Low Dimensional Group Cohomology as Monoidal Structures

Low Dimensional Group Cohomology as Monoidal StructuresAuthor(s): Barry MitchellSource: American Journal of Mathematics, Vol. 105, No. 5 (Oct., 1983), pp. 1049-1066Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2374333 .

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LOW DIMENSIONAL GROUP COHOMOLOGY AS MONOIDAL STRUCTURES

By BARRY MITCHELL

Two monoidal structures on a category V are equivalent if there is a bimonoidal (that is, strict monoidal) structure on the identity functor 1 v, using one monoidal structure in the domain and the other in the range. Also if V and V are monoidal categories, then two bimonoidal structures on a functor T: V -+ V are equivalent if there is a monoidal isomorphism T - T using one bimonoidal structure in the domain and the other in the range.

Let G be a group acting on a monoidal category V. This means that for each x e G, there is a bimonoidal equivalence Tx: V -+ V and monoidal isomorphisms

(1) TXTY - T,, T1 lv

making a couple of obvious diagrams commute. Such an action induces an action of G on the abelian group Z*V of automorphisms of 1 v. Let GV denote the category of G-graded objects of V (that is, the direct product of G copies of V). If V has coproducts, we can define a tensor product in GV by the rule

(2) (A 0)B)z= ( Ax ( TxBY' xy - z

Under the assumptions that the tensor product of V preserve coproducts and epimorphisms and that the unit of this tensor product be a generator for V, we show:

THEOREM 1. The equivalence classes of monoidal structures on GV using the tensor product (2) are in 1 - 1 correspondence with the elements

Manuscript received February 9, 1981. Manuscript received revised May 17, 1983. Work supported by NSF grant MCS-7703645.

1049



1050 BARRY MITCHELL

of H3(G, Z*V). Moreover, the equivalence classes of bimonoidal structures on 1GV, using any one of the above monoidal structures in both domain and range, are in 1 - 1 correspondence with the elements of H2(G, Z*V).

Two symmetric monoidal structures on a category V are equivalent if there is a symmetric bimonoidal structure on lv making the monoidal structures equivalent. Now for GV with the tensor product (2) to have a symmetric monoidal structure at all, one must assume that V is symmetric monoidal and that G is abelian and acts trivially on V (that is, Tx = 1 v for all x e G with the isomorphisms (1) identities). In this case the monoidal structure on GV using the tensor product (2) and the trivial 3-cocycle will be referred to as the trivial monoidal structure. Then again with the above blanket assumptions on V, we show:

THEOREM 2. If G is abelian and V is symmetric monoidal, then the equivalence classes of symmetric structures on the trivial monoidal structure on GV are in 1 - 1 correspondence with the equivalence classes of bilinear, antisymmetric maps f : G X G -+ Z*V, where two such maps f, f' are equivalent if there is a 2-dimensional cocycle h such that

f'(x, y) -f (x, y) = h(x, y) - h(y, x)

for all x, y e G. An immediate consequence of the above theorems, using the fact that

the group of integers has cohomological dimension one, is that if K is a commutative ring, then up to equivalence there is precisely one monoidal structure on the category of Z-graded K-modules (with the usual graded tensor product), and the symmetries for this structure are in 1 - 1 correspondence with the elements k e K such that k 2 = 1. In particular, if K is a domain, we find that the only symmetries are given by

ap (0 bq - bq (0 ap and ap (? bq _ (-1)Pqbq (? ap.

Finally, if we start with an abelian group K on which a group G acts, then we can take V to be SetsK (so that Z*V - K as G-modules), in which case Theorem 1 gives new interpretations of the cohomology groups H3(G, K) and H2(G, K).

1. Preliminaries. If S and T are functors with common domain and range, then Nat(S, T) denotes the class of all natural transformations from



LOW DIMENSIONAL GROUP COHOMOLOGY 1051

S to T. The class of all endomorphisms of the identity functor 1 Aof a category A is called the center of A and is denoted ZA. Then composition gives ZA the structure of a commutative monoid (neglecting the fact that it may not be a set). The units of A are the automorphisms of 1A. They form a subgroup of ZA, denoted Z*A.

