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Monadicity, Purity, and Descent Equivalence
.A t hesis submitted to the Faculty of Graduate Studies
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
Graduate Programme in Mathematics and S tatistics
York Cniversity
Toronto, Ontario
National Library 191 of Canada Bibliothèque nationale du Canada
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Monadicity, Purity, and Descent Equivalence
by Xiuzhan Guo
a dissertation submitted to the Faculty of Graduate Studies of York University in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
E 2000 Permission has been granted to the LIBRARY OF YORK UNIVERSITY to lend or sel1 copies of this dissertation. to the NATIONAL LIBRARY OF CANADA to microfilm this dissertation and to lend or seIl copies of the film. and to UNIVERSITY MICROFILMS to publish an abstract of this dissertation. The author reserves other publication rights, and neither the dissertation nor extensive extracts from it may be printed or othewise reproduced without the author's written permission.
Abstract
This dissertation is intended to study monadicity, to introduce the notion of
descent equiwlence, and to initiate descent theory in locally presentable categories.
.At first. we give tn70 monadicity results %*hich do not involre explicitly checking
presenation of certain coequalizer diagrams, as well as a type of "monadicity lift-
ing' theorem. Shen ive summarize the fundamentals of descent theory in order to
lay the groundwork for deveioping the new notion of descent equivalence and for
inrest igating its properties. Finally, ive study closedness of (effective) codescent
rnorphisms under directed colimits and t heir connect ion wit h pure monomorphisrns
in locally presentable categories.
Acknowledgement s
I would like to thank Professor Walter Tholen: my supervisor, for his constant
guidance. support, and encouragement dunng my study at York University. I have
benefited greatly fiom the courses he taught, from numerous discussions with him.
and from conferences. workshops? and seminars he encouraged me to attend. During
the preparation of this dissertation. he provided me with many motivating ideas and
deep insights: valuable comments and suggestions: and helpful references. Without
his help, it would never have been possible to finish this dissertation.
T also owe Professor S.O. Kochman an ovenvhelming debt for h
ment and financial support. As iveil: 1 am grateful to Professors J.W
Janelidze, and 31. Sobral for their encouragement and comments.
is encourage-
. Pelletier. G.
Finallp, 1 wish to express my gratitude t,o D. Promislow and J. Wu. the directors
of the graduate program, and to Mrs. P. Miranda. for their help.
O. Introduction
Table of Contents
1.1. Revielr- of monads and T-algebras
1.2 . So te on Beck's theorem
1.3. -4 monadicity result in abelian categories
1.4. Transformations of monads
2. Preliminaries for Descent Theory
2.1. Fibrations and descent theory
9.2. Interna1 categories, actions: indexed categories.
and descent theory
3. Descent Equivalence
3.1. The functor Eq
3.2. Descent equivalence
4. Purity and Effective Descent
4.1. Pure morphisms are effective descent for modules
4.9. Locally presentable categories? accessible categories,
and pure morphisms
4.3. Descent theory in locally presentable categories
References
vii
O . Introduction.
Descent Theory plays an important role in the development of modern algebraic
geometry by Grothendieck [13] 114): Giraud [12Io and Demazure [IO]. Generaliy
speaking, it deals with the problem of ahich morphisms in a given b*structured''
category allow for change of base under minimal loss of information. and hou- to
compensate for the occuring loss. such morphisms are called effective descent mor-
phzsrns. In commutative algebra, the descent problem may roughly be described
as follows: giren a morphism f : R i S of commutative unital rings. ahen does
t here exist. for each S-module dl with descent data (see 2.1.3). an R-module S such
that -11 Z S S R S? Putting the problem into more precise terrns. it is n-el1 knoan
ivhich morphisms f in the opposite of the category of commutative monoids in sup-
lattices are effectize descent morphzsms. with respect to the fibration induced by
the indexed categov nhich assigns to each commutative rnonoid -4 in sup-lattices
an -4-module: precisely the pure rnonomorphisms (see [XI). In their monograph
pl], Joyal and Tierney showed that Grothendieck's techniques of descent developed
for commutative unital tings can be estended to Locale theory: open surjections are
efect iue for descent. not only in locale theorg but also in topos theory. In topol-
ogy. Reiterman and Tholen [27] completely characterized effective descent maps of
topological spaces. Janelidze's Galois theory has a strong connection with descent
1
theory, as is s h o m in [16], [l'il, 191: and [20]. Janelidze and Tholen gave a very
motivating introduction to descent theory in their paper [la]: to which we refer the
reader, as well as to its successor [19].
A s is well k n o m . the category DesE(p) of descent data relative to a morphzsm
p can be defined in the general contevt of a Ebred categoy E (see 2.1.3) or of a
C-indeued category A (see 22.4). However, in the case of a bifibmtion satisfying
the Beck- Chevalley condition (see 2.1 .a). descent problems can be converted into
monadicity problems. Hence Beck's SIonadicity Theorem tums out to be estremely
useful. But. preservation of some kind of coequalizer diagtarns. the crucial require-
ment of Beck's Theorern. remains difficult to check in some cases. including the
module case. In Chapter 1: ive gke two monadicity results (Theorems 1.2.4 and
1.3.6) which do not inïolve checking presenation of some kind of coequalizer dia-
grams. In this chapter, we also give a "monadicity lifting' type result (Theorem
l.-i.-l).
Chapter 2 is devoted to a sumrnary of some definitions and resuits rrhich we
shall use later on. They include: descent data Aith respect to a fibration. the Beck-
Chevalley condition, Bénabou-Roubaud's Theorem and Beck's Theorem: indexed
categories and their actions, descent data with respect to an indexed categop: some
theorems of Janelidze and Tholen.
.As illustrated in [18] and [19]. finding sufficient (and necessr'.) conditions for a
morphisrn in a given category to be effective descent can be a challenging problern.
On the other hand, a natural question is:
How to compare morphzsrns (or bundles) in the descent sense.?
More clearly, which c o n d z t z o ~ can guamntee that two morphzsms have the same
descent structure? In Chapter 3, Fe begin with a study of the induced equivalence
relation functor Eq (Propositions 3.1.2 and 3.1.4). Then we give the notion of
descent equivalence (De finition 3.2.3) and study its propenies (Propositions 3.2.2
and 3.2.5, Theorerns 3.3.2. 3.2.3' and 3.2.4).
The concept of a locally presentable categoq is due to P. Gabriel and F. Ulmer
[Il]. This notion has tumed out to be eutremel- useful [2]. Tt is broad enough to
include: varieties and quasivarieties of many-sorted ranked algebras. Hom classes
of relational structures. and presheaf categories. It is strong enough to show that a
locally presentable category is complete. cocomplete, wllpowered. and cowellpo~v-
ered: and has a strong generator: moreover. pure morphiçms have good properties
[Il. We feel that locally presentable categories (and even the more general accessible
categories d t h pushouts) provide a suitable environment for tackling descent prob-
lems. In Chapter -1: we give a first outline of descent theory in locally presentable
categories and study closedness of (effective) codescent morphisms under directed
colimitç (Theorem 4.3.2, Proposition 4.3.4).
1. Monadicity
As mentioned in Chapter 0, the preservation of some kind of coequalizer di-
agrams, the crucial requirement of Beck's Theorem. remains difficult to check in
some cases. Hence? in this chapter, u7e wish to gzvr some monadicity criteria uihich
do not involve checking preservation o/ some kind of coequalizer diagrams.
F e also u i sh to revisit the "lifting problern!' for rnonadicity. Hence. we consider
a commutative (up to natural isomorphisrn) square of categories and functors:
Our second problem is then to jnd conditions u~hzch can paran tee that rnonadicity
of G' implzes rnonadicity of G.
1.1. Review of Monads and Their Algebras
1.1.1. In algebra' a monoid M is a semigroup with an identity elernent. It may be
viewed as a set with two operations
such t hat
and
are commutative, where the object 1 is the one-point set {O}: the morphism 1 is an
identity map, and where q and x? are projections.
Definition. d monad T = < T, q: p > in a category C consists of an endofunctor
T : C i C and two natural trans/ormations
such that
and
TT qT
C
i-, are commutative. where I : C + C zs the identity functor.
Recall that an adjunction from C to B is a triple < F, G. ; >: C -i B' where F
and G are functors:
; is a function which assigns to each pair of objects C E C. B E B a bijection of
sets
; = i j ~ . ~ : B(FC. B ) S C(C. GB)
n-hich is natural in C and B.
By i-4: p.83, Theorem 21: each adjunction < F. G. ; >: C i B is cornpletely
determined by
(Y ) functors Fy G and naturai transfomations 7 : lc i GF and E : FG + lB svch
that Gr - qG = lG and EF - Fq = IF.
Hence we often denote the adjunction < F. G: ; > by < F: G: q. 5 >: C + B.
If < F! G: 7: 5 >: C -t B is an adjunction, then < GF. q. GcF > is a monad
on C (see [24], p.138). In fact? eve- monad anses this way (see 1.1 2 ) .
1.1.2. Definition. Let < T. r ) , p > be a monad in C7 a T-alegbra < C, ( > is a
pair consisting of an object C E C and a rnorphism ( : TC -t C in C such that
T2C Tt
-TC
and
are commututiue. -4 morphism f : < C? < > + < C'? ,E' > of T-algebras zs a
morphism f : C i C' of C such that
TC Tf
-TCr
cornmutes.
