THE NORMALISER DECOMPOSITION FOR p-LOCAL FINITE GROUPS

ASSAF LIBMAN

Abstract. We construct an analogue of the normaliser decomposition for plocal finite groups (S, F, L) with respect to collections of F-centric subgroups and collections of elementary abelian subgroups of S. This enables us to describe the classifying space of a p-local finite group, before p-completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms.

1. The main results

For finite groups Dwyer defined in [5] three types of homology decompositions of classifying spaces of finite groups known as the “subgroup”, “centraliser” and “normaliser” decompositions. These decompositions are functors F : D → Spaces, where D is a small category which is constructed using collections H of carefully chosen subgroups of G. The essential property of these functors is, that given a finite group G, the spaces F (d) have the homotopy type of classifying spaces of subgroups of G. Moreover the category D is constructed using information about the conjugation in G of the subgroups in H. We say that D depends on the fusion of the collection H of G.

The purpose of this note is to construct an analogue of the normaliser decomposition for p-local finite groups in certain important cases. Throughout this note we will freely use the terminology and notation that by now has become standard in the theory for p-local finite groups. The reader who is not familiar with the jargon is advised to read §2 prior to this section, and is also referred to [4] where p-local finite groups were initially defined.

It should be noted that the analogues of the “subgroup” and the “centraliser” decompositions for p-local finite groups was already known to Broto, Levi and Oliver [4, §2].

The normaliser decomposition which is introduced in this note enabled the author together with Antonio Viruel to analyze the nerve |L| of p-local finite groups (S, F, L) with small Sylow subgroups S. We prove that these are classifying spaces of, generally infinite, discrete groups [10]. The author also used normaliser decompositions to give an analysis of the spectra associated with the nerve, |L|, of the linking systems due to Ruiz and Viruel in [16] and other “exotic” examples, see [9]. These results will appear separately as they involve techniques that have little to do with the actual construction of the normaliser decomposition.

Date: March 17, 2006.

Key words and phrases. Homology decompositions, p-local finite groups.

The author was supported by the Nuffield Foundation grant NAL/00735/G.

Part of this work was carried out in and supported by Institute Mittag-Leffler.

We now describe the main results of this paper. Throughout we work simplicially, thus a space means a simplicial set. The category of simplicial sets is denoted by Spaces. The nerve of a small category D is denoted Nr(D) or |D|. We obtain a functor | − | : Cat → Spaces where Cat is the category of small categories. A more detailed discussion can be found in §3

1.1. Definition. Let (S, F, L) be a p-local finite group. A collection is a set C of subgroups of S which is closed under conjugacy in F. That is if P ≤ S belongs to C then so do all the F-conjugates of P. A collection C is called F-centric if it consists of F-centric subgroups of S.

1.2. Definition. A k-simplex in a collection C is a sequence P of proper inclusions P_{0} < P_{1} < … < P_{k} of elements of C. Two k-simplices P and P′ are called conjugate if there exists an isomorphism f ∈ Iso_{F}(P_{k}, P′_{k}) such that f(P_{i}) = P′_{i} for all i = 0, …, k. The conjugacy class of P is denoted [P].

1.3. Definition. The category s̄dC is a poset whose objects are the conjugacy classes [P] of all the k-simplices in C where k = 0, 1, 2, . . .. A morphism [P] → [P′] in s̄dC exists if P′ is conjugate to a subsimplex of P.

Recall from 2.8 that in every p-local finite group it is possible to choose morphisms ι_{P}^{Q} in the linking system L which are lifts of inclusions P ≤ Q of F-centric subgroups. The choice can be made in such a way that ι_{Q}^{R} ∘ ι_{P}^{Q} = ι_{P}^{R} for inclusions P ≤ Q ≤ R.

1.4. Definition. Let C be a centric collection in (S, F, L) and let P be a k-simplex in C. Define Aut_{L}(P) as the subgroup of ∏_{i=0}^{k} Aut_{L}(P_{i}) whose elements are the (k + 1)-tuples (φ_{i})_{i=0}^{k} which render the following ladder commutative in L

P_{0} ⟶^{ι_{P_{0}}^{P_{1}}} P_{1} ⟶^{ι_{P_{1}}^{P_{2}}} ⋯ ⟶^{ι_{P_{k−1}}^{P_{k}}} P_{k}
φ_{0}↓  ^{φ_{1}}↓  ↓^{φ_{k}}
P_{0} ⟶_{ι_{P_{0}}^{P_{1}}} P_{1} ⟶_{ι_{P_{1}}^{P_{2}}} ⋯ ⟶_{ι_{P_{k−1}}^{P_{k}}} P_{k}

1.5. Proposition. The assignment (φ_{i})_{i=0}^{k} ↦ φ_{0} canonically identifies Aut_{L}(P) as a subgroup of Aut_{L}(P_{0}). More generally, if P′ is a subsimplex of P in C then restriction induces a monomorphism of groups Aut_{L}(P) → Aut_{L}(P′).

Proof. The second assertion follows immediately from Proposition 2.11. The first follows from the second by letting P′ be the 1-simplex P_{0}. □

Notation. BAut_{L}(P) denotes the subcategory of L whose only object is P_{0} and whose morphism set is Aut_{L}(P).

1.6. Definition. Given a centric collection C in a p-local finite group (S, F, L), let L^{C} denote the full subcategory of L generated by the objects set C.

Frequently, the inclusion L^{C} ⊆ L induces a weak homotopy equivalence on nerves. For example, this happens when C contains all the F-centric F-radical subgroups of S. This fact is proved in [2, Theorem 3.5].

The following theorem applies to all centric collections. The decomposition approximates L if the inclusion L^{C} ⊆ L induces an equivalence as explained above.

Theorem A. Let C be an F-centric collection in a p-local finite group (S, F, L). Then there exists a functor δ_{C} : s̄dC → Spaces such that

(1) There is a natural weak homotopy equivalence

hocolim_{s̄dC} δ_{C} ⟶^{≃} |L^{C}|.

(2) There is a natural weak homotopy equivalence BAut_{L}(P) ⟶^{≃} δ_{C}([P]) for every k-simplex P.

(3) The natural maps δ_{C}([P]) → |L^{C}| are induced by the inclusion of categories BAut_{L}(P) ⊆ L^{C}.

