rnanuscripta

math.

I, 99-I09 (1969)

BAER ADDITION

OF E X T E N S I O N S

Hans-Berndt

Brinkmann*

A B S T R A C T : A c a t e g o r y w i t h O and b i p r o d u c t s for p a i r s carries a u n i q u e s e m i a d d i t i v e structure. If f u r t h e r m o r e identities h a v e n e g a t i v e s , the c a t e g o r y is a d d i t i v e . Using that Y o n e d a e x t e n s i o n s over an a b e l i a n c a t e g o r y ~ form a g r a d e d c a t e g o r y ~, we show that 0 and b i p r o d u c t s of ~ r e t a i n their p r o p e r t i e s in ~. Hence ~ is (graded) additive. We thus defined Baer a d d i t i o n o f e x t e n s i o n s w h i c h is f u r t h e r m o r e unique. The aim of the p a p e r is to d i s p e n s e w i t h all c a l c u l a t i o n s w h i c h are u s u a l l y i n v o l v e d in e s t a b l i s h i n g the prop e r t i e s of Baer a d d i t i o n and to s h o r t e n the proof.

O.

INTRODUCTION:

stract)

is b a s e d on a d 3 o i n t

lations. a more dix

The p r o o f

However,

direct

given below

functors

and avoids

the less c a t e g o r y m i n d e d

p r o o f of the m a i n

(7). The b a s i c

theorem

idea o f the p r o o f

functors

on ideas of M i t c h e l l gratefully

F.W.Lawvere

and B . M i t c h e l l .

include

the case of r e l a t i v e

Ext

in the a p p e n -

b y using

adjoint

[12].

acknowledges

R.Baer,

find

is some years o l d , b u t

to a v o i d all c a l c u l a t i o n s

The a u t h o r

all c a l c u -

reader may

(3.2)

the p o s s i b i l i t y relies

(see the ab-

discussions

B.Mitchell

with

suggested

to

(5) in our treatment.

i. P R E L I M I N A R I E S :

l.i. A D D I T I V E category ~ with

CATEGORIES:

0 together

with

ture on e a c h of its m o r p h i s m position elements.

is d i s t r i b u t i v e If f u r t h e r m o r e

A semiadditive an a b e l i a n

sets

category

semigroup

(classes)

~(A,A) , then all of the ~(A,B)

identity i

A are a b e l i a n

struc-

such that com-

and zero maps b e h a v e each

is a

as n e u t r a l

has n e g a t i v e s

in

groups.

* Ich d a n k e dem F o r s c h u n g s i n s t i t u t f~r M a t h e m a t i k der E T H in Z ~ r i c h u n d dem B a t t e l l e I n s t i t u t in G e n f f~r G a s t f r e u n d schaft und Unterst~tzung.

99

2

BRINKMANN

N o w let ~ be a c a t e g o r y w i t h O. Let Pl (l.l.i)

P2

A (

) C ( ii

contain a product

~ B i2

and a c o p r o d u c t of A and B and let = ~i,j=k

(i.i.2)

Pkij

We call this a b i p r o d u c t of such diagrams products

L o , J &k-

diagram

since,

assuming

for all pairs, ~ has p r o d u c t s

(*) and b y i.i.2 these c o i n c i d e

existence

(x) and co-

as functors

(i.e.

f,g=fxg=:f~g). If ~ has O and b i p r o d u c t

diagrams

for pairs,

then the

composition

(1.1.3)

A

for f+g gonal

~ ~ A e A ~

Be~

(f,g:A ~ B, ~ the d i a g o n a l

(coproduct))

Furthermore

defines

9

6 >B

(product),

a semiadditive

structure on ~.

this is the o n l y s e m i a d d i t i v e

structure w h i c h

can p o s s i b l y c a r r y and it does in p a r t i c u l a r on the choice among e q u i v a l e n t biproducts. in 6. A c a t e g o r y ~ is additive, grams

With r e f e r e n c e

not depend

A p r o o f is g i v e n

if it has O, b i p r o d u c t

for pairs and if all of the ~(A,B)

groups.

