ADDITIONAL INFORMATION ON

“EXTENSION THEORIES FOR MONOIDS” By Charles Wells

Semigroup Forum 16 (1978), 13-35.

CORRECTIONS 1. In the first paragraph of page 29, the category should be the category of split

Leech extensions by a centralizing functor which satisfy the other requirements given.

SE M↓

2. The functor called in Theorem 8 is called F in the proof. YF3. The phrase “ ” just above (6.1) should be changed to read, “

is isomorphic to the nth cotriple cohomology group of W with respect to the cotripl”. Of course, the latter group is in the application that follows.

( , )n nBH W H W Y≈ M

M

nH We

( , , )G ε δ ( , )nBH W Y

PAPERS CITING THIS PAPER Wells, Charles (1980) A triple in CAT. Proceedings of the Edinburgh Mathematical Society

23(3). (This is tangential). Wells, Charles (2001) Extension theories for categories (unpublished).

http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf Leech, Jonathan (1986) The D-category of a monoid-category of a monoid. Semigroup

Forum 34(1) Grillet, P. A. (1991) Commutative semigroup cohomology. Semigroup Forum 43(1) B. Novikov (2008) 0-Cohomology of semigroups. In Handbook of algebra, Volume 5. Editor:

Michiel Hazewinkel. Elsevier.

Semigroup Forum Vol. 16 (1978) 13 -35

RESEARCH ARTICLE

EXTENSION THEORIES FOR MONOIDS

Charles Wells

1. INTRODUCTION

Recently, J. Leech [8J, [9J developed a theory for constructing

and classifying certain extensions E ~ M of a monoid M which gen­

eralizes the Schreier theory for groups as well as several existing

theories for special types of monoids. He also developed a cohomology

theory for the case that the extension has commutative-valued Schutzen­

berger functor which generalizes the Eilenberg-Mac Lane theory for

groups.

On the other hand, in the 1960's J. Beck [3J developed a general

theory which shows how to classify "extensions of an algebra by a

group object" (suitably defined) by simplicial cohomology; this theory

works for any category of triple algebras, which in particular includes

any variety of universal algebras. In the case of groups, the Beck

theory coincides [1] with the Eilenberg-Mac Lane theory of extension

with Abelian kernel.

In this note, the connection between Leech's and Beck's theories

is made explicit. The result is that the two theories coincide (in­

cluding the cohomology groups in higher dimensions) in the commutative

case, but that in the general case every Beck extension is a Leech

extension but not conversely.

Before these results are described, I will sketch the work of

Leech and Beck in sufficient detail for the reader unacquainted with

it to read what follows without having to read [3J or [9]. The reader

will find unexplained categorical terminology in [llJ or [12J and

semigroup term.inology in [7].

13 0037-1912/78/0016-0013 $04.60

©1978 Springer-Verlag New_l<-=-.:o:..:.:rk:.;;.,I.;.;;.;;n;.;;..c';.-. _

WELLS

2. TERMINOLOGY AND NOTATION

The terminology used by semigroup theorists (influenced by uni­

versal algebra) and category theorists (influenced by topology)

diverges widely, so I shall make some effort to sort it out here.

Grp denotes the category of groups and homomorphisms, Mon the

category of monoids and unity-preserving homomorphisms, and Cat the

category of small categories and functors. [Q!:EJ denotes the cate­

gory of groups and equivalence classes of homomorphisms, where f: G-tH

and g:G -t H are equivalent if there is an inner automorphism. i of

H for which f i = g.0

Functions and functors will be written on the right and composed

from. left to right. They will often be written exponentially for

clarity. Group actions are written on the left. I use Mac Lane's

barred arrows: thus, n ~ 2n: Z -t Z is an endomorphism. of the in­

tegers on addition.

A functor F:~ -t Grp (~ any category) is separated if k f. k I

in Ob~ implies that kFnk'F is empty. Any group-valued functor

is naturally isomorphic to a separated functor: For a given F de­

fine F by kF = kF X [k}, transferring the group structures and

homomorphisms via the bijections m H (m,k):kF -t kF; these bijections

then become a natural isomorphism F .... F. These remarks also apply

to other categories (for example, [Grp]) in which the objects are sets

with structure.

If .Q is a category with object M, the ~ category .52 ~ M

has as objects arrows W~ M of .,g (objects over ~) and morphisms

f making this diagram. commute:

W~WI

'\ la' M

In this case f is called a homomorphism. over M. If f:W .... WI over

M and f is an isomorphism. in ~, it is a fortiori an isomorphism.

in .Q ~ M and conversely. The set of homomorphisms from W to WI /

over M is denoted HomM(w,w ).

If M is a monoid, an object of Mon ~ M, i.e. a homomorphism

W~ M, is an extension (or coextension) of M if cr is surjective.

If m E M, the subset clscr(m) = [wlwcr := m} of W is the fiber

over m or the congruence class corresponding to m. If for any 14

WELLS

w,w' in W in the same fiber there are u,v,y,z E W such that , , I I CJ • (j/

W = uw , w = uw, w = w y, and w = wz, then W -+ M 1S an Pf.- exten­

sion (U-coextension).* Since there is some indication [6, p. 243ff. ]

that monoids which cannot be expressed as proper nontrivial U"-exten­

sions can be constructed in terms of posets, U-extensions are well

worth studying.