LEMMA 1.1. Consider functors

F T F' C= A,B= C'

G GI

where T is an equivalence. Then there are bijections

Nat(F, G) Nat(TF, TG)

Nat(F', GI) Nat(F'T, G'T)

given by q F-+ T r, and q' -+ q' * T respectively. In particular we have isomorphisms

ZA = Nat(T, T) = ZB

of monoids. The proof is straightforward. Note that if ZT:ZA -+ ZB denotes the

composite isomorphisms in the lemma, then ZT is given by

Tk= ZT(k)T, k eZA.

It follows easily that if S: B -+ C is also an equivalence, then ZSZT = ZST, and of course Z1A = identity. Now if A: T = T' is a natural isomorphism with T and T' equivalences, and if 1 T = Tk where k e ZA and 1 e ZB, then the diagram

TkA TA -Tk TA

T'A - 1- T'A

commutes whether the bottom morphism is T'kA (by naturality of b) or IT'A (by naturality of 1). Thus 1 T' = T' *k, and consequently ZT' = ZT.



1052 BARRY MITCHELL

Summarizing and ignoring set theory, we can say that Z is a functor from the category of categories and natural isomorphism classes of equivalences to the category of commutative monoids and isomorphisms. Composing with the functor which assigns to each monoid its subgroup of invertible elements, we see that Z* can likewise be described as a functor with values in the category of abelian groups and isomorphisms.

LEMMA 1.2. Let I be a generator for a category V, and let S, T: V -- V befunctors where S preserves epimorphic families. Then the map

Nat(S, T) - V(SI, TI)

defined by q + q/ is injective. In particular

(1) ZV -+ V(I, I)

is an injective homomorphism of monoids.

Proof. Suppose q1I = q. Then for any A e V and any morphism f:I -+ A, we have

?1AS(f) = T(f),q1 = T(f),qj = qAS(f).

Since S preserves epimorphic families, it follows that qA = m4A and since A is arbitrary, this gives q = '.

In the sequel we shall be considering a bifunctor 0: V X V -+ V. For objects A and B, we shall usually write AB in place of A (0 B. However for morphisms f and g we shall always write f (0 g so as not to confuse with composition. An object I e V is a left unit for ?9 if there is a natural isomorphism 1 = 1A :IA -+ A. Likewise I is a right unit if there is a natural isomorphism r = rA :AI -+ A. If I is a left unit and J is a right unit, then we have I = IJ = J, and thus AI = AJ = A. Therefore we may always assume that the left unit is the same as the right unit, providing both exist, in which case we say simply that I is a unit for 0&. We claim that we may then assume rI = l. For if this is not the case, define rA as the composite

AI 1IrI AI rA A.

Then using naturality of 1, we have

rj = rl ll7r7 11, = l.




LEMMA 1.3. If I is a unit for X, then (1) is a split epimorphism of monoids. Hence if I is also a generatorfor V, then (1) is an isomorphism of monoids. Consequently we obtain an isomorphism

Z*V = AutI

of groups.

Proof. If k e V(I, I) then l(k X 1)1'- e ZV. This provides a monoid homomorphism V(I, I) -+ ZV. By naturality of r, the diagram

II -

k(gl ||k

II - ' I rI

is commutative. Since we may assume II r1, it follows that V(I, I) ZV -+ V(I, I) is the identity.

In the following lemma we need only assume that we have a bifunctor (0: A X V -+ A. In this case we can still talk of a right unit for ?, that is, an object I e V for which there is a natural isomorphism r :AI -> A.

LEMMA 1.4. Suppose I is a generator for V and a right unit for 0: A X V A, and that A@_ preserves epimorphic families for all A e A. If 5, T: C A are any functors, then the map k F k 0& 1 induces a 1 - 1 correspondence

Nat(S, T) -+ Nat((S-)(&-, (T4)(&-).

Moreover, kAV = kA 0 1 v for all k e ZA.

Proof. For k:S T, define

,c,v kc0 lv:SC0 V TC? V.

Then q is natural by naturality of k and bifunctoriality of?. On the other hand given q, we can define k by commutativity of

(2) SC I--I r > SC

T7C,i kC

T f

TC (&)I TC



1054 BARRY MITCHELL

Then k is natural by naturality of r and q. Naturality of r also yields that k N-+ q F-+ k is the identity. On the other hand if we start with q and define k by (2), then for fixed C we have that c, v and k c ?) 1 v are natural transformations of functors of V which agree when V = I. Since I is a generator and SC(?- preserves epimorphic families, it follows from Lemma 1.2 that

flc,v = kc(0 1v for all V. Now if k e ZA, then kAI= kA (0 1I by naturality of k. Thus by the

above (with S = T = lA), we have kAV = kA(Dlvfor all Ve V andA E A.

