<<

. 8
( 14 .)



>>

3
Editors™ note: The editors believe that the “a” at the right end of the last line is correct, although
the original shows a ξ.
32 JONATHAN MOCK BECK

Hence
X· · a = X·· · Xµ · a
= X··((aT · θ) —¦(ξT · a))
= (X·· · aT · θ) —¦(X·· · ξT · a)
= (X· · a) —¦(X· · a)
In a group, this implies X· · a = 1.
G a is a map of 1-cocycles, then b˜’1 = X —
It must also be proved that if b: a
Y —¦ b: a˜’1 G a ˜’1 in PO(X, Y ). One must dissect the diagram

(X—Y —¦ b)T
G (X — Y )T
(X — Y )T
π π
 
XT — Y T XT — Y T
ξ—θa ξ—θa
 
GX —Y
X —Y —¦b
X—Y


as in the proof of associativity, and at the end invoke the coboundary relation a —¦ ξb =
(bT · θ) —¦ a . Also one has to use the fact that X — Y —¦ b can be written as X — (πY —¦ πX b).
G Y in A to start with.)
(b is a map X
From all the foregoing we know that ˜’1 is correctly de¬ned with regard to T-
structures, both on objects and on maps. However, we do not quite know yet that
the values of ˜’1 are actually (Y, θ)-principal algebras. We have to verify that the T-
structure and the Y -operation on X — Y in a˜’1 are connected by the multiplicativity
G X — Y in A, that is, e = (e0 , e1 ) where
relation stated in Lemma 4. We take e: A
G X and e1 : A G Y , and recalling that Y operates on the left of the second
e0 : A
factor of X — Y , we have

(y —¦(e0 , e1 ))T · π(ξ — θa) = (e0 T, (y —¦ e1 )T )(ξ — θa)
= (e0 T · ξ, ((y —¦ e1 )T · θ) —¦(e0 T · a))
= (e0 T · ξ, (yT · θ) —¦(e1 T · θ) —¦(e0 T · a)),
(yT · θ) —¦(e0 , e1 )T · π(ξ — θa) = (yT · θ) —¦(e0 T · ξ, (e1 T · θ) —¦(e0 T · a))
= (e0 T · ξ, (yT · θ) —¦(e1 T · θ) —¦(e0 T · a)).
The multiplicativity condition to be satis¬ed by b˜’1 = X — Y —¦ b (Lemma 4) can be
similarly veri¬ed. This completes the proof of Theorem 5.
If there were an exact sequence in the ¬rst variable of the cohomology, all of this might
be avoided. (See [Barr & Rinehart (1966)], for such a possibility.)
33
TRIPLES, ALGEBRAS AND COHOMOLOGY

Modules and Extensions. We shall now consider a special case of the foregoing the-
ory, in which principal objects are interpreted as algebra extensions of a given algebra
by one of its modules. This specialization is the version of triple cohomology outlined in
[Barr & Beck (1966)]. In this part we limit ourselves to discussing tripleable adjoint pairs,
for brevity.
Recall that the “comma category” (AT , X), where X is a T-algebra, has maps Z
G X in AT as its objects, and commutative triangles

GZ
Zc
cc 
cc  in AT
c1 
X

as its maps. This is also called the category of objects over X. (See [Lawvere (1966)] for
the general de¬nition of this useful notation. Nothing depends on our having T-algebras
in this de¬nition. X could be an object in any category. The same is true of the following
de¬nition.)

DEFINITION 5. An X-module is an abelian group object in the category (AT , X).