A group G acts simply-transitively (= regularly) on a set X

on the left if for each x,y E X there is exactly one g E G such

that gx = y. Equivalently: G acts transitively and only the unity

stabilizes any point. Group actions by G on X and Yare equiva­

lent if there is an equivariant bijection ~:X -t Y, meaning

g(x~) = (gx)~ for all x E X and g E G. A simple transitive group

action is equivalent to the left regular representation.

3. THE LEECH THEORY

A monoid M gives rise to a small category ~ with object set

M and arrow set MXMXM, with m = domek, m, n) and kmn = cod(k, m, n).

Composition (left to right) is given by

(3.1) (k,m,n) (k',m',n') (k 'k, m, nn').0

If f:M .... M I

is a monoid homomorphism, set

(3.2) (k,m,n).f~ (kf, mf, nf).

This makes 12: Mon -+ Cat a functor. (With very little trouble, 12 can

be extended to an endofunctor of Cat. Then D turns out to have a

left adjoint and to be tripleable [13J.)

A right semifunctor F:~ .... Grp is roughly speaking a function

inducing a functor F:~ -+ [.Q!:.EJ by composing with the natural map

Grp -t [Grp], and which is itself is a fffunctor except on left trans­

lations!'. Precisely, F must take elements of M to groups and

arrows of MD to group homomorphisms in such a way that

* The disparity in the literature between IT extensionff and ff coexten­sion" is occasioned by the fact that for half the mathematical world an exact sequence 1 .... K-+ E -+ G-+ 1 of groups makes E an extension of G by K, whereas for the other half it is an extension of K by G.

15

-----

0

WJ:;.LL0

(3.3) (1, m, l)F = idmF (m E M)

(3.4) (l,k,m.)F 0 (l,km,n)F = (l,k,mn)F (k,m,n E M)

and

(3.5) (1, m, n)F (k, ron, l)F = (k, m., l)F 0 (1, kID., n)F

(k,m, n)F (k,m, n E N).

If F also satisfies

(3.6) (m, n, l)F 0 (k, mn, l)F = (km, n, l)F (k,m, n E F)

then it is easy to see that F is actually a functor. In general,

however, (3.6) will not hold.

A monoid extension W~ M is an extension of M by F if each

group imF acts simply transitively on the left on cIs (m), with a

the action denoted

a(3.7) (h, w) H h·w (h EmF, w = m.)

(whereas m.ultip1ication in the monoid W is denoted by juxtaposition)

in.such a way that

a(gov)w = g(l,k,m)F • (vw)(3.8) (va= k, w = m, g EkF)

and for every v E cIs (k) there is a group element g E kF (depend­a

ing on v only) such that for every m E M, w E cIs (m) and h EmF,0"

(3.9) v (h • w) = h (k, m, 1)Fog • (vw),

where g is the inner automorphism of (km.)F induced by g(l,k,m)F.

(To fix notation: in any group with elements a and b, the inner

automorphism induced by a takes b to aba-I.)

An extension W~ M will be called a Leech extension if it is an

extension of M by some right semifunctor F: ~ ~ QEE. If mF is

Abelian for all m. E M, W ~ M is an Abelian Leech extension.

16

WELLS

I

Two extensions W q. M and Wi ~ M of M by F are equivalent

if there is a monoid isomorphism ~:W ~ Wi over M, that is, 0" = pc; I, such that

a(3.10) (g • v) P = g • v~ (v E W, v = k, g E kF) •

An extension W~ M of M by F is split if there is a monoid

homomorphism. T:M ~ W for which TO" = id. It is easy to see that m any extension equivalent to a split extension is itself split.

Now let F:~ ~ Grp be a separated right semifunctor (meaning

that F is separated). Let E = U mF and let 'IT = (x EmF) t-+ m: E ~ Mmer-­be the proj ection. A function a: M X M ... E such that

(3.11) (k,l)a = (l,k)a = lkF (k E M),

z (m, n, l)F (k, ron, l)F = (k, m)a(l, km, n)F • z (kIn, n, l)F(3.12) 0

- (I, kID., n)F (k E E F)• (k, m )a , m, n M, z n

and

(3.13) (k,m)ex(l,km,n)F. (km,n)a = (m,n)c/k,mn,l)F. (k,mn)a

(k,m,n EM),

where • denotes multiplication in the group (kmn)F in (3.12) and

(3.13), is a factor set for F.

THEOREM 1. Let F,M,E and 'IT be as in the preceding paragraph

and let ex be ~ factor set for F. Define ~ multiplication on E by

(1, m, n)F (m, n, l)F ( )(3.14) xy=x .y ·m,na (xEmF, yEnF).

Then E is ~ monoid and E ~ M is an extension of M by F, with

action by left multiplication. Furthermore, every extension of M

by F is equivalent to an extension obtained in this way for som.e

factoT set ex.

Proof. That E is a monoid and 'IT a homomorphism is Lemma 3.4

of [9J with left and right reversed (it follows straightforwardly

from. ~ormulas (3.1), (3.3)-(3.6) and (3.11)-(3.13)). The simple tran­

sitive action of mF on cIs (m.) (which is mF) is left multiplica­'IT ­

17

W~LL)jWELLS

tion; (3.8) and (3.9) then follow easily (the g in (3.9) is v it­

self).

