A CATEGORICAL APPROACH TO PROBABILITY THEORY
by Mich~le GIRY (Amiens)
The aim of this paper is to give a categorical definition of random p r o c e s s e s and provide
tools for their study.
A p r o c e s s is meant to d e s c r i b e something e v o l v i n g in time, the history before time t
~,probabilistically,> determining what will happen Iater on. For i n s t a n c e , it may represent a moving point x , being at time t in a space ~t endowed with a c~-algebra ~t ; the problem is then
the e x a c t position of x in ~ t "
In the very particular example of a Markov p r o c e s s , time is running through N and the
position of x in ~ n + l
~n •
n+l if
to [ 0 , 1 ]
[n from
Bn+ 1 at time
only depends on where it was in ~ n " So, for each n, a map
is g i v e n :
fn(c~
is the probability for x to be in
it was on con at time n. fn is thus asked to s a t i s f y the two following p r o p e r t i e s :
for each con,
fn ( " Bn +1 )
[n(con,')
is a probability measure on ( ~ n + l , ~ n + l ) , a n d
for each
Bn+l,
is measurable, fn is called a transition probability [4], or a p r o b a b i l i s t i c map-
ping [3] from ( ~ 2 n , ~ n ) to (F~n+1, ~ n + l ) " The p r o c e s s is then entirely defined by the
13ut if time runs through R , we need a transition probability /ts from (i'2s, ~ s )
(flt,~t)
for each couple
(s, t)
[n's.
to
with s < t. Then, if r < s < t, there are two ways of comput-
Bt knowing it was on cot at time r : forgetting s , which
frt(cor, Bt) ; or considering how z behaved at time s , it seems then reasonable to take
the mean value of f~ (cos, Bt) (f~ cos running through ~ s ) relatively to the probability measure fsr(cor,.) on f t s , which yields to f f ~ ( . , Bt) dfr(cor,.). T h i s integral is shown to deing the probability for x to be in
gives
fine a transition probability from ft r to f~t, called the composite of f ] and f ~ , which is
a s k e d , in (,good,> p r o c e s s e s , to be the same as f~. T h i s equality is called the Chapman-Kolmogoroff relation.
The composition of transition probabilities is a s s o c i a t i v e . T h i s property is equivalent
to Fubini Theorem for bounded functions and its proof, as well as that of s t a b i l i t y of transition
probabilities for this law, is rather t e c h n i c a l . T h e s e results will be c o n s e q u e n c e s of the following one : the transition probabilities and their composition form the K l e i s l i category of a monad
on the category of measurable s p a c e s . A similar monad will be constructed on a category of topological spaces
( g e n e r a l i z i n g the one defined by S w i r s z c z [6] on compact s p a c e s ) .
As F. W. Lawvere already pointed out in an unpublished paper [3 ] in 1962, most problems
in probability and s t a t i s t i c s theory can be translated in terms of diagrams in t h e s e K l e i s l i c a t egories.
In the s e q u e l w e ' l l mainly study projective limits which will lead us to construct probability m e a s u r e s on sample s e t s of p r o c e s s e s .
69
I'd like to express my gratitude to the organizers of the International Conference on
Categorical Aspects o f Topology and Analysis, who gave me the opportunity to give a lecture
and to publish this work, and to warmly thank Andr&e Charles Ehresmann, whose constant
help and understanding made it possible.
I. THE PROBABILITY MONADS.
]. Notations. ~ , ~ is the category of measurable s p a c e s ; an object will be denoted by
and its a-algebra by ~
. The morphisms are the measurable maps9 ~,,l is the category of
Polish spaces (topological spaces underlying a complete metric space) ; again, an object is
called ~ and ~fl is its Borel o-algebra. The morphisms are the continuous maps.
A monad is going to be constructedon both ~lI~ and ~,~ ; the definitions being very
much the same, in the sequel }( will stand for either of them, unless otherwise notified.
29 Construction.
a) Thefunctor 1I (called P i n [ 3 ] ) : If ~ is an object of J{, 11(~) is the set of probability measures on ~2 ( i . e . , the o-additive maps from ~
to [0, 1] sending ~ to 1 ),
endowed :
9 if J{ = ~ ,
with the initial a-algebra for the following evaluation maps, where B
runs through ~ :
P B : [ 1 ( f ~ ) -~ [0,11: P ~ P ( B ) ;
9 if 3"( = ~,~e, with the initial topology for the maps
Q:n(a)
-~ R :
P b flaP,
where f is any bounded continuous map from [1 to R ; II ( a ) is then a Polish space (its
topology is called the weak topology) [5].
If f: f~ -, f~' is a morphism of }(, and P is in I1 ([1), the probability measure on
a ' image of P by f i s defined by
II([)(P)(B') = P('~(B'))
forevery B ' i n
SO"
b) The natural trans[orm 71: ld}( * 11: The characteristic function of an element B
of ~[1 is denoted by XB" For ~o in a , the probability measure concentrated on o is
defined by
71~(co)(B) = XB(CO),
foreach
B in ~ .
e) The natural transform g: [12--> 11: For P~ in f l 2 ( ~ ) , a probability measure on
is defined by:
I~(P')(B)
= fPB dP', for every B in ~ .
These integrals are well-defined for each PB is measurable from II ( ~ ) to [ O, 1 ], hence
is integrable for P ' . The measurability of PB if fl is an object of ~,~s follows from the
fact that, ~ being metrizable, we have
~
ordinal and: ~o is the set of open sets of ~ ,
=
u ~a , where A is the first uncountable
ce<A
70
=[nn~Bn I Bn E ~r I
~a +1
if a is even,
= {Wn}Bn I Bn ~ ~a I if a is odd,
~*B~ I B n e / ~ a O e / 3 I
if a
islimit,
and is proved by induction on the c l a s s of B.
The aoadditivity of g f ~ ( P ' ) i s a consequence of the monotone convergence theorem.
3. Theorem 1. (lI,rl,~L) is amonadon }{.
A . F i r s t let us prove the following properties, valid for any morphism f..~ -, ~ ' of }(,
any P in I I ( ~ ) ,
and O':f~'-,R
P ' in [I2(f~), co in f~, 0 : f ~ R
bounded measurable
functions :
a) fo, dn(f)(P) = f O ' o f d P ,
b) fO dq~(o)) = O(co),
c) if ~:0 is definedby C o ( P ) = fO dP, CO is measurable from lI(f~) to R,
d) fO dtzf~(P') = f ~ o d P '.
