PUBL1CATIONES MATHEM ATIC AE

PUBL1CATIONES MATHEM ATIC AE

2 4 . K Ö T E T

D E B R E C E N

1 9 7 7

ALAPITOTTÄK:

RElNYI ALFRED, SZELE TIBOR ES VARGA OTTO

D A R Ö C Z Y Z O L T Ä N , GYIRES BE-LA, R A P C S Ä K A N D R Ä S , T A M Ä S S Y LAJOS

KÖZREM ÜKÖDESiVEL SZERKESZTI:

BARNA BEI LA

A D E B R E C E N I T U D O M Ä N Y E G Y E T E M M A T E M A T I K A I I N T ^ Z E T E

I N D E X

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tures of a semilocal ring" by A . Rosenberg and R. Ware ". 181 K u m a r , V., Convergence of Hermite—Fejer interpolation polynomial on the extended nodes 31 Lajkö, K.—Szekelyhidi. L.—Daröczy, Z . , Functional equations on ordered fields 173 L a l , H . , On radicals in a certain class of semigroups 9 Lambek, J . — M i c h l e r , G., On products of füll linear rings 123 Leavitt, W. G . — E n e r s e n , P . , A note on semisimple classes 311 L o o n s t r a , F . , Subproducts and subdirect products 129 M a k s a , Gy., On the functional equation f ( x + y ) + g ( x y ) = h ( x ) + h ( y ) 25 Märki, L . — R e d e i , L . , Verallgemeinerter Summenbegriff in der /?-adischen Analysis mit Anwen

dung auf die endlichen p-Gruppen 101 M i c h l e r , G.—Lambek, J., On products of füll linear rings 123 M l i t z , R . , Kurosch-Amitsur-Radikale in der universalen Algebra 333 M o o r e , J. D . — B o w m a n , H . , A method for obtaining proper classes of short exact sequences of

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Mikusihski Operators 229 P a r e i g i s , B . , Non-additive ring and module theory I. General theory of monoids 189

Pareigis, B . , Non-additive ring and module theory II. ^-Categories, ^-Functors and ^ -Morphisms 351

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M . EICHLER, Quadratische Formen und orthogonale Gruppen. 2. Auflage. — H . H . SCHAEFER, Banach Lattices and Positive Operators. — Lie groups and their representations. Edited by I. M . G E L F A N D . — J . DIEUDONNE, Grundzüge der modernen Analysis, II. — G A B R I E L K L A M B A U E R , Mathematical analysis. — R U D O L F L I N D L , Algebra für Naturwissenschaftler und Ingenieure. — W u Y i HSIANG, Cohomology theory of topological transformation groups. — Numerische Behandlung von Differentialgleichungen. Herausgegeben von R. A N S O R G E , I. C O L L A T Z , G . H Ä M M E R L I N , W. T Ö R N I G — N A R A Y A N C . GIRI, Introduction to Probability and Statistics Part II: Statistics. — R. VON R A N D O W , Introduction to the Theory of Matroids. — Die Werke von Jakob Bernoulli. Band 3. Bearbeitet von B. I. V A N DER W A E R D E N , K . K O H L I und J. H E N N Y . — Beiträge zur Numerischen Mathematik 3. Herausgegeben von FRIEDER K U H N E R T und J O C H E N W. SCHMIDT. — A . M . OLEVSKIJ, Fourier Series with Respect to General Orthogonal Systems. — Husemoller, D . FIBRE Bundles. — J O N A T H A N S. G O L A N , Localization of noncommutative rings. — M . B R A U N , Differential Equations and Their Applications. — E . H A R Z H E I M — H . R A T S C H E K , Einführung in die allgemeine Topologie. — J. L . K E L L E Y A N D I. N A M I O K A , Linear Topological Spaces. — A . W E I L , Elliptic Functions according to Eisenstein and Kronecker. — J . W E R M E R , Banach Algebras and Several Complex Variables. — C. J. Mozzocm—M. S. G A G R A T — S . A . N A I M P A L L Y , Symmetrie Generalized Topological Structures. C. J. M O Z Z O C H I , Foundations of Analysis. Landau revisited. — G . P Ö L Y A — G . S Z E G Ö , Problems and Theorems in Analysis. Volume II. — H . J. K O W A L S K Y , Vektoranalysis II. — B. A N G E R — H . B A U E R , Mehrdimensionale Integration. — Beiträge zur Analysis 8. Herausgegeben von R. K L Ö T Z L E R , W. T U T S C H K E , K . WIENER. — NOLTEMEIER, H . , Graphentheorie mit Algorithmen und Anwendungen. — G . N Ö B E L I N G , Einführung in die nichteuklidischen Geometrien der Ebene. — W E R N E R G Ä H L E R , Grundstrukturen der Analysis, I.

