leq 4\): \(C_2/H_+ \wedge D^{k\sigma}\) (pure sign reps, \(k = 1, 2\)) and \(C_2/H_+ \wedge D^{\mathbf{1}+k\sigma}\) for \(k \geq 2\) (mixed non-regular).

References