6 the codia-

to uniqueness,

dia-

are abelian

the a d d i t i o n

is not

taken into the data of ~. From the c a t e g o r i c a l that ~ has b i p r o d u c t s coproducts

point of v i e w we prefer to say

for pairs,

(•

and

(,) for pairs and if these coincide as functors

(f*g=f•

If ~ has O, then this is e q u i v a l e n t

above: For every pair of objects containing

if ~ has p r o d u c t s

select a d i a g r a m

a p r o d u c t and a coproduct.

to the

i.l.i

For fi:A ~ A',

f2:B ~ B' the e q u a t i o n P~'i~f =f.p, i (obvious notation) is K33 KK3 immediate. Let e.g. A ' = O , f 2 = i B. Then P 2 i l = O and P2i2 is an isomorphism,

since p~,i~ o b v i o u s l y

(A'=O!). C h a n g i n g

e.g.

are isomorphisms -I i 2 to i2(P2i2) etc. we can arrange

for 1.1.2.

100

BRINKMANN

3

1.2. N - G R A D E D A D D I T I V E CATEGORIES: (n the natural assignment

numbers)

An N--graded c a t e g o r y

is a c a t e g o r y ~ together w i t h the

f ~-~ ifl s n of a degree to every f and such that

identities have degree 0 and c o m p o s i t i o n (Igfl=igl+Ifl).

The grading h e n c e

adds degrees

is a functor ~ ~ n, where

n is v i e w e d as a c a t e g o r y w i t h one object. N-graded

categories

are s u p p o s e d to p r e s e r v e degrees

c a t e g o r y of N__-graded c a t e g o r i e s

Natural

transformations

a c a n c e l l a t i o n monoid. required.

more general c a t e g o r y monoids)).

(Lawvere

[8] or

are h o m o g e n e o u s

In m o r e general

The c o n d i t i o n

seems natural

if i n t e r p r e t e d

(C,M) of g r a d i n g s m

(n is

cases this m u s t be in the

(M the c a t e g o r y of

C o n s e q u e n t l y w e assign the same grading

dual c a t e g o r y

(The

is thus the comma c a t e g o r y

(~,n) w i t h ~ the c a t e g o r y of c a t e g o r i e s [9;p.13])).

Functors b e t w e e n

(*IfI=Jff) , 0 is c h a r a c t e r i z e d

to the

by requiring

that all of the -Cp (A,O),C- q (O,B) c o n t a i n p r e c i s e l y one ele(C__D(A,B) are the m a p s A ~ B of degree p) and p r o d u c t s

ment

are to be taken h o m o g e n e o u s l y

(.C.p(X,AXB)=%(X,A) x.Cp(X,B) ,

where

the i s o m o r p h i s m m u s t be of degree O, since n has no

other

invertible

elements).

The m o r p h i s m s

of degree 0 of

form a c a t e g o r y w h i c h we denote by C_D. O b v i o u s l y C_D contains all objects

of ~.

An N--graded s e m i a d d i t i v e

category

g o r y ~ together w i t h an a b e l i a n

is an n - g r a d e d

semigroup

cate-

structure on each

of the -Cp (A,B) such that c o m p o s i t i o n is d i s t r i b u t i v e and zero maps b e h a v e as neutral elements. Again, if each identity iA has a n e g a t i v e

in --oC(A,A) , then all of the --PC(A,B)

are a b e l i a n groups. The rest of 1.1 t r i v i a l l y (see 6). C l e a r l y I.•

can

extends

(and need)

to the graded case o n l y be formed,

if

Ifl=Igi. Remark:

The d e f i n i t i o n

and u n i q u e n e s s

of the a d d i t i o n

for m o r p h i s m s

of degree p does only d e p e n d on these and C . --o This m a y be seen from 6 or by letting C_~(A,B)=O for all q&O,p.

This trick using a q u o t i e n t

101

c a t e g o r y was suggested

4

BRINKMANN

b y F.W. L a w v e r e .

1.3. sider

EXTENSIONS:

exact

E:O ~ B ~ C I ~

called

~ A ~ 0 n from A to B and c o m m u t a t i v e

n-extensions E

E t

two

such

is g i v e n

or e x i s t s

~

C~

(Using

may

be

indicates

for

found

adopting

MacLane

rows

E : A ~ E for t h e e l e m e n t s

over ~

or M i t c h e l l

A. ~ = A

form

~

A ~

O

indicates

by ~ defines

defined

below,

some

(Yoneda

in

E"

[3])).

