Theory and Applications of Categories, Vol. 16, No. 28, 2006, pp. 771–784.

PULLBACK AND FINITE COPRODUCT PRESERVING FUNCTORS BETWEEN CATEGORIES OF PERMUTATION REPRESENTATIONS

ELANGO PANCHADCHARAM AND ROSS STREET

Abstract. Motivated by applications to Mackey functors, Serge Bouc [Bo] characterized pullback and finite coproduct preserving functors between categories of permutation representations of finite groups. Initially surprising to a category theorist, this result does have a categorical explanation which we provide.

1. Introduction

For a finite group G, we write G-set_{fin} for the category of finite (left) G-sets (that is, of permutation representations of G) and equivariant functions. We write Spn(G-set_{fin}) for the category whose morphisms are isomorphism classes of spans between finite Gsets. Coproducts in Spn(G-set_{fin}) are those of G-set_{fin} and composition in Spn(G-set_{fin}) involves pullbacks in G-set_{fin}.

According to Harald Lindner [Li], a Mackey functor M on a finite group H is a coproduct preserving functor M : Spn(H-set_{fin}) [figure] Mod_{k}. A functor F : G-set_{fin} [figure] H-set_{fin} which preserves pullbacks and finite coproducts will induce a functor

Spn(F) : Spn(G-set_{fin}) [figure] Spn(H-set_{fin})

preserving finite coproducts. By composition with Spn(F), each Mackey functor M on H will produce a Mackey functor M ∘ Spn(F) on G.

This observation led Bouc [Bo] to a systematic study of pullback and finite coproduct preserving functors F : G-set_{fin} [figure] H-set_{fin}. He characterized them in terms of G^{op} × Hsets A (where G^{op} is G with opposite multiplication). This perplexed us initially, as the category (G^{op} × H)-set of such A is equivalent to the category of finite colimit preserving functors L : G-set_{fin} [figure] H-set_{fin}; these L generally do not preserve pullbacks, while the F generally do not preserve coequalizers. Of course, Bouc’s construction of L from a left H-, right G-set A is quite different from the standard module theory construction of F from A. We shall explain the two constructions.

The authors are grateful for the support of the Australian Research Council Discovery Grant DP0346047, and the first author for the support of an Australian International Postgraduate Research Scholarship, and an International Macquarie University Research Scholarship.

Received by the editors 2006-09-15 and, in revised form, 2006-11-13.

Transmitted by RJ Wood. Published on 2006-11-15.

2000 Mathematics Subject Classification: 58A03, 18A30.

Key words and phrases: topos, permutation representations, limits and colimits, adjunction, Mackey functors.

\begin{matrix}c \\ ◯ \end{matrix} Elango Panchadcharam and Ross Street, 2006. Permission to copy for private use granted.

We put (g, h)a = hag for g ∈ G, h ∈ H and a in the (G^{op} × H)-set A, so that A becomes a left H-set and a right G-set. For each left G-set X, define the left H-set A ⊗_{G} X to be the quotient of the set A × X = {(a, x) | a ∈ A, x ∈ X} by the equivalence relation generated by

(ag, x) ∼ (a, gx), a ∈ A, x ∈ X, g ∈ G.

Write [a, x] for the equivalence class of (a, x) and define h[a, x] = [ha, x]. For A finite, this defines our functor L = A ⊗_{G} − : G-set_{fin} [figure] H-set_{fin} on objects; it is defined on morphisms f : X [figure] X′ by L(f)[a, x] = [a, f(x)]. Certainly L preserves all colimits that exist in G-set_{fin} since it has a right adjoint R : H-set_{fin} [figure] G-set_{fin} defined on the left H-set Y by R(Y) = H-set_{fin}(A, Y) with action (g, θ) [figure] gθ where (gθ)(a) = θ(ag). All this is classical “module” theory.

Now we turn to Bouc’s construction. Again let A be a (G^{op} × H)-set. Rather than a mere G-set X, we define a functor on all (K^{op} × G)-sets B where K, G, H are all finite groups. Put

A ∧_{G} B = {(a, b) ∈ A × B | g ∈ G, ag = a ⇒ there exists k ∈ K with gb = bk}.

This becomes a (K^{op} × G × H)-set via the action

(k, g, h)(a, b) = (hag^{−1}, gbk).

