For each object A of CI there is a morphism id A E Mor(A, A) which acts as left and right identity for the elements of Mor(A, B) and

or

equivalently,

^{that the}

completion

^{of G with} respect ^to the sequence

{Hi}

^is

uthe same" as the

completion

^with respect^{to the full}

family

^{. We leave this}

verification to the reader.

The process of

completion

^is

frequent

in mathematics. For

instance,

^weshall mention

completions

^of

rings

ⁱⁿ

Chapter ^{III, 10;}

^{and in}

Chapter

^{XII we shall}

deal with

completions

^{of fields.}

11. CATEGORIES AND FUNCTORS

Before

proceeding further,

^{it willnow}be convenienttointroduce^{some new}

terminology.

^{We have met}

already

several kinds of

objects:

sets,

monoids,

groups. We shallmeetmany^more,and foreach such kind of

objects

^wedefine

special

^{kinds of} maps between them

(e.g. homomorphisms).

^{Some formal}

behavior will becommonto allof

these, namely

^the existence of

identity

maps ofan

object

^onto

^itself,

^{and the}

associativity

of maps when such maps^occurin succession. We introduce the notion ofcategory^to

give

^a

general setting

^{for all}

of these.

A

category

^CIconsists ofacollection of

objects Ob(Ci);

^andfortwo

objects A,

^BE

Ob(CI)

^aset

Mor(A, B)

^{called the set of}

morphisms

^{of A into}

^B;

^{and for}

three

objects A, B,

^CE

Ob(Ci)

^{alaw of}

composition (i.e.

^a

map)

Mor(B, C)

^x

Mor(A, B)

^-+

Mor(A, C)

satisfying

^the

following

^axioms:

that

f

^{is a}

morphism

^of

(1,

^{i.e. an element of some set}

Mor(A, B)

^{for some}

A,

^BE

Ob(Ci).

By

^{abuse of}

language,

^wesometimes refer^tothecollection of

objects

^asthe

category itself,

^ifitis clear what the

morphisms

^{aremeant to be.}

An

elementfE Mor(A, B)

^{is also}

writtenf:

^{A -.Bor}

A ^-.f B.

A

morphism f

^{is called an}

isomorphism

if there exists a

morphism

g^:B^-+ A such that_g⁰

f

^and

fog

^are the identities in

Mor(A, A)

^and

Mor(B, B)

respec-

tively.

^{If A =}

B,

^thenwealsosaythat the

isomorphism

^isan

automorphism.

A

morphism

^ofan

object

^A into itself is called an

endomorphism.

^Thesetof

endomorph

^{ismsof Ais denoted}

by End(A).

^{Itfollowsatoncefromouraxioms}

that

End(A)

^isamonoid.

Let A bean

object

^ofa

category

^{Ci. We denote}

by Aut(A)

^{the set ofauto-}

morphisms

^{of A. This set is infact a} group, because all ofour definitions are so

adjusted

^{so astosee}

immediately

^{that the}groupaxiomsaresatisfied

(associa- tivity,

^unit

element,

^and existence of

inverse).

^{Thus we now}

begin

^{to see some}

feedback between abstract

categories

^{andmore concreteones.}

Examples.

^{Let S be the}

category

^whose

objects

^are sets, and whose

morphisms

^aremapsbetweensets. We say

simply

^{that S isthe}

category

^ofsets.

Thethreeaxioms CAT

1, 2,

^3are

trivially

^satisfied.

Let

Grp

^{be the}

category

^ofgroups, i.e. the

category

^whose

objects

^aregroups and whose

morphisms

^are

group-homomorphisms.

^Here

again

^{thethreeaxioms}

are

trivially

^satisfied.

Similarly,

^{we have a}

category

^of

monoids,

^denoted

by

Mon.

Later,

^{when we define}

rings

^and

^modules,

^itwill be clear that

rings

^{form a} category, ^{and so do modules over a}

ring.

It is

important

^to

emphasize

here that there are

categories

^{forwhich the set}

of

morphisms

^{is not an} abelian group. Some of the ^most

important examples

are:

The

category eO,

^whose

_objects

^are open ^sets in Rⁿ and whose

morphisms

are continuous maps.

The category ex with the same

objects,

^{but whose}

morphisms

^{are the Coo}

maps.