Given an extension W~ M of M by F, let T:M ~ W be a set

function (it will not necessarily be a homomorphism.) satisfying

(3.15) TO� = i~

(3.16)� l~ = ~

and

kT(h-w) =h(k,m,l)F. (kTw)(3.17)� (k, mE M, wE clSo- (m.), h EmF) •

(Compare (3.9)~) It is possible to choose T this way, for suppose

v E cIs (k) and g satisfies (3.9) for all m,w and h with 0­

w E cIs (k) and h EmF. It is� straightforward to calculate that 0­

taking kT = g-l.v will make (3.17) true. A function T satisfying

(3.15), (3.16) and (3.17) is a normalized transversal* for W~ M

and F.

Define ex by

(3.18 ) (mTnT) = (m.,n)ex.� (mn)T (m, n E M);

ex is well-defined because the action on (mn)F is simple transitive.

Then ex is a factor set for F; (3.11) follows from (3.16), (3.13) Tfrom associativity and (3.12) by calculating kTm~(z· n )

T[(k,m.)ex • (km.)TJ(z· n ) using (3.8) and (3.17).

Now let E ~ M be the extension defined by (3.14) using the fac­

tor set ex just constructed. Define ~:w ~ E by

f3 T(3.19 ) w = w • m (m E M, wo- = m.).

It is a straightforward exercise using practically every formula in

this section to show that f3 is an equivalence of extensions of M

by F. This completes the proof of Theorem. 1.

Let us call extepsions E ~ M defined by (3.14) synthetic exten­

* Note: It is perfectly possible for W~M to be an extension of M by another right semifunctor F' which agrees with F on obj ects but disagrees on arrows because of different gls in (3.9). Such a right semifunctor is Tlnaturally equivalentTl to F in a fairly obvious sense [9, p. 196J. It is not difficult to see that every T satis­fying (3.15) and (3.16) is a normalized transversal for som.e such Fl.

18

sions of M by F; thus Theorem. 1 says that every Leech extension of

M is equivalent to ~ synthetic extension of M.

Now suppose F is an Abelian-group valued right sem.ifunctor, so

that F:~ ~ Grp is in fact a functor. Let W~ M be any extension

of M, not necessarily by F or any other functor. The set Cn(W,F)

of n-cochains with values in F is the set of functions ex:wn~m~

(if n = 0, ex: {lw} ~ 1M!) such that

(3.20 ) (wl, ••• ,wn)ex E [(w1w2••• w )o-JF.n

A coboundary function 5n:Cn (W,F) ~ cn+l(W,F) is defined by

n (3.21) (Wl ,···,w +l )a5 = n

(J(wI' N, l)F n+l (1, M, w )F

(W , ••• ,w + )a + (-1) (W1' ... ,wn)a n+12 n 1

+� ~ (_l)k(w1' • •• , wk' wkwk+l' wk+2' • • ., wn)a k=l

where n

(3.22)� M = II wO�

k=l i�

and� n+l�

(3.23)� N = IT w~.

k=2 l�

In particular, ex E C2 (W,F) is a cocycle if and only if

) (k, mn, I) F ( ) ( ) ( ) (1, km, n) F(3.24) (v,wex - uv,w a+ u,V'Wex- u,vex = 0,

where u E clso-k, v E clsom, w E clson.�

The nth cohomology group obtained by this construction will be�

denoted here by H~(W,F), the Leech cohomology of W with coeffi­

cients in F.

As (3.13) and (3.24) suggest:

THEOREM 2. For Abelian-valued F, H~(M,F) is in one-one cor­

respondence with the set of equivalence classes of extensions of M

by F. Proof. By (3.13), (3.24) and Theorem. 1, there is a mapping of

2-cocycles of M in F onto the set of equivalence classes of ex­

tensions of M by F. It is only necessary to show that cohomologous

19

W1'.JDLJ0

cocycles yield equivalent extensions. This is essentially the content

of [9, §2.4.9J, except for showing (3.10), which follows easily from.

(3.14). (Note: I have left and right interchanged compared to [9J.)

Suppose W~ M is an U-extension. Leech [9, p. 198J shows that

any function ~:M ~ W satisfying (3.14) and (3.15) yields a right

semifunctor F,..:~ -. Grp whose value on an object m E M is the

left Schutzenberger group of cIs (m.).cr

THEOREM 3. Every U-extension W~ M is an extension of M El F for any function ~:M ~ W satisfying (3.14) and (3.15). (In ~ ---- -­

short: EYery U-extension is ~ Leech extension.)

Proof. Omitted. All the calculations for the proof are in

Leech [9, Chapter V]. Theorems 2 and 3 of course imply that there is an injection from

the set of equivalence classes of U-extensions of M by a given func­

tor F into H~(M,F). The injection is not generally surjective

[8, §5.l6J* but since being an U-extension is a property of equiva­

lence classes of extensions, it follows that HiCM,F) contains a

subset of cohomology classes corresponding to ~-extensions. This

subset is not always closed under addition, but it is under taking

additive inverses [8, Theorem 5.15J.

The theory presented in this section is essentially due to Leech

[8J, [9J, but it has been reorganized and supplemented to facilitate

comparison with the Beck theory. In particular for F:M~ -. Ab, Leech

define s H~ (W, F) only for W=M; I shall need the more general defi­

nition in §6. Also, Leech does not define "extension by a right sem.i­

functor tT in the abstract way given here, and so does not state Theo­

rems 1- 3. He is primarily interested in U-extensionsand gets his

desired classification of U-extensions by cohomology through directly

constructing a synthetic extension for each U-extension. Thus the

only extensions he defined are U-extensions and synthetic extensions.

However, Theorems 1 - 3 are due to Leech in spirit if not in precise

fact; they simply represent another way of organizing his work.