77-3523 — Szegedi Nyomda — F. v.: D o b ö Jözsef

Non-additive ring and module theory I. General theory of monoids

By Bodo Pareigis (München)

Much of the present theory of rings and modules depends only on the multi-plicative structure, not on the additive structure of the objects in question. Another part of this theory depends mostly on the additive structure. This last part has exten-sively been studied in the context of abelian (or additive) categories. We want to introduce categorical tools to study just the multiplicative properties. A surprisingly large part of ring and module theory can be recovered in this way, e.g. Morita theorems, Brauer groups of Azumaya algebras, Maschke's theorem about separable group rings etc. Furthermore this generalized theory applies to a series of examples to be discussed later.

The general background of our theory is the notion of a monoidal category (not necessarily Symmetrie or closed or complete or cocomplete). This is a category

in which we can form associative "tensor products" of objects and morphisms, with "unit" such that I®X^X^X®L Rings are generalized to monoids in i.e. objects A£%> together with an associative unitary multiplication JX\ A ^ A ^ A . Modules are generalized to ^-objects, i.e. objects together with an associative unitary multiplication v: A & M - + M . We use a Yoneda Lemma like technique to do most computations elementwise. So the associativity of the monoid A , \ i \ A & A - + A can be expressed by a(bc) = (ab)c for "elements" a , b, c of A (see § 1). Similarly an ,4-morphism / : M - + N of ^-objects Af, N is described by f ( a m ) = af(m).

This coneept may be applied to a series of examples some of which are rings and modules, coalgebras and comodules, Banach algebras and Banach modules, //-module algebras and modules over them for H a Hopf algebra, monoids (in sets) and monoid sets with equivariant maps etc.

In this paper we want to introduce the general background of non-additive ring and module theory, tensor products over arbitrary base-monoids (read: base-rings), and the technique of elementwise computation.

190 B. Pareigis

1. Notation and the universal property of the tensor product

Let <g>,/) be a monoidal category, i.e. a category # with a functor ® : # X # — # and an object and with natural isomorphisms

a: , 4 ® ( £ ® C ) ^ 0 4 ® 5 ) ® C

/: I®A^A

Q: A(S>I = A such that all diagrams

>4®(£®(C®Z))) — (/*®5)®(C®Z>) — ( ( ^ ® ^ ) ® C ) ® Z ) |/4®oc ja®£>

A<8)((B<g)C)®D) * ( ^ ® ( 5 ® C ) ) ® D

A ®(I ®B) *(A®l)®B

A ®3

I<S>I = I G I

and

and

V

I

f

commute. If there is an additional natural isomorphism

such that

and

and

y: A®B ^ B®A

täA®B = (A<8>B—~ B®A A®B)

A»I - f®A

A

A®{B®C)^(A®B)®CC®{A®B) M ® y | a * a y ® £ *

A®(C®B) — (A®C)®B (C®A)®B

commute, then # is called a Symmetrie monoidal category [3].

Non-additive ring and module theory 191

Let ^ always be a monoidal category. For objects A9C€<# we use the notation A ( X ) : = V ( X 9 A ) . So A will be

considered as a contravariant functor from # to the category ¥ of sets or as a "variable set". By the Yoneda Lemma [12] there is a bijection between the morphisms /: A ^ B in # and the natural transformations (p: A(X)-»B(X) by c p = < g ( X 9 f ) 9

which then will be denoted by the same letter: / : A ( X ) - + B ( X ) . Observe then that f ( a ) = f o a for a£A(X).

Consider A ® f? as a "variable set". We want to introduce a universal property for it, which resembles the usual universal property of the tensor product. For this purpose consider a "variable set" A ( X ) X B ( Y ) in the two "variables" X and in Y. A natural transformation / : A(X)XB(Y)-+ffir(X®Y)9 natural in X and X, with a contravariant functor & \ <g + Sf is called a bimorphism. We simply write it as / : A X B - + & .

Clearly

A ( X ) X B ( Y ) = <€{X, A)XC€{Y9 B ) $ ( a 9b)—+a®b£c£{X®Y9 A®B) = A®B(X®Y)

is a bimorphism.

1.1 Lemma. (Universal property for tensor products). F o r each bimorphism f : AXB^»!F there is a uniquely determined n a t u r a l transformation / * : A Q B ^ ^ , . such that

A ®B

® g F u r t h e r m o r e A X B —»A<&B —- !F is a bimorphism for each n a t u r a l transforma

tion g:A®B^^. P R O O F . Let X = A 9 Y = B and f ' = f ( i d Ä 9 i d B ) . Then f'€&(A®B) may be

considered as a natural transformation / * : A < g > B ( Z ) - + t F ( Z ) by the Yoneda Lemma namely = Now let X 9 Y^fß be arbitrary objects. For (a9b)£A(X)X X B ( Y ) we have

A ( A ) X B ( B ) —— &(A®B) \ A ( a ) * B { b ) \&(a®b)

A(X)XB(Y)——3?(X®Y)

commutative, hence for (id^, idB) £A(A)XB(B) we get

&{a®b)of(\dA9 idB) = f o A ( a ) X B ( b ) ( i d A 9 idB) = f ( a 9 b)

hence ( /* o®)(a, b)=f%a®b) = &(a®b)(f') = &(a®b)of(ydÄ9 i d B ) = f ( a 9 b ) . Thus. the diagram of the Lemma commutes.