The

category 801,

^whose

objects

^areopen ^sets in

en,

^{and whose}

morphisms

are

holomorphic

maps. In each^casethe axioms ofacategory^are

verified,

^because for instance for

801,

^the

composite

^of

holomorphic

maps is

holomorphic,

^and

similarly

for the other

types

of maps. Thus ^a

CD-isomorphism

^{is a continuous}

map!:

^{u Vwhich hasa}continuous inverse g: V U. Note thatamap may be a

CD-isomorphism

^{but not a}

Coo-isomorphism.

^For

instance,

^{x x3 is aCo-}

automorphism

^of

R,

^but its inverse is not differentiable.

In mathematics one studies manifolds in anyone of the above

categories.

Thedetermination of the group of

automorphisms

^ineachcategory ^isoneofthe basic

problems

^{of the area of} mathematics concerned with that

category.

^In

complex analysis,

^{onedetermines}

early

the group

ofholomorphic automorphisms

of the unit disc as the group of all maps

. c ^- z

z e'B

_

1 ^- cz

with ()real and ^{c E}

C, I

^c

I

^{< 1.}

Next we considerthe notion of

operation

ⁱⁿ

categories. ^First,

observe that if G isagroup, then the G-sets form^a

category,

^whose

morphisms

^arethe maps

f:

^{S S' such}

thatf(xs)

⁼

xf(s)

^{forx E G and s E S.}

More

generally,

^{we can nowdefinethenotion ofan}

operation

^ofagroup G

on an

object

ⁱⁿ any

category. Indeed,

^{let CI} be ^a

category

^{and AE}

Ob(CI).

By

^an

operation

^{ofGon Aweshallmean a}

homomorphism

^ofG into the group

Aut(A).

^In

practice,

^an

object

^{A is asetwith}

elements,

^{and an}

automorphism

in

Aut(A) operates

^{on A as a} set, i.e. induces ^a

permutation

^ofA.

Thus,

^ifwe havea

homomorphism

p ^: G

Aut(A),

then for eachx E Gwehave an

automorphism p(x)

of A which isa

permutation

of A.

An

operation

^ofagroup G ^{on an}

object

^{A is alsocalled a}

representation

^of

Gon

A,

^andone then says that G is

represented

^{as a}group of

automorphisms

of A.

Examples.

^Onemeets

representations

in many^contexts. In this

book,

^we shall encounter

representations

^ofa group^on finite-dimensional vectorspaces, with the

theory pushed

^tosome

depth

ⁱⁿ

Chapter

^{XVIII. We}shall also deal with

representations

^ofa group^onmodules over a

ring.

^In

topology

and differential

geometry,

^one represents groups ^as

acting

^{on various}

topological

spaces, for instance

spheres.

Thus if X isadifferential

manifold,

^{or a}

topological manifold,

and G isa group, ^one considers all

possible homomorphims

^{of G into}

Aut(X),

where Aut refers to whatever

category

^is

being

dealt with. Thus G may be

represented

^inthegroup of

CO-automorphims,

^orCoo

-automorphisms,

^or

analytic automorphisms.

^Such

topological

^{theories are not}

independent

^{of the}

algebraic theories,

^because

by functoriality,

^an action of G on the manifold induces an

action on various

algebraic

^functors

(homology,

^K-

functor, whatever),

^{so that}

topological

^ordifferential

problems

^{aretosomeextent}

analyzable by

^{thefunctorial}

action on the associated groups, ^vector spaces, ^or modules.

Let

A,

^{B be}

objects

^ofa

category

^{Cl. Let}

Iso(A, B)

^{be thesetof}

isomorphisms

of'A with B. Then the group

Aut(B)

operates ^on

Iso(A, B) by composition;

namely,

^{ifu E}

Iso(A, B)

^{andv E}

Aut(B),

^then

(v, u)

^v0u

gives

^the

operation.

If Uo is ^one element of

Iso(A, B),

^then the orbit of Uo is all of

Iso(A, B),

^so

v v⁰Uo is ^a

bijection Aut(B) Iso(A, B).

The inverse

mapping

^is

given by

u .-+ Uo

Uo 1.