[II~7]

thus:

and in p a r t i c u l a r

We

one

can

[16;

let

terminology

from

and i n t r o d u c i n g

of E x t n ( A , B )

an N--graded

that

with

(IB,...,IA) :E ~ E'.

the u s u a l

of Ext m a y be r e c o r d e d

(n~O)

C t

with ~ and ending

addition

e.g.

category

[I0~3]

~

~ generated

the

Then

~A~O

n (~, .... ~ ) : E ~ E'

a n d E X E'

:=~(A,B).

properties

---

1.3.2 beginning

A short proof

sions

~

E~E ' iff E ~ E" ~ E'

Ext~

.~C

diagrams

~n

B t

relation

Extn(A,B),n)l.

p.574].

~

~

sequences.

s u c h as

equivalence

~C

~1 O

a diagram

show that

We c o n -

... ~ C

I11n ~I

O~B

19 :

(1.3.2) between

category.

sequences

(1.3.1)

The

L e t ~ be an a b e l i a n

1,

the b a s i c

Classes

(possibly

ar-

of n - e x t e n -

illegitimate)

~ and ~ have

the

same

objects. Proofs in

[10]

note

of the r e s p e c t i v e

or

[II]

as c i t e d

properties

above.

For

of Ext m a y be

further

found

reference

we

that

(1.3.3)

(~ ..... ~ ) : E ~ E'

This

implication

what

follows.

statement not use

2.

lemma

(the p r o o f

~E~E'~.

is trivial)

It m a y be n o t e d

of 1.3.3

implies

is b a s i c

that moreover

is e q u i v a l e n t

to ~ E ~

the

E'~,

for all first

but we will

this.

FUNCTOR (2.3)

EXTENSION:

of M i t c h e l l

i For simplicity c l a s s of E.

For use [12]

in 3.1 we r e p r o v e

(lemma

I.I,p.3•

w e u s e no p a r t i c u l a r

102

notation

l.c.).

a The

for the

BRINKMANN reader

who wishes

ly switch

to a v o i d

to 3.1.

Application to a sequence (2.1)

adjoint

He should

b y the c a l c u l a t i o n

given

of an exact

the latter

the e q u i v a l e n c e

letting

~f=Tf

We refer

functor

Now

for f ~ ~, ~

behaves

(oA/A ~ ~)

of m a p s

transformation

S ~ T as well.

resp.

(use 1.3.3).

of T. O b v i o u s l y with

respect

O) c o n s t i t u t e s

Furthermore

such t r a n s f o r m a t i o n s

to

It is i m m e d i -

of B c o n s t i t u t i n g

(of degree

as S ~ T resp.

are

a a

equality

independent

~ ~ T ~ ~ or as S ~ T

S ~ T ~ U.

On the other

hand

exact,

~ , B abelian).

1.3.1,

~ defines

w h i c h b y 1.3.3

let ~:S ~ T be g i v e n Then given

a commutative

implies

TB~

commutes n=O, w e

(S,T:~ ~

E s 6 Extn(A,B),n~i.as diagram

e.g.

(~B ..... ~A) :~E ~ ~E,

that SB(

gree

Furthermore

(~,B abelian).

natural

them

Ext.

functorial

~ ~ ~

of e n v i s a g i n g

sequence

TA ~ 0

extension

transformation

of two

an exact

is a functor ~ ~ ~

natural

and c o m p o s i t i o n

(~,~ abelian)

T E we see that ~ re-

defining

let S , T : A ~ B be exact

ate that a f a m i l y

T:~ ~ ~

... ~ TC n

sequence

relation

extension

the p r o o f of 3.1

in ~ yields

to ~ as to the c a n o n i c a l

the c a n o n i c a l

may immediate-

in 7.

0 ~ TB ~ TC i ~

spects

functors

then r e p l a c e

E such as 1.3.1

in ~. D e n o t i n g

5

in ~. T o g e t h e r

SA

TA

TE

with

the r e s p e c t i v e

see that ~ is a natural

diagram

transformation

for

S ~ ~ of de-

O. This proves 2.3. L E M M A

of a b e l i a n

(Mitchell

cateqories.

[I0]) : Let A~-~q~ be exact

Then

(e,~):S~T:(~,~)

functors

iff

(~,~) : s ~ T: ( ~ , ~ ) . The n o t a t i o n

refers

to adjointness:

103

T is a d j o i n t

to S

6

BRINKMANN

and S is c o a d j o i n t and B.

to T, iff ~(SB,A)--BB(B,TA)

In the g r a d e d

an i s o m o r p h i s m B (B,TA) --p

case the

of graded

for every p).

transformations

isomorphism

sets

natural

in A

is r e q u i r e d

to be

(classes;

i.e. -Ap (SB,A)= we m a y ask for natural

Equivalently

s:i B ~ TS and ~:ST ~ I A such that

the dia-

grams (2.4)

S

S e > STS

~S> S

,

T

s T > TST

iS commute

(e.g.