Everp monad is determined by its T-algebras? as specified by the folloning the-
orem:
Theorem. If< T, r). p > zs a monad in C . then al1 7-a lgebras and their morphzsms
/ o m a cutegory CI. T h e ~ e 2s an adjunction
where GT : cT + C zs the obwiovs forgetful junctor and zs gzven b y
/urthermore. qT = q and z:~,~, = { for each T-algebra < C. < >. The monad
d e f i e d in C by this adjunction is < T? r)' p > .
Proof. See [%LI' pp.140-141.
Examples. a. Modules. Let R be a unital ring. Shen
and
p.4 : ( A 3 R) 3 R i -4 8 R : (a 8 rl) @ - H a 8 (rir2) for every abelian group -4
give a monad (TR: p , p ) on the ca tegov Ab of al1 abelian goups. and .AbTR is the
category Mod-R of right R-modules.
b. Group .Actions. Let G be a group. Then setTc is the category setC of G-sets.
where the monad < Tc, I ) ? p > on Set is defined b>*
and
P X : G x (G x S) -t G x S : (g,, (g2: x)) ci (gig2: 2).
8
More pnerally, every varie- of universal algebra is the category of T-algebras
over Set, where TX is t h e the underlying set of the free algebras over X.
1.1.3. Definition. Let T = (T, 7, p ) and Tl = (Tt: r)'? p') be two monads in a
category C . -4 monad morphism /rom T to Tl is a natural tran@rrnation : T i T'
svch that
and
commute. where a . a is the morphisrn T'a 0 aT.
Remark. There is a bijection between monad morphisms Q : T + Tt and functors
I,' : cT' + cT for which
comrnutes.
For a functor 1- : cT' + Cr ni th ITV = LT': one defines a : T + Tt by
cuc = <cTqb: Khere eC iç given by V(TtC: &) = ( T T ? cc).
9
Conversely, for a monad rnorphisrn ct : T 4 T', V : Cr' + cT is given by
kloreover, one has a functor T:
T (Monad(C))'P - CAT
Xaturally, we have the following question:
Question. W h e n does the functor T reflect zsomorphisms?
Uorita theop shows that one can study a ring R by inrestigating the categop
of al1 R-modules. A more general question is:
Can we characterire T by C and CT? In particular. what is the relationship
between Tl and T2 if cT1 z cT2 and C is "good"?
1.2. Note on Beck's Theorem
Throughout this section, < F, G: q, E >: C + B is an adjunetion. and
T = < GF: 7. GaF > is the induced monad.
1.2.1. Theorem (Cornparison of adjunctionî with algebras). There zs a unique
10
such that GTK = G and KF = :
Proof. See [XI. pp.142-113.
Definition. G is monadic (premonadic) îi the cornparison junctor K is an equiva-
fence of categon'es ( ful l and fazthjut). F 2s (pre)cornonadic if FO* i s (pre)monadic.
where FoP belongs to the adjunct ion < QP. FoP? E . q >: B O P - CoP.
1.2.2. Recall that a splzt fork in a catego- is a diagam
mhich satisfies the conditions
cd = ce. d = lc, ds = Ig! es = tc.
- .A pair of arross d, e : E B is a G-splzt pair if Gd, Ge : G E GB is a
part of a split fork: it is refieziue if d and e have a common right inverse.
Example. Given a monad < T: 7, p > in C: if (C. {) is a T-algebra. then
11
is a fork in C split by
Furt hermore
is a reflexive GT-split pair since pc and Tc have a common right inverse TV=.
In particular. for any B E obB,
GEFGB GtB GFGFGB 1 GFGB -GB
G F G E ~
is a reflexive split fork and
is a reflexive GT-split pair.
We now discuss the monadicity criteria.
Theorem (Beck). 1. G zs prernonadic if and only if E R zs a regular epimorphisrn
for each B E obB.
2. G is monadzc ij and on$ 2f G reflects isomorphisrns. B has ccequalizers O/
repexive G-split pairs, and G presemes them.
Proof. See [3], pp.110-114.
From the proof of Beck's Theorem, we actually have:
G is monadzc ij and only if G refiects isomorphisms, and /or each (C. 5) E c'.
F{: EFC : FGFC =i FC hm a coequalizer, and G preserves it.
1.2.3. Let f : R + S be a morphism in h g , . We define functors
( M ) = \ (restriction of scalars)
f ( 1 ) = Y 8 S (e~rtension of scalars)
for each M E Mod-S and iV E RIIod-R. with the obvious assignrnents for rnor-
phisms. Then f. is a left adjoint of /!. with units and counits given by
for al1 1V E Mod-R, M E Mod-S.
Definition. A morphism n : 1% + & in R-Mod zs pure if and only ijfor each
!C.I E Mod-R,
lnr 8 n : M BR !VI + .C[ aR 5
is a monomorphzsm. Similady, a rnorphzsm n : !VI i -Y2 in Mod-R is a pure
morphism if n 8 1 : ~ ~ is a rnonomorphism for each M in R-Mod.
Since a ring homomorphism f : R + S can be viewed as a morphism in R-Mod.
since
1 % f : X @ R R + ' i 8 R S
is the unit 7 of f. 4 f!: for each :y E Mod-R, and since every rnonomorphism in
Mod-R is regular, we have:
Proposition. f is pure in R-Mod if and only i/ f. is precomonadzc.
1.2.4. Theorem. The jollowing are equivalent:
(2) ( i ) G repects isomorphisms.
(ii) For any morphisms /: g : BI + B2 in B such that
iç a split /ork there 2s a morphism h : B2 + B3 in B u;zth e S Gh. and / : g
has a coequalzzer
14
in B such that Gb is an epirnorphzsm.
(3) G is prernonadzc, and for any rnorphisms f, g : BI i B2 in B such that
is a splzt fork there 2s a rnorphzsm h : B2 + B3 in B such thut e 2 Gh in
GB2/C.
Proof. Bu [3' p.103. Proposition 1 and p-105. Proposition 31. clearly. (1)= ( 2 ) -
( 2 ) - ( 3 ) Since FGQ, E F C ~ : FGFGB + FGB is a G-split pair for each B E obB.
by (ii), it has a coequalizer, say (BI: b). Since E B - C F C B = i~ - FGEB there is the
unique morphism bl : B1 + B such that bIb = a~ :
FGFGB FGE b m B R FGP\ - BI I
Application of G gves the following commutative diagram
GFGFGB G F G E ~ Gb q GFGB \ GBI
l
On the other hand, since S GE^: GB) is the coequalizer of G F G E ~ ) ,
there is the unique morphism c : GB -t GB1 in C such that
c - G E ~ = Gb.
C. Gbl - Gb = cG(blb) = CGE* = Gb.
and Gb is an epimorphism (by (ii))? me have
cGb1 = les,.
On the other hand.
so that Gbl - c = Ica Hence Gbi is an isomorphism: so thar, by (i)' bi is an
isomorphism. Hence 5s is a regular epimorphism. Therefore G is premonadic.
Then (3) holds.
(3)+(1) We check the conditions of Beck's Theorem. Suppose that
is a split fork. By hypothesis. there is a morphism h such that
is also a split fork. Since GT is rnonadic and ( K f7 Kg) is a GT-split pair in Cr.
there is a coequalizer diagram
in cT. Then we have the foliowing commutative diagrams
GFG& GFGh - GFGB3
and
GFGB? GFGh - GFG&
Hence
GE& -GFGh = Gh . GE^: = 0 . GFGh.
Since Gh i e is a split epimorphism. GFGh is an epimorphism. Hence B = GcB3
and t herefore
is a coequalizer diagram in CT. Since K is full and faithful.
is a coequalizer diagram in B and G preserves it since
is a split fork. By Beck's Theorem, G is monadic.
1.3. A Monadicity Result in Abelian Categories
1.3.1. We cal1 k' k' : K t E a joint kernel pair of f g : E B if ci) f k = f k'
and gk = gk'! and (ii) for any pair 1, 1' : L E of morphisms with / 1 = fl' and
gl = 91'. there is a unique morphism u : L -t K such that 1 = kc and 1' = k'c :
X pair j: g : E 7 B is a G-split equiualence relation if G J. Gg : G E =i GB is an
equivalence relation which is a part of a split fork.
Duskin proved:
Theorem. Suppose B has joint kemel pairs and C has kernef pairs o j split epi-
morphisms. Then G is rnonadzc if and on$ zf G reflects isomorphzsms. G-split
equivalence relations have coequalizers, and G presemes them.
Proof. See [3]: pp.304-308. Ci
1.3.2. Recall that a zero object O in a category is an object which is bot h initial and
terminal. If a category A has a zero object O? then to any -4: B E obA the unique
rnorphisms -4 -t O and O + B have a composite O = O$ : -4 -i B called the zero
morphism from A to B. For a morphism f : -1 + B of Al -4 kernel of f is defined
to be an equalizer of morphisms j, O : A i B. The dual notion of kernel is cokernel.
Definition. An abelian category A is a catego y satisfying the follouting conditions:
(1) A hm a zero object,
(9) A hm binary products and coproducts,
(3) everg rnorphisrn in A has a kernel and a cokernel.
(4) e u e y monornorphism 2s a lieniel and evey epimorphism 2s a cokernel.
For esample. Ab' Mod-R, and R-Mod are abelian categories.
Every abelian category is finitely complete, finitely cocomplete. additive. and
exact .
Non we want to see what happens if both B and C are abelian categories in
Duskin's Theorem.
In the rest of this section, we assume that Al and A? are tn-O abelian categories
and < Fr G; q. 5 >: Ai -t A2 is an adjunction.