(4) If P′ is a subsimplex of P then the equivalence 2 renders the following square commutative

\xymatrix{BAut_{L}(P) \ar[r]^{≃} \ar[d]_{B res_{P′}^{P}} & δ_{C}([P]) \ar[d] \\ BAut_{L}(P′) \ar[r]^{≃} & δ_{C}([P′])}

Moreover if P and P′ are conjugate k-simplices and ψ ∈ Iso_{L}(P_{0}, P′_{0}) maps Aut_{L}(P′) onto Aut_{L}(P) by conjugation then the following square commutes

[figure: BAut_{L}(P′); ≃; Bc_{ψ}; δ_{C}([P′]); BAut_{L}(P); ≃; δ_{C}([P])]

Remark. When (S, F, L) is associated with a finite group G one may consider the G-collection H consisting of all the subgroups of G which are conjugate to elements of the F-collection C. Dwyer constructs in [5, §3] a poset s̄dH and a functor δ_{H}^{Dwyer} : s̄dH → Spaces which he calls the normaliser decomposition. We will show in 5.2 that s̄dH = s̄dC and that δ_{C} and δ_{H}^{Dwyer} can be connected by a natural zigzag of mod-p equivalences. That is, a zigzag of natural transformations which at every object of s̄dC give rise to an H_{∗}(−; ℤ/p)-isomorphism.

We now describe the second type of normaliser decomposition that we shall construct in this note. It is based on collections E of elementary abelian subgroups of S.

1.7. Definition. For a k-simplex E in E define Aut_{F}(E) as the subgroup of Aut_{F}(E_{k}) consisting of the automorphisms f such that f(E_{i}) = E_{i} for all i = 0, …, k.

Consider an F-centric collection C in (S, F, L).

1.8. Definition. Fix an elementary abelian subgroup E of S. The objects of the category C̄_{L}(C; E) are pairs (P, f) where P ∈ C and f : E → Z(P) is a morphism in F. Morphisms (P, f) → (Q, g) in C̄_{L}(C; E) are morphisms ψ ∈ L(P, Q) such that g = π(ψ) ∘ f where π : L → F is the projection functor.

Observe that Aut_{F}(E) acts on C̄_{L^{C}}(E) by pre-composition. Thus, h ∈ Aut_{F}(E) maps (P, f) ↦ (P, f ∘ h).

1.9. Definition. For a k-simplex E in E let N̆_{L}(C; E) denote the subcategory of L whose objects are P ∈ C for which E_{k} ≤ Z(P). A morphism φ ∈ L(P, Q) belongs to N̆_{L}(C; E) if π(φ)|_{E_{k}} is an element of Aut_{F}(E).

Recall that the homotopy orbit space of a G-space X, i.e. the Borel construction EG ×_{G} X, is denoted by X_{hG}.

1.10. Proposition. Let E be a k-simplex in E. There is a map

ϵ_{E} : |N̆_{L}(C; E)| → |C̄_{L}(C; E_{k})|_{hAut_{F} (E)}

which is a homotopy equivalence if E_{k} is fully F-centralised. The map is natural with respect to inclusion of simplices.

Proof. This is immediate from proposition 5.3. □

A comment on the categories C̄_{L}(C; E) is in place. If C is the collection of all the F-centric subgroups of S and E is fully F-centralised, then it is shown in [4, Theorem 2.6] that |C̄_{L}(C; E)| has the homotopy type of the nerve of the centraliser linking system |C_{L}(E)|. The categories N̆_{L}(C; E) are more mysterious. Even when E is a 1-simplex E, the category N̆_{L}(C; E) is in general only a subcategory of the normaliser linking system N_{L}(E) because the largest subgroup which appears as an object of N̆_{L}(C; E) is C_{S}(E) which in general is smaller than N_{S}(E). When C_{S}(E) = N_{S}(E) these categories are equal.

The next decomposition result, theorem B, depends on a collection of elementary abelian groups E and a collection C of F-centric subgroups of S. It approximates L if C contains, for example, all the F-centric F-radical subgroups of S. The collection E must be large enough as explicitly stated in the theorem. For example the collection of all the non-trivial elementary abelian subgroups will always be a valid choice.

1.11. Definition. Given a group H and a prime p let Ω_{p}(H) denote the subgroup of H generated by all the elements of order p in H.

Theorem B. Consider a p-local finite group (S, F, L), an F-centric collection C and a collection E of elementary abelian subgroup of S which contains the subgroups Ω_{p}Z(P) for all P ∈ C. Then there exists a functor δ_{E} : s̄dE → Spaces with the following properties.

(1) There is a natural weak homotopy equivalence hocolim_{s̄dE} δ_{E} ⟶^{≃} |L^{C}|.

(2) For a k-simplex E in E there is a weak homotopy equivalences

|C̄_{L}(C; E_{k})|_{hAut_{F} (E)} ⟶^{≃} δ_{E}([E]).

(3) Fix a k-simplex E where E_{k} is fully F-centralised. The equivalences (1) and (2) give a natural map δ_{E}([E]) → |L^{C}| whose precomposition with ϵ_{E} of 1.10 is induced by the realization of the inclusion of N̆_{L}(C; E) in L^{C}.

(4) If E′ is a k-subsimplex of an n-simplex E then the following square commutes up to homotopy

\xymatrix{|C̄_{L}(C; E_{n})|_{hAut_{F} (E)} \ar[d] \ar[r]^{≃} & δ_{E}([E]) \ar[d] \\ |C̄_{L}(C; E′_{k})|_{hAut_{F} (E′)} \ar[r]^{≃} & δ_{E}([E′])}

The homotopy is natural with respect to inclusion of simplices. In addition, the square commutes on the nose if E′_{k} = E_{n}.

2. On p-local finite groups

The term p-local finite group was coined by Broto, Levi and Oliver in [4]. They cropped up naturally in their attempt in [3] to describe the space of self equivalences of a p-completed classifying space of a finite group G. They discovered that the relevant information needed to solve this problem lies in the fusion system of the psubgroups of G and certain categories which they later on called “linking systems”. Historically, fusion systems were first introduced by Lluis Puig in [13].

2.1. Definition. Fix a prime p and let S be a finite p-group. A fusion system over S is a sub-category F of the category of groups whose objects are the subgroups of S and whose morphisms are group monomorphisms such that

(1) All the monomorphisms that are induced by conjugation by elements of S are in F.

(2) Every morphism in F factors as an isomorphism in F followed by an inclu-

sion of subgroups. We say that two subgroups P, Q of S are F-conjugate if they are isomorphic as objects of F.

