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 X
c1 
X

Therefore the unlabelled structural map of E as an object in (AT , X) also has to be
p.
G X on the object p: E G X in the
(2) An operation of the X-module Y
category (AT , X). This means that there is a pairing
в—¦
G (Z, E)X
(Z, Y )X Г— (Z, E)X
G X, ( , )X being the hom
which is (contravariantly) natural in the variable Z
functor in (AT , X). This operation is compatible with the projection p ((y в—¦ e)p = ep),
and is simply transitive: given any two maps
e0
G
GE
Zc
cc e1 

cc 
c1 
X
there exists a unique
y
GY
Zc
cc 

cc 
c1 
X
such that y в—¦ e0 = e1 .
p
G E in A. Thus the extension E G X is split in the
s
(3) There is a section X
underlying category.
One further easily checks that maps of extensions are T-algebra maps over X
f
GE
Ec
cc 

c
p cc  p
1 
X
G X)-operations (and do not need to respect the sections,
which commute with the (Y
which, as with principal algebras, are considered as given parts of the structure).
37
TRIPLES, ALGEBRAS AND COHOMOLOGY

G X as deп¬Ѓned clearly form a category, which we denote by Ex(X, Y ).
Extensions E
Let Ex(X, Y ) be the set of isomorphism classes in this category (which is in fact a
groupoid). Let Aut(Y ) be the automorphisms of the trivial extension, which is just
G X itself. Then Theorem 5 in this context becomes:
the X-module Y

THEOREM 6.
О˜
G Z 1 (X, Y )X
Ex(X, Y )
is an equivalence of categories, inducing isomorphisms

G H 1 (X, Y )X
Ex(X, Y )

G H 0 (X, Y )X
Aut(Y )
Both the extensions and the cohomology are taken relative to the underlying object functor
(U, X): (AT , X) G (A, X).

GX
It is of some interest to make О˜ explicit in this context. An extension p: E
gives rise to a nonhomogeneous 1-cocycle

GY
a
XTc
cc 
cc 
Оѕ c1 
X
G E is the T-structure
which is determined by the formula a в—¦ Оѕs = sT В· П€, where П€: ET
of the extension. Note that the two maps

YX qq
ww qq s
w
Оѕ
ww qq
ww qq
w 5
XT qq YE in (A, X)
ww
qq ww
qq
ww П€
sT qq5 ww
ET
GX
agree when followed by p, so one can be carried into the other by a map into Y (Y
is also an X-module in the underlying category). Conversely, identifying E with Y (as
an object in A) for simplicity, the T-structure on E must be of the form
GE.
П€ = Оё + pT В· a: ET

The sum of these maps exists because they are both compatible with the ever-present
projections into X.
G X to be an abelian group object.
Incidentally, nothing requires the module Y
We have assumed it above only because abelianness is usually present in examples, and
there is no convenient terminology for the other notion.
38 JONATHAN MOCK BECK

A п¬Ѓnal comment concerning the application of these ideas: in practice one starts not
with A and T, but with category B for which one seeks tripleable underlying object
G A. Given one such, the cohomology H 1 (X, Y )X relative to U classiп¬Ѓes
functors U : B
extensions which are split in the underlying category in which U takes values. For instance,
associative K-algebras are tripleable over K-modules, the extensions come out K-linearly
split, and the cohomology which classiп¬Ѓes them turns out to be HochschildвЂ™s. If sets are
taken as the underlying objects of K-algebras, the extensions only have to be split in the
category of sets, i.e., they are not really split at all, and the cohomology which emerges
to classify them is a theory, it turns out, developed by Shukla. But these are subjects
that we will discuss in much more detail in the Examples.

4. Examples
We begin with an archetype of a large number of algebraic examples.
EXAMPLE 1. Groups tripleable over sets. We let G be the category of groups, A the
G A the usual underlying set functor. We have adjointness
category of sets, and U : G
F U , where F is the free group functor. If X is a set, we write the elements of XF
as words spelled by means of formal group operations in generators (x), where x в€€ X.
The empty word is denoted by ( ). By adjointness we get a triple T = (T, О·, Вµ) in A
where XT = XF U is the underlying set of the free group generated by X. One further
G XT is the
veriп¬Ѓes (since these maps are determined by the adjointness) that XО·: X
G XT is the map described as follows. The elements of
map x в‡’ (x), while XВµ: XT T
XT T are words spelled with generators (w), where w в€€ XT . Then XВµ maps (w) в‡’ w
and is extended to other elements by multiplicativity. For example, if w1 = (x1 )(x2 ),
w2 = (x2 )в€’1 , then W = (w1 )(w2 ) в€€ XT T (it can also be written ((x1 )(x2 ))((x2 )в€’1 )), and
W В· XВµ = w1 w2 = (x1 ) в€€ XT . Intuitively, this is the only map Вµ could be; in fact,
this map is correct, because by construction of T, XВµ underlies the counit XF : XF U F
G XF , which is deп¬Ѓned by multiplication in the free group.
We will now show that the underlying set functor is tripleable, by demonstrating that
the category of T-algebras AT is equivalent to the category of groups, G . We do this by
exhibiting a 1вЂ“1 correspondence between T-algebra structures on a set X and group laws
on X. Indeed, one could think of XT merely as a list of all the group operations which
G T is then
could conceivably be performed on elements of X. The T-structure Оѕ: XT
a function which can be thought of as telling us what the values of these operations are.
G X, the
For example, given any two elements x0 , x1 в€€ X, and given a function Оѕ: XT
formal, juxtaposition, product (x0 )(x1 ) exists in XT , and the value of Оѕ on this element,
[(x0 )(x1 )]Оѕ, naturally suggests itself as the deп¬Ѓnition of a binary operation x0 В· x1 . In fact,
G X, we can deп¬Ѓne a candidate for a group law on X:
just using any function Оѕ: XT