Then Bouc defines the (K^{op} × H)-set

A ∘_{G} B = (A ∧_{G} B)/G,

to be the set of orbits orb(a, b) = [a, b] of elements (a, b) of A ∧_{G} B under the action of G. In particular, when K = 1 and B = X ∈ G-set_{fin}, we obtain F (X) = A ∘_{G} X ∈ H-set_{fin}. This defines the functor

F : G-set_{fin} [figure] H-set_{fin}.

1.1. Theorem. [Bo] Suppose K, G and H are finite groups.

(i) If A is a finite (G^{op} × H)-set then the functor

A ∘_{G} − : G-set_{fin} [figure] H-set_{fin}

preserves finite coproducts and pullbacks. (ii) Every functor F : G-set_{fin} [figure] H-set_{fin} which preserves finite coproducts and pull-

backs is isomorphic to one of the form A ∘_{G} −. (iii) The functor F in (ii) preserves terminal objects if and only if A is transitive (con-

nected) as a right G-set_{fin}. (iv) If A is as in (i) and B is a finite (K^{op} × G)-set then the composite functor

\xymatrix{K-set_{fin} \ar[r]^{B∘_{K}−} & G-set_{fin} \ar[r]^{A∘_{G}−} & H-set_{fin}}

is isomorphic to (A ∘_{G} B) ∘_{K} −.

FUNCTORS BETWEEN CATEGORIES OF PERMUTATION REPRESENTATIONS 773

Our intention in the present paper is to provide a categorical explanation for this Theorem.

In Section 2, before turning to the problem of preserving pullbacks, we examine finite limit preserving functors from categories like G-set_{fin} to set_{fin}. We adapt the appropriate classical adjoint functor theorem to this “finite” situation. To make use of this for the purpose in hand, in Section 3, we need to adapt the result to include preservation of finite coproducts and reduce the further preservation of pullbacks to the finite limit case.

Section 4 interprets the work in the finite G-set case. In Section 5 we express the conclusions bicategorically. Implications for our original motivating work on Mackey functors are explained in the final Section 6.

2. Special representability theorem

In this section we provide a direct proof of the well-known representability theorem (see Chapter 5 [Ma]) for the case where “small” means “finite”.

Recall that an object Q of a category 𝒜 is called a cogenerator when, for all f, g : A [figure] B in 𝒜, if uf = ug for all u : B [figure] Q, then f = g.

A subobject of an object A of 𝒜 is an isomorphism class of monomorphisms m : S [figure] A; two such monomorphisms m : S [figure] A and m′ : S′ [figure] A are isomorphic when there is an invertible morphism h : S [figure] S′ with m′ ∘ h = m. We call 𝒜 finitely well powered when each object A has only finitely many subobjects. Write Sub(A) for the set of subobjects [m : S [figure] A] of A.

For each set X and object A of 𝒜, we write A^{X} for the object of 𝒜 for which there is a natural isomorphism

𝒜 (B, A^{X}) ≅ 𝒜 (B, A)^{X}

where Y^{X} is the set of functions from X to Y. Such an object may not exist; if 𝒜 has products indexed by X then A^{X} is the product of X copies of A.

We write set_{fin} for the category of finite sets and functions. A functor T : 𝒜 [figure] set_{fin} is representable when there is an object K ∈ 𝒜 and a natural isomorphism T ≅ 𝒜 (K, −).

2.1. Theorem. (Special representability theorem) Suppose 𝒜 is a category with the following properties:

(i) each homset 𝒜 (A, B) is finite; (ii) finite limits exist; (iii) there is a cogenerator Q; (iv) 𝒜 is finitely well powered. Then every finite limit preserving functor T : 𝒜 [figure] set_{fin} is representable.

Proof. Using (ii) and (iv), we have the object

P = ∏_{[S]∈Sub(Q^{T Q})} S^{T S}.

We shall prove that, for each A ∈ 𝒜 and a ∈ T A, there exists p ∈ T P and w : P [figure] A such that (T w)p = a. The following diagram defines δ, ι and S_{0}.

