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

—¦

G (Z, E)X

(Z, Y )X — (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 —¦ 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 —¦ 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 de¬ned 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.

˜

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 ˜ explicit in this context. An extension p: E

gives rise to a nonhomogeneous 1-cocycle

GY

a

XTc

cc

cc

ξ c1

X

G E is the T-structure

which is determined by the formula a —¦ ξs = sT · ψ, where ψ: ET

of the extension. Note that the two maps

YX qq

ww qq s

w

ξ

ww qq

ww qq

w 5

XT qq YE in (A, X)

ww

qq ww

ww ψ

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.

ψ = θ + pT · 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 ¬nal 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 classi¬es

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 classi¬es them turns out to be Hochschild™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 ∈ X.

The empty word is denoted by ( ). By adjointness we get a triple T = (T, ·, µ) in A

where XT = XF U is the underlying set of the free group generated by X. One further

G XT is the

veri¬es (since these maps are determined by the adjointness) that X·: X

G XT is the map described as follows. The elements of

map x ’ (x), while Xµ: XT T

XT T are words spelled with generators (w), where w ∈ XT . Then Xµ maps (w) ’ w

and is extended to other elements by multiplicativity. For example, if w1 = (x1 )(x2 ),

w2 = (x2 )’1 , then W = (w1 )(w2 ) ∈ XT T (it can also be written ((x1 )(x2 ))((x2 )’1 )), and

W · Xµ = w1 w2 = (x1 ) ∈ XT . Intuitively, this is the only map µ could be; in fact,

this map is correct, because by construction of T, Xµ underlies the counit XF : XF U F

G XF , which is de¬ned 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 1“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 ξ: 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 ∈ X, and given a function ξ: XT

formal, juxtaposition, product (x0 )(x1 ) exists in XT , and the value of ξ on this element,

[(x0 )(x1 )]ξ, naturally suggests itself as the de¬nition of a binary operation x0 · x1 . In fact,

G X, we can de¬ne a candidate for a group law on X:

just using any function ξ: XT

x0 · x1 = [(x0 )(x1 )]ξ (multiplication)

x’1 = [(x)’1 ]ξ (inversion)

1 = [( )]ξ (neutral element)

39

TRIPLES, ALGEBRAS AND COHOMOLOGY

The assumption that ξ is unitary and associative implies that these operations satisfy the

group axioms. For example, here is the proof that the multiplication x0 · x1 is associative.

Let W1 , W2 ∈ XT T be the following words:

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

Pursuing them around the associativity diagram

Xµ

G XT

XT T

ξT ξ

GX

XT ξ

we get

W1 · ξT · ξ = [(x0 · x1 )(x2 )]ξ = (x0 · x1 ) · x2

W2 · ξT · ξ = [(x0 )(x1 · x2 )]ξ = x0 · (x1 · x2 )

whereas W1 · Xµ · ξ = W2 · Xµ · ξ = [(x0 )(x1 )(x2 )], q.e.d. Note that [(x)]ξ = x because of

the unitary axiom

X·

G XT

Xc

cc

cc

cc ξ

Xc cc

1

X

The other group axioms can be veri¬ed in a similar manner.

GX

Conversely, it is quite clear that a given group law on X de¬nes a map ξ: 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

¦

o

AT c G

cc

cc

c1 U

UT

A

G πU which

maps a group π into its underlying set πU with the T -structure π U : πU T

employs the given operations in π to evaluate in πU all the proposed group operations

formally present in πU T . The foregoing procedure of constructing a group out of a T-

algebra de¬nes a functor ¦’1 : AT G G with the properties ¦¦’1 = G , ¦’1 ¦ = AT .

(One gets an actual isomorphism of categories.) The functor ¦, 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 in¬nitary 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 de¬nable 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 restriction”

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 ξ: X

has to be the identity by the unitary axiom.) In the canonical diagram

¦

o

Ac Top

cc

cc

U

A c1

A

we have ¦ = 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,

ˇ

with the usual underlying set functor (the left adjoint is the Stone-Cech compacti¬cation

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 π-sets. Let π be a monoid (group, in the original

Eilenberg-Mac Lane presentation [Eilenberg & Mac Lane (1947)]). Using π we get a triple