When g is an element of S and P, Q are subgroups of S such that gP g^{−1} ≤ Q, we let c_{g} denote the morphism P → Q defined by conjugation, namely c_{g}(x) = gxg^{−1} for every x ∈ P.

We let Hom_{S}(P, Q) denote the set of all the morphisms P → Q in F that are induced by conjugation in S. Also notice that the factorization axiom (2) implies that all the F-endomorphisms of a subgroup P are in fact automorphisms in F. Thus we write Aut_{F}(P) for the set of morphisms F(P, P).

2.2. Definition. A subgroup P of S is called fully F-centralised (resp. fully Fnormalised) if its S-centraliser C_{S}(P) (resp. S-normaliser N_{S}(P)) has the maximal possible order in the F-conjugacy class of P. That is, |C_{S}(P)| ≥ |C_{S}(P′)| (resp. |N_{S}(P)| ≥ |N_{S}(P′)|) for every P′ which is F-conjugate to P.

2.3. Definition. A fusion system F over a finite p-group S is called saturated if

I. Every fully F-normalised subgroup P of S is fully F-centralised and moreover Aut_{S}(P) = N_{S}(P)/C_{S}(P) is a Sylow p-subgroup of Aut_{F}(P). II. Every morphism φ: P → S in F whose image φ(P) is fully F-centralised

can be extended to a morphism ψ:N_{φ} → S in F where

N_{φ} = {g ∈ N_{S}(P) : φc_{g}φ^{−1} ∈ Aut_{S}(P)}.

2.4. Definition. A subgroup P of S is called F-centric if P and all of its Fconjugates contain their S-centralisers, that is C_{S}(P′) = Z(P′) for every subgroup P′ of S which is F-conjugate to P.

2.5. Definition. A centric linking system associated to a saturated fusion system F over S consists of

(1) A small category L whose objects are the F-centric subgroups of S,

(2) a functor π : L → F and

(3) group monomorphisms δ_{P} : P → Aut_{L}(P) for every F-centric subgroup P

of S, Such that the following axioms hold

(A) The functor π acts as the inclusion on object sets, that is π(P) = P for every F-centric subgroup P of S. For any two objects P, Q of L, the group Z(P) acts freely on the morphism set L(P, Q) via the restriction of δ_{P} : P → Aut_{L}(P) to Z(P). The induced map on morphisms sets

π : L(P, Q) → F(P, Q)

identifies F(P, Q) with the quotient of L(P, Q) by the free action of Z(P). (B) For every F-centric subgroup P of S the map π : Aut_{L}(P) → Aut_{F}(P)

sends δ_{P} (g), where g ∈ P, to c_{g}. (C) For every f ∈ L(P, Q) and every g ∈ P there is a commutative square in L

P ⟶^{f} Q
δ_{P} (g)↓  ↓^{δ_{Q}(π(f)(g))}
P ⟶_{f} Q.

Remark. A morphism f ∈ L(P, Q) is called a lift of a morphism φ ∈ F(P, Q) if φ = π(f).

2.6. Definition. A p-local finite group is a triple (S, F, L) where F is a saturated fusion system over the finite p-group S and L is a centric linking system associated to F. The classifying space of (S, F, L) is the space |L|_{p}^{∧}, that is the p-completion in the sense of Bousfield and Kan [1], of the realization of the small category L.

2.7. When S is a Sylow p-subgroup of a finite group G, there is an associated p-local finite group denoted (S, F_{S}(G), L_{S}(G)). See [4, Prop. 1.3 and the remarks after Definition 1.8]. We shall write F for F_{S}(G) and L for L_{S}(G).

Morphism sets between P, Q ≤ S are

F(P, Q) = Hom_{G}(P, Q) = N_{G}(P, Q)/C_{G}(P)

where N_{G}(P, Q) = {g ∈ G : gP g^{−1} ≤ Q} and C_{G}(P) acts on N_{G}(P, Q) by right translation.

A subgroup P of S is, by [4, Prop. 1.3], F-centric precisely when it is p-centric in the sense of [5, §1.19], that is, Z(P) is a Sylow p-subgroup of C_{G}(P). In this case C_{G}(P) = Z(P) × C′_{G}(P) where C′_{G}(P) is the maximal subgroup of C_{G}(P) of order prime to p. Morphism sets of L = L_{S}(G) have, by definition, the form

L(P, Q) = N(P, Q)/C′_{G}(P).

The functor π : L_{S}(G) → F_{S}(G) is the obvious projection functor. The monomorphism δ_{P} : P → Aut_{L}(P) is induced by the inclusion of P in N_{G}(P).

It is shown in [4] after Definition 1.8, that (S, F_{S}(G), L_{S}(G)) is a p-local finite group and that |L_{S}(G)|_{p}^{∧} ≃ BG_{p}^{∧}. It should also be remarked that there are examples of p-local finite groups that cannot be associated with any finite group. These are usually referred to as “exotic examples”.

2.8. In every p-local finite group (S, F, L) one can choose morphisms ι_{P}^{Q} ∈ L(P, Q) for every inclusion of F-centric subgroups P ≤ Q, in such a way that

(1) π(ι_{P}^{Q}) is the inclusion P ≤ Q,

(2) ι_{Q}^{R} ∘ ι_{P}^{Q} = ι_{P}^{R} for every F-centric subgroups P ≤ Q ≤ R of S, and

(3) ι_{P}^{P} = id for every F-centric subgroup P of S. This follows from [4, Proposition 1.11]. Using the notation there, one chooses ι_{P}^{Q} = δ_{P,Q}(e) where e is the identity element in S. Whenever possible, in order to avoid cumbersome notation, we shall write ι for ι_{P}^{Q}.

2.9. From [4, Lemma 1.10(a)] it also follows that every morphism φ : P → Q in L factors uniquely as an isomorphism φ′ : P → P′ in L followed by the morphism ι : P′ → Q. In fact P′ = π(φ)(P)

2.10. It was observed by Broto, Levi and Oliver that every morphism in L is a monomorphism in the categorical sense; cf. the remarks after Lemma 1.10 in [4]. It was later observed in [2, Corollary 3.10] and independently by others, that every morphism in L is also an epimorphism. As an easy consequence we record for further use:

2.11. Proposition. Consider a p-local finite group (S, F, L) and a commutative square in F on the left of the display below

P ⟶^{f} P′
incl↓  ↓^{incl}
Q ⟶_{g} Q′

\begin{matrix}P & ⟶^{f̃} & P′ \\ ^{ι_{P}^{Q}}↓ & & ↓^{ι_{P′}^{Q′}} \\ Q & ⟶_{g̃} & Q′ \end{matrix}

where P, P′, Q and Q′ are F-centric subgroups of S. Then for every lift g̃ of g in L there exists a unique lift f̃ of f in L which render the square on the right commutative in L. We denote f̃ by g̃|_{P}.