[figure: S_{0}; Q^{T Q}; m; Q^{𝒜 (A,Q)}; ι; Q; _{pr_{f}}; A; t; δ; _{pr_{(T f)a}}; f; pb]

Now δu = δv implies fu = fv for all f : A [figure] Q, so u = v by (iii). So δ is a monomorphism. So [S_{0}] ∈ Sub(Q^{T Q}). Since T preserves pullbacks, there is a unique s ∈ T S_{0} such that (T t)s = a and (T m)s transports to 1_{T Q} under T (Q^{T Q}) ≅ (T Q)^{T Q}. Let p transport to (1_{T S})_{[S]} under T P ≅ ∏_{[S]}(T S)^{T S}. Then we can define w to be the composite

\xymatrix{P \ar[r]^{^{pr^{[S0],s}}} & S_{0} \ar[r]^{t} & A}

with (T w)p = a.

Now let K be the equalizer of all the endomorphisms (including 1_{P}) of P (we are using (i)):

[figure: K.; P; l; P; l; e]

Since T preserves limits, there is a unique k ∈ T K with (T l)k = p. Define

θ_{A} : 𝒜 (K, A) [figure] T A

by θ_{A}(r) = (T r)k; this is natural in A. Moreover, θ_{A} is surjective since (K, k) clearly has the same property that we proved for (P, p).

It remains to prove θ_{A} injective. Suppose r and r′ : K [figure] A are such that (T r)k = (T r′)k.

Let n : U [figure] K be the equalizer of r and r′, and let u ∈ T U be unique with (T n)u = k. By the property of (P, p), there exists w : P [figure] U with (T w)p = u.

[figure: K; P; l; P; l; U; w; K; n; l]

From the definition of K, we have lnwl = l. Yet l is a monomorphism (since it is an equalizer), so nwl = 1 and r = rnwl = r′nwl = r′, as required.

For categories 𝒜 and 𝒳 admitting finite limits, write Lex(𝒜 , 𝒳 ) for the full subcategory of the functor category [𝒜 , 𝒳 ] consisting of the finite limit preserving functors.

2.2. Corollary. For a category 𝒜 satisfying the conditions of Theorem 2.1, the Yoneda embedding defines an equivalence of categories

𝒜^{op} ≃ Lex(𝒜 , set_{fin}) , A [figure] 𝒜 (A, −).

3. Finite coproducts

Suppose the category 𝒜 has finite coproducts. An object C of 𝒜 is called connected when the functor 𝒜 (C, −) : 𝒜 [figure] Set preserves finite coproducts. Write Conn(𝒜) for the full subcategory of 𝒜 consisting of the connected objects.

Write Cop(𝒜 , 𝒳 ) for the full subcategory of [𝒜 , 𝒳 ] consisting of the finite coproduct preserving functors. Also CopLex(𝒜 , 𝒳 ) consists of the finite coproduct and finite limit preserving functors. As an immediate consequence of Corollary 2.2 we have

3.1. Corollary. For a category 𝒜 with finite coproducts and the properties of Theorem 2.1, the Yoneda embedding defines an equivalence of categories

Conn(𝒜)^{op} ≃ CopLex(𝒜 , set_{fin}).

Suppose 𝒜 is a finitely complete category and T : 𝒜 [figure] set_{fin} is a functor. For each t ∈ T 1, define a functor T_{t} : 𝒜 [figure] set_{fin} using the universal property of the pullback

[figure: T_{t}A; T A; ι_{t}A; T 1.; T !; 1; t; pb]

Clearly T ≅ ∑_{t∈T 1}T_{t}. Taking A = 1 in the above pullback, we see that each T_{t} preserves

terminal objects. The following observations are obvious.

3.2. Proposition. Suppose 𝒜 is a finitely complete category, T : 𝒜 [figure] set_{fin} is a functor, and the functors T_{t} : 𝒜 [figure] set_{fin}, t ∈ T 1, are defined as above.

(i) If T preserves pullbacks then each T_{t} preserves finite limits. (ii) Each T_{t} preserves whatever coproducts that are preserved by T.

For any small category 𝒞, we write Fam(𝒞^{op}) for the free finite coproduct completion of 𝒞^{op}. The objects are families (C_{i})_{i∈I} of objects C_{i} of 𝒞 with indexing set I finite. A morphism (ξ, f) : (C_{i})_{i∈I} [figure] (D_{j})_{j∈J} consists of a function ξ : I [figure] J and a family f = (f_{i})_{i∈I} of morphisms f_{i} : C_{ξ(i)} [figure] D_{i} in 𝒞.