Relative to a bifunctor 0: A X V A and equivalences T1,

Tn :V -+ V, let us define

Sn :A XVnA

inductively by

SoA = A, Sn (A Vl, . .. ., Vn) = Sn-I(Aq V, * .. *, Vn-1) (9 Tn Vn

COROLLARY 1.5. Under the hypothesis of 1.4, the map k H* k Sn defines an isomorphism ZA = Nat(Sn , Sn).

Proof. Define A 3 V = A ?& Tn V. Since Tn is an equivalence, we have I - TnJ for some J, and it follows that D has a right unit J which is a generator. By induction, k F-+ k Sn-1 is an isomorphism. Therefore by Lemma 1.4,

k + (k *Sn-1) 09 Tn = k * (Sn-1T () T = k*Sn

is an isomorphism.

2. Monoidal categories. Recall that a monoidal category is a category V equipped with a bifunctor 0V X V -+ V, an object I e V, and natural isomorphisms

a = aABc:A(BC) (AB)C




such that the following diagrams commute:

I A (B(CD)) a-- (AB)(CD) '- ((AB)C)D

i?a|' a?l

A((BC)D) a (A(BC))D

II~~~~~~~ A(IB) a (AI)B

101 \ /rl0

AB

If V and V are monoidal categories, then a monoidal functor T: V V is a functor equipped with a natural transformation

b = bAB:TATB -+ T(AB)

and a morphism

u:I-+ TI

such that the following diagrams commute:

III TA(TBTC) '-:%f TAT(BC) - T(A(BC))

4^ Ta

(TATB)TC b-: T(AB)TC -b- T((AB)C)

IV ITA --- TA V TAI TA

u091 Ti lOu 4' Tr

TITA b Vo T(IA) TA TI b,T(AI)

When b and u are isomorphisms, commutativity of II, III, and IV implies that of V. In this case T is called bimonoidal. There is an obvious monoidal



1056 BARRY MITCHELL

structure on the composite of two monoidal functors, called the composite monoidal structure. Also taking b and u identities, we obtain the identity monoidal structure on 1v. All told, monoidal categories and monoidal functors form an illegitimate category.

The definition of a comonoidal functor is had by reversing the direc- tion of b and u. A bimonoidal functor has, of course, both monoidal and comonoidal structure. On the other hand the example of the Alexander- Whitney and Eilenberg-MacLane maps in simplicial abelian group theory shows that a functor can have at once monoidal and comonoidal structure without having bimonoidal structure [6, page 241]. If S is left adjoint to a functor T between monoidal categories, then one can show that the monoidal structures on T are in 1 - 1 correspondence with the comonoidal structures on S.

If T, T': V --> V are monoidal functors, then a natural transformation (isomorphism) q: T -+ T' is a monoidal transformation (isomorphism) if the diagrams

bA VI TATB -3- T(AB) VII I

T'AT'B - T'(AB) TI T'I

commute. When T and T' are bimonoidal and q is an isomorphism, commutativity of VII follows from that of IV and VI. A composite of monoidal transformations is monoidal, and in this way the monoidal functors between two monoidal categories form an illegitimate category.

Two monoidal structures on a category V are equivalent if there exists a bimonoidal structure on 1 v using one monoidal structure in the domain and the other in the range. Two monoidal structures on a functor T are equivalent if there is a monoidal isomorphism q: T -+ T using one monoidal structure in the domain and the other in the range.

In the following section we shall be classifying all equivalence classes of monoidal structures on a certain category using a fixed tensor product 0. It will then suffice to consider a fixed unit object for 0&. For if V is a monoidal category with unitI and u :J -I I is any isomorphism, then defin- ing a' = a and / and r as the composites

IA , IA - A, Ai l AI ' A,




we obtain a monoidal structure on V which is equivalent to the original one using the given isomorphism u and taking b to be the identity.

Example 1. If K is a commutative ring, then we consider Mod K as a monoidal category in the usual way. A homomorphism u :K -+ L of commutative rings induces the restriction functor T: ModL -+ Mod K which is monoidal using the obvious natural transformation

b:A(0KB -A0LB, A,B eModL.