Let g be such that g o ® = / Then gCid^ ® id B) = (g o ®)(id^, id B) ̂ / ( i d ^ , id ß ) = / ' in tF(A<g)B)9 hence the induced natural transformations / * and g from A<2>B to are equal.

192 B. Pareigis

So we have a bijection between bimorphisms / : A X B - + 3 F and natural transformations / * : A<g>B-+lF. In particular for each object and bimorphism

/: A X B - + C there is a unique morphism / * : A < g > B ^ C such that

A(X)xß(Y) - *-A®B(X<*Y)

C(X®Y)

commutes for all X , Y£^. Certain, but usually not all, elements in A<g>B(X®Y) are of the form a®b

with a£A(X)9 b£B(Y). For those elements we have f ( a 9 b)=f*(a®b). Since / * is uniquely determined by / , we want to describe /*(z) for all z£A®B(X) with the use of / . Since / has two variables we introduce the notation

f(zA9zB) :=/*(z).

If we take in particular ®: A X B ^ A Q B as bimorphism, then it clearly factors through i d : A®B^A®B. With the notation just introduced we get z = zA<g>zB

for all z£A®B(X). The elements z which occur as tensor products a®b will be called decomposable tensors9 the others indecomposable.

Since for any bimorphism / : A X B - + C the value f ( z A , zB) is uniquely determined for z—zA%zB9 if one only knows f ( a 9 b) for all ( a 9 b)£A(X)XB{Y)9 we will not make any difference between decomposable tensors a®b and indecomposable tensors zA<g)zB and thus will use the same notation a®b in both cases.

We will list a few interesting examples of monoidal categories, to which we will apply the later results : a) Category of üf-modules for a commutative ring K with the usual tensor product

and jreJC-Mod. b) The dual of K - M o d with the tensor product and K£K-Mod. c) A n y category # which has finite products or coproducts and a final or initial

object, in particular the categories of sets, i?-modules or the category of small categories.

d) The category of endofunctors of a category # with composition and the identity functor.

e) The category of R — i^-bimodules with the tensor product M®RN and R£R-Mod9 where R is a possibly non-commutative ring.

f) The category of real or complex Banach Spaces together with the completed tensor product and R resp. C. The morphisms in this category shall be the linear maps of norm less than or equal to 1.

Observe that the examples a), b), c) and f) are S y m m e t r i e monoidal categories, whereas d) and e) are in general nonsymmetric.


2. Monoids and objects over monoids

A monoidin is an object A^fß together with two morphisms A®A-+A and rj: I - + A , such that the diagrams

A®(A®A) —- {A®A)®A A®A

A®A >A and

I 9 A *1®A» A®A « A

* * > A * I

A

commute. If we write n(a®b) = ab for ( a 9 b)£A(X)XA(Y), then the commutativity of

the first diagram is equivalent to a(bc) = A((x)((ab)c)eA(X®(Y®Z)) for all X , Y, Z£<% and all (a, b9 c)£A(X)XA(Y)XA(Z). As long as it is clear from the context we'll omit A ( a ) and simply write ((ab)c)£A(X®(Y®Z))9 hence (ab)c = a ( b c ) .

In the sequel we shall often omit composites of a, /., Q and y on the side of the domain of a composition of morphisms. This can be done without any harm by the coherence results of [5]. So we'll identify certain morphisms with different domains, or rather pick a fixed domain out of the set of possible uniquely isomorphic domains, using the axiom of choice. Hence when we talk about A(X® Y®Z)9 then X® Y®Z is such a fixed domain.

Since

X ——* A

I®X-^I®A commutes, we get A(?r1)o£(idI®a) = a . Hence the commutativity of the left triangle of (*) is equivalent to A(/r1)(rj(idI)a) = a for all X^€ and all a£A(X)m

Again we will omit A O r1) and obtain r j ( \ d I ) a = a . If we write l=jf(idj), we get

1 *a = a for all a£A(X). The right triangle of (*) is commutative iff a* l = a for all a Z A ( X ) (again omitting A ( Q ~ 1 ) ) .

Observe that for an arbitrary monoidal category ® 9 /) there is no sense in askingifa monoid {A9JJL9Y]) is commutative, since there is no symmetry y:A®A^ ^A®A. If, however, ^ is a S y m m e t r i e monoidal category then a monoid ( A 9 fi9 rj) is called commutative if

A®A

7

AG

13 D

194 B. Pareigis

commutes. This is equivalent to ab—ba for all a£A(X), b£A(Y) (again omitting A ( y ) : A(X®Y)-+A(Y®X)).

Now let A be a monoid. A n A-left-object in ^ is an object together with a morphism v: A & M - + M such that

, 4 ® 0 4 ® M ) — 0 4 ® 4 ) ® M A®M A®\ | v

- M and

commute. Writing v{a®m) = a m 9 this is equivalent to ( a b ) m = a ( b m ) and l -m=ra for all m ( M ( Z ) , a£A(X), b£A(Y). Here again we omit M(a) and A f ^ " 1 ) or Af(A).