T

iT

Eilenberg-Moore

(e,q) : S ~ T: (~,~)

T~

[711,p.382].

in this case

and refer

We w r i t e

to s,D as adjunc-

tions. PROOF

OF 2.3:

preserved

under

The

situation

3. B A E R ADDITION:

3.I.PROPOSITION: with

~ is an N - g r a d e d for pairs.

From

the

first

2.3 it is i m m e d i a t e

constant (O,i.e.

functor

the d i a g o n a l

T:~ ~ ~

(Terminal,

functor

this o b s e r v a t i o n

placed

by N

(trivially

the r e s p e c t i v e in

graded

(~,N)).

is abelian,

~=['})

remains

graded product

has

a biadjoint

valid,

and C__x~ b y

are terminal follows

are a b e l i a n

0 : ~ ~ A and ~ : ~

~ A are b i a d j o i n t s

that ~--N, ~ is the grading, the p r o p o s i t i o n

the same p r o o f [•

2.3:

and all of T , A , O

for T and ~. O b s e r v i n g

in M i t c h e l l

resp.

from

and ~ is the diagonal, Essentially

In the

if I is re-

b y the identity) (these

(~).

iff

and T : ~ ~ ~ and A : ~ ~ A__x~ have bi-

0 and ~. ~ and ~ • Hence

follow-

and that ~ has b i p r o d u c t s

N o w the p r o p o s i t i o n

additive

and ~ are exact.

of a d j o i n t n e s s

A : ~ ~ C__x~ has a b i a d j o i n t

case

given

fo r pairs

that a c a t e g o r y ~ has 0 iff the

graded

adjoints

(~ abelian)

0 and b i p r o d u c t s

definition

an a d j o i n t = c o a d j o i n t )

product

category

of A.

PROOF: ing

We n o w prove

0 and b i p r o d u c t s

are those

given b y e,~ and 2.7 is

~.

for general

In that case,

104

A•215

is proved. products

however,

is

assumptions

BRINKMANN on the exactness

of p r o d u c t s

7

over the indexing

set in ques-

tion are n e c e s s a r y and some care in i d e n t i f y i n g •

x~ with

is involved. From 3.1 w e conclude: 3.2. C O R O L L A R Y

ded additive

(Baer [I], Yoneda

cateqory

(~ abelian).

s c r i b e d b v 3.3 i__~sthe o n l y natural b u t i v e u n d e r composition)

[15]) : A is an N - q r a -

Baer a d d i t i o n addition

as de-

(i.e. d i s t r i -

in ~. ~ o = ~ a_~s additive cateqo -

ries. The first statement

records

tion about Baer a d d i t i o n and 7.3.3,p.174).

MacLane

to d e s c r i b e

the a d d i t i o n of n - e x t e n s i o n s

p l a y of e x t e n s i o n s

informa-

[il~ 7 . 3 . 2 , p . 1 7 3

[10] also u s e d the language of

g r a d e d additive c a t e g o r i e s whether

all the essential

(e.g. M i t c h e l l

of all lengths.

Ext. R.Baer asked

depends on the inter-

The remark in 1.2 shows

that this is not the case. PROOF OF 3.2: B y !.I,• structure.

~ has a u n i q u e s e m i a d d i t i v e

By u n i q u e n e s s ~ =A as a d d i t i v e categories. --o

all

identities

have n e g a t i v e s

tive. This proves

and c o p r o d u c t

in _A=~_o, ~ is N--graded addi-

3.2. Selecting

gonal 6 in this c o n t e x t

pi=IA_X~ m a y be a s s u m e d

the d i a g o n a l

from the p r o d u c t

adjunctions

Since

--

8 and codia-

(8,p):A~:(~x~,~)

(i,6):$ ~ A:(~,A__x~), w h e r e

(I.I.2) , the a d d i t i o n

is further de-

scribed b y the familiar c o m p o s i t i o n (3.3)

8

A

) A~A

w i t h E~E' now a b b r e v i a t i n g wise

E~E I E~E'

6

, BeB

~B

(this is to be taken point-

(2.1)). 3.4. C O R O L L A R Y

(Mitchell

[12]):

I_~fT : ~ ~ B is exact

(~,~ abelian) , then T : ~ ~ B is additive. PROOF: biproducts

Exact

functors

4. THE C O N N E C T E D T : ~ ~ __Ab E

are additive.