Note that b is an isomorphism if u is surjective (or even a ring epimorphism). However T is not bimonoidal unless u is an isomorphism. On the other hand the left adjoint L0K of T is always bimonoidal.

Example 2. This is the nonadditive analogue of Example 1. If K is a commutative monoid and A and B are K-sets, then A (K B is defined as the quotient set A X B! - where - is the equivalence relation generated by relations of the form (ka, b) - (a, kb), k e K, a e A, b e B. This gives the category SetsK a monoidal structure with the obvious morphisms a, 1, and r, and the discussion of Example 1 is valid relative to a homomorphism u :K -+ L of commutative monoids.

If V is any category and G is a monoid, then an action of G on V consists of a family of functors Tx V -+ V, x e G, together with natural isomorphisms

A = Yx: TX TY - TXY I 0:Tj - lv

such that the diagrams

VIII T XTY TZ T TYZ IX TX - T.x

it .Tz | TX 0 X

TXY TZ ' TXYZ TX

are commutative. Note that T1 is an equivalence of categories. Also the diagram

T TZ A - T,z

- Tz \

TZ



1058 BARRY MITCHELL

is commutative. To see this, take x = y = 1 in VIII, and use IX. We obtain

AM(T ,) = ,u(,u T.) = 0. TZ)

Since A is an isomorphism, this gives T1 * A = T, * 0 * Tz, and so since T1 is faithful this gives A = 0* Tz. If G is a group, then all of the Tx are equivalences of categories, and from the discussion at the end of Lemma 1. 1, we get an action of G on the abelian group Z*V. The action is given explicitly byTxk = (xk)Tx.

If V is monoidal, then the monoid G acts on the monoidal category V if it does so in the above sense, with the additional conditions that the functors Tx be bimonoidal, and that the natural isomorphisms A be monoidal, where Tx Ty is given the composite monoidal structure. Since T1 * 0 = A and

A is monoidal, it follows that 0 is also monoidal if 1 v is understood to have the identity monoidal structure.

Example 3. Let G be a monoid acting on a commutative ring K. Then each element x e G induces a ring endomorphism of K, and hence a bimonoidal functor Tx Mod K -+ Mod K given by

TxM = Kx OK M

Here Kx is K as K - K bimodule, with the ring element k acting on the left of the module element a by ring multiplication ka and on the right by a(xk). (See Example 1, with L = K.) The obvious choices of A and 0 then give an action of G on the monoidal category Mod K. When G is a group,

T. is naturally isomorphic to the restriction functor Mod K- Mod K determined by the ring isomorphism x-1 :K - K, and A and 0 may be taken to be identities. In this case the action of G on Z(Mod K) = K agrees with the original action of G on K.

Example 4. The discussion of Example 3 is valid with K a commutative monoid and Mod K replaced by SetsK. Here, of course, K may be an abelian group, hence a G-module, and if G is a group, then Z(SetsK) = K as G-modules.

Example 5. If X is a topological space and K is a commutative ring, let ShKX denote the category of sheaves of K-modules over X. If G is a group acting on X, then for x E G we can define

TX:ShKX -+ ShKX




by pulling back along the homeomorphism of X induced by x-1. This yields an action of G on the monoidal category ShKX. We note that this example differs from Examples 3 and 4 in that the unit of the tensor product is not a generator.

3. Graded categories. Throughout this section, V will be a monoidal category with a fixed monoidal structure (D, I, a, 1, r, and G will be a monoid acting on V as a monoidal category. We shall also assume that V has coproducts and that (0 and the Tx preserve them. Thus, in particular, V has an initial object 0 (the empty coproduct) and (D and Tx preserve it. We let GV denote the product of G copies of V. Then an object of GV is a G-tuple A = (Ax), x e G, and is called a G-graded object of V. We define a functor 0: GV X GV -+ GV by the rule

(1) (AB)z = (0 AxTxB xy =z

Then 0 preserves coproducts in GV, and so since any object A of GV can be considered as the coproduct of the objects A x (that is, A x in dimension x and 0 in other dimensions) and any morphism f :A -+ B in GV can be considered as the coproduct of morphismsfx :Ax -+ B., it follows that any natural isomorphism A(BC) -+ (AB)C in GV is determined by a family of natural isomorphisms

AxTx(ByTyCz) -+ (AXTXBy)TxyCz

in V. Now one such family is the family ?exyz of composites

(2) ATx(BTYC) - A(TxBTxTYC) A(TxBTxyC) a

(ATXB)Txy C

where we have suppressed subscripts on the A's, B's, and C's. The verifica- tion that the pentagon (diagram I) commutes relative to a is given on the following page, where the unmarked panels commute by naturality of a or b or bifunctoriality of (?.