There is a functor

𝒵_{𝒞} : Fam(𝒞^{op}) [figure] [𝒞 , Set]

defined by

𝒵_{𝒞} (C_{i})_{i∈I} = ∑_{i∈I} 𝒞 (C_{i}, −)

which is fully faithful. So Fam(𝒞^{op}) is equivalent to the closure under finite coproducts of the representables in [𝒞 , Set].

We write Pb(𝒜 , 𝒳 ) for the full subcategory of [𝒜 , 𝒳 ] consisting of pullback preserving functors. Also CopPb(𝒜 , 𝒳 ) has objects restricted to those preserving finite coproducts and pullbacks.

3.3. Proposition. Suppose the category 𝒜 is as in Corollary 3.1. The functor 𝒵_{𝒜} induces an equivalence of categories

Fam(Conn(𝒜)^{op}) ≃ CopPb(𝒜 , set_{fin}).

Proof. Clearly Fam(Conn(𝒜)^{op}) is a full subcategory of Fam(𝒜^{op}) and 𝒵_{A} restricts to a fully faithful functor

Fam(Conn(𝒜)^{op}) [figure] [𝒜 , set_{fin}].

It remains to identify the essential image of this functor as those T : 𝒜 [figure] set_{fin} which preserve finite coproducts and pullbacks. However, we have seen in Proposition 3.2 that such a T has the form T ≅ ∑_{t∈T 1}T_{t} where each T_{t} preserves finite coproducts and is left

exact. By Corollary 3.1, we have

T_{t} ≅ 𝒜 (C_{t}, −)

where each C_{t} is connected.

4. Application to permutation representations

A permutation representation of a finite group G, also called a finite left G-set, is a finite set X together with a function G × X [figure] X , (g, x) [figure] gx, called the action such that

1x = x and g_{1}(g_{2}x) = (g_{1}g_{2})x.

If X and Y are such G-sets, a (left) G-morphism f : X [figure] Y is a function satisfying f(gx) = gf(x). We write G-set_{fin} for the category of finite left G-sets and left Gmorphisms.

The terminal object of G-set_{fin} is the set 1 with only one element with its unique action. The pullback in G-set_{fin} of two morphisms f : X [figure] Z and k : Y [figure] Z is given by {(x, y) ∈ X × Y |f(x) = k(y)} with componentwise action g(x, y) = (gx, gy). So G-set_{fin} has finite limits.

Since the set G-set_{fin}(X, Y) is a subset of the set set_{fin}(X, Y) = Y^{X}, it is finite.

The set 𝒫G of subsets of G becomes a G-set by defining the action as

gS = {h ∈ G | hg ∈ S}

for S ⊆ G and g ∈ G. For each x ∈ X ∈ G-set_{fin}, we can define a G-morphism χ_{x} : X [figure] 𝒫G by

χ_{x}(z) = {h ∈ G | hz = x}.

If x, y ∈ X ∈ G-set_{fin} then χ_{x}(x) = χ_{x}(y) implies 1 ∈ χ_{x}(y), so y = x. It follows that 𝒫G is a cogenerator for G-set_{fin}.

Subobjects of X ∈ G-set_{fin} are in bijection with sub-G-sets of X. So G-set_{fin} is finitely well powered.

From Section 2, we therefore have:

4.1. Corollary. Every limit preserving functor T : G-set_{fin} [figure] set_{fin} is representable. The Yoneda embedding induces an equivalence of categories

G-set_{fin}^{op} ≃ Lex(G-set_{fin}, set_{fin}).

Recall that a G-set X is called transitive when it is non-empty and, for all x, y ∈ X, there exists g ∈ G with gx = y.

For any G-set X and any x ∈ X, we put

stab(x) = {g ∈ G | gx = x}

which is a subgroup of G called the stabilizer of x. We also put

orb(x) = {gx | g ∈ G}

which is a transitive sub-G-set X called the orbit of x. We write X/G for the set of orbits which can be regarded as a G-set_{fin} with trivial action so that orb : X [figure] X/G is a surjective G-morphism. If u ∈ X/G, we also write X_{u} for the orbit u as a sub-G-set of X. So every G-set is the disjoint union of its orbits.