~ preserves

O and

if T does.

SEQUENCE:

Given any a d d i t i v e

functor

(N-graded a b e l i a n groups) , that is a c o n n e c t e d

105

8

BRINKMANN

sequence

of a d d i t i v e

functors

provides

an e q u i v a l e n c e

~ ~ Ab,

n.t.[Ext(A,-)

(the left side

ve n a t u r a l

transformations).

Ext t o g e t h e r

with

immediately Mitchell known more

their

from this.

[i•

exact

between

5. R E L A T I V E

sequences

satisfying

12.4,p.369],

1.3.3 holds. functors

lar ~ neda

[13~

If exact ~

(~,~)

is N - g r a d e d

6,p.363]

APPENDIX:

is ~

the de-

MacLane

As b e f o r e

Furthermore

such that T ~ C ~,

everything case.

be full nor

we

relative

(additive)

faithful

by

In p a r t i c u -

and the r e s p e c t i v e ~ ~

[iO~

one arrives

T : ~ ~ ~ are r e p l a c e d

~:~

of

in ~ and

[5] or e.g.

(~)o=~.

exact

is a class

to imitate

to the r e l a t i v e

Obviously

6. PROOF

OF i.i,

Yo-

for e v e r y

(see M i t c h e l l

1.2:

Let C be a (graded)

(*) for pairs.

category

with

Let +, ~ b__~etwo b i n a r y

sets

i__%s d i s t r i b u t i v e

and zero m a p s

Then

[4],

having

on the m o r p h i s m

ments.

Further-

for conditions).

6.i. LEMMA: products

are

of short

sequences

5,p.359]).

additive

but this need n e i t h e r

allow

only

functors

in 2-4 extends

lemma holds.

[•

which

(Buchsbaum

category ~

T:(~,~)

(e.g.

transformations

class

c a t e g o r y ~. That

axioms

Mitchell

at an m - g r a d e d

follow

lemma

sequences

natural

for

(on ~)

and 7.5,p.178).

additive

of Ext but c o n s i d e r i n g

compositions

additi-

sequences

properties

Let ~ be a proper

in an a b e l i a n

h a v e done

The c o n n e c t e d

7.2,p.267

natural

sequences.

sequences

their

groups

(homogeneous)

the c o n n e c t e d

induce

EXT:

for

functorial

(l.c.

functors

abelian

By the B u c h s b a u m - S c h a n u e l

the r e s p e c t i v e

finition

stands

7.4.1,p.•

to be exact

lemma

,T]~TA

of N - g r a d e d

in A and T

the Y o n e d a

(classes)

O and co-

operations

of C such that c o m p o s i t i o n act as two sided neutral

+ and + are the same,

commutative

ele-

and associa-

tive. PROOF given

(Puppe

[i4~

(and let all

2.6~p.248]) : Let

four maps

106

have

the

fl,f2,gl,g2:A same degree).

~ B be

BRINKMANN Let then

f,g:A*A ~ B be d e f i n e d

i 2 are the c o p r o d u c t v i t y we see that equals

fij=fj,gij=gj,

A ~ A,A.

the c o m p o s i t i o n

we have

commutativity

by

injections

(fi~f2)+(gi•

f2=gi=O,

9

but also

Using

i•

distributi-

(f+g)o(i1~i2):A

~ A,A ~ B

(fi+gl)~(f2+g2).

Letting

that + and ~ are the9 same.

follows

where

from f i = g 2 = O

Using

this,

and a s s o c i a t i v i t y

from

f2=O. To finish that,

of i.i,

1.2 it suffices

if ~ has O and b i p r o d u c t s

binary two

the p r o o f

operation

which

sided neutral

are n a t u r a l

is d i s t r i b u t i v e

elements.

But this

transformations!).

material

we refer

Mitchell

Ill;

for pairs,

product

we h a v e

in ~.

B~B'

Since,

contains

assertion.

given

i ~

gonal,

E~E'

A~

one element.

given

the d i r e c t n~•

(terminal)

a product inverse

This proves

B ~ B~B'

the

~' B' in ~,

maps

') g i v e n b y El ~(pE,p'E) ,

sum of E and E' Let

O,

first

(D:A ~ A ~ A the dia-

"pointwise").

We

E e e Extn(A,B~B').

(p' ..... i) :E ~ p'E yield

(I ..... ~) :E ~ pE~p'E.