In the case where 0 is of the form XB, this follows from the definitions; by l i n e a r i t y o f f
this is still true if 0 i s a simple function. The general case is a consequence of the monotone convergence theorem and the fact that 0 is the increasing pointwise limit of a sequence of simple functions.
If / : f~ -, f~ ' is in }(, s o i s
when } ( = ~ g ,
11([): When }(
~ l l ~ , m e a s u r a b i l i t y of II(f) isobvious;
continuity of 11([) follows from formula a . Now gl is a functor because, if
g: ~2'-~ f l " is another morphism of }( ,
-1
-1 -1
( g o f ) ( B " ) = f (g ( B " ) ) for B ' E ~ a . .
qfZ is clearly in }( (by definition in ~1l.~,~and by formula b in ~ g ). So is g ~ :
- if}(=~l~,
if }( = ~ J ,
-
this will follow from property c a b o v e , a p p l i e d t o I I ( f ~ ) , 0 = PB and P';
it is a consequence of formulad .
The diagramms
f
r/~
n(a)
, a'
lr/f~,
ncf)
~ n(a'~
n2(ft)
and
. rI2(f)
,a[
n(a)
._n2(a ')
[,fl,
nc[)
, n(n'~
commute: Only the second one requires some work: if P ' is in II2(f~) and B ' in ~f~,,
( I I ( f ) otzf~(P')) ( B ' ) = g f ~ ( P ' ) ( ' f l ( B ' ) ) = f pTI ( B , ) d P '
and
( g f ~ , o I I 2 ( f ) ( P ' ) ) ( B ') = f P B , d l I 2 ( f ) ( P ' )
= fPB,OlI(f) dP'
from a . But since P-fl(B') = p B,o 11I( f ) , the commutativity follows.
U n i t a r i t y o f 7/ is easy. L e t ' s p r o v e the a s s o c i a t i v i t y of #. If P " is in II3(f~) and B
is in Sf~ ,
71
(Izl2 oII (t,~2 ) ( P ' ) ) ( B ) = f p B o l ~ a d P "
= f (pBdP "
from a and definition of ~ , and
(gf~ ~
Equality follows from d.
= fPBd#H(f~)(P").
A
4. The Kleisli category of ( H , r / , g ) .
The K l e i s l i category a s s o c i a t e d to ( 1I, rI , t~ ) is c a l l e d 9~-. If ]', g are morphisms
in 93", we write
gKf
The canonical functor from J{ to 9~q- sends
~2
h
,-a'
to
A K l e i s l i morphism f : f t ~
~
~
a',
where ,~=
ft' is a transition probability in the following way: let
us define
F:a•
Then F ( c o , . )
by
F(co,B')=f(co)(B').
is a probability measure on [~' and F ( . , B ' )
= pB,of
is measurable, so F
is a transition probability.
If J{ = N ~ - , the transformation f ~ F is a one-to-one c o r r e s p o n d e n c e between 93and the c l a s s of transition probabilities.
Moreover, the composition in 93- is the usual composition of transition p r o b a b i l i t i e s , since :
(gK f)r
)(B)
= f P B o g df(oo ) = f g(. ),fB) df(o.> )
= fg(.,B)dF(oo,.)
with the above notations.
So, Chapman-Kolmogoroff relation means that a <,good,> p r o c e s s
( cf. the introduction) is defined by a functor from the ordered set R to 9 ~ .
II. PROBLEMS OF PROJECTIVE LIMITS IN 9,'~.
In the study of a p r o c e s s , an important problem i s to find a probability measure on
the sample set, compatible with the given transition probabilities. This sample s e t being
the projective limit of the s e t s of <,histories before a time t,>, the e x i s t e n c e of a limit for
a p r o j e c t i v e system of probability measures soon a r i s e s , which can be translated in terms
of preservation by K of p r o j e c t i v e limits in }( (which still r e p r e s e n t s ~li,~,~ or 9 , ~ ). Let
us recall that ~ , ~ admits all limits and 9 , ~ all c o u n t a b l e limits.
In the sequel, we c o n s i d e r the following situation : f~ is the projective limit of a
functor F from a filtered set ( I , > )
grams, for i > j :
to }{ , and is c h a r a c t e r i z e d by the commutative diao
72
we want to know how .K p r e s e r v e s the limit, that i s : when and how can the dotted arrow of
the following diagram be filled up ?
g'
,~
i.
~t ~
(D)
! jl
Notice that commutativity of the outer diagram means that H (fS.)(gi(a~')) = gj(a)') for
e a c h co ', so that the families (gi (a)'))i are projective s y s t e m s of probability m e a s u r e s .
Hence, the first problem will be the e x i s t e n c e of a limit g(~o') for such a system, and the
s e c o n d one the fact that the so defined g is a morphism of ~ .
The results will be of a
different nature in ~R~ and in ~og.
1. Theorem 1. l[ }{ = ~I~ and the qi's are onto: (qi)iE 1 is universal among the cones
(gi)iel with basis w F such that:
(*) For all increasing sequence (in) n of 1 and each (Bin)n in [In~{1in such that
c~ }qiln (Bi ) = ~' the sequence pBi o gi
n
n
pointwisely converges to O.
n
A . L e t ~o' be in {1'. (gi( ~176 b e i n g a p r o j e c t i v e s y s t e m of probability m e a s u r e s
one can define an application
Sf~ to [0,1] by
g ( c o ' ) from the algebra
~' =
u q~l(${1 ) generating
iEl
"
g(~o' )(qi-1 ( B i ) ) = gi(~
Thanks to condition ( * ) , g(~o') i s a probability measure on ~ ' and can s u b s e q u e n t l y
be uniquely extended to ~{1 (cf. [ 4 ] ) . It remains to show that g is measurable from
{1' to H ( { 1 ) , which is equivalent to the measurability of all the PB o g for B in ~{1.
But the set of B~s for which PB o g is measurable c o n t a i n s ~ ' and is a monotone
c l a s s , so that i t ' s ~ i t s e l f .
A
Remark. Condition ( * ) was n e c e s s a r y for the e x i s t e n c e of the g ( o o ' ) ~ s . MeasurabiLity
of g did not require any further hypothesis.
2. Theorem 1 bis. If }( = ~ J ,
( l , 2 ) = (N,> ) and the qn 's are onto, {1 is the projec-
tive limit of K F.
h.