Given a lift f̃ for f, if there exists a lift g̃ for g rendering the square on the right commutative, then it is unique.

Proof. The first assertion follows immediately from [4, Lemma 1.10(a)] by setting ψ = incl_{P′}^{Q′}, ψ̃ = ι_{P′}^{Q′} and ψ̃φ = g̃ι_{P}^{Q}. The second assertion follows immediately from the fact that ι_{P}^{Q} is an epimorphism. □

2.12. Fix a p-local finite group (S, F, L). Given a subgroup P of S, there are two important p-local finite groups associated with it (a) the centraliser of P and (b) the normaliser of P. Both were defined by Broto Levi and Oliver in [4].

The centraliser fusion system C_{F}(P), where P is fully F-centralised, is a subcategory of F. As a fusion system it is defined over the S-centraliser of P denoted C_{S}(P). Morphisms Q → Q′ in C_{F}(P) are those morphisms φ : Q → Q′ in F that can be extended to a morphism φ̄ : P Q → P Q′ in F which induces the identity on P. The objects of the centric linking system C_{L}(P) associated to C_{F}(P) are the C_{F}(P)-centric subgroups of C_{S}(P). The set of morphisms Q → Q′ in C_{L}(P) is a subset of L(P Q, P Q′) consists of those morphisms f : P Q → P Q′ such that π(f) induces the identity on P and carries Q to Q′. It is shown in [4] that (C_{S}(P), C_{F}(P), C_{L}(P)) is a p-local finite group.

Now fix a subgroup K ≤ Aut_{F}(P) where P is fully normalised in F. The K-normaliser fusion system N_{F}^{K}(P) is a subcategory of F defined over N_{S}(P). The objects of N_{F}^{K}(P) are the subgroups of N_{S}(P). A morphisms φ ∈ F(Q, Q′) belongs to N_{F}^{K}(P) if it can be extended to a morphism φ̄ : P Q → P Q′ in F which induces an automorphism from K on P. The fusion system N_{F}^{K}(P) is saturated. When K = Aut_{F}(P) we denote this category by N_{F}(P) and call it the normaliser fusion system of P. The centric linking system N_{L}(P) associated to N_{F}(P) has the N_{F}(P)-centric subgroups of N_{S}(P) as its object set. The set of morphisms Q → Q′ is the subset of L(P Q, P Q′) consisting of those f : P Q → P Q′ such that π(f) carries Q to Q′ and induces an automorphism on P.

3. Homotopy colimits and the Grothendieck construction

Throughout this paper we work simplicially, namely a “space” means a simplicial set. For further details, the reader is referred to [1],[12], [7] and many other sources. In this section we collect several results from general simplicial homotopy theory that we shall use repeatedly in the rest of this note.

Homotopy colimits. Fix a small category K and a functor U : K → Spaces. The simplicial replacement of U is the simplicial space ∐_{∗} U which has in simplicial dimension n the disjoint union of the spaces U(K_{0}) for every chain K_{0} → K_{1} → ⋯ → K_{n} of n composable arrows in K. The homotopy colimit of U denoted hocolim_{K}U is the diagonal of ∐_{∗} U regarded as a bisimplicial set. See Bousfield and

Kan [1, §XII.5].

Consider a functor F : K → L between small categories. For a functor U : L → Spaces there is an obvious natural map, cf. [1, §XI.9].

hocolim_{K}F^{∗}U → hocolim_{L}U.

For an object L ∈ L, the comma category (L ↓ F) has the pairs (K, L ⟶^{k∈K} F K)

as its objects. Morphisms (K, L ⟶^{k} F K) → (K′, L ⟶^{k′} F K′) are the morphisms x : K → K′ such that F x ∘ k = k′. Similarly one defines the category (F ↓ L) whose object set consists of the pairs (K, k : F K → L). Compare [11].

3.1. Definition. The functor F : K → L is called right-cofinal if for every object L ∈ L the category (L ↓ F) has a contractible nerve.

The following theorem was probably first proved by Quillen in [14, Theorem A]. See also Hollender and Vogt in [8, §4.4] and Bousfield and Kan in [1, §XI.9].

3.2. Cofinality Theorem. Let F : K → L be a right cofinal functor between small categories. Then for every functor U : L → Spaces the natural map hocolim_{K}F^{∗}U → hocolim_{L}U is a weak homotopy equivalence.

Associated with a functor U : K → Spaces there is a functor F_{∗}U : L → Spaces called the homotopy left Kan extension of U along F. It is defined on every object L ∈ L by

F_{∗}U(L) = hocolim ((F ↓ L) ⟶^{proj} K ⟶^{F} Spaces).

See [8, §5], [6, §6]. The following theorem is originally due to Segal. See e.g. [8, Theorem 5.5].

3.3. Segal’s Pushdown Theorem. Fix a functor F : K → L of small categories. Then for every functor U : K → Spaces there is a natural weak homotopy equivalence

hocolim_{L}F_{∗}U ⟶^{≃} hocolim_{K}U.

The Grothendieck construction. Recall that a small category K gives rise to a simplicial set Nr(K) called the nerve of K. Its n-simplices are the chains of n composable arrows K_{0} → K_{1} → ⋯ → K_{n} in K. See e.g. [7, Example 1.4] or [1, §XI.2]. We shall also use the notation |K| for the nerve of K.

Given a functor U : K → Cat Thomason defined in [17] the translation category K ∫ U associated to U as follows. The object set consists of pairs (K, X) where K is an object of K and X is an object of U(K). Morphisms (K_{0}, X_{0}) → (K_{1}, X_{1}) are pairs (k, x) where k : K_{0} → K_{1} is a morphism in K and x : U(k)(X_{0}) → X_{1} is a mor-

phism in U(K_{1}). Composition of (K_{0}, X_{0}) ⟶^{(k_{0},x_{0})} (K_{1}, X_{1}) and (K_{1}, X_{1}) ⟶^{(k_{1},x_{1})}

(K_{2}, X_{2}) is given by

(k_{1}, x_{1}) ∘ (x_{0}, k_{0}) = (k_{1} ∘ k_{0}, x_{1} ∘ U(k_{1})(x_{0})).