A monoid-morphism f : A - + B is a morphism such that

and

4 ' ^ £

commute, equivalently

f ( a b ) = f ( a ) f ( b ) and / ( l ) = 1

for all a£A(X)9 b£A(Y) and all X 9 Y^. A n A-morphism f : M - + N is a morphism such that

A®f A®M > A®N

M • N

commutes, equivalently f { a m ) = af(m) for all a£A(X)9 m£M(Y) and all X 9 Y ^ . Thus we get a category A%> of y4-objects and >4-morphisms for each monoid A and a category M o n (#) of monoids and monoid-morphisms.


If the tensor product of % is the coproduct and / the initial object of then the unique morphism I - + X will be denoted by e. The diagrams

/ £ *~ A v •\ \ x A 2? I I A

&Jl A » A1A

hence

commute. So the diagram

I I A

A1A

A M I I J I A

III A ^ A Hl

d e t e r m i n e s a u n i q u e JA: A]LA-+A w h i c h is a m u l t i p l i c a t i o n w i t h 2-s ided u n i t . It is easy t o see t h a t t h e c o d i a g o n a l , is a s s o c i a t i v e . So each o b j e c t i n (ß c a r r i e s a u n i q u e s t r u c t u r e o f a m o n o i d . Since t h e c o d i a g o n a l is a n a t u r a l t r a n s f o r m a t i o n e a c h m o r p h i s m is a m o n o i d m o r p h i s m . Thus M o n ( ^ ) ^ ^ .

• On the o t h e r h a n d l e t ® , / ) a S y m m e t r i e m o n o i d a l c a t e g o r y t h e n M o n (<#) is a S y m m e t r i e m o n o i d a l c a t e g o r y w i t h the t e n s o r p r o d u c t of A, B£ Mon (#) t h e t e n s o r p r o d u c t i n In f a c t i f .4, 2 ? £ M o n ( # ) , t h e n nA®B: (A®B)®(A®B)^ ^(A®A)®(B®B)*tA®ßD-*A®B w h e r e the i s o m o r p h i s m is the u n i q u e i s o m o r p h i s m d e f i n e d b y the c o h e r e n c e t h e o r e m o f S y m m e t r i e m o n o i d a l c a t e g o r i e s : cp: (A®B)® ®(A'®B')^(A®A')®(B®B'). So iiA®B(a®b®a'®b') = aa'®bb'. Now i t ' s e a sy t o c h e c k t h a t A®B is a m o n o i d . Furthermore y: A®B^B®A is a m o n o i d m o r p h i s m h e n c e M o n (#) is a S y m m e t r i e m o n o i d a l c a t e g o r y .

If w e f o r m M o n (Mon (^)) t h e n th i s i s a g a i n a S y m m e t r i e m o n o i d a l c a t e g o r y . Let ( A , f i A : A®A-+A) b e i n Mon (Mon ( # ) ) a n d le t ßA: A®A-*A be t h e m u l t i p l i c a t i o n o f A in M o n t h e n jxA i n p a r t i c u l a r i s a m o n o i d m o r p h i s m , i .e .

(A®A)®(A®A) (A®A)®{A®A) ßA®HA

t*A®»A A®A

A®A

PA

13*

196 B. Pareigis

commutes or if nA(a(g)b)—: aob and ßA(a<g)b)=\ ab, then ( a a / ) o ( b b / ) = ( a o b ) ( a / o b / ) . Observe now that rj: I - + A as morphism in M o n (#) is a monoid morphism. Let rj: I - ^ A be the unit of A in M o n (#). Hence

/ - /

f \ /? A

commutes or // = /]. So the units for A in Mon (#) and for y4 in Mon (Mon (^)) are the same. Let l£A(I) be the unit and a ' = \ = b , then

tfo6' = (tfl)o(lZ/) = (col)(lofe /) = ab', hence JU X =

Furthermore a b = { \ a ' ) ( b \ ) = ( \ b ) { a ' \ ) = ba', hence the multiplication fiA was commutative to Star t with. Conversely if flA is commutative then ßA is a monoid morphism, i.e. the commutative monoids are precisely the monoids in M o n (#). Hence M o n (Mon ( ^ ) ) ^ C M o n (#), where C M o n (#) is the füll subcategory of M o n (#) of commutative monoids.

N o w it's easy to see that the tensor product in C M o n (#) is a coproduct for C M o n ( ^ ) . The injections are

A A<g)I A®B.

Let / : ^ - C and g: B ^ C be morphisms in C M o n ( ^ ) . Then /?: A®B^C defined by h ( a < g > b ) = f ( a ) g ( b ) is a morphism in C M o n ( ^ ) since h ( a a ' %bb') — = f ( a a ' ) g ( b b ' ) = f ( a ) f ( a ' ) g(b) g(b') = f ( a ) g(b) f { a ' ) g ( b ' ) = h ( a ® b) h (a' ® 6') and A( l® 1 ) = / ( 1 ) £ ( 1 ) = 1 . If h ' \ A®B^C reduces to / on A and g on 5 then h'(a®l)=f(a), h'(\®b)=g{b) and h'(a®b)=h'(a® 1)(1 ®b)=f(a)g(b)=h(a®b) hence /?=//. We have proved

2.1: Theorem. 1) Le/ # # Symmetrie monoidal category then M o n (Mon (#)) = = C M o n (<<f) /7ÖT5 //7^ tensor product of as product.