The {1n'S and f~ being Polish~ a probability measure in the s e n s e we use is
a l s o a measure in the Bourbaki sense, Hence all the s y s t e m s (gn(oj '))neN have a
limit g(co ' ) [1]. ( T h i s would be true even if I had only a cofinal countable subset, )
Now we must show that g is continuous from {1' to I I ( { 1 ) . We*ll use the following
result [5] :
A s u b s e t X of 11 ({1) is r e l a t i v e l y compact iff it is uniformly tight, which means
that, for all e> 0 there i s a c o m p a c t
K E in {1 such that P(K e)> 1-e for all P in X.
73
Let
(con)n
R , bounded by
c o n v e r g e to co' in fl ' : we only n e e d to show t h a t , for a n y c o n t i n u o u s map 0 to
1, fO dg(co n)
gp(con)=PPn,
gp(co')= PP,
1 o We get that the s e t
fO dg(co'),
c o n v e r g e to
g(con)=Pn
We'II u s e the following n o t a t i o n s :
and
{ PP I nEN I is uniformly tight
g(co')= P.
s i n c e i t i s t h e image by the c o n t i -
n u o u s map gp of the r e l a t i v e l y c o m p a c t s e t { con I n E N 1.
2o
subset
The set { Pn I n e N I is also uniformly tight :
K 1 of [21 s u c h that PIn(K 1)> 1 - L for e a c h
l e t ' s fix e in R ~_, T h e r e i s a c o m p a c t
K 1 ..... Kp
n. Suppose
2
k < p , Kk
s u c h t h a t , for e a c h
isacompact
s u b s e t of a k , c o n t a i n e d i n
are c o n s t r u c t e d
(~.l?l(Kk)
(for
k > 1 ) and s a t i s f y i n g :
Pkn(Kk)> 1 - ( ~ + . . .
for e a c h n ; t h e r e i s a c o m p a c t
K~+ 1
+~-~)
in f~p+l with
infPP+l(K~+l)> 1- e
.
n
2P+1
Kp +1 = K~ +I n (f~ + l,rl ( Kp ) s a t i s f i e s
T h e compact
inf
).
n pp+l (KP+I) > 1- (e. +... + ~
2
2p+l
So a s e q u e n c e
9
(Kp)p c a n be i n d u c t i v e l y c o n s t r u c t e d , s u c h t h a t Kn
./
t"
(fPp-1) (Kp-1) s a t i s f y i n g inf PP (Kp) > 1-e. T h e
a p c o n t a i n e d in
tion K of t h e
@I(Kp)
infPn(K)> 1-e. It i s
rt
K, qp (~1) i s a n u l t r a f i I t e r
is s u c h that
compact : i f ~1 i s an u l t r a f i l t e r on
h e n c e c o n v e r g e s to a cop in
Kp.
The
i s a compact s u b s e t of
(decreasing) intersec-
enough now to prove t h a t K i s
on
Kp (since qp
i s o n t o ) and
fP being continuous
fPq(cop) =coq
t h e r e f o r e there i s an co in f~ s a t i s f y i n g
foreach
qp (co) = cop
P2 q;
for e a c h p , which m e a n s that co i s
in fact in K . As f~ i s the t o p o l o g i c a l p r o i e c t i v e limit of the f~p Is, i t ' s then e a s y to prove
t h a t ~1 c o n v e r g e s to t h i s co ; c o m p a c t n e s s of K follows.
3~ L e t
Opoqp I p e N ,
A -- {
Op:f~p
-. R c o n t i n u o u s bounded I. A i s a s u b a l g e b r a
of the a l g e b r a of c o n t i n u o u s bounded maps from ~2 to R ; it c o n t a i n s the c o n s t a n t maps and
it s e p a r a t e s t h e p o i n t s of f~ ; so S t o n e - W e i e r s t r a s s Theorem e n s u r e s that A is d e n s e in
t h i s a l g e b r a , endowed with the topology of uniform c o n v e r g e n c e on compact s u b s e t s .
4 ~ We are now a b l e to prove the c o n v e r g e n c e of
( f O dPn) n
to
fO dP.
Let
e> 0
be fixed; by 2 t h e r e e x i s t s a c o m p a c t s u b s e t K of f~ s u c h t h a t
inf({ Pn(K) In ~ N l v { P ( K ) I ) > 1 "8'
hence
(1)
I fOdPn.fOdPI
< IfyOden.fKOdPl+~
"
4
By 3, there e x i s t s a c o n t i n u o u s map Op : ~p -~ R ( t h a t c a n be c h o s e n bounded by 1 ) s u c h
that:
s~p I O(co) Op oqp(co) I < 5e ' .
.
it f o l l o w s :
74
(2)
I fKOdP n - ~KO dP I <- I f K Opoqp dPn" fKOp~
< t fOpoqp
dPn= fOpoqp
-
dP I +~
dP [ + ~ + L .
4
4
But since
PPn = [ I ( q p ) ( P n )
PP= I I ( q p ) ( P ) ,
and
the second member above is
I fop dp~ - yop dppl + 2 '
so that, since PPn converges to PP, it is l e s s than ~ i f
for n > N ,
I f O dP n - fO d P I
n is greater than a N i n N. Hence,
is l e s s than e, from i n e q u a l i t i e s ( I ) and ( 2 ) . So g is a
m o ~ h i s m of ~os filling up diagram ( D ) , and the proof is complete.
A
3. A general process on discrete time is described by a s e q u e n c e ( f n ) of t r a n s i t i o n proba• s n to ~2n+l , Welll see that
b i l i t i e s , fn being from the set of ~,histories before n ~ ~21 •
this sequence i n d u c e s a functor from (N,_<) to ~.~j- and, u s i n g the above theorems, w e ' l l
construct a transition probability from each s l • 2 1 5
to the sample set f i f t h , compatible
n
with the fn ' s . This result, which c o n t a i n s Ionescu T u l c e a Theorem [4], is a corollary of
Theorem 3 below, i t s e l f a c o n s e q u e n c e of the following
Theorem 2. Here }( = ~ .... If ( l , > )
is a filtered ordered set and F
a functor ( l , >) -* ~
whose projective limit is given by the commutative diagrams
qi ~---'~ ~ i
with the further property
( s m ) For a n y i n c r e a s i n g sequence ( i n ) n in l and any (O~in)n in l]~inn such that:
fin+ l ~
tn"
t~~
= Cjtn t h e r e i s a oj in ~ satisfying qin(CO)=op.tn,
and if G is a functor ( l , < ) - * ~ verifying:
a) For i > ] , G ( i , ] ) = g ] : i ft]~ , a i is left inverse to 7]i
b) For i>- io , ~ %. E ~io , B i t $ ~ i '
c~ .