This category is also called the Grothendieck construction of U and the notation Tr_{K}U is also used. Thomason shows in [17] that there is a natural weak homotopy equivalence

(3.4)  η : hocolim_{K} |U| ⟶^{≃} |Tr_{K}U|

A natural transformation U ⇒ U′ gives rise to a canonical functor Tr_{K}U → Tr_{K}U′. The induced map |Tr_{K}(U)| → |Tr_{K}(U′)| corresponds via η (3.4) to the induced map hocolim_{K} |U| → hocolim_{K} |U′|. Furthermore, for every object K in K the natural map

|U(K)| → hocolim_{K} |U|

corresponds under (3.4) to the inclusion of categories

(3.5)  U(K) → Tr_{K} U,  wehere X ↦ (K, X) and x ↦ (1_{K}, x).

Consider now a functor F : K → L of small categories. Given U : L → Cat there is a naturally defined functor

(3.6) F_{!} : Tr_{K}F^{∗}U → Tr_{L}U,  where  \{ \begin{matrix}F_{!}(K, X ∈ F^{∗}U(K)) = (F K, X) \\ F_{!}(k, x) = (F k, x). \end{matrix}

The functor F_{!} is a model for the map hocolim F^{∗}|U| → hocolim |U| in the sense that the following square commutes

|Tr_{K}F^{∗}U| ⟶^{η} hocolim_{K}F^{∗}|U|
|F_{!}|↓  ↓
|Tr_{L}U| ⟶^{η} hocolim_{L}|U|

3.7. Definition. For a functor U : K → Cat define F_{∗}U : L → Cat by

F_{∗}U(L) = Tr((F ↓ L) ⟶^{proj} K ⟶^{U} Spaces)

The maps η (3.4) provide a natural weak homotopy equivalence |F_{∗}U| ⟶^{≃} F_{∗}|U|. The equivalence in the pushdown theorem can be realized as the nerve of a functor between the transporter categories as follows.

3.8. Proposition. The functor F_{#} : Tr_{L}F_{∗}U → Tr_{K}U defined by

\begin{matrix}F_{#} : (L, (K, F K → L), X ∈ UK) ↦ (K, X) \\ F_{#} : (L ⟶^{ℓ} L′, K ⟶^{k} K′, U(k)(X) ⟶^{x} X′) ↦ (k, x). \end{matrix}

renders the following diagram commutative where the arrow at the top of the square is an equivalence by the pushdown theorem.

(3.9)  \xymatrix{hocolim_{L} |F_{∗}U| \ar[dr]^{η}_{≃} & hocolim_{L} F_{∗}|U| \ar[r]^{∼} \ar[d]_{∼} \ar[l]_{∼} & hocolim_{K} |U| \ar[d]_{∼}^{η} \\  & |Tr_{L}(F_{∗}U)| \ar[r]_{|F_{#}|} & |Tr_{K}(U)|}

It is useful to point out that if ⋆ : K → Cat is the constant functor on the trivial category with one object and an identity morphism, then Tr_{K}(⋆) = Nr(K).

4. EI categories

Fix an EI category A, namely a category all of whose endomorphisms are isomorphisms. We shall assume that the category A is finite. We shall also assume that A is equipped with a height function, namely a function h : Obj(A) → ℕ such that h(A) ≤ h(A′) if there exists a morphism A → A′ in A and equality holds if and only if A → A′ is an isomorphism. Clearly, if A is an EI-category then so is A^{op}. The finiteness condition also implies that if A is heighted then so is A^{op}.

We can always choose a full subcategory A_{sk} of A which contains one representative from each isomorphism class of objects in A. We say that A_{sk} is skeletal in A. Clearly the inclusion A_{sk} ⊆ A is an equivalence of categories. In the language of Słominska [15] A_{sk} is an EIA category.

Throughout we let \underline{k} denote the poset \{0 → 1 → ⋯ → k\} considered as a small category.

4.1. Definition. The subdivision category s(A) is the category whose objects are height increasing functors A : \underline{k} → A, namely h(A(i)) < h(A(i + 1)) for all i < k. Morphisms A → A′ in s(A) are pairs (ϵ, φ) where ϵ : \underline{k′} → \underline{k} is a strictly increasing function and φ : ϵ^{∗}(A) → A′ is a natural isomorphism of functors \underline{k′} → A. Composition of (ϵ, φ) : A → A′ and (ϵ′, φ′) : A′ → A′′ is given by (ϵ ∘ ϵ′, ϵ′^{∗}(φ) ∘ φ′).

Note that ϵ is determined by the heights of the values of A namely ϵ(i) = j if and only if h(A′(i)) = h(A(j)).

We shall further assume that A contains a subcategory I which is a poset with the property that every morphism φ : A → A′ in A can be factored uniquely as φ = ιφ′ where φ′ is an isomorphism in A and ι is a morphism in I. The ladder

[figure: ⋯; A(n − 2); φ_{n−1}; (φ′_{n}∘φ_{n−1})′; ≅; A(n − 1); φ_{n}; φ′_{n}; ≅; A(n); ⋯; A′(n − 2); ι; A′(n − 1); ι; A′(n)]

shows that the the full subcategory s_{I}(A) of s(A) consisting of the objects A in which all the arrows A(i) → A(i + 1) belong to I is a skeletal subcategory of s(A). We obtain two skeletal subcategories of s(A)

(4.2)  s(A_{sk}) ⊆ s(A)  and  s_{I}(A) ⊆ s(A).

We observe that Hom_{s(A)}(A, A′) has a free action of Aut_{s(A)}(A′) with a single orbit. Also every φ : A → A′ in s(A) gives rise to a natural group homomorphism upon restriction and conjugation with the isomorphism φ : ϵ^{∗}A ≈ A′

φ_{∗} : Aut_{s(A)}(A) → Aut_{s(A)}(A′).

4.3. Proposition. There is a right cofinal functor p : s(A) → A defined by

p(A) = A(0), (A : \underline{k} → A).

Proof. Słominska shows in [15, Proposition 1.5] that the restriction p : s(A_{sk}) → A_{sk} is right cofinal (3.1) hence so is p : s(A) → A. □

4.4. Definition. The category s̄(A) has the isomorphism classes [A] of the objects of s(A) as its object set. There is a unique morphism [A] → [A′] if there exists a morphism A → A′ in s(A). There is an obvious projection functor

π : s(A) → s̄(A), A ↦ [A].