The empty coproduct is the empty set 0 with its unique action. The coproduct of two G-sets X and Y is their disjoint union X + Y with action such that the coprojections X [figure] X + Y and Y [figure] X + Y are G-morphisms. So every G-set X is a coproduct

X ≅ ∑_{u∈X/G} X_{u}

of transitive G-sets (the orbits X_{u}).

Each subgroup H of G determines a transitive G-set

G/H = {xH | x ∈ G}

where xH = {xh | h ∈ H} is the left coset of H containing x, and where the action is

g(xH) = (gx)H.

Every transitive G-set X is isomorphic to one of the form G/H; we can take H = stab(x) for any x ∈ X.

The G-morphisms f : G/H [figure] X are in bijection with those x ∈ X such that H ≤ stab(x). The G-sets G/H and G/K are isomorphic if and only if the subgroups H and K are conjugate (that is, there exists x ∈ G with Hx = xK).

We provide a proof of the following well-known fact.

4.2. Proposition. A finite G-set X is transitive if and only if X is a connected object of G-set_{fin}.

Proof. A G-set X is non-empty if and only if G-set_{fin}(X, 0) is empty; that is, if and only if G-set_{fin}(X, −) preserves empty coproducts.

A morphism G/H [figure] Y + Z is determined by an element of Y + Z stable under H; such an element must either be an element of Y or an element of Z stable under H. So transitive G-sets are connected.

Assume X is connected. We have already seen that X is non-empty so choose x ∈ X. Then

X = orb(x) + U

for some sub-G-set U of X. We therefore have the canonical function

G-set_{fin}(X, orb(x)) + G-set_{fin}(X, U) [figure] G-set_{fin}(X, X)

which is invertible since X is connected. So the identity function X [figure] X is in the image of the canonical function and so factors through orb(x) ⊆ X or U ⊆ X. Since x ∉ U, we must have orb(x) = X. So X is connected.

We have thus identified Conn(G-set_{fin}) as consisting of the transitive G-sets. This category has a finite skeleton 𝒞_{G} since there are only finitely many G-sets of the form G/H. Corollary 3.1 yields:

4.3. Corollary. The Yoneda embedding induces an equivalence of categories

𝒞_{G}^{op} ≃ CopLex(G-set_{fin}, set_{fin}).

Let N : 𝒞_{G} [figure] G-set_{fin} denote the inclusion functor and define the functor

Ñ : G-set_{fin} [figure] [𝒞_{G}^{op}, set_{fin}]

by ÑX = G-set_{fin}(N−, X).

4.4. Proposition. The functor Ñ induces an equivalence of categories

G-set_{fin} ≃ Fam(𝒞_{G}^{op}).

Proof. We first prove that N is dense; that is, that Ñ is fully faithful. Let θ : ÑX [figure] ÑY be a natural transformation. For each u : C [figure] D in 𝒞_{G} we have a commutative square

[figure: G-set_{fin}(D, X); G-set_{fin}(D, Y); θ_{D}; G-set_{fin}(C, Y).; −∘u; G-set_{fin}(C, X); −∘u; θ_{C}]

Since the single-object full subcategory of G-set_{fin} consisting of G is dense (G^{op} [figure] G-set_{fin} is a Yoneda embedding), by restricting C and D to be equal to G ∈ 𝒞_{G}, we obtain a Gmorphism f : X [figure] Y defined uniquely by f(x) = θ_{G}(x̂)(1) where x̂ : G [figure] X is given by x̂(g) = gx. Then, for all w : D [figure] X and d ∈ D, the above commutative square, with C = G, yields

θ_{D}(w)(d) = (θ_{D}(w) ∘ d̂)(1) = θ_{G}(w ∘ d̂)(1) = θ_{G}(ŵ ∘ d)(1) = (f ∘ w)(d).

So θ_{D} = G-set_{fin}(D, f) for a unique G-morphism f.