~ f !to ( Bi ) implies g to
~ (~io)(Bi)
then for each i o in [ there is a unique morphism gio : ~io ~
K K gio = gio
i
qi
= O,
~ such that
for every i > i o
and
gi o (~ o ) ( B ) = 0 f o r e v e r y (o9io , B ) i n
Moreover, if I is totally ordered, the diagrams
~io x ~
with ozto. ~ q i o ( B ) .
~ i ~ - - . . . . _ gi
]>i
]
commute,
J
75
A. L e t us first notice that property a for G means that
io . This, together with the
F to { i E l I i 2 io I, i m p l i e s
commutes for i 2 ]'2
fact that ~ is also the p r o j e c t i v e limit of
the restriction of
that the e x i s t e n c e a n d unicity of
proved, using Theorem 1, as soon as we have shown that
ftio
theorem. Indeed, if it has not, there is a cot0. in
and s e t s
Bin
in
~[~in
Suppose we have constructed
Since
gio
will be
has property ( * ) of this
(in) n
, an i n c r e a s i n g s e q u e n c e
in l
such that:
~ q i 1n ( B i n ) =
in
in
"
p+I
gip = gip+ 1 K gl~P
lira gioin (coio (Bin)> O.
and
(coio ..... COip)
q> 1, f!q~q-l(c~
For
(gioi)i>i~
i satisfying:
g
i
)=coiq
for n 2 p + l ,
in ~i0 •
lira
i ng ~qn(co q ) ( B i n ) > O "
/ and
we have
in
gip(coip)(Bin )= fPBin~
"
n
~p+/
i
dg.p+l (co i ).
~p
P
Compatibility of integral and pointwise i n c r e a s i n g limit implies that
flinm
in dg! p + l
. )
PBin~ gip+ 1 ~p
(co~p
>0,
and h e n c e that
gi~+l(o~,p)({
9
Condition b then gives a
c~,
co~p+l
9 E ~tip+/
in ft i
f(p+l
tp
(COip+l
The s e q u e n c e
(coin)n
~
in
Bin)>O I ) > 0 .
gip+l(O>ip+l)(
such that
p+l
) =
I l/rn
9
in
lira
n gip+l(coip+l)(Bin)>
and
w e ' v e just constructed i n d u c t i v e l y s a t i s f i e s
for each n " so c o n d i t i o n ( s i n ) provides us with a co in a such that
in
gin(coin)(Bin)>
every n. Now s i n c e
absurd for
t3nq~ln (Bin
So we have our
set of
gebra
O, ~O~n" is
in
Bin
and co in
O.
fl n+l.o (coin+l) = coin
qi n(co)
=
q~l(Bin
).,~
9 for
T h i s is
) was s u p p o s e d to be empty,
gio 9 to
show that
gio (coio)(B)
= 0 if
. ~ qi ~ ( B ) ,
63 t O
remark that the
8 for which this is true ( f o r a fixed COl0 ) is a monotone c l a s s c o n t a i n i n g the al-
i>iou q ~ l ( ~ f t i ) "
which g e n e r a t e s ~f~
It remains to c o n s i d e r the totally ordered c a s e and to show that, for
i
gi o = gi K gio.
Fix i > i
o.
For ]> i> k>
i o,
i > io ,
the following diagrams commute:
we have
76
~io x
i
gio
gi
-~ f~i x
~~
f~io x
i
gio
" ~i "~
gi
" f~
Hence
i
< ~ (gi ~ g!t o ) = gki o ,
qj ~ (gi ~ gio ) = gito and (~ik ~ q i ) ~ (gi ~ gl o) = 'qk
so that for every ] 2 i , ~ j K ( g i ~ giio ) = gio ; from uniqueness of gio , we get the expected
commutativity. A
4. Application to processes.
a) Theorem 3. Let ( Ea j a e l be a family of objects of ~ -
(l,>),
indexed by a well-ordered set
~a (resp. ~ ) theproduetof (E/3)/3< a (resp. ( E a ) a e l ) . Given a family ( f a ) a e l
where fa : ~a ~
E a + l, there is, for each aa in I, a unique gao : ~ao ~e--'~ ~ such that:
For COaoe~ao , Bao e ~ ao ' (Fao+i)l<-i<-n el>i>
l] n ~ a o +i ,
gao r176 ) ( B ) = XBao (COao) fF
ao+l
alia~ (c~
fF
ao +2
d f a ' +1r176 ' xa" +1)""
"" fFa o +ndfa~ +n'1(~176 'xa~ +l . . . . . Xa~ +n'l)
where
B = B ao x
II F
l<i<n ao+i•
[I
E
a"
A. The projections from fl to ~a and fla to ~/3 if a .2>fl are respectively denoted
by qa and q ~ . The q a ' s a r e o n t o a n d s a t i s f y ( s m ) .
a) For each ~ in I and 0J/~ in ~/3, let us call g~+l(co/3) the probability measure
on ~/~+1 = ~ / 3 •
gfi
(~)(Bfi
product of q~/3(oj/~) and [/3(o~/~) , given by
•
= )lB
(o~fi).f[3(~)
(F~+I)
forlB fie ~ 2 f i and
tF~+ le ~Efl+l.
~+~
g/3
is measurable : t h e s e t { B e ~
~+1
[ PBog~
is measurable } is a monotone class
and contains the disjoint unions of the sets B~ • F~+ 1 which form an algebra generating
~ ; so it's ~f~ itself.
The same techniques prove that g~fl+l(o~fl)(B/3+l ) = 0 if co/3 ~ q ~ + l ( B f l + l ) .
b) Let us consider the couples ( ] , G]), where J is a beginning section of l and G]
a functor ( ] , < ) - ~ ~ , such that G ] ( ~ , a ) = h ~ : ~ / 3 ~ ' ~ a if / 3 < a satisfies:
( i ) h~ i s l e f t inverse to ~ .
(iii)
h ~ ( o j ~ ) ( B a ) = O if cot~ ~ q ~ ( B a) , for ~ / 3 e ~ / 3 and Ba e $f~ a.
77
The obvious order on these couples is inductive and, from Zorn*s Lemma, there is a maximal
element (I 1,GII). L e t l s d e n o t e Gil(a,tg)
by g ~ ( a > _ / 9 ) and prove that l 1 is I i t s e l f .
If 11 is strictly included in I , then a = in[{ y~ I f y ~ l l l
e x i s t s . 12 will be the set obtain-
ed by adding a to 11~ Weql consider the two possible c a s e s :
1. If a has a predecessor a *, let*s denote
gya = gaa 'K gya ' for each y < a ,
and denote by GI2 the map send•ng
GI2 is a functor (12, <) --, ~
(/3,y)e12,
gaa = 71~
/3<y
to g ~ , I t ' s then easy to prove that
K
and that g~ is left inverse to q~ for /3 <_.y in l 2 . Now we
want to show that g~(co/3)(Ba) = 0 if w/3 ~ qa~(Ba): W e k n o w i t is true if /3 = a ' .