When D is a full subcategory of s(A) one obtains a sub-poset D̄ of s̄(A) whose objects are the isomorphism classes of the objects of D.

Clearly s̄(A) is a poset and it should be compared with Słominska’s construction of s_{0}(A) in [15, §1]. Also note that s̄_{I}(A) = s̄(A) because s_{I}(A) is skeletal in s(A). Similarly s̄(A_{sk}) = s̄(A).

4.5. Lemma. Let J : C → D be a functor of small categories with a left adjoint L : D → C such that L ∘ J = Id. Then J is right cofinal.

Proof. Fix an object d ∈ D. We have to prove that the category (d ↓ J) has a contractible nerve. Let (d ↓ D) denote the category (d ↓ 1_{D}). It clearly has a contractible nerve because it has an initial object. The functors J and L induce obvious functors

J_{∗} : (d ↓ J) → (d ↓ D), (c, d ⟶^{f} Jc) ↦ (Jc, d ⟶^{f} Jc)

L_{∗} : (d ↓ D) → (d ↓ J), (d′, d ⟶^{f} d′) ↦ (Ld′, d ⟶^{f} d′ ⟶^{η} JLd′).

It is obvious that L_{∗} ∘ J_{∗} = Id. Furthermore the unit η : Id → LJ gives rise to a natural transformation Id → J_{∗} ∘ L_{∗}. Therefore J_{∗} induces a homotopy equivalence on nerves so |(d ↓ J)| ≃ |d ↓ D)| ≃ ∗. □

4.6. Proposition. For every functor F : s(A) → Cat and every A ∈ s(A) there is a functor

Tr_{BAut(A)}F (A) → (π_{∗}F)([A])

which induces a weak homotopy equivalence on nerves. It is natural in the sense that every φ : A → A′ gives rise to a square

[figure: Tr_{Aut(A)}F (A); Tr_{φ∗} F (φ); (π_{∗}F)([A]); π_{∗}([φ]); ⇐^{τ}; Tr_{Aut(A′)}F (A′); (π_{∗}F)([A′])]

which commutes up to a natural transformation τ which is functorial in φ. Here φ_{∗} : Aut(A) → Aut(A′) is the homomorphism induced by restriction and conjugation

by φ : ϵ^{∗}A ≈ A′. The square commutes on the nose if F (A) ⟶^{F (φ)} F (A′) is the identity.

Proof. Fix an object A : \underline{k} → A in s(A). Let Π_{A} be the full subcategory of (π ↓ [A]) consisting of the objects (A′, [A′] ⟶^{=} [A]). It is isomorphic to the full subcategory of s(A) consisting of the isomorphism class of A. The inclusion J : Π_{A} → (π ↓ [A]) has a left adjoint L where

L : (B, [B] → [A]) ↦ ϵ^{∗}B, where [ϵ^{∗}B] = [A] for ϵ : \underline{k′} ↪ \underline{k}.

Clearly ϵ, if it exists, is unique. There is an obvious natural map B ↦ ϵ^{∗}B induced by the identity on ϵ^{∗}B under the bijection s(A)(B, ϵ^{∗}B) ≈ s(A)(ϵ^{∗}B, ϵ^{∗}B). We obtain a natural transformation Id → JL which gives rise to bijections for every object (A′, [A′] → [A]) in (π ↓ [A])

Hom_{(π↓[A])}(B, JA′) = Hom_{s(A)}(B, A′)

≈ Hom_{s(A)}(ϵ^{∗}B, A′) = Hom_{Π_{A}}(LB, A′).

Thus L is left adjoint to J and we apply Proposition 4.5. By definition Π_{A} is a connected groupoid with automorphism group BAut_{s(A)}(A). Therefore upon realization, the functor

Tr(BAut_{s(A)}(A) → s(A) ⟶^{F} Cat) ⟶^{restriction}

Tr(π ↓ [A]) → s(A) ⟶^{F} Cat) = (π_{∗}F)([A])

induces a weak homotopy equivalence. Also, for a morphism φ : A → A′ we get an obvious φ_{∗} : Π_{A} → Π′_{A} by restriction and conjugation by the isomorphism φ : ϵ^{∗}A → A′. It gives rise to the following diagram

\xymatrix{BAut_{s(A)}(A) \ar[r]^{incl} \ar[d]^{φ_{∗}} & Π_{A} \ar[d]_{φ_{∗}} \ar[r]^{J} & (π ↓ [A]) \ar[d]^{[φ]_{∗}} \\ BAut_{s(A)}(A′) \ar[r]^{incl} & Π_{A′} \ar[r]^{J} & (π ↓ [A′])}

The morphism φ provides a canonical natural transformation [φ]_{∗}∘J∘incl → J∘incl∘ φ_{∗}. This provides the natural transformation τ in the statement of the proposition

and its naturality with φ. If F (A) ⟶^{F (φ)} F (A′) is the identity, then F (τ) becomes the identity and the square in the statement of the proposition commutes. □

5. Proof of the main results

Fix a p-local finite group (S, F, L) and a centric collection C. Choose a subcategory I ⊆ L^{C} of distinguished inclusions, cf. 2.8. Note that L^{C} possesses a height function, see §4, by assigning to a subgroup P in C its order. Also every morphism in L^{C} factors uniquely as an isomorphism followed by a morphism in I.

We claim that (definitions 1.3 and 4.4)

s̄dC = s̄(L^{C}).

To see this recall that s_{I}(L^{C}) is a skeletal subcategory of s(L^{C}), see (4.2), hence s̄(L^{C}) = s̄_{I}(L^{C}). The functor π : L → F gives a functor s̄_{I}(L^{C}) → s̄dC because it maps the morphisms ι ∈ I to inclusion of subgroups of S. It is an isomorphism of categories because conjugation (1.2) of two k-simplices P_{0} < … < P_{k} and P′_{0} < … < P′_{k} induced by an isomorphism φ_{k} ∈ Iso_{F}(P_{k}, P′_{k}) can be lifted to an isomorphism of the corresponding objects in s_{I}(L^{C}) by lifting the isomorphism φ_{k} : P_{k} → P′_{k} to L and using 2.11 to obtain the commutative ladder in L^{C}:

\xymatrix{P_{0} \ar[r]^{ι} \ar[d]_{≅} & P_{1} \ar[r]^{ι} \ar[d]_{≅} & ⋯ \ar[r]^{ι} & P_{k} \ar[d]^{φ̃_{k}} \\ P′_{0} \ar[r]_{ι} & P′_{1} \ar[r]^{ι} & ⋯ \ar[r]^{ι} & P′_{k}}

We remark that a k-simplex P_{0} < ⋯ < P_{k} in C can be identified with the object P_{0} ⟶^{ι} ⋯ ⟶^{ι} P_{k} of s(L^{C}). Under this identification we clearly have

Aut_{L}(P) = Aut_{s(L^{C})}(P).