19
1.3.3. Proposition. 1. F is right ezact and G is left ezact.
2. F and G preserve bzptoducts.
Proof. By [6, vo1.2, Propositions 1.3.4 and 1.11.2].
1.3.4. In a category,
21 f P-A- B
2:
is said to be an ezact sequence if ( u t L') is the kernel pair off and f is the coequalizer
Proposition. In an abelzan cutegory. the follou~zng are equitiaf ent :
is a short ezact sequence.
Proof. See [6]: vo1.2, p.95.
1.3.5. Basically. the folloming Proposition is i3' p.305, Lemma 31. Here a e give a
complete proof.
Proposition. Let < F, G; 7, E >: C + B be an adjunction. I j G ts premonadzc,
then G refiects equivalence relations.
Proof. Let d. e : E B be a pair of arrows such that
is an equivalence relation. If x : y : S =i E is a pair of arrows with
dx = dy and ex = ey.
t hen
Gd-Gx=Gd-Gy and Ge.Gz=Ge-Gy.
and so
Gz = Gy,
Gd. Ge : GE monic pair. G is faithful. x = y follows.
Hence. d: e : E =i 3 is a relation.
For any object S. ive want to prove t h a t
Rx = {(dx: er)lz E hom(S1 E ) }
is an equivalence relation on the set hom(S. B).
For any object C' the induced maps
homiC, GE) hom(C. GB)
21
is an equivalence relation in Set, and by adjointness. so is
horn(FC, E) hom(FC, B).
Since G is premonadic,
F G E ~ E X
FGFGX LFGX i- S EFGX
is a coequalizer diagram. Kow we consider the follouing commutative d i a g a m :
horn(S. d ) 1 1 horn(St e) hom(FGS? e) 1 1 hom j FGS. d )
For any a E horn(S, B)? aEm E hom (FGS. B). Since
is an equivalence relation in Set, there is a rnorphism b : FGS + E such that
d b = E X and eb = a s x .
Since
and
we have
bFGix = b ~ ~ ~ ~ .
Then there is a unique morphism c : ,Y -+ E such that
Hence
and t herefore
d c = a . e c = a .
Thus. (a, a ) E Rx. Then Rx is reflexive.
CLAIM 2. Rx is symmetric.
If (a. b) E RXy, then there is a rnorphism xo : S i E such that
a = dzo and b = ezo,
and so
Hence
Since
hom(FGS. E ) =i hom(FGS. B )
is an equivalence relation: ive get
( b ~ ~ : E X ) i RFGx.
Then there is a morphisrn y0 : F G S -+ E such that
and
t hat
FGFGS
Hence
YOFGCY = YOCFCX,
and therefore there is a unique morphism zl : S -t E such that
Tt follows that
Shen
and so
(b. a ) E RX.
Hence Rs is symmetric.
CLAIM 3. Rx is transitive.
If (a. 6 ) . jb. c ) E R.Y, then
( a ~ , ~ , bcx). (ks. E X ) E
and so
(aax, E ~ ) E RFGA.
Hence there is a morphism zo : FGK i E such that
and
and so
FGcx = Z O E F ~ X .
Hence there is a unique morphism 21 such that
FGFGS FGS I
A- 3' :O ; allc
E' e -. B
Therefore
(a. c ) E Rs
Hence Rw is transitive, as desired. @
1.3.6. Recall that an equivalence relation d' e : E t B is effective if (d. e ) is
the kernel pair of some morphism q. If an equivalence relation is effective and has a
26
coequalizer then it is the kernel pair of its coequalizer (see [3]: p.49). Since abelian
categories are exact in the sense of Barr' every equivalence relation is effective in
abelian cat egories.
Theorem. If G is premonadzc. then the following are eguivalent :
(<Il G is monadic.
(2) For each rejlezive G-split pair d. e : E =i B. if
is a coequalizer diagram. then Gc zs an epimorphzsm.
(3) For each G-split equivalence relation d. e : E B. i/
d c E-B - C
e
zs a coequalizer diagram. then Cc ia an epirnorphisrn.
Proof- (1) - (2) . By [3. p.103. Proposition 1 and p.105. Proposition 31.
(2) * (3). Suppose that d. e : E B is a G-split equivalence relation. Then
Gd? Ge : GE =i GB is an equivalence relation. and so is d. e : E i B by
Proposition 1.3.5. Since Al is abelian. d. e : E B is effective. If
d c E-B - C
e 27
is a coequalizer diagram, t hen
is a pullback diagram, and therefore d. e : E B is a reflexiïe G-split pair as one
sees by considering the following diagram
By (2 ) . Gc is an epimorphism.
(3) => (1). Clearly. G = QI\' reflects isomorphisrns. Let d. e : E = B be a G-split
equivalence relation. Then Gd. Ge : GE Y GB is a split equiïalence relation. By
Proposition 1-35! d. e : E 7 B is an equivalence relation. S h e n it is effective.
and so there is a morphism g : B -+ 1' such that
is an exact sequence. Hence, by Proposition 1.3.4'
is a short exact sequence. By Proposition 1.3.3,
is exact. By condition (3):
is exact and t herefore, by Proposition 1.3.4 again.
is a coequalizer diagram. Thus, by Duskin's Theorem, G is monadic.
From Theorem 1.3.6, ive immediately have:
Corollary. If G preserves epimorphisms. then G is rnonadic if and only z/ G zs
premonadic.
1.3.7. If f : R + S is a morphism of Rng,, then we have an adjoint pair:
Recall that the category of al1 right modules over a ring is abelian. and in such
a category the epimorphisms are precisely those morphisms which are surjective
on the underlying sets, and epimorphisms are al1 regular. Hence, f! presemes epi-
morphisms. Clearly. /! is premonadic since each is an epimorphism (see 12.3).
Then. by Corollary 1.3.6, me have:
Proposition. f! is nonadic.
Surprisingly, Mod-S is an Eilenberg-Moore category over Mod-R rvhenever
there is a ring homomorphism f : R t S.
1.4. Transformations of Monads
In [31], Tholen studied adjoint morphisrns and their properties. In this sec-
tion. we consider a special case of adjoint morphisms. Throughout this section.
< Fr G: q: a >: C i B and < F'! G': 11'. E' >: C' + B' are two adjunctions.
T = < GF: 7: GcF > and Tt = < G'F'. $, G'I'F > are the two induced rnonads.
1.4.1. Definition. Let I,' : B + B' and Il- : C i Cr be tmo functors:
< F. G; q, E >: C -t B zs transformable into < Fr' G': q'. z' >: C' -? B' under V
and W z[ there are naiural isomorphzsrns û : TT'G -r G'1' and 3 : F'IT' i I'F such
that
aF t
CVGF - G'VF
and
cornmute.
1.4.2. Let C be a category and B E ob C. Recall that the comma category ofC over
B is the category C / B whose objects are pairs (E. p) with C-morphisms p : E i B:
and ivhose morphisms g : ( E : p ) + (Er : pl) are C-morphisms g : E + Er such that
commut es.
Let C be a category nith pullbacks, and let p : E i B be a morphisrn in C.
Then we have the follonring adjoint pair:
where p!(D, s) = ps, p8(C? r ) = which is given by the following pullback
diagram:
The unit and counit of p! -i p' is @en by r l ( s : c - ~ ) =< S. lc >: C T E x~ C
Examples. Let C be a category with pullbacks. If
is a pullback diagram in Ct then:
1. p! i p' : C/B -, C I E is trans/onnable into pt! -! p" : CIB' - CIE' under t !
2 . p'! i pl* : C/B' i CIE' is t rans fonab le into p! 4 p' : C / B i C I E under
and s!: P !
C I B ' 1 CIE
t' and s*:
P*
t ! 1 p' ! S . r
i
C/B' - _t C / E t
PI*
1.4.3. Proposition. Suppose that Ç F: G; t): e >: C -r B zs transformable znto
< FI, G': q', z' > : C' -t B' under Il' and W. Then there 2s a hnctor
such that
cornmutes (up to zsomorphism).
Proof. Define L : C' + cm' by
L(C, () = (Cvc IV< cu;bG'3c) for each (C: ,C) E cT.
33
(qc = Ic and (GFC = <GzFC:
and so? by applying LV,
W7fIVm = lrvc and CI;{LTiGF< = W < I V G E ~ ~ .
But
cornmutes.
By the naturality of a? 3: and 2 . for any object C of C.
comrnutes. Hence (WC! Wf - ~4&G'3~) E cc* and therefore L is aell-defined.
For any C E obC,
and
Hence
cornmutes.
For any (C: <) E cT.
For any B E obB,
LKB = L(GB, Gês) = (FVGB, I . ~ ' G E ~ ~ & G ' ~ ~ ~ ) .
and
Hence
since
K'VB = (G'bwB? G ' E ~ , - ~ ) .
cornmutes.
1.4.4. Theorem. Suppose that < F. G: r)' a >: C i B is transformable info
< F'. Gr: r)', E' >: Cf + B' under V and W. If
(1) B has the coequalirers of G-split pairs and C* presenes them.
(2) G (or V) refiects isomoiphisrn.
(3) W reflects split forh in to coequalzzer diagram.
(4) Gr zs monadzc.
then G zs monadzc.