If
and 0)/3 ~ q ~ r B a ) ,
~<a'
,K g/3 )(~oB)( B a ) = f ga,(. )( B a) dg~'(cofl).
The set
a
X a ' = [Wa'tf~a' I ga'(~
0}
i s i n c l u d e d i n qa,(Ba),
b y d e f i n i t i o n o f ga,=
Sa
a
a
_%%1. So q~ '(Xa, ) C q~{Ba). Hence,
if ~ofl ~ q ~ ( B a ) ,
coil ~ q ~ ' ( X a , ) and f r o m ( i l l ) , g ~ ' ( o j ~ ) ( X a , ) = 0 . It follows that the
above integral is zero.
2. If a is a limit ordinal, Oa is the projective limit of the (~/3)/3<a and conditions
( i ) , ( i i ) , ( i i i ) satisfied by Gll allow us to use Theorem 2; for each /3 < a, there i s a unique g~ such that the diagram
g~...1~
Ira
qy
a
y
K~
commute for /3 _< y <_ a. If /3 = y , we get that g ~ i s left inverse to q/3
g}(co/3)(Ba)=O
if
Moreover,
o~/3~iq}(Ba) ,
and, 11 being totally ordered, g ~ = g ya K g~ if /3< y _ < a .
So in both c a s e s we have constructed a functor Gl2 satisfying ( i ) ,
i i ) , ( i i i ) . This
contradicts the maximality of (I 1 , Gll ), which proves that l I is l itself.
c) We can apply Theorem 2 to the functor G = Gll itself, which gives us, for each ao
in l , a unique gao such that
~
t
a
~
~a
78
commutes for a 2 ao 9 To complete the proof, w e ' l l show that
ga has the e x p e c t e d form, by
induction on n. There is no problem if n = 0. Suppose that gao s a t i s f i e s the equality given in the theorem for B ' s o f t h e
form Bao Xl<_i<_pHFao+i•
B ' = B ao x
a . Let
li
F +ix
II
E 9
l<i<p+l ao
a>ao+p+l a
Then
gao (~176
= ga~ +p+l (COa~ ) ( B a ~ •
II
F +i )
l < i < p + l ao
= g : : +P+P+I K ga;+P(coao)(Bao xi<_i~p+lFao+i)
=
P+l(.)(Bao •
l]
l<_i<p+l
F
ao
+i)dga~ +P(
The induction hypothesis gives the form of ga 2+p(ooao) on the s e t s Bao •
).
[I F
l<_i<_p ao+{"
From this, we can deduce that, for any c h a r a c t e r i s t i c function, h e n c e for any simple function
and finally for any bounded measurable function X from aa0 +p to R :
f X dga~ +P = f dfao(eOao )... f dfa ~ +p.l(COao ,Xao +l . . . . . Xao +p-1)X"
ao ++p
p+l (')(Bao •
If we apply this to X = ga0
X(c~
11 f +iJdefined by:
i<_p+l ao
+p) = X(~176 'Xao +1 . . . . . Xa o +p)
= X B a ~ (Oao)X Fa o + l ( x a o + l ) ' ' ' X F a o + p ( x a o + p ) f ( e ~
we get
(
)
XBao ~176
~ a o +1
dfa~ (~176176 fF
ao +p+l
T h i s is what we e x p e c t e d and the proof i s complete,
d fa o +p(~
. . . . . Xao +p )"
A
b) Remark. If l = N , the e x i s t e n c e of the gao (e~
rem as stated in [d].
g i v e s back I o n e s c u - T u l c e a Theo-
c) Theorem 3 bis. Theorem 3 is still valid with ~o~ instead of ~
and ( N , > )
in-
s t e a d o f (1,>_).
A. 1~ With the notations of Theorem 3, we first show that gn+l is a morphism of ~
n
that is, is continuous. It will follow from the more general r e s u l t :
Proposition. Let [11 and Q2 be objects of ~,~ and define
0:FI(~l)X11(~t2)'II(ftlX~2):
('product probability measure). Then 0 is continuous.
(P1,P2)I*P1xP
2
3. The set
A ={ f: f~lxf~2 * R t f ( o l , C O 2 ) = i =~l ai fli(~176)f2i(~176 fi]: fti ' R continuous
bounded
}
79
is an algebra containing the c o n s t a n t maps and s e p a r a t i n g the points of ~21 •
; hence, by
Stone-Weierstrass Theorem, it is d e n s e in the s e t of continuous bounded maps from f~l •
to R, endowed with the topology of uniform c o n v e r g e n c e on compact s u b s e t s .
Now let ((P~,Pn2))n converge to (P1,P2)' [ be a continuous map from ~2l •
to R
bounded by 1 and e be a fixed p o s i t i v e real number. For i = 1, 2, the (Pn) n are uniformly
t i g h t ; so there i s a c o m p a c t Ki such that pn(Ki)> 1 "~/16 for all n ; if follows
(PT•
xK 2) > 1 -~g
foreach
n.
C h o o s e a g in et such that
KlsuP•
[ f(~176176
[ <8'
bounded by 1. W e h a v e
(3)
l ffd(P 7 •
But s i n c e
so that
•
g has the form
fgd(P~xP
< d_~ + I f g d g P ~ •
dgPlXe2)l.
~ ai gli(~Ol )g2i(o2) , from Fubini Theorem we get
i=l
f g d(Q 1 xQ2 ) = i=I
~ a i f g l i d Q l f g 2 i dQ2,
2) c o n v e r g e s to f g d(P 1 x P 2 ) : it f o l l o w s that there is an N such
that the second member o f ( 3 ) is l e s s than E for n > N. And 0 is continuous.
3
2~ So gnn + l is continuous. Now, if p > n , let us define the continuous map gPn as the
composite
gaP = g ~ - i
~ "'" ~
gn~ + ~ . The commutativity o f all the diagrams
(n <_m ~ p ) is straightforward ( c f . Theorem 3). For every fixed n in N , a is the p r o j e c tive limit of the (f~p)p> n" It follows then, from Theorem 1 bis, that there e x i s t s a gn such
that
ft
commutes for all P2 N.
The computation of ga0 in Theorem 3 s t i l l a p p l i e s here.