*Warning: In [8J an fTabelian coextension" is what would here be called an Abelian Leech U-extension. It is easy to construct Abelian Leech extensions which are not U'-extensions, but Leech does not call those ffcoextensions".

20

The reader will find useful references and discussion in Grillet

[6J, which presents a generalization of Leech's Theory.

4. THE BECK THEORY

Let C be a category. An object Y of C is a group object

if there is a contravariant functor Hy:~ ~ Grp for which

Hy .f~Q!J2

Hom(-, Y) ~ Underlying set\� Set

commutes. (This says "Hom.(-, Y) is group-va1uedfT .) If .f has finite

limits, this requirement on Y is equivalent to requiring the exis­

tenceofarrows ~:YXY"'Y, i:Y-'Y and u:l-+Y (where 1 is the

terminal object of .f) for which the following diagrams commute:

~ X idy YXYXY ) YXY

(4.1) (associativity)li~X~ 1~

YxY ~ )Y

idy X I-l IJ. X idy Y~y X1 :> YXY ~ 1 XY ';tY

(4.2) i~

~

i~

(u is the unity)

Y

idy Xii Xidy Y~YXY~Y

(i takes an elem.ent(4.3) to its inverse)1 1~ 1�

1 u) y< u 1

In addition, Y is an Abelian group object if

p X P YXY 2 ~YXY

(4.4) ~~/~

y

21

VV.L:.JJJJJiJ

commutes, where PI and P2 are the projections of YXY on Y as

a product.

A group object Y in ~ acts simply transitively on an object

E, and E is a Y-principal homogeneous object in ~, if there is a

natural transformation 8:Hom(-,Y) X Hom(-,E) .... Hom(-,E) such that

for each object X of e the component function

X8:Hom(X,Y) X Hom(X,E) .... Hom(X,E)

is a simple transitive group action of Hom(X,y) on Hom(X,E). Again,

if ~ has finite limits this is equivalent to requiring that there

be an arrow B: Y X E .... E such that the following diagrams commute:

Y XY XE I.l Xia; Y XE

(4.5) e1i~X e 1YXE B >E

uX idE idE Xu E ~l X E ) Y X E ( E X I ~E

(4.6) - iC); e iC); ­1E

(4.7) For every object X of e and pair of arrows

X ~ E, X ~ E, there is a-uniqUe arrow a:X .... Y

for which

X

aXf/~

YXE~E

commutes.

Two Y-principal homogeneous objects E and E' are equivalent

if there is an isomorphism ~:E .... E' making (4.8) commute, where B

and 8' are the given actions on E and E'.

idy XP YXE ~ Y XE'

(4.8) Ie e'1) E IE

P

Let ..Q be a category, Q:..Q .... ~ a functor, and E:Q .... Ie a

natural transformation to the identity functor. For each n, -E in­. (n) k n-k n+l nduces natural transformatlons E

k = Q EQ :g .... g for all pos­

nitive integers n, 0 ~ k ~ n, where G is the nth iterate of G;

the component of E~n) at an object X of .f is XQk.EQn-k (in n-kother words, the result of applying G to the component of E at

xgk ). These natural transformations ~atiSfY the simplicial identities

(n) (n-l) (n) (n-l)(4.9 ) Ei Ej = Ej+lE i (0 ~ i =::;: j ~ n).

The resulting simplicial complex

(0) (1)

(4.10) X EO~ XG EO~ XG2 t XG3 4-· XGn

4­

: XGn+l .(1) - - 4-·· 4­or-

El

denoted XQ*, is the standard resolution of X (with respect to g). If y is an Abelian group object in ..Q, the Abelian group

Hom(xgn+l,Y) is the group of n-cochains with coefficients in Y.

One obtains a cochain complex

dl 2 d2 dn 1 (4.11) 0 .... Hom(XQ, y) .... Hom(Xg, Y) ........... Hom.(Xgn, Y) .... Hom(xgn+ , Y) ...... •

by setting

(4.12) d f (-l)~om(XE~n),y).

n k=O

The Beck cohomology, which I will now describe, is the cohomology

of this complex for the special case where ~ is a comma category of

a certain type: Suppose M is an object of a category ~ and

~:~ .... 12 is a functor with left adjoint !::12 .2; then .!! and !: in­

duce functors .QM:..Q ~ ~ ....12 i M.:Q and FM

:12 i MU ~ t M taking W ~ M to a TT 9: I gF I • [6 ]wu ~ MU and X ~ X to X!: -+ X! respectlvely. Beck 2, p. 0

2322

---

shows that EM is le~t adjoint to ~M and that if (E,~) is triple­

able (= monadic ell, p. 139], then so is (~M'~M).

Set QM = ~~M:~ ~~. Then Q is a functor part of a cotriple

(G, E, 5), and for Y ~ M an Abelian group obj ect in ~ ~ M, and

W -+ M any object of ..Q ~ M, the cohomology with coefficients in

Y ~ M obtained from the complex (4.11) (with Q for Q, E:G ~ id M

the counit of the cotriple, W~ M for X and Y ~ M for Y) is

the Beck cohomology ~(W, Y)M' n = 0,1, 2, • ••• A Y ~ M-principal

homogeneous object E ~ M of C ~ M is an extension of M by Y ~ M,

or simply a Beck extension of M. Beck also imposes a splitting con­

dition on E ~ M which will automatically be satisfied in the cases considered here.

Denote the set of equivalence classes (equivalent as in 4.8) of extensions of M by Y -+ M by EX(M,Y).

