C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES
J OHN L. M AC D ONALD
A RTHUR S TONE
The tower and regular decomposition
Cahiers de topologie et géométrie différentielle catégoriques, tome 23, n
o2 (1982), p. 197-213
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
THE TOWER AND REGULAR DECOMPOSITION
by John L. MACDONALD
andArthur STONE
CAHIERS DE TOPOLOGIE ET
GÉOMÉTRIE DIFFERENTIELLE
Vol. XXIII - 2
(1982)
3e COLLOQUE
SUR LESCATÉGORIES DÉD/É A
CHARLES EHRESMANNAmiens,
Juillet 1980Composition
and itsinfinitary analogues
are among thepartially
defined
operations
on themorphisms
of a category. Thus agiven morphism
may have an
infinitary decomposition corresponding
to agiven
ordinal.Furthermore,
such adecomposition
may have certain universalproperties.
VG’e first examine the
regular decomposition
ofmorphisms
describedby
Isbell[13] (cf. Kelly [14 )
andpoint
out the universalproperties
ofthat
decomposition.
Anexample
suggests therelationship
betweenpartially
defined
algebraic
structures andregular decomposition length.
Secondly,
in the samelight,
we consider theadjoint
tower decom-position
of anadjunction.
Thisdecomposition
isalready implicit
in Beck’s[16]
factorization of anadjunction through
its category ofalgebras
andis more
explicitly
described inApplegate -Tiemey [11 (cf. Day [5] ).
Theuniversal
properties
of thisdecomposition
areentirely analogous
to thoseof the
regular decomposition. However,
theadjoint
tower is not ingeneral
a
regular decomposition,
as shownby
anexample
in the second section.Finally,
wegive
a class ofexamples illustrating
thenontriviality
and interrelatedness of these two
decompositions
for each ordinal. Theexamples
areessentially algebraic
in the sense ofFreyd [6]
and showhow the
regular length
of the counit is related to the essentiallength
ofthe tower.
1. CHAIN COMPOSITES AND REGULAR LENGTH.
In this section the chain
composite f
of afamily
of
morphisms
is defined forgiven
ordinal A. Then theregular epic
com-ponent (or largest regular epic « bite»)
of amorphism f
is described andconditions are
given
for its existence as well as for that of the canonicalregular epic decomposition
off ,
the latterbeing
aspecifically
definedchain
composite,
for a certain ordinalk ,
ofregular epics
followedby
amonic
(cf.
Isbell[13] ; Kelly [14]).
The ordinal À(when
itexists)
is call-ed the
regular length
off .
We
identify
anordinal X
with the ordered set(considered
as a cat-egory)
of ordinals less than X and leta B
denote theunique morphism
froma to
B
when aB .
D E F INIT ION 1.0 .A
chain
ofob j ect length
A in a categoryC
is adiagram
( = functor) X -* C
where A is an ordinal. Themorphism length
of a chainis the order type of the
family
ofmorphisms c a B
with(3
A and6
=a+1.So
morphism length equals object length
when this is 0 or a limitordinal;
otherwise
morphism length equals object length
minus one. A cochain ofobjects colength
À in C is a con travariantfunctor x - C .
Note that in an ordinal k
(regarded
as acategory)
eachobject
yis the colimit of the
diagram
ofmorphisms a/3
with ay and B = a + 1.
Call a chain cocontinuous
(of course)
if it is cocontinuous as a functor.For each
object Ca
in a cocontinuous chain * thefamily
ofmorphisms
c
ea ( 0 a )
forms a colimit cone.Note that finite chains are
always
cocontinuous.DEFINITION 1.1. A
morphism f
is achain composite
of themorphisms
f a B (ctA? B = a + 1 )
if there is a cocontinuous chainC
ofobject length
k + 1 with
When h is finite chain
composite equals composite.
Recall that an
epimorphism
e is :( 1 ) regular
if for som e set{x a’ Y} a E I
of orderedpairs
ofmorphisms satisfying e x a
=e ya
we have thath xa
=h ya (for
alla ) implies
theexistence of a
morphism d
withd e
=h ;
and
( 2 ) strong
if wheneverb e
= m a with mmonic,
there is a«diagonal»
d with d e = a and m d =
b.