The proof of the equivalence of categories will be completed by characterizing the essential image of Ñ as finite coproducts of representables in [𝒞_{G}^{op}, set_{fin}]. If F ∈ [𝒞_{G}^{op}, set_{fin}] is a finite coproduct of representables then we have a finite family (C_{i})_{i∈I} of objects of 𝒞_{G} and an isomorphism F ≅ ∑_{i}𝒞 (−, C_{i}). We have the calculation:

∑_{i} 𝒞 (−, C_{i}) ≅ ∑_{i} G-set_{fin}(N−, C_{i})

≅ G-set_{fin}(N−, ∑_{i} C_{i})

≅ Ñ(∑_{i} C_{i}).

So F is in the essential image of Ñ. Conversely, every X ∈ G-set_{fin} is a coproduct X ≅ ∑_{i}C_{i} of connected G-sets. So the same calculation, read from bottom to top, shows

that Ñ(X) is a finite coproduct of representables.

4.5. Corollary. There is an equivalence of categories

G-set_{fin} ≃ CopPb(G-set_{fin}, set_{fin})

taking the left G-set C to the functor

∑_{w∈C/G} G-set_{fin}(C_{w}, −)

where C_{w} is the orbit w as a sub-G-set of C.

There is an isomorphism of categories

ℓ_{G} : G^{op}-set_{fin} [figure] G-set_{fin}

which preserves the underlying sets. If A is a right G-set then ℓ_{G}A = A as a set with left action ga in ℓ_{G}A equal to ag^{−1} in A. As a special case of the construction in the Introduction, for a right G-set A and a left G-set X, we have

A ∧_{G} X = {(a, x) ∈ ℓ_{G}A × X | stab(a) ≤ stab(x)} and
A ∘_{G} X = (A ∧_{G} X)/G.

Each (a, x) ∈ A ∧_{G} X defines a G-morphism

θ_{X}(a, x) : ℓ_{G}A_{u} [figure] X

where u = orb(a) and θ_{X}(a, x)(ag^{−1}) = gx (which is well defined since ag_{1}^{−1} = ag_{2}^{−1} ⇒ g_{2}^{−1}g_{1} ∈ stab(a) ⇒ g_{2}^{−1}g_{1} ∈ stab(x) ⇒ g_{1}x = g_{2}x). This defines a function

θ_{X} : A ∧_{G} X [figure] ∑_{u∈G\A} G-set_{fin}(ℓ_{G}A_{u}, X)

naturally in X ∈ G-set_{fin} (where G\A is the set of orbits of the right action). Clearly

θ_{X}(a, x) = θ_{X}(b, y) if and only if orb(a, x) = orb(b, y).

This proves:

4.6. Proposition. For all A ∈ G^{op}-set_{fin}, the natural transformation θ induces a natural isomorphism

θ̄: A ∘_{G} − ≅ ∑_{u∈G\A} G-set_{fin}(ℓ_{G}A_{u}, −)

between functors from G-set_{fin} to set_{fin}.

4.7. Corollary. There is an equivalence of categories

G^{op}-set_{fin} ≃ CopPb(G-set_{fin}, set_{fin}) , A [figure] A ∘_{G} −.

5. A bicategory of finite groups

The goal of this section is to consolidate our results in terms of a homomorphism of bicategories which is an equivalence on homcategories. We construct a bicategory whose objects are finite groups and whose morphisms are permutation representations between them. This bicategory is the domain of the homomorphism. The codomain is the 2category of categories of the form G-set_{fin} and pullback-and-finite-coproduct-preserving functors between them.

Suppose G and H are finite groups. There is a monad H × − on G^{op}-set_{fin} whose Eilenberg-Moore algebras are the (G^{op} × H)-sets. For the endofunctor

H × − : G^{op}-set_{fin} [figure] G^{op}-set_{fin},

we simply regard H as a trivial G^{op}-set_{fin} (hg = h) and take A to H × A in G^{op}-set_{fin}. The unit A [figure] H × A and multiplication H × H × A [figure] H × A for the monad are defined by a [figure] (1, a) and (h_{1}, h_{2}, a) [figure] (h_{1}h_{2}, a).