When G is a discrete group we let BG denote the category with one object and G as its set of morphisms. For every k-simplex P in C we identify BAut_{L}(P) with the obvious subcategory of BAut_{L}(P_{0}).

5.1. Theorem. Let C be a centric collection in a p-local finite group (S, F, L). Then there exists a functor δ̃_{C} : s̄dC → Cat with the following properties

(1) There is a naturally defined functor Tr_{s̄dC}(δ̃_{C}) → L^{C} which induces a weak homotopy equivalence on nerves.

(2) For every k-simplex P there is a canonical functor BAut_{L}(P) → δ̃_{C}([P]) which induces a weak homotopy equivalence on nerves. If P′ is a subsimplex of P then the following square commutes

\xymatrix{BAut_{L}(P) \ar[r] \ar[d]_{_{res{P \atop P′}}} & δ̃_{C}([P]) \ar[d] \\ BAut_{L}(P′) \ar[r] & δ̃_{C}([P′])}

(3) The natural inclusion BAut_{L}(P) ⊆ L^{C} is equal to the composition

BAut_{L}(P) → δ̃_{C}([P]) ⊆ Tr_{s̄dC}(δ̃_{C}) → L^{C}

(4) An isomorphism of k-simplices ψ : P′ ⟶^{≈} P in s(L^{C}) induces a commutative square

\xymatrix{BAut_{L}(P) \ar[r] \ar[d]_{c_{ψ}} & δ̃_{C}([P]) \ar[d] \\ BAut_{L}(P′) \ar[r] & δ̃_{C}([P′])}

Proof. We have seen that s̄dC = s̄(L^{C}). Let ⋆ : s̄(L^{C}) → Cat denote the constant functor on the trivial small category with one object and identity morphism. Use the projection functor π : s(L^{C}) → s̄(L^{C}) to define

δ̃_{C} = π_{∗}(⋆)

According to proposition 4.6 we have a canonical functor

BAut_{L}(P) = Tr_{BAut_{L}(P)}(⋆) → π_{∗}(⋆)([P]) = δ̃_{C}([P])

which induces a weak homotopy equivalence. Since ⋆ is constant, the square in the statement of proposition 4.6 commutes and we obtain the naturality assertions in point (2) and (4). The natural functor of (1) is defined using 3.8 and 4.3 by

Tr_{s̄dC}(δ̃_{C}) = Tr_{s̄(L^{C})}(π_{∗}(⋆)) ⟶^{π_{#}} Tr_{s(L^{C})}(⋆) = s(L^{C}) ⟶^{p} L^{C}.

It induces a weak homotopy equivalence by the pushdown theorem 3.3, proposition 3.8 and the cofinality theorem 3.2. Whence point (1). Inspection of the functor π_{#}, the inclusion BAut_{L}(P) ⊆ (π ↓ [P]) = π_{∗}(⋆)([P]) and 3.5 yield point (3) □

proof of Theorem A. Apply theorem 5.1 above and define δ_{C} = |δ̃_{C}|. □

5.2. We now relate the construction in theorem A to Dwyer’s normaliser decomposition [5, §3]. We will show that the two functors are related by a zigzag of natural transformations which induce a mod-p equivalence.

Fix a finite group G and the p-local finite group (S, F, L) associated with it. A collection C of F-centric subgroups of S gives rise to a G-collection H of p-centric subgroups of G (cf. [5, §1.19], 2.7) by taking all the G-conjugates of the elements of C. We let T^{H} denote the transporter category of H. That is, the object set of T^{H} is H and the morphism set T^{H}(H, K) is the set N_{G}(H, K) = {g ∈ G : g^{−1}Hg ≤ K}. We also let T^{C} denote the full subcategory of T^{H} having C as its object set. Almost by definition T^{C} is skeletal in T^{H}. We also obtain a zigzag of functors (see 2.7)

T^{H} ← T^{C} → L^{C}.

Dwyer defines in [5, §3] a category s̄dH whose objects are the G-conjugacy classes [H] of the k-simplices H_{0} < … < H_{k} in H. There is a unique morphism [H] → [H′] in s̄dH if and only if H′ is conjugate in G to a subsimplex of H. It follows directly from the definition of H as the smallest G-collection containing C and from the definition of F = F_{S}(G) that s̄dH = s̄dC. We obtain a commutative diagram (see 4.1)

[figure: s(T^{H}); π_{2}; s(T^{C}); π_{1}; ⊇; s(L^{C}); π; s̄(T^{H}); s̄(T^{C}); s̄(L^{C}).]

Fix a k-simplex P = P_{0} < ⋯ < P_{k} in C. Note that (π_{2} ↓ [P]) is isomorphic to the subcategory of T^{H} of the objects P′ which admit a morphism to P. It contains a full subcategory Π_{P} of the objects of s(T^{H}) that are isomorphic to P; cf. the proof of proposition 4.6. By inspection Π_{P} is the translation category of the action of G on the orbit of P, that is it is the transported category of the G-set [P] in H thought of as a functor BAut_{L}(P) → Sets, cf. [5, §3.3]. Thus

δ_{H}^{Dwyer}([P]) = EG ×_{G} [P] = Nr(Π_{P}).

The inclusion J : Π_{P} ↪ (π_{2} ↓ [P]) has a left adjoint L : (P′, [P′] → [P]) ↦ ϵ^{∗}P′ where ϵ^{∗}P′ ≈ P in s(L^{C}). Compare the proof of 4.6. Lemma 4.5 implies that J is right cofinal. We obtain a zigzag of functors

δ_{H}^{Dwyer} ⟶_{≃} |(π_{2})_{∗}(⋆)| ⟵_{≃}^{incl} |(π_{1})_{∗}(⋆)| ⟶^{mod-p} |π_{∗}(⋆)| = |δ̃_{C}| = δ_{C}.

The third map induces a mod-p equivalence by the following argument. For any object [P] we obtain a map (π_{1})_{∗}(⋆)([P]) → π_{∗}(⋆)([P]) which by proposition 4.6 is equivalent to the map

BAut_{G}(P) → BAut_{L}(P).