The class of
regular epics
in a category need not be closed undercomposition (Herrlich -Strecker [11],
p.262). However,
aregular epic
iss tron g and
P ROPOSITION 1.2. A
chain composite o f strong epics
isstrong epic.
P ROOF.
By
induction on thelength
of the chain. For the induction step consider thediagram
where e’ and e are strong
epic, m
ismonic,
andb e e’
= m a . We have adiagonal
d’ since e’ is strongepic.
Then the strongepic
egives
us adiagonal d
for the squarem d’
=b e .
At a limit ordinal the(infinite)
dia-gram is
similar,
and thediagonal
d existsby
the definition of colimit. 0 DEFINITION 1.3.The
regular epimorphism f 01
is aregular epic component
ofmorphism f
iff = f1 , f01
for some(necessarily unique) f 1
and wheneverf = A.g
withg a
regular epic,
there is a(necessarily unique) j with j , g
=f01 .
Note that the usual
«uniqueness» proof applies,
so thatwhen g
isanother
regular epic
component off , above, j
is anisomorphism.
A reg-ular
epic component
factors out off
as much as we can get in oneregular
epic
« bite».PROPOSITION 1.4.
I f f 01
is acoequalizer morphism for
akernel pair for f , then f 01 is
aregular epic component o f f .
P ROO F .
Let p , q be a kernel
pair
forf ,
withcoequalizer morphism f 01 ,
andf= f1 , f01 . Let g
be aregular epic
for thefamily {xa’ ya}a E I
withf = h . g .
Thenf . xa
=f-ya ( a E I)
soby
the definition ofkernel pair
weget a
family {ha} a E I satisfying
Since f01 . p - f01. q
thisgives
usf 01 . xa = f01 . Ya
and hence a mor-phism j.
oDEFINITION 1.5. A
canonical regular epic decomposition
of amorphism f : Sce f - Tgt f
in acategory C
is a chainin the comma category
C/ Tgt f
in which(0) f= f0,
( 1 ) when B = a + 7
themorphism f a B
is aregular epic
componentof f a
( 2 )
the chain iscocontinuous,
and( 3 ) f a
is non-monic for ak ; f À
is monic.Such a
decomposition
is easy to constructgiven enough
limits andcolimits,
forexample
we have:P ROP OSITION 1.7.
Let th e category C have kernel pairs, coequalizers o f
kernel pairs and col imits o f (possibl y large) chains o f regular epimorphisms.
Then
everymorphism f in C has
acanonical regular epic decomposition.
PROOF.If f
ismonic,
thenf
has a canonicalregular epic decomposition
in which the chain has
morphism length
0 . Whenf
is notmonic, proceed by
inductionusing
1.4 and colimits at limit ordinals. We get a chain com-posite
ofcoequalizer morphisms fag
as in 1.6 whichproceeds through
theordinals.
However,
if y is least ordinal withcardy > card C
thenf a
=f B
for some
pair (a, B)
with aB
y . It isstraightforward
to show thatf a
=f B
for aB implies f a
monic. Let k be the smallest ordinal such thatf k
is monic.PROPOSITION 1.8.
Cano nical regular epic decompositions o f mo rphi sms (when they exist)
areunique
up tounique isomorphism of diagrrxms.
P ROO F . Fo llo w s
inductively
from the notefollowing
1.3.The
regular length
of amorphism f
is themorphism length
of acanonical
regular epic decomposition
off. (The regular length
is undefined when there is no suchdecomposition.)
Theregular length o f C ,
if it is def-ined,
is the supremum of theregular lengths
of themorphisms
ofC .
So
monomorphisms
haveregular length
zero.Categories
with(reg-
ular
epic)-monic
factorizations(and
at least onenon-monic)
haveregular length
one.It can be shown that
Cat
hasregular length
2. A basictypical example
of aregular length 2 ,
strongepimorphism
inCat (in
notation sug-gesting
it to be the value of the counit of anadjunction)
is the natural func-tor
E o3 : G°3 - 3 .