If AT has pullbacks (¬bered products) the addition in an X-module will be represented
by a binary operation
GY
Y —X cY

cc

cc
c1 
X
(The ¬bered product is the ordinary cartesian product in (AT , X). It exists if ¬bered
products exist in A.) The identity map X is always terminal in (AT , X), since an object
G X admits the unique map
p: Z
p
GX
Zc
cc 

c
p cc  X
1 
X

Since a terminal object is a 0-fold product, the nullary operation consisting of the zero
G X will be represented by a map of the terminal object:
element in an X-module p: Y

GY
s
Xc
cc 
cc p
X c1 
X

Thus an X-module always has a zero section, which splits the projection, sp = X, in the
category AT . In e¬ect, our procedure is to identify X-modules with split extensions.
34 JONATHAN MOCK BECK

Now let A F G AT G A be a tripleable adjoint pair. Then we get adjoint functors
U

on comma categories

(F,X) (U,X)
G (AT , X) G (A, X)
(A, X)

where X = (X, ξ) is a given T-algebra. (U, X) is the obvious forgetful functor. (F, X) is
the functor which on objects:

Z (ZT, Zµ)
CQ
p pT


X (XT, Xµ)
ξ

(X, ξ)

This is the usual free algebra functor lifted up to the comma categories. The composition
(F, X)(U, X) induces a triple (T, X) in (A, X) given by

(·,X) (µ,X)
G ZT G ZT
Zc ZT T
c
cc cc
 
 
c cc
p cc  
pT T ·ξT ·ξ c1
 pT ·ξ  pT ·ξ
1
X X
p
G X. The adjoint pair (F, X)
on a typical object Z (U, X) is tripleable, that is, the
canonical functor
¦
(A, X)(T,X) o (AT , X)
cc 
cc 
cc 
cc 
cc  (U,X)
(T,X)
cc 
U
1 
(A, X)
GX∈
is an isomorphism. For it is a triviality to verify that a (T, X)-structure on p: Z
|(A, X)|
GZ
θ
ZTc
cc 

cc
 p
pT ·ξ c1 
X
p
G (X, ξ), i.e., an object of (AT , X).
is precisely equivalent to a T-algebra map (Z, θ)
We have a cohomology theory

n ≥ 0,
H n (Z, Y )X ,
35
TRIPLES, ALGEBRAS AND COHOMOLOGY

de¬ned for Z a given T-algebra over X (suppressing the algebra structures and the projec-
G X from the notation) and Y a given X-module. The cohomology is relative
tion Z
to the underlying object functor (U, X): (AT , X) G (A, X). The subscript X is put
in as a reminder that all algebras, . . . are being considered with given projections into
X. The complexes used to de¬ne these groups resemble the usual ones (homogeneous or
nonhomogeneous), but all cochains have to be maps over X.
By tripleableness, H 0 (Z, Y )X is the abelian group of maps
GY
Zc
cc 
cc  in (AT , X) ·
c1 
X
G Z over X. We will return to this
H 1 (Z, Y )X classi¬es Y -principal T-algebras E
in a moment, giving a separate interpretation to the case Z = X. First we point out that
there is a sort of co-Shapiro lemma by which one can in e¬ect always assume that Z = X,
that is, there is a cohomology isomorphism

H(Z, Y p’1 )Z G H(Z, Y )X ·

This results from the adjoint pair

(AT ,p)
G
T
(AT , X)
(A , Z) o
p’1


where if Y is an algebra over X, Y p’1 is the pullback (existing, as remarked above, if
A has pullbacks). The coadjoint is just composition with p, the ¬xed structural map Z
G X. Since p’1 is an adjoint, it preserves abelian group objects:
± 
 Y p’1 • • • •G Y 
 
 
 
 
 
Z-module X-module
 
 
 
 
 GX 
Z 
p


This explains the appearance of Y p’1 as coe¬cients above. Once everything is de¬ned,
the proof of the co-Shapiro lemma is trivial (the two complexes involved are isomorphic
under the above adjointness). In this sense it is su¬cient to consider H 1 (X, Y )X , which
evidently provides a cohomology classi¬cation for the objects described in the following
de¬nition:

G X (in the category (AT , X))
DEFINITION 6. An extension of X by the X-module Y
G X)-principal T-algebra over the terminal object X G X, which is trivial-
is a (Y
ized relative to the underlying object functor (U, X): (AT , X) G (A, X).
36 JONATHAN MOCK BECK

Explicitly, such an extension consists of the following data:
p
G X.
(1) A T-algebra map E

The de¬nition insists on our giving a T-algebra in (AT , X) together with a projection
map p into the terminal object:
p
GX
Ec
cc 

cc in AT ·

<<

. 8
( 14 .)



>>