A
III. RANDOM TOPOLOGICAL ACTIONS OF CATEGORIES.
In the above study, p r o c e s s e s were always defined by a functor from an ordered
set, representing time, to ~ - .
One could also imagine that between two times
t < t',
s e v e r a l actions on the <,moving point~> were p o s s i b l e , in which c a s e the p r o c e s s could
be determined by a functor from a certain category C to ~ .
T h i s i s the reason we now
80
define the random topological actions, which are the ~,non-deterministic~) analogues to
the topological actions of a category on a topological space E. I t i s hoped they lead to applications in ~,non-detetministic, optimization problems similar to those studied by A. Ehresmann [2] in the ~detetministic, c a s e ; this notion should as well be useful for a probabilistic
generalization of stochastic automata.
Throughout this Part III, C is a category object in ~ d , that i s , a category C endowed with a Polish topology for which
C2
comp.
dora
,. C1
codom ~"CO
are continuous.
1. Definitions. Let E be an object of ~,ff.
a) A random topological action (abbreviated in rta) (resp. a topological action ( t a ) ) of
C on E i s a functor " from C to ~
(resp. to ~ o f ) s a t i s f y i n g :
( i ) There i s a continuous map p . from E onto Co with ~ = ~ / [ e l
= E e f o r a n y e~ Co.
( i i ) If C*E is defined by the pullback
C*E
1
C'
in ~,~ then, in ~ ,
,. E
dora
l
~ Co
the m a p ( f , x ) 1-" 7(x) defines a morphism from C*E to E.
b) Remarks. 1o A topological action is equivalent to an intemal diagram in ~ o f ,
2~ In the c a s e o f a r t a ,
[ ( x ) is in fact in l I ( E c o d f ) , but this space ishomeomorphic
to, and identified with, the subspace of 1I ( E ) of those probability measures with support
the fibre Ecodf.
c) Examples.
lo If ." i s a ta, its composite with .K ~ d - , ~ is a rta ( o f C on E ).
f~n+l is a r t a of the order
( N , > ) on thecoproduct E of the state spaces f~n ; indeed, ." maps ( n , n + l ) on fn"
More generally, if several actions were possible between time n and n+l, the category act2~ A Markov process given by the morphisms fn: t2n ~
ing on E would still have N as set of its objects but there would be sev er al morphisms
between integers m and n, m> n (cf. [2] for an example of s u c h a category),
2. Topological action associated to a random topological action.
The domain of the internal diagram (or ~,cat~gofie d'hypermorphismes ,~) a s s o c i a t e d
to a t a o n E is C*E with composition
(f,x)(g,y)=(fg,
y)
iff x = ~ ( y ) .
Looking for the corresponding notion in the c a s e of a r t a naturally leads to make C act on
probability measures on E , which is possible thanks to the canonical functor:
-, ~--,
~,e,(n,,
0 ~ a')
~ (n(a)
ga'~176
[[(~,) ).
a) In this section, we suppose given a rta 7 of C on E, and we denote by p the as-
81
s o c i a t e d s u r j e c t i o n p-: E-~ Co , by E ' t h e s u b s p a c e
W I I ( E e ) of
e 9 Co
H ( E ) . T h i s union b e i n g p a i r w i s e d i s j o i n t , one c a n d e f i n e a map p ' from E ' onto Co by:
p'(P) = e
iff
P E[I(Ee)
u
eE Co
(iff
II(~l{e])
=
P ( E e ) =1 ).
b) P r o p o s i t i o n l. E ' is closed in II ( E ) (hence is polish) and p" is continuous from
E ' to Co 9
A. L e t ( P n ) n be a s e q u e n c e of E ' c o n v e r g i n g to P in I I ( E ) and
en = p ' ( P n ) .
1 ~ T h e s e t { Pn I n 9 N I w[ P } i s c o m p a c t , h e n c e uniformly t i g h t ; in p a r t i c u l a r t h e r e
K such that P ( K ) >
is acompact
1 / 2 and P n ( K ) > 1 / 2 for e a c h n . If w e c h o o s e a n x n
in e a c h K c~ Een , the s e q u e n c e ( x n)n h a s a s u b s e q u e n c e ( X n k ) k w h i c h c o n v e r g e s to an
x in K ; then
e = p ( x ) = lirnn enk. I f Pnk i s d e n o t e d by Qk and enk by e ~ , the s e q u e n -
c e s ( Q k ) k and ( e ~ ) k r e s p e c t i v e l y c o n v e r g e to P and e . Welll now prove t h a t P ( E e ) =
w h i c h will imply t h a t P 9 I I ( E e ) c E ' . If t h i s w a s not t r u e , t h e r e would b e a n
that P(Ee)
1
9 > 0 such
< 1- E. E e being c l o s e d in E m e t r i z a b l e , it h a s a n open n e i g h b o r h o o d U s a t -
isfying P(U) < l-e.
But U i t s e l f b e i n g a G 8 in E nounal, i t s c h a r a c t e r i s t i c map XU
i s the p o i n t w i s e d e c r e a s i n g limit of a s e q u e n c e o f c o n t i n u o u s m a p s from E to [ 0, 1 ] . H e n c e
thereis
such amap
r
with v a l u e 1 on U such that f(o dP < 1 - e . But, s i n c e ( Q k ) k con-
fq5 dP k < 1 - e for every k > m .
v e r g e s to Q, t h e r e i s an ra in N s u c h t h a t
Let K' beacompact
subspaceof
E satisfying Qk(K')>l-e
i n t e r s e c t i o n with E e l . If n o n e of t h e s e t s
Yk e K k,
for e v e r y k, a n d K k its
K k (k > ra ) w a s c o n t a i n e d in U , w e could find an
Yk ~ U, for e a c h k , a n d the s e q u e n c e ( Y k ) k would h a v e a s u b s e q u e n c e c o n v e r g i n g
to a n y in K with p ( y ) = e. So y would be in E e , and h e n c e i n U , w h i c h i s a b s u r d s i n c e
( Y k ) k i s a s e q u e n c e of the c l o s e d complement of U. T h e r e f o r e t h e r e is a K k ( k > ra ) cont a i n e d in U ; it f o l l o w s t h a t
1 . 9 < Q k ( K k ) <_ Q k ( U ) ~ f r
We h a v e r e a c h e d a c o n t r a d i c t i o n , which m e a n s t h a t
2~ Suppose
dQk < 1 - e .