Here 3 is the ordinal number À =3regarded
as a cat-egory and
G ’3
is thedisjoint
union of sixcopies of 2 ,
one for each mor-phism
of 3 . Ifa (x )
denotes the copy of a in 2 indexedby x
in 3 thenE°3
mapsLet
C1 3
differfrom 3 by having
an additionalmorphism
introducedby requiring
that02 #
12.01 . It is astraightforward
exercise to show that6°3
has a canonicalregular epic
factorization2. THE TOWER AND MONADIC LENGTH.
In this section we show how an
adjunction
has adecomposition
(when enough
colimitsexist)
with the same type of universalproperties
as
regular decomposition.
Tobring
out thispoint
we use well known results of Beck[16]
onadjunctions
andmonadicity
and ofApplegate -Tierney [1]
on towersfreely throughout omitting proofs
of standard results.More
specifically,
the monadic component (orlargest
monadic«bite» )
of an
adjunction
is described and conditionsgiven
for its existence as wellas for that of the canonical monadic
decomposition,
the latterbeing
a spe-cifically
defined chaincomposite,
for a certainordinal 3 ,
of monadic ad-junctions (the
successive« bites »)
followedby
anidempotent adjunction
with invertible unit
(cf. Applegate -Tierney [1]).
The ordinal 8 is calledthe monadic
length
of theadjunction.
When 8 is a successor ordinal it
happens
in some(but
notall)
cases that in the canonical monadic
decomposition
thecomposite
of thelast
adjunction
(with invertibleunit)
with thepreceding
monadicadjunction
is itself
idempotent.
Then we say that theessential length
k is one lessthan the monadic
length 6 .
Otherwise we let k = 8 .This
decomposition
is not the same as the canonicalregular epic decomposition
as we showby presenting
anexample
of a monadic compon-ent which is not
epic,
even in a rather weak sense.NOT ATION . An
adjunction
N consists of left andright adjoint
functorsF and U
together
with a choice of unit andcounit 77 and f. (The
choiceof one
of 71 and c
determines theother.)
The monad =triple
determinedby N
isFollowing
MacLane[16]
we take the source and target ofN
to be those of F and say that N is thecomposite N B A N a of!f/3
andNu
whenThen
we have:LEMMA
2.1-If N = NB ANa,
thenIn the
adjoint
tower thecomposite
of certainadjunctions satisfy
the
hypotheses
ofLEMMA 2.2.
When
Call an
adjunction N
monadic( = tripleable)
if itsright adjoint
Uis monadic. Recall
(e.g.,
MacLane[16]):
P ROP OSITION 2.3.
I f the adjunction N
ismonadic,
thenthe
counit c isregular epic;
it i s aco equaliz er for E G and G E (where G
= FU ).
DE FINITION 2.4. A monadic
adjunction N01
is a first monadiccomponent
of an
adjunction N
ift
r
somePROPOSITION 2.5
(c f.
Beck[16]). I f A
hascoequalizers, then
every ad-junction has
afirst monadic component.
PROOF.
Let
XI
denote the category ofEilenberg-
Moorealgebras X 1 = X,
a>where a :
T X ->
X is anEilenberg -Moore
structuremorphism in X ,
and letFol
andU 01
be theEilenberg -Moore
free andunderlying
functors(thus F01 X = U F X, U E F X>). Choosing
the unitby setting n01 = n
fixesthe
adjunction.
The counit(01
mapsX, a> l-> a (regarded
now as amorphism in X 1 ) .
The Beck
comparison
fu nctorU1 maps A l-> U A , UE A > .
Then clear-ly FO 1
=U’F.
It is then routine toverify
thatN
=N1 A N01 using
the well known[16] description
of the leftadjoint FI
ofUl
and its associat- ed unit7y
and counitfl.
0We have the
following uniqueness
property for theadjunction N’
of 2.4:
P ROP O SITION 2.6.
I f
with N01 monadic,
11 =11°1 and Fol = V1 F
=VY F, then there
is aunique
natural isomorphism F1 -+ FY (and
aunique natural isomorphism U Y , Ul ).