THEOREM 4 (Beck). With the notation of the 12receding two para­

graphs, if (X,.Q) is tripleable, then ~ is ~ bijection1EX(M, Y) ~ H (M, Y)M.B

If ~ = Grp, and ~ is the underlying set functor, then a group object Y -+ M is any split extension

(4.13) 1 -+ K ~ Y -+ M -+ 1

with Abelian kernel K, and an extension of M by Y -+ M in the

sense of this section is any extension in the usual sense of M by

K with the action of M on K induced by (4.13). Such extensions

are classified by the second Eilenberg-Mac Lane cohomology group

H~(M,K), so that Theorem 4 implies that there is a bijection (it is

in fact a group homomorphism) ~(M, Y) ~ H~(M, K). This suggests the following theorem which is proved in [lJ:

THEOREM 5 (Barr-Beck). ~(M'Y)M is isomorphic to H;l(M,K) for all n ~ -1.

(Here the cohomology groups in dimensions -1 and 0 must be properly defined.)

Note: If ~ is any variety of universal algebras and ~ is

the underlying set functor, then (E,~) (where E is the free alge­

bra functor) is tripleable [12, Theorem. 3.l.26J and furthermore if

infinitary operations are allowed the converse is true by a theorem. of Linton [lOJ, [12, Theorem.l.5.40J.

24

W~.L.L0

5. THE TWO THEORIES COMPARED

Let M be a monoid. A right semif'unctor F: ~ ..... Grp is cen­

tralizing if for all m, n E M the im.age of' (1, m, n)F: mF -+ mnF cen­

tralizes the image of' (m,n,l)F:nF -+ mnF in the group mnF. It

follows immediately from. (3.12) and the remark surrounding (3.6) that

~ centralizing right semifunctor which has ~ factor set must be ~

functor. Thus when treating extensions it is only necessary to consi­

sider centralizing functors F:~ ..... Grps. For such F, the g

disappears from. (3.9) in the definition of "extension by FIT.

THEOREM 6. An extension y:q M of monoids is ~ group object in

Mon ~ M if and only if it is ~ split extension of M ~ ~ centraliz­

igg functor FY:~ ..... Q!:E.

Proof. First I will make some general observations about Mon~M.

The terminal object of Mon~M is easily seen to be M14 M• If ~~ M

and X ~ M are in Mon~~ their product is given by the pullback

P(a,x)~X

(5.1) 1 lx W~M

regarded as an object P(a,x) ..... M of Mon~M by either path in (5.1). P(a,x) will always be realized here as

(5.2) Pea, X) = U (cIs (m.) X cIs (m.»)mEM (J X

crwith the map p(cr,X) -+ M being (w,x) 1-+ m if w xX = m. pea, X)

is a monoid by componentwise m.ultiplication.

Now suppose Y ~ M is a group object in Mon~M. Then there are

homomorphisms ~: p( TI, 11) -+ Y, i: Y ..... Y and u: M -+ Y, all ~ M,

meaning

(5.3) (y.y')TT = m (yTI = Y ITT = m)

where I write (y,y')~ = y.y/,

( -1)11' TI(5.4) Y = Y (y E Y)

25

where I write y-l i:for y , and

(5.5) UTI m ==m. (m. E M).

In the :first place, (5.5) implies that y ~ M 1:E!i!: spEt ~_

~ o:f M. Also, the diagram. (4.1) becomes

~Xi~

p( IT, TI, n) ~ pCTI, n)

(5.6) ~/i~ X ~ , M I ~

/~

P(n,TT) ~ ,.y

where P(TT, TT, TT) = U [cIs (m) X cIs (m) X cIs (m) ] and the map to MmEM IT n IT is (y,y',y") -+ m i:f y,y',y" E cIs (m). However, in :fact M and

n the arrows to it may be omitted :from (5.6) since it is automatic that

the triangles with M as a node commute. (This :follows :from the :fact

that ~ is a morphism. in ~~M, not merely in ~, i.e. it :follows :from. (5.3).)

In the sam.e way, (4.2) becomes

y ~p( TT, i~) ld.f Xu) p(TT, TT) < u Xi'\ p( TT, i~) ~~y

(5.7) i~ 1~ idy

y

where the maps to M are omitted by the s~e argument. Note that a

typical element o:f P(TT,i~). is (y,m) where yn = m, and the iso­

morphism. shown is y ~ (y,m). Finally, (4.3) become,s

·~X· ·X·'\ In 1 1 1 1 ITT.P(TTl:)~

(5.8)

M U ,y( U M

where the map i~Xi takes y to (y,yi), and so on.

(Observe

26

~at in Mon~M the unique map :from. Y ~ M to the terminal object

ri~M iS~.)

Now (5.6), (5.7) and (5.8) imply that every fiber (congruence

~lass) of TI is a group. For y,y' E cls (m), the multiplication is - ---- TT

g;iven by

(5.9)I

y • y' = (y,y')~,

~i = y-l, and the unity is m U • For each m E M, define mFy to

be cls (m.)11

with this group structure.

For k,m,n E M, define (k,m,n)Fy:mFy 4 kmnFy by

'(5.10) (k,m, n)F

y Y = kUynU (yll = m.)

I,where juxtaposition denotes the monoid multiplication in Y. It is

,easy to see that Fy is then a functor. Because ~ must be a monoid

ihomomorphism, it follows that

(xTT(5.ll) (x·x') (y.y') = (xy) • (x/y') = x/ll, yTT= y/11).