Proof. For any (C. <) E CT, by Example 1.2.9:
is a split fork. By (1): F<: EFC : FGFC =i FC has the coequalizer. say (DI c ) :
and
is a coequalizer diagram. which is equivalent to saying that
F ' ( W < ~ U & G ' ~ ~ ) 1-c- 3= F' G' F'FC Z F'tl'C I'D
is a coequalizer diagam since
I'FGFC 1'5
commutes. where t = 3GFcF'a;~F'G'3c. But (WC. 11'<a;$'3,-) E C? by
Beck's Theorem and (4): F'(Wfa,kG'3c), $,,,, : F'G'F'WC i F ' T C has the
coequalizer. say (DI: c') (s (VD. V c 3& and
is a split fork, mhich amounts to saying
is a split fork since
cornmutes. where rn = G'F'G'~~'G'F'~~~G'~G~~~FCFC~ n = G'Jz'a~c.
rn
GF( GFGFC
Gc X F C -GD
is a coequalizer diagram. Shen, by Beck's Theorem. G is monadic.
t Gr F ' ( ~ < C & G ~ ~ ) i Gf(Ix-~3c) t
G' F'G' F f W C L i G ' Ff CI/C G' -1 - G'I'D
F'WC
WGi jrc
Combining Theorem 1 ..)A and Example 1.4.2.1 yields the following corollan
n
which was shorvn in [79] for the first time.
QD
Corollary. Let C be a category with pullbuch. and let
be a pullback diagram in C. I j (pl)' 2s monadict so zs p'.
Proof. If bl: b2 : (Cl? r l ) -t (C2, r2) is a p'-split pair in C I B . then we have
a (p')'-split pair bl, : (Cl, hl) -t (C2, t - ) in CIB'. Since (pl)' is monadic, by
Beck's Theorem, 61: b2 has a coequalizer, say (c, (C, r)). in C I B'. Clearly t here is
a unique morphism b : (C. r ) -t ( B . t ) in C / B t such that
Soa it is easy to check (c. (C, 6)) is a coequalizer of
in C / B , which is preserved by t!. Clearly, s! reflects split forks into coequalizer
diagrams and p* reflects isomorphisms. Hence. by Theorem 1.4.4. p* is monadic. O
2. Preliminaries for Descent Theory
In this chapter. me recall the basic notions and results of descent theory which
niIl be used in Chapter~ 3 and 4. They are: fibrations and descent theory. the
Beck-Chevalley condition: interna1 categories and their actions. indesed categories
and descent theory: some theorems of Janelidze and Tholen.
2.1. Fibrations and Descent Theory
This section is devoted to the presentations of the fundamentals of fibrational
descent t heory.
2.1.1. Let P : E -+ C be a functor and p : E i B a morphism of C. The fibre of P
at B is the non-full subcategory E(B) of E ahose objects are in P L B (i.e.. those
objects -4 of E Mth P A = B) and whose morphisms f : .4 - -4' are E-morphisms
such that P f = lg. Let S E E(B). a morphirm d,S : p8S i S of C is a cartesian
lifting over p ut X if
C2. for an- morphisrn z: : 1' -t X of E and any morphism h : P Y + E in C
satisfying ph = P u , there is a unique zc : Y + p8S in E such that
ô,S . w = v and Pw = h.
X functor P : E + C is called a fibration if for any rnorphism p : E i B in C
and every object S in E(B) there is a cartesian lifting ( p ' S ' 3,S) over p at S.
Examples. 1. For every category C. the identity functor Ic is a fibration.
3. If C is a category with pullbacks. and if C2 is the rnorphism categop of C.
so that
- objects are morphisms f : E + B in CI
- morphisms from f : E i B to f' : E' + B' are pairs (u. c ) of morphisms in C
such that f'u = z. f: where u : E + E' and ü : B + B' are C-rnorphisms:
then the codornain functor
is a fibration, called the basic fibration of C : for any morphism p : E i B in C and
an object (x : ,Y + B) in ZL(B): the following pullback diagram:
yields a morphism (dPdy: p) : ( j : d'S + E) i ( x : ,Y + B) in c'. n-hich is a
cartesian lifting over (p : E + B ) at (x : S -t B ) .
On the other hand, El is an opfibration (Le.. aQD iis a fibration). In fact. for any
morphism p : E + B in C and any object (I : S + E) in 8- ' (E) . an opcartesian
lifting over ( p : E -t B) at ( x : S -+ E ) is given by
Hence a is a bzfibration.
3. Let MOD be the category defined as follon-s:
- An object of MOD is a pair (R. M)! where R E CRng,. A4 E Mod-R.
- .A rnorphism (1. u ) : (R? M) + (RI' .LIr) has f : R' + R a morphism of CRng,
and u : .\.I + JIr Bfl R a rnorphism of Mod-R.
- If (f. u ) : ( 1 ) + ( R I r ) (g, v) : (RIt LW') i (Ru: ll") are morphisms of
SIOD: t hen
(g: v) 0 (/, u) : (R. .If) + (Rn, .LI")
44
Then the hnctor mod: MOD -t (CRng,)OP given by
( R ) ct R
is a fibration. In fact. for any unital commutative ring homornorphism f : R + S
and any (R. M ) E rnod-'(R), (f! liLIORS) : (S. Al aR S ) -f (R: M) is a cartesian
lifting over / at (R: M ) .
4. Let F : Top + Set be the forgetful functor. Then F is a fibration: for any
morphisrn p : E + B in Set and for B E Top. if E is equipped with the coarsest
topologp nhich makes p : E + B continuous? then p : E -t B is a cartesian lifting
over p at B. F is also an opfibration: for any morphism p : E + B in Set and for
E E Top, p : E -t B is a cocartesian lifting over p at E if B is equipped with the
finest topology nhich makes p : E -r B continuous. Hence F is a bzjibratiorr.
2.1.2. If P : E -t C is a fibration and p : E + B is a morphism in C' then n-e have
the inverse-image functor
p . : E(B) -t E(E)
gven by -4 H p'A and obvious assignments on morphisms, and a cieavage
6, : JEpm + JBt
45
where JB : E(B) + E is the inclusion functor and Pap = A p : AE i AB is the
constant natural transformation. Hence one gets a pseudo-junctor
CAT
which is given by
since t here are the uniquely determined natural equivalences
such that
and
for any p : E + B and q : .Y i E in C by the definition of the cartesian lifting.
This means that ( ) * : COp + C.4T is a C-indesed categop- (see 2.2.3).
For example, the basic fibration yields the basic C-indesed catego- giren by
the sliced categories and pullback functors (see 2.2.3). On the other hand. given a
C-indesed category A : C a p + CAS: by the Grothendieck construction, one may
construct a fibration (see [4] or 2.2.3 below). Xny C-indexed category essentially
anses in this way (see [NI).
2.1.3. Let C be a category with pullbacks, p : E -t B a morphism in C and
P : E + C a fibration. Descent data (C, f) for C E E(E) (relative to p ) are given
by certain morphisms ( : p;C + p;C in E ( E x~ E ) such that
cornmutes at C, and
cornmutes. nhere ( p l , p2) is the kernel pair of p and
T = C P 1 7 i l : p2T2 >: ( E X B E ) X E ( E X B E ) + E X B E
is given by the following pullback diagram:
: C + piC is such that Pb* = b and 1 9 ~ ~ C - 6 , = lc (6' : E -+ E x B E is the
"diagonal" ) the canonical isomorphisms j and j* arise from the identities
These descent data. with morphisms h : (C. c) + (Cf. <') given b~ rnorphisms
h : C -t C' of E(E) such that
cornmutes, form the category D e s ~ ( p ) .
For any -4 E E(B), p 4 A is provided nith canonical descent data
Hence p* can be Iifted to the cornparison functor @ p such that
cornmutes.
p is called an E d e s c e n t (effective E-descent) morphism if W is full and faithful
(an equivalence of categories) .
2.1.4. Let P : E + C be a bifibration (i.e., both P and Pop are fibrations). Then,
dually to the inverse image functor p* and the cleavage t J p . ive have a direct image
functor
and a cocieavage
dp : JE + J B p ! .
Hence t here exist uniquely determined natural transformations
which serve as unit and counit of the adjunction p! 4 p'. Furthermore. we have the
Beck transfomation
(P?)!P; P'P! ~ q t h ( J E J ~ ) ( ~ ~ ~ ~ P ; ) = ( J E P ~ ) ~ ~ ~ .
One says that P satzsfies the Beck-Chevalley condition /or p i l 3, is a natural
equivalence.
Example. The basic fibration d : C2 + C (see Example 21.1 (2)) satzsjies the
Beck-Checalley condition for any p : E -t B.
49
In fact, for any (x : S + E ) E 8-l ( E ) :
(p?)!p;(x : X -t E ) = (p2crl : E X B ,Y + E )
is given by the following pullback diagrams:
On the other hand.
is @en by the following pullback diagram:
Clearly, there is an isomorphism (p201 : E x B S + E ) n (5, : E x B S i E ) .
Given E-descent data 5 : pîC i p;C for C E E(E): one has a bijective corre-
spondence (< tt (e)*) by the universal property:
If P : E + C is a bifibration satisbing the Beck-Chevalley condition for p:
then 3, is a natural equivalence, and so it yields a bijective correspondence between
(<)" : (p2)!p;C + p'p!C and the algebra structures 0 : pap!C i C trith respect to
the monad induced by p! i p*:
This essentially establishes the bijective correspondence (< tt 8). Hence:
Theorem. (,Bénabou-Roubaud [SI: Beck) For E bzjïbred over a category C tcith
pullbacks and for p : E -+ B in C such that the Beck-Chevailey condition holds
jor p. DesE(p) zs zsomorphzc to the Eilenberg-Moore category of the monad giuen by
p! 4 p'. Hence p is an (effective) E-descent morphism iJ and only i fp ' is prernonadzc
(monadic) .