P ROO F . If
Na
ismonadic,
thenca
and 1 are acoequalizer morphism
andobject
for thepair Gaca, fl Ga by
2.3. Leftadjoints Fl
andFY
pre-serve the
coequalizer diagram.
Ifthen the
resulting pair
when a = 0 1 isSo
Fl
andFY
arecoequalizer objects
for the samepair.
oA first monadic component
N01
of anadjunction N
is maximal(in
an obvious
sense)
among the monadicadjunctions Na through which factors;
it takes outof N
as much as can be had in one monadic« bite », just
as theregular epic
component of amorphism f
factors out as much aswe can get in one
regular epic
« bite ».P ROP O SITION 2.7.
L et N - N1 A N 01 = N A N a with N a monadic and
N01
afirst monadic component o f N. Then there
is anadjunction N8 with
N01
=N8ANa and (up to isomorphism) NB = N1 A N8, provided Xl has
coe qual iz ers.
P ROO F . For
simplicity
assume that thecategories Xl
andXB
are(and
not
just equivalent to)
theEilenberg-Moore categories
for the monadsTo’
and
ra,
and letU1
be(not just isomorphic to)
thecomparison
functor.If
N
=NBANa
then there is a natural transformation(which
is in fact amorphism
ofmonads).
Theright adjoint ve
mapsis a
Ta-algebra ;
if the mor-phism
h in X is aT-homomorphism,
then it is also aTa-homomorphism).
The
comparison
functor forNa : X - X f3
is theidentity.
Thus forY
inwe have Hence
and Thus
In
particular,
Furthermore
by
definition ofUO , using
iThe natural transformation r and the left
adjoint Fd
are definedby
thiscoequalizer diagram
in(XB, X1 ) :
and the counit
6
is theunique morphism
from thecoequalizer object F d Ud
so that
E8.T U0 = E01 .
The unitTJO
is theunique morphism
from the co-equalizer object
1 =Tgt Ea
for whichTJ (). f1
=U8T .nB Ga
so,by
theproof that f1
is acoequalizer, 770
is determinedby
We remark that the usual
«uniqueness» proof applies,
so that ifN Q is another first monadic component of
N ,
thenNO
is anisomorphism
(or more
accurately,
there is anadjunction N8
such that theright adjoints
6 and U%
areinverses ).
Proposition
2.6 shows that a first monadic component of anadjunc-
tion has certain characteristics of an
epimorphism. However,
monadic ad-junctions
are notnecessarily epic,
not evenpseudo-epic : N a
monadic andNB A Na = Ny 0 Na
need notimply FB isomorphic
toF y .
Forexample, in
let
Na
be monadic and let thealgebras
of B haveexactly
twice the al-gebraic
structure of thealgebras
ofA .
To be concrete let thealgebras
ofA
have,
say, one unaryoperation
(noaxioms)
and let thealgebras
of Bhave two. Then there are
non-isomorphic
monadic(somewhat forgetful)
func-tors
U,8
andUY
for which thecomposites
with theforgetful
functorUa
are
equal.
A monad T =
(T,n, 03BC )
is said to beidempotent
if n is an isomor-phism
andtrivial if,
inaddition,
-q is anisomorphism.
Anadjunction
N isidempotent or trivial
if its associated monad is.P ROPOSITION 2.8.
The adjunction
N isidempotent i f and only i f
any oneo f the natural transformatiohs
is an
isomorphism, i f and only i f
any twodistinct natural trans formations
2.9
with th e
sarnetarget and
source areequals (say E F U
=F U E ).
In
particular
N i sidempotent
if c is monic(since
c.(F U
=E .FU E ).
The
proof
of theproposition
usesonly repeated applications
of theadjunc-
tion
equations,
thenaturality of 71 and c
and the fact thatDEFINITION 2.10. A
canonical monadic decomposition
of anadjunction
N
is a chainin the comma category of
adjunctions
over A in which :(0) N =N0
-
(1)
when13
= a + 1 theadjunction Nai3
is a first monadic componentof N a ,
henceFaB
=UB Fa
andn aB = na ;
(2)
when K is a limitordinal X K
is the limit inCat
of thediagram
of
right adjoints U aB (
aB K ) ,
with(3)
N a is non trivial fora 6 ; N 6
is trivial.VG’e refer to the
adjunctions ¡ya¡3
in 2.11 asmonadic components
of N . Monadicdecompositions
areexamples
ofadjoint
towers. Conditions for the existence of auch adecomposition
aregiven by
thefollowing
result(cf. Applegate -Tiemey [1]).