It is easy using (5.10) and (5.11) to calculate that

(1, k,m)F (k, m, l)F (k, m, l)Fy (1, k, m)F TT TTy y y(5.12) x .y =y ·x = xy (x =k, y =m)

whence F is a centralizing functor. Each group mF acts on cls my y 11

(which is the sam.e as a set) simply transitively by left multiplica­

tion, and for x TI = x'TI = k, yTI = m~

(x·x')y = (x.x/)(m~·y)

U I = Xln. • X Y

(1, k, m.)Fy I = x • X Y

and similarly for x TT = k, yTI. = Y /11 = m,

, (k, m, l)Fy I

x(y·y ) = y • (xy ),

so by (3.8) and (3.9), y ~ M is a split extension of M by Fy •

(The splitting map is u.)

Conversely, if Y ~ M is a split extension of M by a central­

izing functor F:~ ~ Grp with splitting homomorphism. u:M 4 y, then

the factor set a induced by u via (3.18) is trivial «m., n)a =1 F mn 27

- -

WELLS

for all m)n EM). (It is easy to see that u must satisfy (3.15), (3.16) and (3.17).) It follows from. Theorem 1 that Y ~ M is equiva-

I

lent as an extension of M by F to a synthetic extension yl ~ M

in which Y = U mF and monoid multiplication is given bymEM

(1, m, n)F (m, n, l)F(5.13) xy=x .y (x EmF, y E nF).

Since yl ~M is isomorphic in Mon~M to Y ~ M and since for any

category an object isomorphic to a group object is a group object, it

is sufficient to assum.e that y!l M is itself ~ synthetic extension

of M by F with multiplication given by (5.13) and n the natural

projection (x EmF) ..... m..

De:fine ~: p( n, n) -+ Y and i: Y -+ Y by

(5.14) (y,y')1J. = Y • yl y,y' E Y, yTI = y/TI)

and

(5.15) yi Y -1 (y E Y).

That (5.6) commutes is a straightforward consequence of the associa­

tivity of multiplication in each group mF. To show that the left utriangle in (5.7) commutes, one must show that x.m. = x for any

x E clsn(m). First observe that for u,v E~, it follows from.

(5.13) that

(l,l,l)F (l,l,l)F(5.16) uv = u • v = u • v

because (l,l,l)F is the identity automorphism. of ~. Thus the mon­

oid multiplication and the group multiplication must coincide in the

fiber over ~ (and incidentally because F is centralizing this

fiber must be commutative). Since (~)u = ly' (lM)U is the unity

in 1M!. But for any m. E M, (m)l,l)F:~ -+ mF is a group homomor­Uphism, whence (ly)(m,l,l)F = m • Then by (5.13),

x'mu = x(l,m,l)F • (ly)(m,l,l)F = xly = x,

as required. A similar argument shows that the righthand triangle of -1 -1 u(5.7) commutes. It now follows immediately that y.y =y .y =m, so

that (5.8) commutes. Hence Y ~ M is a group object in Mon~M. This

completes the proof of Theorem. 6.

A homomorphism. from. a group object Y ~ M to a group object, y' l"4M in Mon~M is a monoid homomorphism Y -+ y' over M for

which the induced function HO~(X,y) -+ HomM(x,y') is a group homo­

morphism for every object X -+ M of Mon~M. This can be translated

into diagram.s as in §4. The group objects of Mon~M and their mor­

phisms form a category GOtM. (It is not a subcategory of MontM

since Y ~ M might be a group object in more than one waYe) The

centralizing separated fUnctors F:~ -+ Grp, with their natural trans­

formations, form a category CS(~,Grp). Finally, the split Leech

extensions in MontM, with a given splitting, with homomorphisms over

M which preserve the splitting, form. a category SE~M. The proof of

Theorem. 6 can be extended to show that in fact GO~M = SE~M and that

GO~M is isomorphic as a category to CS(~,Q!£). The details will be omitted here.

THEOREM 7. An extension E ~ M is ~ Beck extension of M El the group obj ect y ~ M if and only if it is ~ Leech extension of M

by Fy , where Fy is the functor corresponding to Y constructed in

the proof of Theorem.~. Proof. The details of the proof are sim.ilar to those of the

proof of Theorem 6, so the proof will only be sketched here. Let

E ~ M be an extension of M by the group object Y ~ M. Without

loss of generality, let Y ~M be the synthetic extension of M by

the centralizing functor F = Fy:~ -+ Q!£. It follows in a manner

similar to the proof of Theorem 6 that (4.5) and (4.6) yield an action

by the fiber mF on cIs (m) for each ill. E Mj this action will be cr

denoted (g,w) H g·w for g EmF, wf::J = m.. Let wand Wi both lie

in cls (m). Let X be the free monoid with generator x, let f::J

f: X -+ E be the m.ap induced by x .... w, and let g: X -+ E the m.ap in­

duced by x ~ WI • Let a: X -+ Y be the unique map given by (4.7).