2.2. Int ernal Categories, Actions, Indexed Categories, and Descent The-
ory
As rnentioned in Chapter 0: the category of descent data relative t o a morphism
p can also be defined in the contevt of a C-indexed category A. namely the category
of actions of the equivalence relation Eq(p) induced by p on k It is convenient
to work with indesed categories instead of fibrations in some cases [see [19]). In
rhis section. ive summarize the notions of interna1 categories. actions. and indered
categories and sorne related results. Throughout this section. C denotes a category
with pullbacks.
2.2.1. - In intemal categoq D in C is given by a diagam
which satisfies
I l . de = ID, = ce.
12. dm = dx2, cm = ml,
13. m(lD, x m) = m ( m x ID,):
14. rn < ID,: ed > =Io, = m < ec. ID, >:
mhere Do, Di E obC, d , c, e, m E MorC, and D2, T I , ?i2 are given by the folloning
pullback diagram in C:
.-ln interna[ junctor f : D -t D' between two internal categories D. D' in C is
$ven by two morphisms fo : Do + DO: f l : Dl i D: of C such that
Composition of internal functors is given by the obvious way. Hence we obtain
cat(C)-the category of al1 internal categories and internal functors in C. cat (C) is
actually a 2-category (see [19]) since one can define an internal natural transforma-
tion a. : / + g of internal functors f: g : D i D': mhich is given by a morphism
a : Do + Di in C such that
Let /? g, h : D D' be internal functors and let cu : / -i g and i3 : g i h be
internal natural transjonnation. one defines the composition 3a : f + h to be the
morp hism
m'< 9, (Y >: DO -t D::
and the identity interna1 natural transfomation lf : f -t f to be the morphism
effo : Do i Di.
An intemal functor f : D i D' of C is an internal categoy equivulence if there
is an internal functor g : D' i D such that
gf 2 l D and f g 2 lDt.
For esample. if p : E i B is a morphisrn in C. then
is an internal category in C' where e =< lE: l E >' (aL' Q) is the kemel-pair of p. q2
This inremal category is denoted by Eq(p). If B E ob(C), then B can be viewd as
a discrete internal category B of C:
Clearly Eq(le) is isomorphic to the above discrete internal categop B.
2.2.2. Let D be an internal category in C. An action of D in C is given by a triple
(C. 7. E) such that
where C E obC, :J : C + Do and < : Dl x~~ C i C are morphisms in C. and
Dl x ,, C is gven by the pullback diagram
.Ilorphisms h : (C. ni- <) i (Cf. r'. (') between two actions of D are giren by
C-rnorphisms h : C i C' over Do (-i'h = 7) such that
A11 actions and morphisms between actions of D form the category cD- actions
Example. Let D be an internal category in C. One defines functors
F ~ ( C J ) = ( D l x ~ , C,c- proj,,m xo, 1) for each (C.:;) E obC/Do.
and the obvious assignments for morphisms. Then FD is a left adjoint of CD. By
Theorem 1.2.4 (3): CiD is monadic.
2.2.3. A C-indezed catego y A is a pseudo-functor h : CoP + CAT such that for
e v e n / : E i D: g : D + C in C there are natural isomorphisms:
which sat isfy t hat
and
commute, where = A(D) and x* denotes A(z) : AD -t k iE for any morphism
x : E + D i n C .
For example,
A : COP + CAT
given by B c+ C / B and (/ : E -+ B ) ci f ' : C / B 4 C I E . the puilback functor
along f , is a C-indeued category, and ne cal1 it the basic C-indesed category.
Grothendieck Construction. Given a C-indexed category A : COP i CAT. one
defines a category G(C7 A) as follom:
- An object of G(C: A) is a pair (C. z). where C is an object of C and r is an
object of h(C).
- .A morphism (f, u) : (C. z) -t (Cf. x f ) consists of a C-morphism f : C i C'
and a A(C)-morphisrn u : 2 -+ A( f )(x').
- If ( f 7 u ) : (C. 2 ) i (Cl! 2'): (g, L') : (Cf: y ) i (Cu. I") are morphisms of
G(C. A): then
(g: U ) 0 (f' U) : (C. x) + (Cf" 1")
(9. 4 0 (f. 4 = ig /! A(f)(c) 4.
The projection functor P : G(C: A) -t C given by
is a fibration. This process is called the Grothendieck constmction (see [?]).
Clearly, mod: MOD -+ (CRngJOP is given by the Grothendieck construction ap-
plied to the (CRngJ0~-indexed category R ~ M o d - R , in Example 2.1.1(3). Hence
it is a fibration.
2.2.4. Let A : COp i CAT be a C-indesed category and D an interna1 category in
C as in 2.2.1. One defines AD to be the category -+th
- objects: pairs of (C. ,C). shere C E obkiDO and < : dmC i c8C is a morphism in
AO! such that
and
cornmute, in f i and ~~2 respectively. The above natural *2'"s arise from I l and
12 in 2.2.1 and Definition 2.2.3.
- morphisms: h : (C, <) + (Cf- cf ) of are g i~en by morphisms of h : C -t Cf of
A ~ O such that d'h
d'C - d*Cf
cornmutes in AD' .
2.2.5. Janelidze and Tholen (191 proved:
Theorem. For every C-indexed category W : COP i CAT. the u t e n s i o n
given by the assignment D -i AD. is a pseudo-functor 012-categon'es.
They also proved:
Lemma. For every C-indezed category A. and every interna1 category equicalence
f : D i Df of C . the junctor f' : A ~ ' i is an eçuicalence O/ categories.
2.2.6. Let A be a C-indexed category and p : E i B a morphism of C. The
category
DesA@) = flEqbi
is called the category of A-descent data relatiue to p.
a9
Since the discrete functor p : E + B c m be factored as
p, we have a commutative diagram (up to natural isomorphisms) in CPIT:
p is called an A-descent (eflectiue A-descent) morphism if the cornparison functor 4 p
is full and faithful (an equivalence of categories). p is called an absolutely (efect ice)
descent rnorphzsm if it is an (effective) A-descent morphism for every C-indesed
category A.
For A defined by a fibration P : E + C? the category DesA(p) is the category
DesE@) in 2.1.3. and the notion of (effective) A-descent is equivalent to the notion
of (effective) Edescent in 2.1.3 (see [19]: Section 3.3. for reasons).
Janelidze and Tholen [19] also showed:
Theorem. -4 morphism in a categoq uri-th pullbacb zs an absolutely effective rnor-
phzsrn if and only i/ it is a split epimorphzsrn.
3. Descent Equivalence
Throughout this chapter, C is a c a t e g o ~ with pullbacks. and A : COP i CAT
is a C-indeued category. Let B E obC, for a given morphism q : [E? p ) i (A-' ;)
in C / B , one may ask:
Which conditions can guarantee that bundles p and 9 have the same A-descent
structure?
In this chapter! we give the notion of descent equivalence and study its properties.
3.1. The Functor Eq
3.1.1. For any rnorphism q : ( E : p) -i (S: ;) in CIB. Janelidze and Tholen [19]
constructed the follo~ving commutative diagram in cat(C):
They called it t h e basic equiualence diagram (BED) induced by t h e morphism
Q : ( E , p) -t (,Y, y). They proved:
Proposition. Each BED is a pullback diagmm in the ordznary category cat (C) , and
it is also a pvshout diagram in cat(C) if q is a pullback-stable regular epirnorphzsm
Recall that for each morphisrn / : D + C in a categon a i t h binary products
and for each object X in the category.
and
are pullback diagams.
Since products in C j B are given by pullbacks in C. we have:
3.1.2. Lemma. If p : (E ' p) -t (S. 7) is a morphism of C / B . then. for any
(1- y) E ob(C/B),
is a pullback diagram, where ~1 and ;il are giuen by the follouiing pullback diagrams:
and
& also a pullback diagram.
If
is a commutative diagram in C! then we have the following commutative diagram
in cat (C):
where (%)* = y, (çJl = y x , y which is given b-
here the front face and the back face are pullback diagrams in C' and the ot her faces
are commutative.
Note that the morphism y x , y can be decomposed as
Proposition. 1. If (3) zs a pullback diagram. then
(i) the left face and top face O/ ( 4 ) are pullback dzagrams 2.n C.
(ii) (3) zs a pullback diagram in cat (C) .
2. lf y is a regular epirnorphzsrn and y x, y zs an epirnorphism. then the left face
of (4) zs a pushout diagram in C .
iCloreover. if (2) zs a pushout diagmm in C . then (3) is a pushout diagram in
cat (C) .
Proof. 1. ( i ) If (2) is a pullback diagram, then. by the pullback diagram composition
lai-: in diagram (4, the left face + the back face = the front face + the right face
is a pullback diagram. But the back face is a pullback d iapam. so. by the pullback
diagram cancellation law, the left face of is a pullback diagram. Sirnilarly. the top
face of (4) is a pullback diagram, as desired.
(ii) Let f : FI7 -t Eq@), g : I.V -t S be rnorphisms in cat(C) such that
m f 1 = P& = xgt-
Hence there are a unique morphism ho : CVo -+ Y such that
ph0 = fo; (?ho = 90 :
and a unique rnorphism hl : Wt + 1' x x Y such that
It is easy to check that (ho: h l ) : D i Eq(q) is a morphism of cat(C). Hence (3) is
a pullback diagram.
2 . Let w i : Y i R.' and w2 : E x B E -r II* be morphisms in C such that
yu1 = yu2 ' L'tu1 = îL ' !U2.