T H EO R EM 2 012.
1 f A
hascoequalizers
andcolimits of (perhaps large)
chains of strong epimorphisms,
then everyadjunction N : X -> A has
a ca-nonical monadic decomposition.
P ROO F .
By
induction on a - Whenj3
= a + 7apply
2.5. when K is a limitordinal we must define
N aK
andN K
in such a way thatN a = N K A N a K ,
( 2 )
holds. We presentonly
a few of theimportant
details. Anobject X K
ofX K
is an indexedfamily X a > a K of objects
of thecategories X a
sat-= =
isfying U aB XB = Xa (a B K). Furthermore, UaK( X8 >8k ) =Xa
and the
limit comparison UKA - Ua A > aK° Furthermore, FaK = tJK Fa
and
F"is
a colimit of thediagram
of
(companentwise)
strongepimorphisms.
Thus if we do not stop when
Na
becomes trivial we canpush
the de-composition
2.11through all ordinals a .
But we have assumed that thecategories
of this paper are at mostlarge (relative
to a universe U[18] ).
Consequently
there are at most2card U
distinct naturaltransformations (a
in 2.11. Let A be the first ordinal for which we have
Then
by 2.2 (1) F B n BF kB
= 1 and(since E B F B
is a one-sided inversefor
F B n B)
So
E À F À U À =
1 - whichby
2implies that N À
isidempotent.
If,
on the otherhand,
we let À be the first ordinal for whichN k
isidempotent,
then6
isisomorphic
toE B
for allj8
> À. Thisfollows, by induction,
from theequation
and the fact that
F B E a BU B
is acoequalizer when B =a+1
for thepair Gaca, EQGa - which
is non-distinct whenÀ a.
So N B
isidempotent for B >
X.Claim :
if N X
isidempotent
andB
= À + 1 thenN B
is trivial. For ifCXFX
is anisomorphism,
then so isB À B (since
it is acoequalizer
of thepair GÀEÀB, EÀBGÀB = UBEÀFÀUÀB -with common-one-sided
inverseFÀBnÀBUÀB). Hence,
so areUÀFBEÀkB and nkUkB (one-sided
inverse ofUÀBEÀB) -
whichby
definition ofnB
haveUxt8riP
as theircomposite.
Since
UkB
isfaithful, TJ f3
is anisomorphism.
The ordinal 8 of 2.10 is À
if N À
istrivial; otherwise 6 = À + 1 .
oThe
uniqueness
of canonical monadicdecompositions (up
to iso-morphism)
followsinductively
from the remarkfollowing
2.7. Inparticular:
PROPOSITION 2.13.
In
2.11the ordinal 8
isuniquely determined the X a
are
determined
up toe quivalence of categories and
once theXa
arechosen
the T a (associated
toNa )
aredetermined
up toisomorphism o f monads.
DEFINITION 2.14. When an
adjunction N
has a canonical monadic de-composition,
theessential length (monadic length)
ofN
is the smallestordinal
A in 2.11 such thaty-
isidempotent (trivial).
In the
proof
of 2.12 we havealready
seen that for agiven adjunc-
tion monadic
length equals
either essentiallength
or essentiallength plus
one. An
example
with essentiallength
1 and monadiclength
2 is the usualadjunction
from sets to torsion free abelian groups.The torsion free abelian groups is
typical
of ageneral phenomenon :
where X and 8 denote the essential and monadic
lengths,
all the « visible »operations
of thealgebras in X 6 already
appearin XX.
PROPOSITION
2.15. I f A
is at mostlarge then the
monadiclength of
anadjunction N : X -> A (when defined) has cardinality
at mostthat of the
universe U .
The
proof
calls for a reexamination of theproof
of 2.12.There,
toavoid
clutter,
wespoke
of natural transformationsca
inplace
ofmorphisms E a A .