2) If is a monoidal category which has the coproduct as tensor product then C M o n ( ^ ) ^ M o n ((ß)^<ß, i.e. each object of ^ carries a unique monoid structure, which is commutative.

A special case of this is the fact the usual tensor product is a coproduct in the category of commutative AT-algebras, also the usual tensor product is a product in the category of cocommutative A^-coalgebras.

This uses the fact that the (commutative) monoids in K - M o d resp. K-Mod°p

(examples a) and b)) are the (commutative) A'-algebras resp. the (cocommutative) ÄT-coalgebras. A % is then the category of ^(-modules for the Ä-algebra A resp. the category of /f-comodules for the ^T-coalgebra A . For the last assertion observe that for M , N£AK-Modop = ^ we have / £ ^ ( A f , TV) iff

N M \ \

A®N-A®M commutes in A^-Mod.


For the example c) in the special case (Sets, X , {0}) the monoids are precisely the (abstract) monoids and A

cß is the category of sets on which the monoid A acts, together with the equivariant maps. For (#, X , E ) with E a final object one obtains the categorical monoids.

In example d) the monoids are precisely the monads Jf. A n example for a left-Jf-object, if comes from a pair of adjoint functors J * : and (§\ cß [12], is the functor 2?. The functor J5* is a right -object, and J f 7 ^ ® as two-sided -objects.

The monoids for example e) are the Ä-rings in the sense of P. M . C O H N [see also SWEEDLER 15].

For example f) the monoids are precisely the Banach algebras.

2.2. Theorem. The for gelful functor A<6^<ß is monadic.

PROOF. Since A is a monoid, the functor A®— : <ß-+(ß is a monad with \i\ (A<g)(A(&)-»A<g) and 7®-*,4® induced by /<: A®A-+A and rj: I - * A . Furthermore 7® is identified by /. with Id#.

The "algebras" for the monad A® are [12] pairs ( M , v) where M£<ß and v: A & M - + M is a morphism in <g such that

A&CAQM)

A&M —

commute. This clearly is equivalent to the definition of an v4-object. A n "algebra morphism" for A® is a morphism / : M — N , such that

A % M A®N 1 / *

M > N

commutes. Again this is equivalent to / : M - * N being an ,4-morphism. Hence /ß-*cß is monadic with left-adjoint (ß5M-»A®Me/ß.

2.3. Corollary: / is a monoid in (ß and the category {ß is equivalent to <ß.

PROOF. Let 7 ® / — / be /. or Q and //: 7—/ be the identity. Then by the coherence theorem [5] / is a monoid. Since 7® ^ I d ^ by both have equivalent categories of algebras, ß for the functor 7® and (ß for the functor Id.*?.

A special instance of this is the fact that the category of abelian groups " i s" the category of Z-modules, since (Ab, ®, Z) is a monoidal category.

2.4. Corollary: The forgetful functor $U\ /ßcreates limits and isomorphisms. If cß is complete so is A<ß and the limits a r e formed in °U creates those colimits which a r e preserved by A®. If <ß has a generator then /ß also has a generator.

Although we will not use the following facts they should be stated. The straight forward proofs are left to the reader.

198 B. Pareigis

2.5. Proposition: The forgetful functor %\ M o n ( ^ ) - * # creates limits and isomorphisms. I t even creates difference cokernels of Ql-contractible pairs.

The functor <?/: M o n however, has in general no left adjoint, e.g. the category of finite sets with the product as (g>.

3. Inner hom-functors and tensor products

Let A , B be monoids in the monoidal category cß. A n A—B-biobject is an object M $ f 6 together with morphisms A & M - + M and M<g>B-»M suchthat M becomes an v4-left-object, a i?-right-object and the diagram

- M<g>B

A®M M

commutes, i.e. a(jmb) — ( a n i ) b . A morphism of A — 5-biobjects is a morphism in which is an ,4-left and i?-right morphism, i.e. f ( a m ) = a f { m ) and f ( m b ) =

= f ( m ) b . The category of A — 5-biobjects is denoted by A

( € B . Clearly there are forgetful functors °U\ A € B - ^ A ^ r e s P - ^ : A ^ B ^ B - These functors are monadic in the same way as the functor in Theorem 2.2. Observe that °ll does not in general preserve colimits.

Assume that (ß and jß have difference cokernels. Let M ^ BCC A and N£/6,

then we consider the two morphisms in B

c 6

f : M®(A<8>N) M@ViV. M<g>N and

g : M®(A®N)-^ (M®A)®N-^^~ M®N.