There are u, u' : LT + Y x x Y such that
But
= 'U'lU2.
Since y is a regular epimorphism, there is a unique u: : E i II- such that
On the other hand,
- wn1(y x, y) = myTl = z q ~ l = w2(y x r y).
Hence W'IC! = u.-2: as desired.
Let f : Eq(p) -t D. g : S + D be morphisrns in cat(C) such that
/cy = 94-
If (2) is a pushout diagram, then. there is a unique ho : B -t Do and a unique
hl : B + Dl such that
and
Since y is an epimorphism: x is an epimorphism by the pushout stabilitr of epimor-
phisms. X o s it is routine to check that cho7 hl) : B + D is an interna! functor.
Hence (3) is a pushout diagram in cat(C). @
For a morphism q : (E. p ) + (S: p) in CIB: by considering the commutative
diagram:
with (3) we get the commutative diagram:
Moreover, one can also look at
3.1.3. Let C be a category +th pullbacks. Recall that for a morphism p : E -t B
in C we have the interna1 category Eq(p) (see 2.2.1). Then. for a fised object B of
C: the assignrnents:
( E . p) * &(P) and (q : ( E , P) -t (X, 4) * @:
define a functor:
E q : C / B i cat(C).
3.1.4. Proposition. Let C be a category with pullbacks and B a fked object of C .
Thm the functor
preserues those limits which are preserved by the forgetful functor CIL3 i C.
Proof. Let p : E + B be the limit of p, : Et i B in CIB:
such that lim Ei = E. We claim that Eq(p) = lim Eq(pi) in cat(C).
For any I-cone ( u i : ci) : D + Eq(pi) in cat(C): since Iim E, = E. there is a
unique x : Do i E such that
Sirnilarly, since lim Ei x B Ei = E x B E, and lim Ei x B Ei x B Ei = E x E x E:
there are a unique morphism y : DI + E xB E with
and a unique rnorphism
for al1 i
Note that. for ail il
and
~ l y = XC, e z = yf: and .;;?y = zd.
I l
Similady, we can check that
cornmutes serially. Therefore lim Eq(p,) = Eq@) in cat(C).
3.1.5. Corollary- Let C be a categoy with pullbacks and B a Jxed object of C.
Then
preserves pullback diagrams and znoerse lirnits.
Proof. It is easy to check that the forgetful functor C I B i C preserves pullback
diagrams and inverse limits. By Theorem 3.1.4: the Corollary is clear. t3
3.2. Descent Equivalence
3.2.1. Let C be a c a t e g q with pullbacks and A: COP i CAS a C-indesed
category. one has a pseudo-functor
A : cat(c)OP -r CAS
of -categories, rvhich extends the C-indexed category A : C O P i CAT (see 2 . 2 3
- r2
or [19]). Recall that for a morphism p : E -t B in C, one defines
Hence, for any fixed object B of C: DesA( ) = AoEq becomes a pseudo-functor:
cat ( C ) O P
Proposition. Let A : COP + CAT be a C-zndezed cutego y such that
A : cat ( C ) O p -+ CAT preserves pushout diagrczms (directed coZirnits).
Then so does DesA( ).
Proof. By Corollary 3.1 3, Eq: (Cf B ) O P i (cat ( C ) ) " P preserves pushout diagrams
(directed coiimits) . so does
Desa( ) = A 0 Eq.
For a morphisms q : (E : p ) + (S. ;) in C/B, Reiterman. Sobral. and Tholen
[96] considered the following diagram in cat (C):
Applying A to the above diagram ( 7 ) ? one obtains the following commutative
diagram (up to natural isomorphisrns) in CAT:
where CF = (bx)*. 1 - = ( i ) @P = @)*. and Desr(q) = (g)'.
3.2.2. Definition. Let q : ( E . p ) i (S. 7 ) be a morphzsm in CIB. W e c d 1 q
an A-descent equivalence (Adescent preequivalence) zj Des&) is an equivalence
of categories (full and faithful). W e cal1 q an absolute descent equivalence (absolute
descent preequivalence) if DesA(q) Zs an equivalence of categories (full and faithful)
for every C-indezed category A
By the functoriality of DesA( ), immediately we have:
(1) I j two of q: s, and sq are A-descent equiualences. so is the third one.
(2) If s is an A-descent pre-equivalence. then q is an A-descent pre-equiwlence i j
and only if sq is a n A-descent pre-equivalence.
The following result shows us why A-descent (pre-)equivaIence is a useful notion.
Invariance Theorem. Let q : ( E , p) i (S: 9) be an A-descent pre-equivalence
(A-descent equivalence) in CI B. Then p is an A-descent (effective A-descent) mor-
phism if and only if 9 is an A-descent (eflective A-deseentj morphisn.
Proof. By Diagam 3.2.1 (7): Desn(q)9* = W (up to natural isomorphism). If q is
an Adescent equivalence. then Des&) is an equivalence of categories and therefore
is an equivalence of categories if and only if @P is an equivalence of categories.
Hence p is an effective kdescent morphism if and only if ,- is an effective Adescent
morphism.
Suppose non- that q is an A-descent preequivalence. Then Des&) is full and
faithful. If p is A-descent pre-quivalence morphism, then OP = Dese(q)W (up to
isomorphism) is full and faithful. Hence p is an A-descent morphism.
On the other hand, if p is A-descent morphism. then DesA(q)Oz = 9 p (up to
isomorphism) is faithful and so is W .
For any Bi, Bz E A* and any morphism h : W ( B l ) i W ( B 2 ) in DesA(S' q) :
DesA(q) (h) : D e s A ( q ) W ( B I ) + D e ~ ~ ( q ) O p ( & ) is a morphism of D e s ~ ( p ) . Since the
following diagram
cornmutes (up to natural isomorphism), tvithout loss of generality ive may a; -sume
DesA(q)9;(Bi) = W ( B , ) ? i = 1 2 . Since OP is full. there is a morphism 1 : B1 -t B2
of such that
QP(f) = Des*(q)(h).
That is
Des,(q)Wf = D e s ~ ( d ( h ) *
Since Des:,(q) is faithful. @Y([) = h. Hence 9' is full. Then is an A-descent
pre-equivalence. a
3.2.3. Absolutely effective descent rnorphisms are precisely the split epimorphisms
(see 2.2.6 or (191). However, for absolute descent equivalences, ive have:
Sheorem. Let q : ( E , p ) + ( X , 7 ) be a rnorphism in C I B .
1. IJq is a split epimorphwm in C' then p zs an absolute descent equzz~alence.
2. If q zs a split monomorphzsm in CIB, then q zs an absolute descent equiüalence.
Proof. 1. Suppose qs = lx for some morphism s : X -t E in C. Shen. clearly,
cjS = lEq(;). On the other hand. we claim frG S lEq(vi and the internai natural
transformations
are given bu
a = < IE? sq >: 3 = < sq, I E >: E - r E x g E in C.
By Definition 3.3.1:
and
Since
and
me obtain
7i13 << SQ, l E >: < 1 ~ : sq >>= esq
Hence 3a = lj4 : Sq -t $4 and a8 = llEq(pl : lEq(Pl + lEqbP Then the interna1
functor 6 : Eq(pj + Eq(9) has an inverse S : Eq(9) i Eq(p!. Therefore. by
Lemma 2.2.5. Desa(q) is an equivalence of categorier.
2. Since q is a split monomorphism in C/B. there is a morphism s in C/B such
that sq = 1. By 1' s and 1 are absolute descent equivalences. Hence. q is an absolute
descent equivalence by Proposition 3.5.2(1).
Recall that a morphism in Set is a split epimorphism if and only if it is surjective.
Hence, in Set. epimorphisms? elqremal epimorphisms. regular epimorphisms. uni-
versa1 epimorphisms, and split epimorphisms are al1 the same. Recall also that Set
has the (epi, mono)-factorkation system. Then, in Set. a $yen morphisrn p : E i B
can be decomposed into m e with a monomorphism m and an epimorphism e:
Since e : ( E , p ) + (S, m) is an absolute descent equivalence, we have
Hence. in Set. we need only consider the computability of DesA(p) with monomor-
phisms p.
However. an absolute descent equivalence does not need to be a split epimorphism
or a split monomorphisrn. See the following:
Example. Let B = {bl' b?}. 1,. = {y}. and S = {zl' z2}. U e define
and
Then
Hence q : (Y:p) -t (X', p) is a split monomorphism in Set lB .
For any morphism f : E + 1- with IEl 3 2 in Set. f is an epimorphism, so
q f : ( E , p f) + (,Y, 9) is an absolute equivalence. But q f is neither a n epimorphism
nor a monomorphism in Set.
3.2.4. Recall that a functor F : B -t D is called essentially surjective if for each
object D of D there is an object B of B such that FB r D.
Theorem. Let q : (EI p ) -t (S. 9) be a morphism in C / B . lfDesA(q) is essentially
surjective for every C-indexed categoy A. then there is a rnorphism s : S i E in
C such that
In particdar. if p is an absolute descent equivalence. then there is a morphism
s : X i E in C such that
Psq = P.
1. If p is a rnonomorphzsm and if q is an absolute descent equivalence. then q is a
split monomorphzsm.
2. I f ;. is a rnonomorphisrn. then there zs a rnorphzsm s : ,Y i E in C such that
Proof. We define a C-indexed category 4 as folion-s:
c 0 p 4
CAT
where $ = C(S: E) carries an equivalence relation given by
and t' : C()'- E ) + C(X: E) is the composing functor rvith t . Since
the object 1~ of %E hm a descent structure < : ii;(lE) -i ;T&). where (si. 3) is
the kernel pair of p. Hence?
