TRIPLES, ALGEBRAS AND COHOMOLOGY

JONATHAN MOCK BECK

Contents

Title page 3

0 Introduction 4

1 Triples and Algebras. 4

2 Cohomology 12

3 Interpretation of Cohomology in Dimensions 0 and 1 21

4 Examples 38

Editors™ Preface

It is with great pleasure that the editors of Theory and Applications of Categories make

this dissertation generally available. Although the date on the thesis is 1967, there was a

nearly complete draft circulated in 1964. This thesis was a revelation to those of us who

were interested in homological algebra at the time.

Although the world™s very ¬rst triple (now more often called “monad”) in the sense of

this thesis was non-additive and used to construct ¬‚abby resolutions of sheaves ([Gode-

ment (1958)]), the then-prevailing belief was that the theory of triples had a use in ho-

mological algebra only via additive triples on abelian categories, typically something like

Λ —Λ—Λ ’, on the category of Λ-Λ bimodules. In fact, [Eilenberg & Moore (1965b)] went

so far as to base their relative homological algebra on triples that were additive and pre-

served kernels. Thus there was considerable astonishment when Jon Beck, in the present

work, was able not only to de¬ne cohomology by a triple on the category of objects of

interest (rather than the abelian category of coe¬cient modules) but even prove in wide

generality that the ¬rst cohomology group classi¬es singular extensions by a module. Not

the least of Beck™s accomplishments in this work are his telling, and general, axiomatic

descriptions of module, singular extension, and derivation into a module. The simplicity

and persuasiveness of these descriptions remains one of the more astonishing features of

this thesis.

Transmitted by R. Par´, J. Stashe¬, and R.J. Wood. Reprint published on 2003-06-17.

e

2000 Mathematics Subject Classi¬cation: 18C15,18C20.

Key words and phrases: Cotriple cohomology, Beck modules.

c Jonathan Mock Beck, 1967. Permission to copy for private use granted.

1

2 JONATHAN MOCK BECK

We hope that this publication will provide an informative and useful resource for

workers in our ¬eld. We have left the thesis unchanged except for (surprisingly few)

typographical corrections. In addition, the editors have made a few notes in places where

we have updated references, corrected the original manuscript, or, in one place, clari¬ed

things somewhat. But what you have before you is basically the thesis presented in 1967.

Acknowledgments. One of the gratifying aspects of this reprint e¬ort was that within

24 hours of asking for volunteers for the retyping project, there were more volunteers

than we could use. Eventually, we chose 11, more or less at random, and asked each one

to type 10 pages of the 109 pages (plus bibliography). There were logistical problems

getting the pages to the volunteers, but not one of them took more than a few days to

do his or her bit once they had it. The typists we thank are Robert Dawson, Robert

L. Knighten, Francisco Marmolejo, Shane O™Conchuir, Valeria de Paiva, Dorette Pronk,

Robert Rosebrugh, Robert Seely, Andrew Tonks, Charles Wells, and Noson Yanofsky. In

addition we would like to thank Jack Duskin, Je¬ Egger, and Maria Manual Clementino

who were equally prepared to volunteer their labor. Joan Wick-Pelletier gave us a copy

of the original thesis, without which we could not have proceeded. Finally, we thank

Donovan Van Osdol for a masterly proof-reading job that went beyond the ¬nding of

variations from the original, but also found errors in the original.

While working on this we were reminded how we used to type mathematics in those

days. Putting in symbols by hand, sometimes fabricating them by overtyping two char-

acters and so on. Mathematics was called “penalty copy” and linotypists would charge

double or more to do it. So we also owe a big debt of gratitude to Donald Knuth, to

A

Leslie Lamport, the LTEX 2µ team, Kris Rose, and Ross Moore (the creator and current

developer, resp., of X -pic).