Since P_{0} is p-centric then C_{G}(P_{0}) = Z(P_{0}) × C′_{G}(P_{0}) where C′_{G}(P_{0}) is a characteristic p′-subgroup of C_{G}(P_{0}) and Aut_{L}(P_{0}) = N_{G}(P_{0})/C′_{G}(P_{0}). Therefore Aut_{G}(P) → Aut_{G}(P)/C′_{G}(P_{0}) induces a mod-p equivalence as needed.

We shall now prove theorem B. Fix an F-centric collection C and a collection E of elementary abelian subgroups in (S, F, L). Recall from 1.8 and 1.9 the definitions of C̄_{L}(C; E_{k}) and N̆_{L}(C; E) where E is a k-simplex in E.

5.3. Proposition. Fix a k-simplex E in E, namely a functor E : \underline{k} → F^{E}. There is a functor

ϵ : N̆_{L}(C; E) → Tr_{BAut_{F} (E)}(C̄_{L}(C; E))

which is fully faithful and natural with respect to inclusion of simplices. If E_{k} is fully F-centralised, its image is also skeletal and in particular induces homotopy equivalence |N̆_{L}(C; E)| ⟶^{≃} |C̄_{L}(C; E_{k})|_{hAut_{F} (E)}.

Proof. The objects of H := BAut_{F}(E) ∫ C̄_{L}(C; E) are pairs (P, f) where P ∈ C and f ∈ F(E, Z(P)). Morphisms are pairs (φ, g) where φ ∈ L(P, P′) and g ∈ Aut_{F}(E) such that f′ = π(φ)∘f ∘g (see §3). Since f, f′ are monomorphisms, g is determined by φ. Define ϵ : N̆_{L}(C; E) → H by

ϵ(P) = (P, E ⟶^{incl} Z(P) ≤ P) (5.4)

ϵ(P ⟶^{φ} P′) = (P ⟶^{φ} P′, π(φ)|_{E_{k}}^{−1}).

It is well defined and fully faithful by the definition of N̆_{L}(C; E). Naturality with respect to inclusion of simplices is readily verified. Consider an object (P, f) ∈ H. Note that f(E_{k}) ≤ Z(P), hence for g := f^{−1} ∈ Iso_{F}(f(E_{k}), E_{k}) we must have (see 2.3) N_{g} ⊇ C_{S}(f(E_{k})) ⊇ P. By axiom II of 2.3 we can extend g to an isomorphism h : P → P′ in F. Clearly P′ is in C because the latter is a collection. Fix a lift h̃ ∈ L(P, P′) for h. We have π(h̃) ∘ f = h ∘ incl_{f(E_{k})}^{P} ∘ g^{−1} = incl_{E_{k}}^{P′}. Therefore

(id_{E_{k}}, h̃) is an isomorphism (P, f) ⟶^{≅} (P′, incl_{E_{k}}^{P′}) in H. This shows that ϵ embeds N̆_{L}(C; E) into a skeletal subcategory of H and the result follows. □

proof of Theorem B. For every P ∈ C define ζ(P) = Ω_{p}Z(P), see 1.11. Note that if f : P → P′ is an isomorphism in F then f^{−1} : ζ(P′) → ζ(P) is an isomorphism in F. Also if P ≤ P′ in C then Z(P) ≤ Z(P′) because P and P′ are F-centric so their centres are equal to their S-centralisers. It easily follows that this assignment forms a functor

ζ : L^{C} → F^{E}^{op}.

Fix E ∈ E. Since every homomorphism f : E → Z(P) factors through ζ(P) we see that C̄_{L}(C; E) = (ζ ↓ E). In particular (3.7)

(5.5)  ζ_{∗}(⋆) : E ↦ C̄_{L}(C; E).

We now observe that F^{E} is an EI-category. The assignment E ↦ |E| gives rise to a height function in the sense of §4. Furthermore the set I of inclusions in F^{E} forms a poset where every morphism in F^{E} factors uniquely as an isomorphism followed by an element in I. We conclude that

s̄E = s̄_{I}(F^{E}) ≈ s̄(F^{E}).

There is an isomorphism τ : s(F^{E}^{op}) → s(F^{E}) which is the identity on objects while on morphisms set between E : \underline{k} → F^{E} and E′ : \underline{n} → F^{E}

Hom_{s(F^{E}^{op})}(E, E′) = Iso_{F^{E}}^{op}(ϵ^{∗}E, E′) =

Iso_{F^{E}} (E′, ϵ^{∗}E) ⟶_{≈}^{f↦f^{−1}} Iso_{F^{E}} (ϵ^{∗}E, E′) = Hom_{s(F^{E})}(E, E′).

The functor p : s(F^{E}^{op}) → F^{E}^{op} of 4.3 fits into a commutative diagram

[figure: s(F^{E}^{op}); p; τ; ≈; F^{E}^{op}; s(F^{E}); μ; _{F^{E}_{op}}]

where μ(E) = E_{k}. Since τ is an isomorphism, μ is right cofinal. We obtain a zigzag of functors

s̄dE ⟵^{π} s(F^{E}) ⟶^{μ} F^{E}^{op} ⟵^{ζ} L^{C} ⟶^{⋆} Cat.

Define

δ̃_{E} = π_{∗} ∘ μ^{∗} ∘ ζ_{∗}(⋆), and δ_{E} = |δ̃_{E}|.

Since μ is right cofinal, the cofinality theorem 3.2 and the pushdown theorem 3.3 imply a weak homotopy equivalence

hocolim_{s̄dE} δ_{E} ⟶^{≃} |L^{C}|.

This is point (1) of the theorem. Inspection of (3.5), 3.8 and 3.6 shows that this equivalence is given as the realization of a functor Tr_{s̄dE} δ̃_{E} → L^{C} where

(E, E_{k} ⟶^{f} P) ↦ P
(E → E′, g ∈ Aut_{F}(E′), P ⟶^{φ∈L} P′) ↦ φ.

Point (2) follows from (5.5) and proposition 4.6. Point (3) follows from proposition 1.10, inspection of ϵ in proposition 5.3, of (5.5) and the functor Tr_{s̄dE}(δ̃_{E}) → L^{C}. Point (4) is a consequence of proposition 4.6. □

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Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE , Scotland, U.K.

E-mail address: assaf@maths.abdn.ac.uk