In the case G = 1, we have the monad H × − on set_{fin}. This lifts pointwise to a monad H × − on [G-set_{fin}, set_{fin}]. There is a canonical isomorphism

[figure: G^{op}-set_{fin}; [G-set_{fin}, set_{fin}]; [G-set_{fin}, set_{fin}],; H×−; G^{op}-set_{fin}; H×−; ≅]

which is compatible with the monad structures, where the horizontal functors are both A [figure] A ∘_{G} −. The component

A ∘_{G} (H × X) ≅ H × (A ∘_{G} X)

of the isomorphism at X is induced by the G-set isomorphism

ℓ_{G}A × (H × X) ≅ H × (ℓ_{G}A × X) , (a, (h, x)) [figure] (h, (a, x)).

Notice that, for (h, x) ∈ H × X, stab(h, x) = stab(x). It follows that A [figure] A ∘_{G} − induces a functor

(G^{op} × H)-set [figure] [G-set_{fin}, H-set_{fin}].

If F : G-set_{fin} [figure] set_{fin} preserves pullbacks and finite coproducts, so does H × F −. Also, the forgetful functor H-set_{fin} [figure] set_{fin} creates both finite coproducts and pullbacks. Therefore, Corollary 4.7 yields:

5.1. Theorem. There is an equivalence of categories

(G^{op} × H)-set ≃ CopPb(G-set_{fin}, H-set_{fin}) , A [figure] A ∘_{G} −.

Parts (i), (ii) and (iii) of Theorem 1.1 follows from Theorem 5.1. It is also clear, in the setting of Theorem 1.1(iv), that there exists a (K^{op} × H)-set C such that

A ∘_{G} (B ∘_{K} −) ≅ C ∘_{K} −

since the composite of pullback and finite coproduct preserving functors also preserves them. It remains to identify C as A ∘_{G} B. We do this directly.

5.2. Proposition. If A, B and Z are respectively (G^{op} × H)-, (K^{op} × G)-, and K-sets then

A ∘_{G} (B ∘_{K} Z) ≅ (A ∘_{G} B) ∘_{K} Z , [a, [b, z]] [figure] [[a, b], z]

is an isomorphism of H-sets.

Proof. To say (a, [b, z]) ∈ A ∧_{G} (B ∘_{K} Z) is to say that ag = a implies g[b, z] = [b, z]; that is, there exists k ∈ K such that gb = bk and hz = z. In particular, this means that (a, b) ∈ A∧_{G} B. We need to see that ([a, b], z) ∈ (A∘_{G} B)∧_{K} Z. So suppose [a, b]k = [a, b]. Then there exists g ∈ G with a = ag and bk = gb. The former implies there is k_{1} ∈ K such that gb = bk_{1} and k_{1}z = z. Then bk = gb = bk_{1}, so bkk_{1}^{−1} = b. Since (b, z) ∈ B ∧_{K} Z, we have kk_{1}^{−1}z = z; so kz = z. One also sees that

[a, [b, z]] = [a′, [b′, z′]] ⟺ ∃g ∈ G, k ∈ K : a = a′g, gb = b′k, kz = z′
⟺ [[a, b], z] = [[a′, b′], z′].

This proves the bijection. Clearly the H-actions correspond.

Let CopPb denote the 2-category whose objects are categories admitting pullbacks and finite coproducts, whose morphisms are functors which preserve these, and all natural transformations between such functors. This is a sub-2-category of the 2-category Cat of small categories. Let CopLex be the sub-2-category of CopPb consisting of the categories which also have terminal objects (and so are finitely complete), and the functors which preserve them.

We can now summarize the results in:

5.3. Theorem. There is a bicategory Bouc whose objects are finite groups, whose homcategories are

Bouc(G, H) = (G^{op} × H)-set,

and whose composition functors are

Bouc(G, H) × Bouc(K, G) [figure] Bouc(K, H) , (A, B) [figure] A ∘_{G} B.

There is a homomorphism of bicategories

Φ : Bouc [figure] CopPb

defined on objects by ΦG = G-set_{fin} and on homcategories by the equivalences of Theorem 5.1. The restriction of the morphisms A : G [figure] H of Bouc to those G^{op} × H-sets which are connected as right G-sets is a sub-bicategory Boucc of Bouc, and Φ restricts to a homomorphism

Φ_{C} : Boucc [figure] CopLex

which is also an equivalence on homcategories.