But Des), (q) is essentially surjective. so there is (g, p ) E Des:, (S. ;) such t hat
and t herefore
That is
O V ( g : p ) ( l E : <').
But is just a lifting of q' (see 2.2.6).
q8L;'(g, p ) 6'0qC'P(g: p ) 2 b ' ( l E : f ) .
Hence there is a s E = C(S. E) such that
and t herefore
as desired.
1. If q is an absolute descent equivalence. then Desn(q) are always essentially
surjective for a11 ih's. Hence there is a morphism s : X + E in C such that psq = p
and therefore sq = 1 since p is a monomorphism.
2. Since p = ;q: r q s q = psq = ;q.
3.2.5. In what follows we inrestigate the relation between th-descent pre-equivalence
(A-descent equivalence) and A-descent (effective A-descent ) .
Proposition. T h e morphism p : ( E , p) + ( B : lB) in C / B 2s an A-descent
pre-equiualence (A-descent equiuaLence) if and only if p 2s an b d e s c e n t (effective
A-descent) rnorphisrn.
Proof. Applying Eq to the followïng commutative diagram:
we obtain the commutative diagram:
A@ is an equivalence of categories o is an equivalence of categories.
as desired.
Remark. From a theorem of Reiterman. Sobral. and Tholen [26], ne have the
following:
Let E : B ct IE(B) be the subfibrution of the basic fibration O/ a category C @o.en
by a pullback-stable class E of rnorphisms in C! and q : (E' p ) i (S. ;) a morphism
83
in C/B. I j g is a pullback stable regular epimorphzsm of C , then q : E i S zs an
Edescent (effective Edescent) morphism in C implies q : ( E , p ) i (S: q ) zs an
Edescent pre-equivalence (%descent epuzvalence).
By the above Rernark and Theorem 3.2.2. immediately we have the following:
Proposition. Let q : ( E . p) i (S. 9) be a morphisrn in a category C u ~ t h
pullbacks. and let IE : B ct IE(B) be the subfibration of the basic fibration of a
category C gzven b y a pullback-stable clms E of morphisrns in C . Then the foilowing
are equzvalent:
1. q: E i S is an E-ciescent (effective Edescent) rnorphism in C:
2. For each morphism z : S i B in C. q : ( E . xq) i (S: I) 2s an E-descent
pre- equinalence @-descent equiaalence) morphism:
3. For each morphism x : X i B in C . xq zs an E-descent (effective adescent)
rnorphism in C if and only if z zs an E-descent (effective IE-descent) morphism in
Saturally, the following question may be interest in;:
Question. For which C-indexed categories A. is an A-descent (effective A-descent)
morphism q : ( E . p) -t (S' 7) also an A-descent pre-equivalence (A-descent eguia-
aience) ?
Non. Ive corne back to the commutative diagram (5):
in C I B . which. as we already sam. induces the following commutative diagram ( 6 )
Applying h to the last diagam, we get the commutative diagam (up to isomor-
phism) in CAT:
Clearly, an answer to the above question as well as pullback stability of (effective)
bdescent morphisrns largely depend on the top face and the front face of diagram
! 9). respect ivelu.
3.2-6. Reiterman. Sobral, and Tholen [26] proved the followïng composition and
canceIlation t heorem:
In a category with pullbaclis. both the class 2) of descent morphisms and the
class E of effective descent morphisms with respect to the basic fibration satzsfy the
jollowzng composition-cancellation mie: giuen a composite p = ; - q . then
i f q E V , t h e n p ~ D o ; E 2)
and
Let ;7 and q be two morphisms of C such that v - q is defined. If p = 9 q is a
86
descent (an effective descent) morphism Nith respect to the basic fibration. then we
form the following pullback diagram:
Since si is a split epimorphism, 7il is an effective descent morphism. On the other
hand. by the pullback stability of descent (effective descent) morphisms [29]? h2 is
a descenr (an effective descent) morphism. Then. by the the above composition-
cancellation rule. q and ; are descent (effective descent) morphisms. and so tve have
the follotving proposition.
Proposition. Let p and q be two rnorphisms of C J U C ~ that 9. q is dqîned. Then
F q is a descent (an effectaue descent) morphism W t h respect t o the basic fibration
if and only (1 q and q are descent (eflective descent) morphisms.
4. Purity and Effective Descent
h theorem of Joyal and Tierney [XI states that pure morphisms are effective
descent in the opposite category of commutative monoids in sup-lat t ices r i t h respect
to the fibration induced by the indexed category which assigns to each commutative
monoid -4 in sup-lattices an A-module. Slesablishvili (251 gave a proof that pure
morphisrns are effective descent in the opposite category of commutative unital rings
with respect to the fibration induced by the indexed category R H Mod-R for each
commutative unital ring R. In this chapter, we want to study the relationship
between purit? and effective descent.
4.1. Pure morphisms are effective descent for modules
Recall that for a unital ring homomorphism f : R i S. we have the adjoint pair
f. i f! (see 12.3):
In this section. our objective is to answer the folloning question:
For which morphisms .f : R + S of unital rings: are f. comonadic?
4.1.1. Recall that Q/Z is an injective cogenerator for Ab. If .Il is a ripht R-module,
85
then it is a bimodule &IR. Hence the left R-module structure on hornz(M: Q/Z)
is given by rf : m H f ( m ~ ) . One calls this left R-module the character module of
M: and denotes it by CR(M). Hence, one has the representable functor
Similarly. one has the functor:
CR : (RM0d)Op + Mod-R.
4.1.2. We need the following propenies of the above functor CR:
Proposition. 1. CR zs jaithful.
is exact zj and only i j
zs exact.
3. CR is exact.
1. CR presemes and rejiects coequalzzer dzagrams.
89
Proof. 1. Since Q / Z is a cogenerator of Ab.
2. See [28]: p.87, Lemma 3.51.
3. Clearly CR is additive. By (2) and 16, ~01.2: p.50, Proposition 1.11.31, it is
clear.
4. Since CR is exact' CR presenres coequalizer diagrarns. Clearly. CR reflects
isomorphisms. Hence. by [l5, Theorem 24.71. it also refiects coequalizer diagrams.
In commutative unital rings. B. Mesablishvili [Z] shorved that CR( f) is a split
epimorphism in R-Mod for each pure morphism f of R-Mod. As lIesablishrili did
in [X]. ive have the following:
Lemma. Let .\Il: -1- be h o R-birnodules. Suppose that f : JIl i .I& is a pure
morphism of R-Mod. Then
1. C R ( f ) is a split epimorphism in R-Mod.
2. The natural transformation
Proof. 1. By adjointness of 8 and hom. one has the follou-ing commutative diagam:
90
in nhich both vertical morphisrns are isomorphisms. Since J : .\.Il i .Cf2 is pure in
R-Mod, 1 8 f is a monomorphisrn. Since Q/Z is injective in Ab. CR( f 3 1) is an
epimorphism, so is h ~ r n ~ ( C & \ f ~ ) ~ CR( f )). Therefore IC,(,~,) produces a retraction
for CR ( / ) .
2. Clearly a is a natural transformation. Again. by adjointness of 8 and hom,
one has the follo~ving commutative diagram:
for each E Mod-R.
By 1: C R ( f ) has a right inverse, Say h' which gives a right inverse of a,lf and a
is natural in M.
4.1.3. Theorem. Let f : R i S be a rnorphzsm of rings. Then f. is comonadzc if
and only zf f is a pure rnorphism in R-Mod.
Proof. +: By Proposition 1.2.3.
e: If f : R + S iç a pure morphism in R-Mod, the unit 7 : ~ : .V i N BR S
of f. -i f! : Mod-S -+ Mod-R is a monomorphism for each !Y E Mod-S, then f.
is precomonadic. For any ( M , 8) E (Mod-S)jaf!, consider the following equalizer
diagram:
in Mod-R: where i : 3 = {m E AI, O(m) = vzr(m)} + M is an inclusion map, and
a split fork:
in Mod-R. Then we have the following commutative diagram
for some morphisrn 1 : Y + Ad in Mod-R. Applying CR to the abore, ive get the
following commutative diagram in R-Mod:
in which the bottom row is a coequalizer diagram and the top row is a split fork. If
and CR(qM) (Y CR(libf 8 f)) have right inverses s, t respectively, such that
cornmutes serially. Then there is a morphism k : CR(X) + CR(-\[) in R-Mod such
t hat
Since the top row in diagram (10) is a split fork. by an easy diagram chasingo
the bottom row:
is also a split fork. Applying the functor hOmR(S, -) to (11) and by adjointness of
8 and hom. we get the follonring split fork:
By Proposition 4.1.2 (4):
is an equalizer diagram. Hence, by Beck's Theorem, f. is cornonadic.
4.1.4. Let B be a subcategory of Rng, such that
(*) CRng, C B and for any morphzsm f : R + S of B. f i s descent (effective
descent) mith respect t o the fibration induced by Bop-indezed category R c, Mod-R
if and only if f. is precomonadic (comonadic).
By the same process as one applied in the proof of Theorem 4.1.3. ive have the
following:
Let f : R + S be a rnorphism in B. Then f is effective /or descent uith respect
tu the fibrution induced by Bop-indexed category R ci Mod-R i/ and onfg if/ is pure
rnorphisrn in R-Mod.