x0 В· x1 = [(x0 )(x1 )]Оѕ (multiplication)
xв€’1 = [(x)в€’1 ]Оѕ (inversion)
1 = [( )]Оѕ (neutral element)
39
TRIPLES, ALGEBRAS AND COHOMOLOGY

The assumption that Оѕ is unitary and associative implies that these operations satisfy the
group axioms. For example, here is the proof that the multiplication x0 В· x1 is associative.
Let W1 , W2 в€€ XT T be the following words:

W1 = ((x0 )(x1 ))((x2 )), W2 = ((x0 ))((x1 )(x2 ))В·

Pursuing them around the associativity diagram
XВµ
G XT
XT T

ОѕT Оѕ

 
GX
XT Оѕ

we get
W1 В· ОѕT В· Оѕ = [(x0 В· x1 )(x2 )]Оѕ = (x0 В· x1 ) В· x2
W2 В· ОѕT В· Оѕ = [(x0 )(x1 В· x2 )]Оѕ = x0 В· (x1 В· x2 )
whereas W1 В· XВµ В· Оѕ = W2 В· XВµ В· Оѕ = [(x0 )(x1 )(x2 )], q.e.d. Note that [(x)]Оѕ = x because of
the unitary axiom
XО·
G XT
Xc
cc
cc
cc Оѕ
Xc cc
1
X
The other group axioms can be veriп¬Ѓed in a similar manner.
GX
Conversely, it is quite clear that a given group law on X deп¬Ѓnes a map Оѕ: XT
by just performing the indicated group operations using the group law, and this map will
be unitary and associative. More precisely, the canonical functor
О¦
o
AT c G

cc

cc 
c1  U
UT
A
G ПЂU which
maps a group ПЂ into its underlying set ПЂU with the T -structure ПЂ U : ПЂU T
employs the given operations in ПЂ to evaluate in ПЂU all the proposed group operations
formally present in ПЂU T . The foregoing procedure of constructing a group out of a T-
algebra deп¬Ѓnes a functor О¦в€’1 : AT G G with the properties О¦О¦в€’1 = G , О¦в€’1 О¦ = AT .
(One gets an actual isomorphism of categories.) The functor О¦, leaving underlying sets
unchanged, simply interchanges two equivalent formulations of the notion of a group
structure on a set.)
The example of groups is typical. It is known that all algebraic categories in the
sense of [Lawvere (1963)] are tripleable over sets, with respect to their usual underlying
40 JONATHAN MOCK BECK

set functors. [Linton (1966)] has shown that over sets this is almost the whole story:
admitting inп¬Ѓnitary operations one gets equational categories of algebras, and over the
base category of sets tripleableness is equivalent to equationality.
Over other base categories, tripleableness does not seem to have any such standard
interpretations. It is the proposal of this paper that tripleableness be regarded as a new
type of mathematical structure, parallel to but not necessarily deп¬Ѓnable in terms of other
known types of structure, such as algebraic, equational, topological, ordered, . . . .
Inasmuch as it has been insinuated all along that tripleableness is a restrictionвЂ”
EXAMPLE 2. A non-tripleable adjoint pair. Let Top be the category of spaces, A the
G A the usual forgetful or underlying set functor. Left
category of sets, and U : Top
adjoint to U is the discrete space functor, XF = the topological space of underlying
set X with the discrete topology. The composition F U is just A. Indeed, the triple
T in A induced by the adjoint pair F U is just the identity triple consisting of the
identity functor with its identity natural transformation as unit and multiplication. The
GX
corresponding category of algebras is nothing but A. (An algebra structure Оѕ: X
has to be the identity by the unitary axiom.) In the canonical diagram
О¦
o
Ac Top
cc

cc
U
A c1 
A

we have О¦ = U , not an equivalence.
As we have mentioned, tripleableness implies some sort of algebraicity, and tends to ex-
clude topological structures. However, replacing Top by the category of compact4 spaces,
Л‡
with the usual underlying set functor (the left adjoint is the Stone-Cech compactiп¬Ѓcation
of the discrete space), the resulting adjoint pair is tripleable [Linton (1966)].
We shall now give some examples involving groups and monoids, and corresponding
cohomology theories which come from our general theory.
EXAMPLE 3. The category of ПЂ-sets. Let ПЂ be a monoid (group, in the original
Eilenberg-Mac Lane presentation [Eilenberg & Mac Lane (1947)]). Using ПЂ we get a triple
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