Next, consider I as an object of GV concentrated in dimension 1 (0 in other dimensions). A natural isomorphism IA -+ A in GV is determined by a family of natural isomorphisms

IT1Ax -Ax

in V. One such family is the family Xx of composites

(3) ITA Ax T1Ax AxX



1060 BARRY MITCHELL

Likewise we can define p :AI -+ A by taking px, to be the composite

(4) AXTI XI Ax Ax.

The following diagram then shows that II commutes relative to al, X, and p:

ATx(ITIB) 31 A(TxITxTlB) -A(TxITx,B) a

(ATxI)TxB

| IV

ATxT1B - A(ITxT1B) - A (ITx1B) - (AI)T B

Ix II

ATXB

Thus we have exhibited a monoidal structure on GV. Henceforth we assume that G is a group, and we add to our blanket

hypothesis on V that 0 preserves epimorphisms and I is a generator. This puts us in the position to apply Corollary 1.5 and Lemma 1.3. If U :A(BC) -+ (AB)C is a natural isomorphism in

ATx(BTy(CTDD)) ---

ATX(B(TyCTyTD))

ATx(B(TyC7TD)) I A(T

A(T,BT,T,(CT,D)) --

A(TxBT,(TyCTyTD)) A(TxBT,(TyCTy,D))

VI III

A (T, BT,, (C TzD)) A(T>B(TxTCTxTvTD)) A(TxB(TxTCTxTy,D))

A (T, B(T,, CT,, Tz D))

(A T,B)Tl,,,(CT, D) A((TxBTxTC)T TyT.D) A((T.,BTxTC)TxTy,D) A(T,(BTyC)T TyzD)

(AT,B)(T,,CT,,,T D) A((TxBT,,C)TXVTzD)

5 f ~~~~~Villl

A(T B(T,,CT,,,D))

aI I a ((AT,B)T,,,C)T.,,D xJ (A(T,BT,,,C))T,,,,D - (A(T.,BTlTC))T.,,:D - (AT,(BTyC))T,y,D




GV, then a -

is a natural automorphism of (ATXB)TXyC in V, and so from Corollary 1.5 we obtain

= k(x, y, z)a

for some k(x, y, z) E Z*V. Then k: G X G X G -+ Z*V is a 3-cochain relative to the cohomology of the group G with coefficients in the G-module Z*V. Then using the fact that the pentagon commutes relative to ae and writing the group operation (composition) in Z*V additively, one checks easily that the condition that the pentagon commute relative to -a is the cocycle condition

(5) k(x, y, zw) + k(xy, z, w) = xk(y, z, w) + k(x, yz, w) + k(x, y, z).

Similarly if X :IA -- A and - :AI -- A are natural isomorphisms in GV, then Xx = c(x)XX and PX = d(x)px for some c(x), d(x) E Z*V. The condition that II commute relative to a-, X, and - is then

(6) d(x) + k(x, 1, z) = xc(z)

Note that if we start with any 3-cocycle k and define c and d by

(7) c(z) = k(l, 1, z) and d(x) = -k(x, 1, 1),

then (6) holds because of the cocycle condition ak(x, 1, 1, z) = 0. Now suppose that (k, c, d) and (k, c^, d) determine monoidal struc-

tures on GV. A bimonoidal structure on GV from the monoidal structure determined by (k, c, d) to that determined by (k, c^, d) is given by automorphisms

h (x, y): :Ax TxBy +A x TxBy, v VJ I

which may be identified with elements of Z*V by 1.5 and 1.3. Commu- tativity of III and IV is then equivalent to

(8) k(x, y, z) - k(x, y, z) = xh(y, z) - h(xy, z)

+ h(x, yz) - h(x, y)

= ah(x, y, z)



1062 BARRY MITCHELL

(9) h(l, z) = c^(z) -c(z) - v

respectively. In particular if we start with a monoidal structure (k, c^, d) and define k = k and c and d by (7), then from (6) we obtain

c^(z) - c(z) = c(z) -k(1, 1, z) = d(l).