Y

The editors of Theory and Applications of Categories

Triples, Algebras and Cohomology

Jonathan Mock Beck

1967

Submitted in Partial Ful¬llment of the Requirements

For the Degree of Doctor of Philosophy,

In the Faculty of Pure Science,

Columbia University

4 JONATHAN MOCK BECK

0. Introduction

This thesis is intended to complete the exposition in [Eilenberg & Moore (1965a)] with

regard to certain points. In §1 we recall the de¬nitions of triple, algebra over a triple, and

give our main (original) de¬nition, that of tripleable adjoint pair of functors. In §2 we

show how to obtain a cohomology theory from an adjoint pair of functors. In §3, when

the adjoint pair is tripleable, we prove that the cohomology group H 1 classi¬es principal

homogeneous objects. When coe¬cients are in a module, principal objects are interpreted

as algebra extensions. §4 is devoted to examples. Many categories occurring in algebra are

shown to be tripleable. The corresponding cohomology and extension theories, ranging

from groups and algebras to the classical Ext(A, C), are discussed. Many new theories

arise.

A method for proving coincidence of triple cohomology with certain standard theories

has been given by [Barr & Beck (1966)]. That paper contains a summary of the present

work.

I should like to express my most profound gratitude to Professor S. Eilenberg, with

the help of whose energetic criticism and encouragement these results were obtained.

1. Triples and Algebras.

DEFINITION 1. T = (T, ·, µ) is a triple in a category A if T is a functor A ’ A, · and

µ are natural transformations A ’ T and T T ’ T respectively, and the diagrams

T· ·T

G TT G TT

Tc Tc

c c

cc cc

cc cc

cc cc

µ µ

T ccc T ccc

cc cc

1 1

T T

Tµ

G TT

TTT

µ

µT

GT

TT µ

commute. Thus · is a right and left unit for the multiplication µ, and µ is associative.

Dually, G = (G, , δ) is a cotriple in B if G: B ’ B is a functor, : G ’ B and δ: G ’ GG

5

TRIPLES, ALGEBRAS AND COHOMOLOGY

are natural transformations, and

G GG G GG

δ δ

Gc Gc

cc cc

cc cc

cc cc

c c

G ccc G ccc

G G

cc cc

1 1

G G

G GG

δ

G

δ δG

G GGG

GG Gδ

commute.

Triples and cotriples usually arise from adjoint functors. We recall that an adjoint pair of

G B, U : B G A together with a natural isomor-

functors consists of functors F : A

phism

±

G (AF, B)

(A, BU )

where A ∈ |A|, B ∈ |B|. The functor U is the right adjoint (or adjoint), and F is the left

adjoint (or coadjoint). The relation between F and U is often symbolized by ±: F U.

G B de¬nes a natural transformation : U F G B.

Taking A = BU , (BU )±: BU F

G AF U de¬nes a natural transformation ·: A G F U.

Taking B = AF , (AF )±’1 : A

· and are called the unit and counit of the adjointness, and satisfy the relations [Kan

(1958)]

·F U·

G F UF G UF U

Fc Uc

c cc

cc cc

cc cc

cc c

F ccc U ccc

F U

cc cc

1 1

F U

The adjoint pair ± induces a triple T = (T, ·, µ) in the category A, de¬ned by

±

GA

T = FU : A

GT

T ·: A

GT

µ = F U : TT

and a cotriple in the category B, de¬ned by

±

GB

G = UF : B

GB

G :G

G GG

δ = U ·F : G

6 JONATHAN MOCK BECK

Indeed, T ··µ = F U ··F U = F (U ·· U ) = F U = T and ·T ·µ = ·F U ·F U = (·F ·F )U =

F U = T dispose of the unitary axiom. For associativity, T µ · µ = F U F U · F U =

F (U F · )U and µT · µ = F U F U · F U = F ( U F · )U . These coincide by naturality of

. The proof for G is dual.

The triple T, with its unit and multiplication, is something like a monoid. The next

de¬nition formalizes the intuitive idea of such a monoid™s operating on an object of the

category A [Eilenberg & Moore (1965a)].

DEFINITION 2. (X, ξ) is a T-algebra if ξ: XT ’ X is a map in A and the diagrams

X· Xµ

G XT G XT

Xc XT T

cc

cc

cc

c

X ccc

ξ ξT ξ

cc

1

GX

X XT ξ

commute. ξ is called the T-structure of the algebra and the above diagrams state that ξ

is unitary and associative. f : (X, ξ) ’ (Y, θ) is a map of T-algebras if f : X ’ Y in A and

is compatible with T-structures:

fT

GYT

XT

ξ θ

GY

X f

T-algebras form a category which we denote by AT .

We have a canonical adjoint pair of functors

FT UT

G AT GA

A