^{This trivial formalism is very}

basic,

^{and is}

applied constantly

^to

eachone of the classical

categories

mentioned above. Ofcourse, ^we also have

asimilar

bijection

^ontheother

^side,

but the group

Aut(A) operates

^{on the}

right

of

Iso(A, B) by composition. Furthermore,

ifu:A B is an

isomorphism,

^then

Aut(A)

^and

Aut(B)

^are

isomorphic

^under

conjugation, namely

w uwu-^l is an

isomorphism Aut(A) Aut(B).

Two such

isomorphisms

^differ

by

^{an inner}

automorphism.

One may visualize this system ^{via the}

following

commutative

diagram.

u B

!

^uwu-I

U B A

w!

A

Let p ^: G

Aut(A)

^and

p':

^G

Aut(A')

^be

representations

^{of a}group G

on two

objects

^{A and A' in the same} category. ^A

morphism

^of

pinto p'

^{is a}

morphism

^{h: A} A' such that the

following diagram

is commutative for all

x E G:

h

A'

!P'

^(x)

A' A

P(x)!

A

h

It is then clear that

representations

^ofa group G in the

objects

^ofa

category

^C1 themselves form a category. ^An

isomorphism

^of

representations

^{is then an}

isomorphism

^{h : A--+ A'}

making

^{the above}

diagram

commutative. An isomor-

phism

^of

representations

^isoften calledan

equivalence,

but I don't liketo

tamper

with the

general

system^of

categorical terminology.

^Notethat if h isanisomor-

phism

^of

representations,

then instead of the above commutative

diagram,

^we

let

[h]

^be

conjugation by ^h,

^{and we} may ^use the

equivalent diagram

Y

Aut(A)

G

P

!

^{[h J}

Aut(A ')

Considernextthe case where C1 is thecategory^ofabelian groups, which ^we may denote

by

^{Ab.Let Abean}abelian group and G^agroup. Given^an

operation

of G on the abelian group

A,

^{Le. a}

homomorphism

p ^: G

Aut(A),

let us denote

by

^{x · athe element}

Px( a).

^{Then we see} that for all_x, y ^E

G,

^a, b E

A,

^{we have:}

e^.a ⁼ a,

x.(a

⁺

b)

^{= x.a +}

x.b,

x^.0 ⁼ O.

x^.

(y

^.

a)

⁼

(xy)

^.a,

Weobservethatwhen agroup G

operates

^{on itself}

by conjugation,

^thennot

only

^doesG

operate

^{onitselfas asetbut also}

operates

^{onitselfas an}

object

^inthe

category

^ofgroups, i.e.the

permutations

^induced

by

^the

operation

^are

actually group-automorphisms.

Similarly,

^we shall introduce later other

categories (rings, modules, fields)

andwehave

given

^a

general

definition ofwhat it meansforagroup^to

operate

on an

object

in anyoneof these

categories.

Let CI be a

category.

We may take ^as

objects

^{of a new}

category

^{e the}

morphisms

^{of CI. If}

f:

^{A -.Band}

f':

^{A' -.B' are two}

morphisms

^{in CI}

(and

thus

objects

^of

e),

^{then we define a}

morphism f

^-.

f' (in e)

^{to be a}

pair

^of

morphisms (qJ, 1/1)

^{in CI}

making

^the

following diagram

commutative:

A ^f B

j j

A^I B^'

f'

In that _way, it is

clear

that e isa

category. Strictly speaking,

^aswithmaps of sets, ^we should index

(qJ, 1/1) by f

^and

f' (

otherwise CAT 1 is not

necessarily satisfied),

^{but such}

indexing

^{is omitted in}

practice.

Therearemanyvariationsonthis

example.

^For

instance,

^wecould restrict

ourattentionto

morphisms

^{in CIwhichhaveafixed}

object

^of

departure,

^orthose

which havea fixed

object

of arrival.

Thuslet A bean

object

^of

CI,

^{and let}

CIA

^{be the}

category

^whose

objects

^are

morphisms

f:

^{X-. A}

in

CI, having

^{A as}

object

of arrival. A

morphism

ⁱⁿ

CIA

^from

f:

^{X-.A to}

g^: Y^-.A is

simply

^a

morphism

h:X-.Y in CIsuchthat the

diagram

is commutative:

X

h)

_Y

\}

A