If A is at mostlarge,
then for eachobject
A there are at mostcard U distinct
morphisms E a A .
Now in theproof
of 2.12 letÀA
be thesmallest ordinal k for which we have
E À A = E B A
for someB
> À and let À =sup A E A À A .
0Note that in the
proof
of 2.12 we do not need all thecocomplete-
ness that we ask for in the statement of the
proposition.
We needonly
thecoequalizers T aB X B
ofpairs Fa U a B E a B X B, E a Fa Ua B X B
for Eilen-berg-Moore algebras X B (B = a +1 )
and colimits of theresulting
chainsfor limit ordinals
K
À.REMARK 2.16. Monadic
decompositions
are a refinement of theDay
factor-izations
[5]
ofadjunctions, analogous
to the refinement of strongepic-
monic factorizations
by regular decompositions.
3. AN EXAMPLE.
A
simple example
that willproduce adjunctions
of any desiredlength
X andepimorphisms
ofregular length
X is obtainedby letting
A =
Ah
be the category ofalgebras A
for a sequence ofpartial operations
m (B X)
wheremB
defined at a c Aiff ma a
= a for all aB.
( So m = m0
iseverywhere defined. ) Homomorphisms f : A -> B
are func-tions
satisfying ma f (a) = f (ma a )
wheneverma a
is defined.For consideration of this
example,
which isessentially algebraic
in the sense of
Freyd [6],
let the a-sequence of an element a (when def-ined)
be the sequenceand call this sequence
free
if its elements are distinct. There is anadjunc- tion Nay: Aa-> AY a y )
in whichvaY forgets
theoperations m ( a 8 y )
andF aY A
is thealgebra
A with a new free a-sequence ad-joined
at a for each element asatisfying me a
= a( 8 a
It is easyto see that the
Eilenberg-Moore
category forN a y
isisomorphic
toAB
where
8 =
a + 1 - and thatN0 À
has essential and monadiclength
X .The
example
can be modified togive
monadic
length =
essentiallength
+ 1 .For
example
wemight impose
upon thealgebras
A of Aa( 0 a )
the re-quirement
thatthey satisfy
an axiom of the form(m0
leaves fixed the elements ofIm me ) for a
=8 + 1 ,
and for a a limitordinal we
might
let the axiom be( m0
leaves fixed the elements of02 a Im me ) .
We next show how
epimorphisms
ofregular length
X arise as valuesof the counit c of
N 0 À.
The unitn ay
ofN aY (a y )
consists of the ob-vious
embeddings
Aa ->Uay FayAa .
The counitmorphisms
maps
a l->a
fora E A B
and mapsmn m aa l-> mn m aa (defined freely
onthe left and
using
the structure ofAB
on theright)
wheneverma is
def-ined at a .
For A = 2 the canonical monadic
decomposition
andregular
conuitdecomposition
look as follows :Using
this we construct anexplicit length
2regular decomposition
of the
counit c
at theobject 2
ofA 2
described as follows : theunderlying
set
l2l
of 2 is that of thecategory 2 ,
which isjust
the set10, 11 .
Theoperations
of 2 areand E 2 has
regular decomposition
In the same way for an ordinal A an
object A
of/4"
may be des- cribed for which E A haslength
Aregular decomposition.
Moreexplicitly,
the
underlying
setl À I of À
is that of thecategory k ,
in otherwords,
theset of ordinals less
than À. Furthermore,
apartial operation ma
is def-ined
on ) A) i
for each a À asfollows : given B
cAj I
theelement m a B
isundefined
for 8
a,An
adjunction model induced by
a set L ofcardinality
m of ob-jects
of A is one whoseright adjoint U:
4 -+Ens
mapsIn this
terminology
we see thatN0 k: A0-> Ay
is model inducedby
the setL whose
only
member isF0Y
of the one element set,i. e,,
the free 0-sequence on one element.
J .
MAC DONALD: Mathematics DepartmentUniversity
of British ColumbiaVANCOUVER, B.C. V6T 1W5 CANADA
A. STONE: Mathematic s
Department University
ofCalifornia,
Davi sDAVIS, CA 95678 U. S. A,.
213
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