The difference cokernel of / and g in ß will be denoted by f „CO

M®(A®N) M®N—- M®AN. If we consider / and g as morphisms in wegeta difference cokernel co:M<g>N-+ - + M < g > A N through which Ba> factors uniquely:

Observe that we use the same notation for the objects of the difference cokernels in # and in ß though <%(M<g>AN)&M®AN in general for <?U\ ß ^ the forgetful functor. If, however, B® : preserves difference cokernels then <%(M<g>AN)^ 9±M<g>AN. This is the case if tensor products in ^ preserve difference cokernels, so for example in Symmetrie closed monoidal categories which are discussed later on in this paragraph. If in addition N£/ßc then Bcoc: M < g > N ^ M < g i A N denotes the difference cokernel of / and g in ßc.


Let M£B<gÄ9 N€A<g and P£jß. A bimorphism / : M x N - P is called a B-left A-bimorphism if f ( m a , n) = f ( m 9 a n ) and f ( b m 9 n ) = b f ( m 9 n) for all m£M(X)9

n£N(Y), a£A(Z)9 b£B(U).

3.1. Proposition: Let M£B<8A9 N£A<S and P£B<£. F o r each B-left A-bimorphism f : M X N - + P there is a unique morphism g: M®AN-»P in ß such that the diagrams

M(X) xN(Y) - ®-+M®N(X<i>Y) -JLLm*AN(X*Y) v P(X®Y)

commute. PROOF. By Lemma 1.1 / may be factored uniquely through <g> by a morphism

/*: M®N-+P. f * i s in B%> becauseof b f ( m , n ) = f ( b m 9 n ) . Now f ( m a 9 n ) = f ( m 9 a n ) implies f*(ma®n)=f*(m®an) hence f*o(vM®N)oa=f*o(M®vN). So / * can be uniquely factored through Bco.

Let ® , /) be a monoidal category. If the functor X®: (6^Y-+X®Y<i<6 has a right adjoint it will be denoted by [X, —], so that

V(X®Y9Z)~V(Y9[X9Z])

natural in Y and Z . If X® has a left adjoint it will be denoted by (X, so that

V ( Y 9 X®Z) - V ( ( X 9 Y ) 9 Z )

natural in Y and Z . If ß9 ®9 I ) has right adjoint functors to X® for all Xttf, then # is called a (left-) closed monoidal category. If ®91) has left adjoint functors to X® for all X£f€9 then # is dual to a closed monoidal category, which we shall call a ffe/f-J coclosed monoidal category.

The most interesting example for this is the dual of the category of modules over a commutative ring, which is used for studying coalgebras.

If ®X has a right adjoint for all X$fß9 then ^ is called a ( r i g h t - ) closed monoidal category. If cß is Symmetrie, then # is left-closed iff cß is right-closed. We will only consider monoidal categories which are left-closed and call them closed monoidal categories.

Let M9N^jß9 X $ f e . Then M®X is again in A<ß. So M ® i s a functor. We want to find a right adjoint to it, which will be denoted by A [ M 9 TV] hence

A

C€(M®X9 TV) s* <€{X9 A [ M 9 TV]) = „[M, TV] ( X ) .

We observe that there are two morphisms w, <£(M9 N)^(£(A®M9 TV), namely f - * f o v M and f-+vNo(A®f). By definition / i s i n /ß(M9 TV) iff these two maps coincide on / iff

^(g)M /l(g)TV

I V M " I V N f

M »N

200 B. Pareigis

commutes. Hence we get jßiM, TV) as a difference kernel:

A V ( M 9 TV) (ß(M9 TV) =r <g(A®M, N ) .

Let ^ be a closed monoidal category with difference kernels. Let u\ v'\ [ M 9 N ] ( X ) ^ ^ß(M®X9N)=^((A®M)®X9N)^[A®M\N](X). Let K(M, N) be the d i f ference kernel of u'9 v'': [ M 9 N]^[A®M, TV]. Then we get a commutative diagram of difference kernels

< e ( M ® X } N ) £ ( M ® X N ) U t . KCCAQMfaX^) A ' V

Wl Wl Wl

<e(X, K(MtN)) - WXJMM) f €(X,[A*MtNJ).

Hence M®\ <g-+A<8 has a right adjoint, so K ( M , N ) ^ A [ M 9 TV], i.e.

Ä [ M 9 TV] — [M, TV] =*[A®M9 TV]

is a difference kernel. Here we use the fact that (ß is left closed. In fact if the O p e r a t i o n of A on

M is right sided, then we cannot form %A(M®X9 TV). For this we need commutativity of

M®A®X - N®A® X or of M®X®A - N®X®A I I 1 1

M® X TV® X M®X N®X

but in the first diagram the upper horizontal morphism and in the second diagram the two vertical morphisms cannot be defined in general.

Dual to the above construction we get a difference cokernel

(A®M9 TV> =* <Af, TV> — A ( N 9 M ) 9

although we cannot immediately dualize the above construction, since A then becomes a comonoid. But the only thing we have used is that there is a morphism A®M^M9 not that A is a monoid.