6. Application to Mackey functors

Given a finite group G, there is a bicategory of spans in the category G-set_{fin} [Bé]. We write Spn(G-set_{fin}) for the category obtained by taking isomorphism classes of spans as morphisms. This category is compact closed and the coproduct in G-set_{fin} becomes direct sum (that is, it is also the product) in Spn(G-set_{fin}). These matters are made explicit in [PS].

We fix a field k and write vect for the category of finite dimensional vector spaces over k with linear functions as morphisms. A finite dimensional Mackey functor on G is a functor M : Spn(G-set_{fin}) [figure] vect which preserves finite direct sums. We write Mky_{fin}(G) for the category of finite dimensional Mackey functors on G; the morphisms are natural transformations. It is an abelian k-linear category.

Each pullback preserving functor F : G-set_{fin} [figure] H-set_{fin} induces a functor Spn(F) : Spn(G-set_{fin}) [figure] Spn(H-set_{fin}) (since composition of spans only involves pullbacks). If F also preserves finite coproducts then Spn(F) preserves direct sums. In that case, we obtain an exact functor

F̄ : Mky_{fin}(H) [figure] Mky_{fin}(G)

defined by F̄(N) = N ∘ Spn(F) for all N ∈ Mky_{fin}(H). Moreover, F̄ has a left adjoint

Mky_{fin}(F) : Mky_{fin}(G) [figure] Mky_{fin}(H)

defined by

Mky_{fin}(F)(M) = ∫ ^{C} Spn(H-set_{fin})(F (C), −) ⊗ MC

where C runs over the connected G-sets as objects of Spn(G-set_{fin}) and the tensor product is that of additive (commutative) monoids.

Let AbCat_{k} denote the 2-category of abelian k-linear categories, k-linear functors with right exact right adjoints, and natural transformations.

6.1. Corollary. There is a homomorphism of bicategories

Mky_{fin} : Bouc [figure] AbCat_{k}
(A : G [figure] H) [figure] (Mky_{fin}(A ∘_{G} −) : Mky_{fin}(G) [figure] Mky_{fin}(H)).

Actually, Mky_{fin}(G) is much more than an abelian k-linear category. It is monoidal under the Day convolution tensor product [Da] using the monoidal structure on Spn(G-set_{fin}) coming from cartesian product in G-set_{fin}. In fact, Mky_{fin}(G) is ∗-autonomous in the sense of [Ba]; details can be found in [PS].

If F : G-set_{fin} [figure] H-set_{fin} is left exact and finite coproduct preserving then Spn(F) is strong monoidal (=tensor product preserving). It follows (for example from [DS]) that Mky_{fin}(F) is strong monoidal. It also preserves the ∗-autonomous structure.

Let ∗-AbCat_{k} denote the 2-category of ∗-autonomous monoidal abelian k-linear categories, ∗-preserving strong-monoidal k-linear functors with right exact right adjoints, and natural transformations.

6.2. Corollary. The homomorphism of bicategories in Corollary 6.1 induces a homomorphism

Mky_{fin} : Boucc [figure] ∗-AbCat_{k}.

References

[Ba] M. Barr, ∗-Autonomous Categories, Lecture Notes in Math., 752 (Springer-Verlag, Berlin, 1979).

[Bé] J. Bénabou, Introduction to bicategories, Lecture Notes in Math., 47 (Reports of the Midwest Category Seminar) (Springer-Verlag, Berlin, 1967), 1–77.

[Bo] S. Bouc, Construction de foncteurs entre catégories de G-ensembles, J. Algebra, 183 (1996), 737–825.

[Da] B. J. Day, On closed categories of functors, Lecture Notes in Math., 137 (Springer, Berlin, 1970), 1–38.

[DS] B. Day and R. Street, Kan extensions along promonoidal functors, Theory and Applic. of Categories, 1 (1995), 72–77.

[Li] H. Lindner, A remark on Mackey functors, Manuscripta Math., 18 (1976), 273–278.

[Ma] S. Mac Lane, Categories for the Working Mathematician, Lecture Notes in Math., 5 (Springer-Verlag, New York, 1971).

[PS] E. Panchadcharam and R. Street, Mackey functors on compact closed categories, submitted.

Centre of Australian Category Theory, Macquarie University, New South Wales, 2109, Australia

Email: elango@maths.mq.edu.au
      street@maths.mq.edu.au

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