Then clearly xa·w = Wi, and xa is unique in this respect since

if g·w = Wi the map a':X -+ Y induced by x t-+ g would satisfy

(4.7). Hence the action of mF on cIs (m) is simple transitive. cr

It remains to verify (3.8) and (3.9) (the latter with g omitted). crLet v,w E E with vcr = k, w = m) and let h EmF. First note that

by the definition (5.13) of monoid multiplication in Y and the fact

that F is a functor to Q!£,

28 29

(5.l7) h(k,m,l)F = (~oh)(k,m,l)F = lkmF0h(k,m,l)F

l(l,k,m.)F.h(k,m,l)F _ 1 h kF - kF

Thus

(5.18) v(hew) (lkFev) (hew)

(lkFh) • (vw) = h (k, m, 1)F• (vw )

as required. The second equality in (5.18) comes from. the fact that

the map 8: Y X E -+ E m.ust be a monoid homomorphism. This verifies

(3.9). The veri~ication o~ (3.8) is similar. Thus E ~M is a Leech extension of M by F •

y

Conversely, suppose that E ~M is an extension of M by

F = Fy • Assum.e that E ~ M is a synthetic extension of M by F

with m.ultiplication given by (3.14). Define the extension y ~ M

as in the proof of Theorem. 6 with multiplication (5.13). Thus E and Y have the same underlying set m.~MmF, and as set maps 11 = (]. De­fine e: p( TT, cr) -+ E by

(5.19) (u, U I ) e = u. U I (u, U I EmF).

It is strai~htforward to verify that e is a monoid homomorphism and

that the group object y ~ M acts on E ~ M via e in accordance

with the definitions of §4. I will only verify the simple transitive

part: ~or every extension W4 M and pair o~ homomorphism.s ~:W -t E,

g:W -+ E over M, there is a unique homomorphism, a:W -+ Y over M for which

(5.20) axV~\g

p(n,cr)-.4E

commutes (this is the diagram corresponding to (4.7)) 0 For w E W,T a

w = m, let w be the unique element of mF such that

(5.21) f gwa·w = w

This m.akes sense because, f and g being homomorphisms over M 30

f g(i.e. in Mon~M), w and w must both lie over m.. It is straight­

forward to verify that a is a monoid homomorphism. over M; for 'T 'T

example, suppose u = k, w = m. Then

a a ()f a a f fu w uw = U W ·u we

(ua.uf ) (wa·wf )

gugw == (uw)g

(the second equality because e is a monoid homomorphism.). Since

(uw)a is by definition the unique element g of Y such that

ge(uw)f = (uw)g, this shows that a preserves multiplication. It

preserves inverses by a sim.ilar argument. Finally, (5.20) commutes

because for w~ = m,

waXfoe a f) a f g(w,w 8 = w ew = w,

and a is obviously unique in this respect. This completes the proof

of Theorem. 7. Abelian-valued right semifunctors F:~ ~ Grp are actually func­

tors, and are trivially centralizing. Furthermore, it is not diffi ­

cult to see that an Abelian group object Y ~ M corresponds via

Theorem. 6 to an Abelian-valued functor F:~ -+ Q!E and conversely.

Thus Theorem.s 6 and 7 together imply that the Abelian Beck extensions

are precisely the Abelian Leech extensions, and so by Theorem 3 every

Abelian ii-extension is ~ Beck extension. The following is then a con­

sequence of Theorems 2 and 4: If F: M12 -+.9:El2 is an Abelian valued

functor with corresponding group obj ect y -+ M, then there is ~ bi­

jection H~(M,F) ~H~(M'Y)M. Actually, as I shall prove in §6, this

holds in all dim.ensions and the bijection is a group isomorphism..

The situation with non-Abelian extensions is more involved.

There are non-Abelian centralizing functors and so there are non-Abel­

ian Beck extensions. This is in contrast to the situation in Grp

where every Beck extension is Abelian (use the centralizing property

to prove that if Y -+ M is a group object in Mon~M, then the fiber

over \1 must be Abelian - but if M is a group, all the fibers are

isomorphic). There are also non-centralizing functors, so every Beck

extension is ~ Leech extension but not ·conversely.

31

I

One thing I do not know is whether there are any ~-extensions

with non-Abelian centralizing functor. In other words, is every Beck

~-extension Abelian?

6. COHOMOLOGY

In this section I shall show that with a dim.ension shi:ft the Beck

cohomology and the Leech cohomology are the sam.e:

THEOREM 8. Let Y -+ M be an Abelian group object in Mon~M

with corresponding functor Fy:~ .... Ab. Then there is ~ natural iso­

morphism

H~(W, Y) ~ H~+I(W,Fy)

for all n ~ 0 and ~ object W .... M of Mon~M.

Proof. Observe that for each n, CnCW,F) is actually a contra­

variant functor from, Mon~M to Ab; given a morphism.

W~WI

\/M

in Mon~M, CnC y,F): CnCW',F) .... CnCW,F) is composition with the in­

duced map yn:vr -+ (W,)n. Furthermore, the differential 5n : Cn(W,F)

-+ Cn+lCW,F)� is a natural transformation. Thus in fact Cn(_,F) is

a cochain complex of functors from, Mon~M to Ab.

will now state a contravariant version of Proposition 1l.2 of

[2J suitable for the present application. According to that proposi­

tion, if ~ is a category and (Q,E,5) a cotriple in ~, and

o .... E .... E(O)� .... E(l) ....

is a cochain complex of contravariant functors from. ~ to Ab with

nth cohomology group HnW for W an object of ~, then the follow­

ing two conditions together imply that H~ ~H~(W'Y)M:

WGE Cn =0) (6.1) i\wg) = 0{

(n>O)

32

(6.2)� For each n:::: 0 there is a natural transform.ation

e :GE(n) .... E(n) such that for each object W of

c~ (W€)E(n) owe = WE(n). (Here (W€)E(n) de­

~otes the effec~ of applying the functor E(n) to

the component of the natural transformation E

at W.)