1.3.3

shows

9 9 Extn(A,B) xExtn(A,B').

E ~(pE~p'E)~. The

yield

(p ..... (IA,O)) :E~E' ~ E. This

by •

and p ( E ~ E ' ) ~ w E ( i A , O ) ~ ~ argument

d u a l i t y we need

A~

for every given A and n

(p ..... i) :E ~ pE and

respective

[2;7-8],

n

... O

precisely

Then,

need o n l y c o n s i d e r

(E,E')

and r e l a t e d

any E e 9 Extn(A,O),n>O,

E x t n ( A , B ~ B ,) ~ E x t n ( A , B ) • (E,E')

Using

"'"

show that we have m u t u a l l y

(E~E,)~-~

(~ and 6

[6] or to e.g.

of ~ act as p o i n t

O ~ 0 ~ O ~

we

as

diagram

E:O ~ 0 ~

first

and has zero m a p s

a

i.18].

a commutative

Extn(A,O)

defines

For m o r e d e t a i l s

to E c k m a n n - H i l t o n

s h o w that O resp.

resp.

I.•

is trivial

7. PROOF OF 3.• BY C A L C U L A T I O N : only

to show

applies

to E'

107

Now

let

first p r o j e c t i o n s shows

E since Since

p(E~E')~E(IA,O) (•

6=IA.

The

i.i.2 o b v i o u s l y

I0

BRINKMANN

remains

valid

(8,p) : A 4 ~ : ( ~ • previous

proof

in ~,

the p r o p o s i t i o n shows

in this

is p r o v e d

that we h a v e

special

(In fact

only r e f o r m u l a t e d

the

case).

References I.

R.Baer, E r w e i t e r u n g von G r u p p e n und ihren men, Math. Z. 3~8, 375 - 416 (•

2.

H . - B . B r i n k m a n n und D.Puppe, Springer L e c t u r e Notes

3.

H . - B . B r i n k m a n n , E q u i v a l e n c e of n - e x t e n s i o n s , Arch. Math. (Basel), !SS [IS88}0 to appear,

4.

D . A . B u c h s b a u m , A note on h o m o l o g y in categories, Ann. Math. 6__99, 66 - 74 (1959).

5.

D . A . B u c h s b a u m , Satellites and u n i v e r s a l Ann. Math. 7_~I, 199 - 209 (1960).

6.

B . E c k m a n n and P.Hilton, G r o u p l i k e structures in general c a t e g o r i e s I (Multiplications and c o m u l t i p l i c a tions) , Math. Ann. 145, 227 - 255 (1962).

7.

S . E i l e n b e r g and J.C.Moore, A d j o i n t triples, Illinois J. Math. ~,

8.

F.W.Lawvere, F u n c t o r i a l semantics of a l g e b r a i c ries, thisis, C o l u m b i a U. 1963.

9.

F.W.Lawvere, The c a t e g o r y of c a t e g o r i e s as a foundation for M a t h e m a t i c s , Proc. Conf. Cat. Alg. (La Jolla 1965), I - 20, Springer 1966.

10.

S.MacLane,

Homology,

Kategorien i__88 (1966).

Springer

u n d Funktoren,

functors,

functors and 381 - 398 (1965). theo-

1963.

Ii. B.Mitchell, T h e o r y of categories, (second printing). 12.

Isomorphis-

Academic

Press

1967

B.Mitchell, On the d i m e n s i o n of objects and c a t e g o r i e s I. Monoids, J. A l g e b r a ~, 314 - 340 (•

13. B.Mitchell, On the d i m e n s i o n of objects and c a t e g o r i e s II. Finite o r d e r e d sets, J. A l g e b r a ~, 341 - 369 (1968) . 14. D.Puppe, S t a b i l e H o m o t o p i e t h e o r i e 243 - 274 (•

108

I, Math.

Ann.

i69,

BRINKMANN •

N.Yoneda, Sci.

On the h o m o l o g y t h e o r y of m o d u l e s , J. Fac. Univ. Tokyo, Sec. I ~, 193 - 227 (1954).

16. N . Y o n e d a , On Ext and e x a c t sequences, J. Fac. Univ. Tokyo, Sec. I 8, 507 - 576 (1960).

Hans-Berndt

Brinkmann

Mathematisches

Institut

D 69 H e i d e l b e r q Tiergartenstrasse und Forschungsinstitut der E T H C H 8006

li

fur M a t h e m a t i k

ZOrich

{Received January 8, 1969)

109

Sci.