If M®\ cß^/ß has a right adjoint, then we get the adjunction morphisms X - * A [ M 9 M®X] and M®A[M9 TV] —TV the last morphism in A<g. It induces a bimorphism M X A [ M 9 TV]-*TV which we write as ( m , f ) - + { m ) f . Since the morphism M® A [ M 9 TV]— TV isan ,4-morphism we get ( a m ) f = a ( ( m ) f ) for a£A(X)9 m£M(Y) and f£A[M9N](Z). Conversely if f£[M, TV](Z) and ( a m ) f = a ( { m ) f ) for all m$M(Y)9 a£A(X) then f€A[M9 TV](Z) by the difference kernel property.

Since the diagram

M C x ) x [ M a l i e r ) t » N(x®Y)

Wl Ii

' A

with ( p ( m 9 f ) = (X®Y M®Y TV) commutes, we get that ( m ) f = —fo(m®Y) identifying along the natural adjointness isomorphism.


3.2. Theorem: Let £: M(X)-+N(X®Y) be a n a t u r a l transformation in X com-patible with the action of A on M resp. N . Then there is a unique f ^ . A [ M 9 N ] ( Y ) suchthat £(m) = ( m ) f for a l l X^(€ and m£M(X).

PROOF. Without taking into account the y4-action, the Yoneda Lemma gives us a unique ; ( i d ) M = / € # ( A f <g> Y) such that C(m)=/o(/w® Y) = ( m ) f for all X$fß and all m£M(X). Now £ ( a m ) = a£(m)imp\iesa((m)f) = ( a m ) f so thatf£A[M, N ] ( Y ) .

3.3. Corollary: There is an associative multiplication

A [ M , N ] ® A [ N , P ] - ~ A[M9 P ]

with unit I - A [ M . M ] for a l l M 9 suchthat (I®A[M9 N ] - + A [ M 9 M]® A [ M 9 N ] - + - * A [ M , N]) = A and ( A [ M 9 N] ® I - + A [ M 9 N] ® A [ N 9 N] - + A [ M 9 N]) = g.

PROOF. Define the multiplication by (m)n(f®g) = ( ( m ) f ) g and write fi(f®g) = —fg. This defines a morphism by Proposition 3.2. Then ( m ) f g = ( ( m ) f ) g hence ( m ) ( f g ) h = ( ( m ) f g ) h = ( ( ( m ) f ) g ) h = ( m ) f ( g h ) and by Proposition 3.2 we have ( / g ) / 7 = / ( g / ? ) , the associativity.

Define 1 — + A [ M , M ] as morphism corresponding to gÂ<ß(M®I9 M ) . Then for m£M(X) we have (m)rj = g(m®I) = m (again identifying X and X®I via g ) 9

hence rjf=f and grj=g-

3.4. Corollary: F o r each M^jß we get a monoid A [ M 9 M ] .

3.5. Corollary: For MÂ

(ß9NÂ

(ßB we have A [ M 9 N ] e < g B .

PROOF. The map

M ( X ) X A [ M 9 N ] ( Y ) X B ( Z ) 1 ( m 9 f 9 b) ~ ({m)f)b£N(X®Y®Z)

induces a bimorphism

A [ M 9 N ] ( Y ) X B ( Z ) $ ( f 9 b) ~ f b e A [ M 9 N)(Y®Z)

by Proposition 3.2 such that ( ( m ) f ) b = ( m ) ( f b ) . Hence A [ M , N] becomes a right 5-object.

3.6. Corollary: A / £ / 6 induces afunctor A \ M 9 —\.A

cßB-+(ßB.

3.7. Corollary: Let MÂ

(ßB and N£/ß. Then A [ M 9 N ] ^ B ^ .

PROOF. M ( X ) X B ( Y ) X A [ M 9 7V](Z)3(w, b9fy+{mb)f£N(X® Y®Z) induces a bimorphism

B ( Y ) X A [ M 9 N ) ( Z ) $ ( b 9 f ) - b f e A [ M 9 N](Y®Z)

by Proposition 3.2 such that ( m ) ( b f ) = ( m b ) f Hence A [ M 9 N ] becomes a left B-object.

3.8. Corollary: M G Ä induces a functor A [ M 9 — Y A ^ - ^ B ^ -

3.9. Corollary: If MZA<gB9N£A<gB then A[M9N]£B%C.

202 B. Pareigis

3.10. Proposition: F o r Q^B

(ßA9 M ^ f l and N^^fß there is a n a t u r a l isomorphism A V ( M , B [ Q 9 N ] ) ^ B < 8 ( Q ® A M , N ) if B [ Q , N] and Q ® A M exist.

PROOF. The difference cokernel in B

(6 Q®A®M=zQ®M-»Q®AM induces a commutative diagram of difference kernels

The only problem is the commutativity of the right lower Square (with respect to the multiplication vQ:Q®A^Q). Now for f£B<#(Q®M, N)9q£Q (X),a£A(Y) and m ^ M ( Z ) have

( q c i ) l F ( f ) ( m ) = f{qa®m) = fo(vQ®M)(q®a®m) = (q) V(fo(yQ ®M))(a ®m)

hence the claimed commutativity. A n object Q£BßA is called A-coflat i f for all monoids C and objects M ^ / 6 C

the difference cokernel Q®AM^B

(ßA exists and if the natural morphism Q® A

(M®N)-+(0®AM)®N in induced by the associativity of the tensor product, is an isomorphism for M£A9?C and N£<ßD.