P ( E e ) = 1.
en = p ' ( P n ) does not c o n v e r g e to e = p ' ( P ) : t h e r e i s a n e i g h b o r h o o d V
of e which c o n t a i n s no point of a s u b s e q u e n c e ( e n k ) k of ( e n ) n. p b e i n g c o n t i n u o u s , t h e r e
is, foreach
x in E e , an o p e n n e i g h b o r h o o d F x in E which i n t e r s e c t s no Een k ( c h o o s e
Fx
s u c h that P ( V x ) C V ) . The u n i o n U of the V x i s an o p e n n e i g h b o r h o o d of E e , so that t h e r e
i s a c o n t i n u o u s map r from E to [ 0 , 1 ]
U c~
Eenk
with v a l u e 1 o n E e and 0 o n t h e c o m p l e m e n t of U.
b e i n g empty, we h a v e
f r dPnk = 0
and
f r dP = 1 ,
w h i c h i s a b s u r d s i n c e ( P n k ) k c o n v e r g e s to P . H e n c e p ' ( P n )
is continuous.
A
c o n v e r g e s to p ' ( P ) ,
and p'
82
c) Theorem 1. If
" is a random toQological action on E, its composite with the functor
~ ~,~ is a topological action . on E'.
7: ~
A . D e f i n i n g p ' as in a , c o n d i t i o n ( i ) of D e f i n i t i o n 1 is s a t i s f i e d from P r o p o s i t i o n 1 . It
r e m a i n s to show that ( f , P )
~ f ( P ) i s c o n t i n u o u s from C * E ' ( d e f i n e d a s the o b v i o u s pullb a c k ) to [ I ( E ) . I r i s enough to prove that, for each c o n t i n u o u s map 0: E - * [ O , 1 ] , t h e map
( f , P ) i+ fO J [ ( P )
T: C * E ' + [ O ' , J ] :
i s c o n t i n u o u s . Indeed, from Formula a of the proof of T h e o r e m 1 ( I - 3 ) :
fO d r ( P ) = fEe~OO~dP
e = d o N ( f ) . T h e map ( f , z )
with
C * E of C •
to [ 0 , 1 ] , i t
= fEe[fOdT(.)]
dP,
~ fO d ' f ( z ) being c o n t i n u o u s from the c l o s e d s u b s e t
c a n be e x t e n d e d to a c o n t i n u o u s map 0 ' : C •
Now
O: C x H ( E ) + [ 0 , 1 ] : ( f , P ) ~. f O ' ( f , . J dP
i s a n e x t e n s i o n of T to C x 1 1 ( E ) .
Let
H e n c e it i s enough to show i t i s c o n t i n u o u s o n C * E ' -
((fn,Pn)Jn c o n v e r g e to ( f , P )
and e be a fixed real p o s i t i v e n u m b e r ; for e a c h n ,
tO(fn, PnJ-O(f,P)l
Since •(f,.)
<_ I f O ( f n , P n ) - O ( f ,
is c o n t i n u o u s ( f o r 0 ' ( f , . )
Pn)l + I O ( f , P n J ' O ( f , P ) l .
i s ) , there i s a n N s u c h t h a t
[ O(f, Pn)-|
for n > N .
<~
2
T h e s e t of P n ' s b e i n g uniformly tight, t h e r e i s a c o m p a c t K in E such t h a t
inf P n ( K ) > 1 r
n
8
Then,
0' b e i n g bounded by 1,
IOcfn'PnJ'O(f'Pn)]
-< fK t~
dPn +~'4
0 ' i s uniformly c o n t i n u o u s on the c o m p a c t K ' = ({fn I n ~ N l u l f i )
that O ' ( f n , . ) uniformly c o n v e r g e s to O ' ( f , . ) on K ; s o t h e r e i s
She I o ' ( f n , ~ ) - O ' ( f , z ) l
< ~-
tO(fn,Pn)'~)(f,P)l
< e
z~K
4
•
which i m p l i e s
N ' such that
for
n>m'.
Then
I t follows t h a t 0 i s c o n t i n u o u s .
d ) Remark. T h e functor
"
for n> s u p ( N , N ' ) .
A
of T h e o r e m 1 t a k e s i t s v a l u e s in the c a t e g o r y 5 of free
a l g e b r a s of ( 1 1 , 7 / , / x ) , so t h a t not any t a o n a
s e t E ' s u c h t h a t E ee = 1I(E e ) , with
( E e ) e ~ Co a p a r t i t i o n in c l o s e d s u b - s p a c e s of a P o l i s h s p a c e E , a c t u a l l y c o m e s from
a rta. In fact, it c a n be s h o w n that only t h o s e ta ~ which f a c t o r i z e through ~ do, t h a n k s
to the i s o m o r p h i s m b e t w e e n ~ and 2~-.
3. The category of random topological actions of C .
a) N o t a t i o n s . 1o T h e o b j e c t s o f t h e c a t e g o r y
~ are the c o u p l e s ( E , p )
where E i s a
P o l i s h s p a c e a n d p a map from E onto Co 9 A morphism ~ : ( E , p)-, ( F , q) i s a morphism
qS: E*e
, F s u c h that
83
E ,r
&
-~F
Co
commutes~ C o m p o s i t i o n i s d e d u c e d from ~ g .
2 ~ ~l,~,,a h a s for o b j e c t s the random t o p o l o g i c a l a c t i o n s o f C , If " is a rta on E and "
a r t a o n F , a m o r p h i s m ~b: " ~ " i s a morphism ~ : ( E , p - ) ~
( F , p - ) in ~ such t h a t the
family (v5 e)e~Co ' where Ce: EeK .., F e is t h e r e s t r i c t i o n o f r
to the f i b e r s o n e, con-
s i d e r e d with values in l l ( F e ) , d e f i n e s a natural t r a n s f o r m a t i o n " => ? . T h i s l a s t c o n d i t i o n
m e a n s that r
commutes with the a c t i o n s . Composition i s again d e d u c e d from ~ .
b) Theorem 2. The forgetful functorfrom ~ d
over ( E , p )
to ~ has a left adjoint. The free object
is the rta o.p on the topologicalsubspaee C*E of C x E
C*E
defined by the pullback
, E
l
C
l
dom
, Co
0
given by
gP(f,x ) = ~C,E(gf, x)
A . Let us first prove that
fff domg = codomf.
op. ( d e n o t e d here .~ ) i s a r t a o n
po : C*E * Co : ( f , x ) - ,
C*E.Themap
eodom(f)
i s onto and c o n t i n u o u s , and for each e of Co, w e have o = P~1 { e} = ( C ' E ) e. So con0
d i t i o n ( i ) of Definition 1 i s s a t i s f i e d . For any g: e-, e' in C, w e can s e e g
phismin ~',
as a m o r -
0
from ( C * E ) e to ( C * E ) e , . It is e a s y then to s h o w that . : C - . ~ Y i s a
functor. At l a s t , the map
C*(C*E) , C'E:
(g,(f,x))
0
~g(f,~)
i s c o n t i n u o u s s i n c e c o m p o s i t i o n in C and O C * E are.