Therefore if we take h = 0 and v = d(1), we see that (k, c, d) and (k, c^, d) define equivalent monoidal structures. In other words, any monoidal structure is equivalent to one satisfying (7). But if (k, c, d) and (k, c^, d) are any monoidal structures satisfying (7) and if k and k differ by a coboundary, say k - k = Ah, then in particular

c^(z) - c(z) = k(1, 1, z) - k(1, 1, z)

= h(1, z) - h(1, z) + h(1, z) - h(1, 1),

so that if we define v = -h(l, 1), then we obtain (9). Summarizing, we have shown that the equivalence classes of monoidal structures on GV using the tensor product (1) are in 1 - 1 correspondence with the elements of H3(G, Z*V).

Now let a fixed monoidal structure on GV be given by the triple (k, c, d). Taking k = k and c^ = c in (8) and (9), we see that a bimonoidal structure on 1GV must be given by a 2-cocycle h and an element v E Z*V satisfying h(1, z) + v = 0 for all z. But this last equation is automatic for any 2-cocycle h by the cocycle condition ah(l, 1, z) = 0 if we take v = -h(l, 1). An isomorphism of 1GV to itself is given by a family of natural isomorphisms g(x):A, -+ Ax, that is by a 1-cochain g:G -+ Z*V, and from the diagram VI we see that the bimonoidal structures on 1GV given by cocycles h and h' are equivalent via g if and only if

(10) h(x, y) - h'(x, y) = xg(y) - g(xy) + g(x) = ag(x, y).

This proves that the equivalence classes of bimonoidal structures on 1 GV

using the same monoidal structure (k, c, d) in domain and range are in 1 - 1 correspondence with the elements of H2(G, Z*V). This completes the proof of Theorem 1.

Remark. One can also interpretH1(G, Z*V) as follows. Letg be a 1- cochain giving rise to a monoidal automorphism of 1GV relative to a fixed




bimonoidal structure given by a 2-cocycle h. Putting h' = h in Equation (10), we find that g is a 1-cocycle. If we define the monoidal automorphisms given by two 1-cocycles g, g' to be equivalent if there is an element w E Z*V such that g'(x) - g(x) = xw - w for all x E G, then we find that the equivalence classes of monoidal automorphisms of 1GV are in 1 - 1 correspondence with the elements of H1(G, Z*V).

4. Symmetries. Recall that a symmetric monoidal category is a monoidal category V equipped with a symmetry, that is, a natural transformation

t = tAB :AB -+ BA

such that the diagrams

X A(BC) - (AB)C - C(AB) XI AB t - BA

lO(t a t

A(CB) a 3-- (AC)B tQ31 (CA)B AB

commute. Diagram XI implies, of course, that t is an isomorphism. If V and V are symmetric monoidal categories, then a symmetric monoidal (bimonoidal) functor T: V -+ V is a monoidal (bimonoidal) functor such that the diagram

XII TATB - 1-. T(AB)

t | | Tt

TBTA ,10 T(BA)

is commutative. Two symmetric monoidal structures on a category V are equivalent if there is a symmetric bimonoidal structure on 1v using one symmetric monoidal structure in the domain and the other in the range. Now if (0, I, a, 1, r) and (g, I, a-, 1, r) are equivalent monoidal structures on V via isomorphisms

b: ( ,

-A& , J +I



1064 BARRY MITCHELL

then given a symmetry t for the first structure we can define t^ by commutativity of

A B -b ,A ?B

b B OA -, B?A

and it is easy to verify that f is a symmetry for the second structure. Clearly this sets up a 1 - 1 correspondence between symmetries for the two structures, and consequently in classifying all the equivalence classes of symmetric monoidal structures on a category V, it suffices to take a single rep- resentative from each equivalence class of monoidal structures and classify all equivalence classes of symmetric monoidal structures using it.