3.11. Proposition: Let Q^ßj^ be A-coflat. Then there is a n a t u r a l isomorphism

for M£A<ß9N£B<ß and QdB<ßA.

PROOF. The isomorphism is given by the following commutative diagram

II? Ulf MV

flOI, B[Q,N1) —£(M> JQ.N]) ^ X t C t t o M , JQ.NJ)

where the left isomorphism results from the universal property of difference kernels.

V : B I Q ® A M 9 N ] ~ A [ M 9 B [ Q 9 N ] \

* ( C B [ Q ® A M 9 N]) — <6{C9 A[M, B[Q9 N]\)

B ß{{Q®AM)®C9 N ) C* B<ß(Q®A(M®C)9 N ) - A<ß{M®C9 B [ Q 9 N } ) .

3.12. Proposition: The following diagram is commutative

[C, A M , B [ Q , N ] ] ] X - A [ M ® C , B [ Q , N ) ] I Q ® A ( M ® C ) , N ]

[ C , B [ Q ® A M , N ] ] fil [{Q®AM)®C, N ]

for C€<g,M£A<g,NeB<& and for Q€B^A A-coflat.


PROOF. Invest the definitions of *F to get a commutative diagram

t ( D , fC, A C M , 5 [ Q , N W ) • Ä ö , ALH»C, a l Q - N i l ) t ( D , 8 [ Q O / M 0 C ) , Kl] )

t(C9ö J M , R L Q , N]]) AtCCM*C)öb, J G N J )

x t ( M < 9 ( C * D ) , ß [Ö. NJ)

A

<2> t ( b , I C , ulQ<2> M, NJJ) t { D , A Q < * M ) 9 C , NJ]

The outer frame of this diagram implies the assertion of the proposition.

The examples of § 1 have the following properties. The example a) is closed, com-plete, cocomplete, and abelian. Example b) is coclosed, complete, cocomplete and abelian. Example c) may be closed (e.g. category of sets with the product), complete, cocomplete and abelian (e.g. K - M o d with the product) or not. Example d) is in general not closed. Example e) is closed, complete, cocomplete and abelian, but not S y m m e t r i e . Example f) is closed b y [X, Y] the set of continuous linear maps [14]. Furthermore there are difference kernels and difference cokernels, but the category is not additive.

Bibliography

[1] H . BASS, The Morita Theorems (mimeographed notes). U n i v e r s i t y o f O r e g o n 1962. [2] H . BASS, Algebraic K-Theory. N e w Y o r k 1968. [3] S. EILENBERG and G . M . K E L L Y , Closed Categories. P r o c . Conf. C a t . A l g . , L a J o l l a

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A m . M a t h . Soc. 50 (1975), 55—60. [5] G . M . K E L L Y and S. M A C L A N E , Coherence in Closed Categories. J. P u r e a n d A p p l i e d A l g e b r a

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359—370. [7] B . I. P. L I N , Morita's Theorem for Coalgebras. C o m m . A l g e b r a 1 (1974), 311—344. [8] H , LINDNER, Morita — Äquivalenz von Kategorien über einer geschlossenen Kategorie. D i s

s e r t a t i o n . Düsseldorf, 1973. [9] S. M A C L A N E , Categories for the Working Mathematician. N e w Y o r k — H e i d e l b e r g — B e r l i n , 1971.

[10] K . M O R I T A , Duality for Modules and its Application to the Theory of Rings with Minimum Condition. Sc. Rep. T o k y o K y o i k u D a i g a k u Sect. A , N o . 150 (1958), 84—142.

[11] D. C. N E W E L L , Morita Theorems for Functor Catesories. T r a n s . A m . M a t h . Soc. 168 (1972), 423—434.

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[12] B. PAREIGIS. Categories and Functors. Academic Press, New Y o r k — L o n d o n , 1970. [13] M . RIEFFEL, Morita Equivalence for C*-Algebras and W*-Algebras. J. P u r e and Applied A l g .

5 (1974). 51—96. [14] Z. SEMADENI, Banach Spaces of Continuous Functions. Volume 1. P W N . Warszawa 1971. [15] M . E . S W E E D L E R . Groups of Simple Algebras. Puhl. I . H. E. S. 44 (1974), 79—190. [16] M . E . SWEEDLER, The Pre-Dual Theorem to the Jacobson—Bourbaki Theorem. P r e p r i n t 1975. [17] M . WISCHNEWSKY, On Linear Representations of Affine Groups I. Algebra—Berichte N r . 29.

U n i - D r u c k München, 1975. [18] H . M . HASTINGS, Stabilizing tensor products. Proc. Amer. M a t h . Soc. 49 (1975), 1—7.

(Received November 2 0 , 1 9 7 5 . )

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