We now d e f i n e
np: ( E , p ) - ' ( C * E , p o ): x [~ ~ C . E ( P ( X ) , X ) .
Let
" b e a rta on F , and r
(E,p)
-, ( F , p ^) a n y m o r p h i s m if 2 . If a morphism
from .o to " in ~,~.~a i s such that
(E,p)-
np
,(C*E,po)
( F , p -)
commutes, it s a t i s f i e s n e c e s s a r i l y ~ ( p ( x ) ,
9 ( f , ~ ) = (~, , o f ) r
x ) = cb(x), and, from the d e f i n i t i o n o f ~,~,~a,
= (/,, r
= (},, ~ ) ( ~ ) ,
for every ( f , x ) in C*E. So d) i s unique i f it e x i s t s , To show t h a t the a b o v e d e f i n e d
f i t s , it remains to c h e c k it i s c o n t i n u o u s . But s i n c e
~(f,x) = [#FeodfO~(})](~(x)),
84
continuity of 9 follows from continuity of ( f , x ) I-+ (f, qS(xJ) and of
Of, P) I--> V F c o d f O I I ( f ) ( P ) = fCP)
(cf. 2 - c , Theorem 1).
A
C) Remark. Let ~d be the subcategory of ~ with the same objects but with morphisms
only the deterministic ones (of the form ~ with 0 in Y d ) , and g , ~ t the subcategoryof
~,~,~a with objects the ta (Example 1, c, lo ) and morphisms the deterministic maps between
those. Then, by restriction, op is still the free object over ( E , p ) for the forgetful functor
from ~ , ~
to ~d ; this is already known.
At last, we prove the following result, similar to the one obtained in the case of topological actions :
d) Theorem 2. ~,~,~d is (isomorphic to)the Eilenberg-Moore category of the monad generated by the above adjunction.
A. Let us call this monad ( P , n , m ) . For any 6: ( F , q ) - ~ ( E , g ) ,
P(q~) is the only
morphism in ~ , ~ a such that np • ~5 = P(~b) K nq. Via the comparison functor from ~o~,t
every rta " on E becomes an algebra: the structural arrow is given by
h ( f , x ) = "[(x)
forany ( f , x ) e C*E.
In particular, mp is defined by
m p ( g , ( f , x ) ) = q C . E ( g f , x) for ( g , ( f , x ) ) ~ C * ( C * E ) .
Every morphism in ~,~-,a becomes a morphism of algebras as well. We now wanna prove the
converse. Let us consider
~:(F,q)-,(E,p),
k:(C*F, poq)-,(F,q),
a moThism of algebras; we denote k ( f , y )
h:(C*E,pop)-,(E,p),
by f ( y ) for ( f , y ) ~ C*F and h(f, x) by 7(x)
for ( f , x ) r C*E. Using the notations of Theorem 3 his ( I I - 4 , c), we get the
Lemma. For ( f , y ) in C*F such that q ( y ) = e, we have:
a)
b)
c)
d)
(np K ~ ) ( y ) = r l c ( e ) x ~ ( y ) .
P(E)(f,Y) = ~c(f) xE(Y).
(~,~ k ) ( f , y ) = (~ ,~ ~ ) ( y ) .
(hK P ( ~ ) ) ( f , y ) = ( 7 ~ 5 ) ( Y ) .
3. C*E being a closed subset of C x E , every element of I I ( C * E ) can be seen as an
element of II (C x E), hence is determined by its values on the subsets B C x B E ' where
BC r $C and BEe B E. For such a s u b s e t :
a) (np K ~C)(y)(Bc•
fnp(.)(BcXBE)dE(y)
= f r l C . E ( e , . ) ( B c X B E ) d E ( y ) (for ~ ( y ) is concentrated
onE
= 71c(e ) ( B c ) . ~ ( y ) ( B E ) .
b) P ( E ) ( f , y J ( B C x B E ) = f T P ( . , . ) ( B c X B E ) d ( n p K E ) ( y )
e
)
85
= f~
)(Bc•
d~C(y)
from a and Fubini Theorem
= frlC*E(f,')(Bc•
c) (~K k ) ( f , y )
= rlC(f)(Bc)'~(Y)(BE)"
=(gEoIl(~})(k(f,y))
d) ( h K P ( r
= (gE~
= (~ K f ) ( Y ) .
= fh(.,.)(BE)dP(r
= ff(.)(BE)
= fh(f,.)(BE)dr
d~(y) = (Te:~)(y)(BE).
Applying c and d of this Lemma to an algebra ( ( E , p ) , h )
(F,q) = (C*E,pop),
~ = h,
8
with
k = mp ,
we get
(h K m p ) ( f , ( g , x ) )
= (ht(~
= h(fg, x) = ~g(x)
and
(hK P(h))(f,(g,x))
= (7:< h ) ( g , x )
= (gEoll(7))(~(x))
= (~K ~ ) ( x ) ,
so that 7 is a functor to ~ - (the fact that it sends units to units follows from the equality
hK np= r1 E ) , and hence a r t a o n E.
To prove that any morphism of algebras comes from a morphism in ~o~4, we use c and d
of the Lemma again, applied this time to such amorphism ~ between algebras ( ( F , q ) , k )
and ( ( E , p ) , h ) ,
considered as rta ." and ? respectively. F r o m c a n d d , ~ commutes with
the actions and so is a morphism in ~,~,~a.
A
REFERENCES.
1, N. BOURBAKI, Intdgration, Chapitre 1X, Hermann, Paris.
2, A. EHRESMANN ( - B a s t i a n i ) , Syst~mes guidables et probl~mes d'optimisation, Labo. Auto.
Th$orique Univ. Caen, 1963"65.
3. F. W. LAWVERE, ?'he category ofprobabilistic mappings, Preprint, 1962.
4. J. NEVEU, Bases mathdmatiques du calcul des probabilitds, Masson, Paris, 1970.
5. K. R. PARTHASARATHY, Probability measures on metric spaces, Academic Press, 1%7.
6. SEMADENI, Monads and their Eilenberg-Moore algebras in Functional Analysis, QueenWs
papers in Pure and Applied Math. 33, Qu eenls Univ., Kingston, 1973.
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