Now let V be a symmetric monoidal category satisfying the blanket hypothesis of the preceding section, namely, that I be a generator, that V have coproducts, and that X preserve coproducts and epimorphisms. We shall also let G be an abelian group acting trivially on V, that is, Tx = 1 v for all x with i and 0 identities. In this case the tensor product (1) of Sec- tion 3 takes the relatively simple form

(AB)z= AxBy, Xy =Z

and the monoidal structure xe, X, p is given by

el = a:Ax(ByCz) -+ (AxBY)Cz

fx = 1:IAX - AX, Px = r:AXI -+ AX.

We shall refer to this monoidal structure as the trivial monoidal structure on GV. One symmetry for this structure is given by T :AB -+ BA where

Txy = t:AXBY ByAX

Any other symmetry must be of the form I-r = f (x, Y)Txy with f (x, y) E

Z*V, where, writing down the condition that X and XI commute relative to ae and T, we have




(1) f(xy, z) = f(x, z) + f(y, z)

(2) f(x, y) + f(y, x) = 0.

But these are just the conditions that f be bilinear and antisymmetric. Two such maps f,f ' will determine equivalent symmetric monoidal structures if and only if there is a 2-cocycle h: G X G -+ Z*V such that (diagram XII)

f'(x, y) -f (x, y) = h(x, y) - h(y, x).

This proves Theorem 2.

Remark. Theorem 2 may be sharpened as follows. Let G be any abelian group and let K be an abelian group with G acting trivially. If h: G X G -+ K is a 2-cocycle, then the map f: G X G -+ K defined by

f(x, y) = h(x, y) - h(y, x)

is, of course, antisymmetric, and it is also bilinear, as one sees by expand- ing the right side of

0 = ah(x, z, y) - ah(z x, y) - ah(x, y, z).

Moreover, if h is a coboundary, say h ag, then

f (x, y) = [g(y) - g(xy) + g(x)]- [g(x) - g(yx) + g(y)] = 0.

Thus if BA(G, K) denotes the abelian group of all bilinear, antisymmetric maps f :G X G -+ K, then we have a map

H2(G, K) -- BA(G, K),

and Theorem 2 says that in the case where K = Z*V, the equivalence classes of symmetric monoidal structures in question are in 1 - 1 correspondence with the elements of the cokernel of this map.

Example. If G is the abelian group Z of integers, then H 3(G,-) = 0, and so by Theorem 1 the only monoidal structure on the category of Z-graded K modules (K a commutative ring) with the usual



1066 BARRY MITCHELL

graded tensor product is, up to equivalence, the trivial one. Moreover, since H2(G, -) = 0, by Theorem 2 the equivalence classes of symmetric monoidal structures are in 1 - 1 correspondence with the bilinear, antisymmetric maps f: Z X Z -+ K* where K* is the multiplicative group of units of K. But such a map can be identified with a map Z X Z = Z -K, hence with an element k e K which, by symmetry, satisfies k 2 = 1. When K is a domain, this means k = ? 1, and we obtain the usual symmetries

t(ap?(bq) = bq?(ap

and

t(ap ? bq) = (l)Pqbq ( ap.

When G = Z X Z, so that we are talking about bigraded modules, then we still have H3(G, -) = 0, and there is still only one monoidal structure. However there are more possibilities for the symmetries.

RUTGERS UNIVERSITY

REFERENCES

[1] J. Benabou, Categories avec multiplication. C. R. Acad. Sci. Paris, 256 (1963), 1887- 1890.

[2] S. Eilenberg, and G. M. Kelly, Closed categories. Proc. Conf. on CategoricalAlgebra (La Jolla, 1965), Springer-Verlag 1966, 421-562.

[3] D. B. A. Epstein, Functors between tensored categories, Invent. Math., 1 (1966), 221- 228.

[4] F. Foltz, C. Lair, and G. M. Kelly, Algebraic categories with few monoidal biclosed structures, or none. Sydney Category Seminar Reports, May 1979.

[5] G. M. Kelly, On MacLane's conditions for coherence of natural associativities, commu- tativities, etc. J. Algebra, 1 (1964), 397-402.

[6] S. MacLane, Homology. Springer, Berlin. 1963. [7] S. MacLane, Natural Associativity and Commutativity, Rice University Studies, 49

(1963), No. 4, 28-46. [8] A. Pultr, Extending tensor products to structures of closed categories. Com. Math. Univ.

Carolinae 13, 4 (1972), 599-616.



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Low Dimensional Group Cohomology as Monoidal Structures