To apply this Proposition, define

(6.3)� WE = HO~(W,y)

and

(6.4)� WE(n) = cn+l(W,F).

The components of the natUl'al transformation E(n) .... E(n+l) required

by the proposition will be the Leech differential given by (3.21). To

see what the map E'" E(O) should be, observe that by (3.21)

ex E CI CW, F) is a 1- cocycle if and only if

(1, w~, W~)F (w~, w~, l)F(6.5) (wl w2)ex = wICX + w2cx

which means by (5.13) that a E HO~(W, Y). (I assum.e, without loss of

generality, that y is the synthetic extension of M by Fy .) Hence

HO~(W,y) = Z~(W,F) is naturally included in WE(O) = C'(W,F), and

that inclusion is the component of the m.ap E ... E(O) at W. Such

maps a are called derivations [9, §6.2.7J because when Wand Y

are groups such m.aps are derivations in the classical sense. (To see

this, one must identify the various groups mF - which are all iso­

morphic when M is a group - with III via the isomorphisms

(l,l,m)F, whereby all arrows (k,m,n)F becom.e automorphisms.)

It now follows that the cohomology of the complex (E(n)Jn~o is

given by

HOmM(W, Y) n=O

(6.6)� HD(W) = { H~+l(W,F) n>O

Condition (6.1) is immediate from (6.3) and (6.6), and Theorem. (2.6.8) of [9J. (WQ is the free monoid on the elements of W, and Leech's

33

concept of a monoid being fl semifree on X with inverses in XItT re­

duces to being free on X when X' is empty.)

Some notation is needed to define e for (6.2). If W'" M is n

a monoid over M, WQ is the free monoid generated by the elements of

M. An element s of WQ is a string [w1, ••• ,wkJ of elements of W.

The counit WE:WQ'" W is multiplication: [wl, ••• ,wkJ H wl ••• w • Thekother cotriple map We: WG ... WGG takes the string s to [s ] E WGG.

n+l( ) - -- n+l. ( -­If a E C W,F then a:W ... Y 1S a set map remember I am.

assuming Y is the synthetic extension of M by F ). It followsy that the effect of (WE)E(n) on a is given by

(6.7) (Sl,···,sn)[a.(W€)E(n)] = (ns1, ••• ,TISn)a

where TIS denotes the product of the elements of the string si ini w. (The subscript on s is not the index of the product.) Define

e as follows: For ~:(WG)n~~ Y given, define n� ­

(6.8Y (wI' • • •, W ) [~. we J ([WI J, ... , [wn])~· n n

Then� clearly a.[(WE)E(n)oWe J = a, so (6.2) is verified. This con-n

eludes the proof of Theorem 8.

7. NOTES ANTI ACKNOWLEDGMENTS

Duskin [4J, [5J has given a general construction of non-Abelian

triple cohomology which has as a special case Dedeckert s work on the

non-Abelian cohomology of groups (see the references in [5J). Thus

there must be a Dedecker-like theory for monoids. Duskin's work in­

cludes a general obstruction theory which has as a special case the

H3 2fact that in groups (Beck's H ) classifies obstructions to ex­3tensions. Leech [9J has an H -obstruction theory for monoids which

one would imagine is also an instance of Duskin's theory. At this

writing I do not understand the relation between Duskints theory and

the theories of Leech or Gri1let [6J.

The work reported in this article was done while I was a guest of

the Forschungsinstitut fUr Mathematik E.T.H., ZUrich. I am. grateful

for helpful conversations with M. Barr, J. Duskin and T. Fox.

34

WJ:I.,LLu

REFERENCES

1.� Barr, M. and J. Beck, Acyclic models ~ triples, Proc. Confer­ence on Categorical Algebra 1965, Springer-Verlag (1966).

2.� Barr, M. and J. Beck, Homology and standard constructions, Semi­nar on Triples and Categorical Homology Theory, Springer Lecture Notes in Math. 80 (1969), 245-335.

3.� Beck, J. Triples, Algebras and Cohomology, Dissertation, Columbia University (1967). University Microfilms #67-14,023.

4.� Duskin, J., K( TIt n)-torsoTs and the i,nterpretation of "triple"� cohomology, Proc. Nat. Acad. Sci. (USA) 71 (1974), 2554-2557­

5·� Duskin, J _, Simplicial methods and the interpretation of tT tr:lple" cohomology, Memoirs A.M.S. l63-rl975J.

6.� Grillet, F., Left coset extensions, Semigroup Forum. 7 (1974),200-263. ---- ----­

7·� Howie, J., An Introduction to Semigroup Theory, Academic Press� (1976) .�

8.� Leech, J., U'-coextensions of monoids, Mem.. A.M.S. 157 (1975).

9.� Leech, J., The cohomology of monoids, preprint.

10.� Linton, F., Som.e aspects of equational categories, Proe. Confer­ence on Categorical Algebra 1965, Springer-Verlag (1966).

ll.� Mac Lane, S., Categories for the Working Mathematician, Springer­Verlag (1971).

12.� Manes, E., Algebraic Theories, Springer-Verlag (1976).

13.� Wells, C., ~ triple in Cat, preprint.

Case� Wes tern Reserve Un ivers i.ty Cleveland, Ohio 44106

Received March 10, 1976. Final form mailed by author November 11, 1977 but lost in mail. Copy received by editors February 21,1978.

35