Clearly, letting B = C h g , in the above, by [6. vo1.2. Theorem 4.7.41. . . n e obtain
the t heorem of Mesablishvili [XI.
4.2. Locally Presentable Categories, Accessible Categories, and Pure
Morphisms
This section collects sorne basic notions and results of locally presentable care-
gories, accessible categories, and pure morphisms. It is the preparation for the nest
section.
4.2.1. Recdl that a non-empty poset is directed if each pair of its elements has
an upper bound, and directed colimits are colimits of diagrams whose schemes are
directed posets.
Definition. An object K O/ a category K zs called finitely presentable if
hom(K, -) : K -t Set
preserues directed colzmits.
For euample, an object K in Set is f i i t e l y presentable if and only if IKl < No.
.An object G in Grp is finitely presentable if and only if G is isomorphic to the
quotient g o u p of the free group F(q: - - -:z,) on n generators modulo a congruence
generated by finitely many equations on F(21: - - ? z n ) The g o u p (2. 0) of integers
is finitely presentable but the group (R, +) of the real numbers is not (see [?]: p.10).
Definition. A category K is called locally finitely presentable if it zs cocomplete
and hm u set S oj f inz te ly presentable objects such that every object is a directed
colimits of objects !rom S.
For esample, Set, Poso and Grp are locally finitely presentable categories.
Theorem. E v e v uariety oJfinîta y algebras zs a locally Jinitely presentable category.
Proof. See [2]' p.141. O
4.2.2. Throughout, X is a regular cardinal. Recall t hat a poset is A -directed if eveq
95
subset of cardinality smaller than A. has an upper bound and A-dzrected colimits are
colimits of diagrams whose schemes are A-directed posets. .An object K of a category
is called Xpresentable if hom(K, -) preserves A-directed colimits. An object is
called presentable if it is A-presentable for some A.
For example, an object in Set is A-presentable if and only if it has cardinality
srnaller than A (see 121, p.19).
Definition. A category is called locally A-presentable 2j i t zs cocomplete and has a
set S of A-presentable objects such that euery object is a A-dzrected colimits O/ objects
from S. -4 category is called locaily presentable if it i s locally A-presentable for some
A.
For example. locally No-presentable categories are precisely the locally finitel-
presentable ones. The catego- Ban of comples Banach spaces and linear con-
tractions is locally NI-presentable (see [2]? p.10. Example 1 &). But the self-dual
category Hi1 of Hilbert spaces and linear contractions is not locally presentable (see
[2], p.51, Example 1.65 (2)).
4.2.3. Definition. A category K zs called A-accessible category i j K hm A-directed
colzrnits and has a set S O/ A-presentable objects such that e u e q object in K is a h-
dzrected colimits of objects from S. -4 categojy i s called accessible i f i t is A-accessible
category for some A.
For example, even locally X-presentable category is X-accessible category. The
category Hi1 of Hilbert spaces and linear contractions is NI-accessible (see [2] : p.70.
Example 2.3 (9)).
Recall that a morphism f : d + B in a category K is X-pure if for every
commutative square
in which -4' and B' are X-presentable objects. the morphisrn u factors through /'.
If K is a X-accessible category? then X-pure morphisms are monomorphisms. If K is
locally X-presentable, then X-pure morphisrns are regular monomorphisms.
Adaniek, Hu, and Tholen (11 showed:
Theorem 1. In a X-accessible category K u;ith pushouts. every X-pure rnorphzsm /
zs a X-dzrected colimit (in K2) of split monomorphisms with the same domain as /.
They also proved:
Theorem 2. In a X-accessible category K ~ z t h pushouts. reguhr monomorphisms
are closed vnder X-dzrected colzmits in K2.
4.3. Descent Theory in Accessible Categories
Let K be a X-accessible catego- with pushouts. In this section. we want to
study descent theory in KOP with respect to the basic fibration. By Theorem 2.1.4
and Example 2.1.4 (1): we have:
a morphzsm p : E -t B in KOP is descent (eflectiue descent) with respect t o the
basic fibratzon zf and only if p. i s precomonadic (comonadic) in the situation p. i p! :
Hence ive need to study comonadicity of p..
4.3.1. For a morphism p : B -t E of K: codescent data for ( K . r ) E ob(E/K) are
@yen by morphisms 0 : K + K +B E in K which make
and
commute, where (1) is a pushout diagram. These are objects of the category Des8(p):
a rnorphism h : ( K , r ; 8 ) + ( K t , f ; 6') of Des8(p) is a K-morphisrn which makes
commute.
The cornparison functor
given by O p ( L ? a) = (L i s E! Q: ol + 1) and the following pushout diagram:
-4 rnorphism p : B + E of K is called a codescent (an eflective codescent)
molphzsm if a, is full and faithful (an equivalence of categories).
Clearly. the category Des'(p) we defined above is the category of Eilenberg-Moore
coalgebrus given by p. 4 p!. Hence? by the dual of Beck's Theorem.
p is a codescent morphism in K
p, is precornonadic
e units of t h e adjunction p. -I p! are regular monomorphisms
e p is a pushout stable regular monomorphism.
Xow, let B E obK,
V B = {ail codescent morphisms in K a i t h t he domain B} y
and
EB = {al1 effective codescent morphisms in K with the domain B } .
4.3.2. Theorem. I/K is a A-accessible category urith pus ho ut^ then Dg is closed
Proof. Let pl : B i E, ( i E 1) be a A-directed d i agam in B/K. p, E DB: and
Consider the following pushout diagrams:
and
for any morphism z : B -t S in K. Taking A-directed colimit in the last pushout
colim = T L .
By Theorem 4.2.3.2: si is a regular monomorphism. Hence p is a pushout stable
regular monomorphism and therefore p E Dg. 0:
By Theorem 4.2.3.1 and Theorem 4.3.2. one has the following:
Corollary [2, Proposition 2.311. In a A-accessible category uith pushouts pure mor-
phisrns are codescent morphisrns and so they are pwhout stable regular monornor-
p hisms.
4.3.3. In [q Borger mentioned the following condition
(B): If eiel is a regular monornorphisrn. so zs e?.
Condition (B) does not hold in general (see [22] and [8] for CO
is true under some conditions (see [7]. 3.5(4) and Proposition
rvhen the category has (Epi, RegMono)-fact orizat ion systern.
u n t eresarnple) . but it
1.2(ii)). for esample,
Theorem. Let C be a category 2~2th pirshouts. If C satisfies ( B ) , then fg E 27
implzes g E D for any rnorphisms 1, g in C such that fg is defined.
Proof. Suppose g : -4 + B , f : B -t C are morphisms such that fg is defined in
C. For any morphism x : -4 + S in C: we take the follouing pushout diagrams (1)
and (2):
Then (1)+(2) becomes a pushout diagram and so
is a regular monomorphism since f g E DC. Hence 7 1 is a regular rnonomorphism by
(B) and therefore g E Dc. il
4.3.4. Sow we consider the effective codescent morphism case. Let K be a Iocally
A-presentable category and let (E. p) be the A-directed colimit of (E,. pl)tEl in
B/K in nhich each pi E EB. %y Theorem 1.3.2, p E II)*. For the effectirity of p. it
suffices to prore that a, is essentially surjective. We have:
Proposition. Let K be a locally A-presentable categorg and ( ( E . p ) . (t&lj a A-
directed colinrit O/ (&, in BIK in which each pi E EB. If (K' r: 8) E Desa(p)
can be vn'tten as a A-directed coliniit of (h;. ri; Bi) in Des8(p) .:-;ch that ail h; S
102
are A-presentable in K, then thereis (L, b ) E BIK such that
Proof. We distinguish the following cases:
Since colimi(K +B Et) = K +B E , there are jo E I and B, : K + K +g El for
al1 j 2 jo such that
cornmutes. It fûllows that
(1 + f j)Ojrt j = B+tj = r2tj = (1 + t j ) xz j :
P
By [2: Proposition 1-62], 1 + t j is a monomorphism. SO Ojrtj = ~ 2 j for al1 j 2 jo.
'ion. it is routine t o check (K: T t j ; 0,) E De~*(pj ) by looking at the following
commutative diagrams:
and
Since (K. T t j : d j ) E DesW(pj) and Qh is an equivalence of categories. there is
(L , , 1,) in B/K such that
QPj(Lj, i j ) ( K , r t j : G j ) ?
where ( L j , 1,) is defined by the following equalizer diagram:
104
in BIK.
Clearly, (Lj, lj)jao is directed. Taking the directed colimit in the above @es a
new equalizer diagram:
It follow-s that
CASE 2 . ( K . r; O ) can be witten as a A-directed colimit o j (h;! ri: Bi) with
A-presentable Ki.
By Case 1' for each i there is (L,, bi) E B/K such that
Since is full and faithful' ( L I : bi ) is Xdirected. Since has the right adjoint. it
preserves colimits. Then
( K , r; 8) = colirn (K*, ri; Bi)
Z colim @(Li, bi)
= @(colim (L i : b i ) )
= W L , b ) ) ,
as desired.
-4 iocally A-presentable category K has a representative set of a11 A-presentable
objects. We denote an. such set by PresAK and consider it as a full category of K.
Clearly, we have:
Proposition. Let B E ob(PresAK). Suppose ( E . p) is a A-dzrected colirnit of
(Ei, pi ) i er in B/K S U C ~ that E. Ei E ob(PresAK). I/ each
4.3.5. Let E be a category and K a locally A-presentable category. Suppose that
the cofibration P : E + K satisfies
(4) DesE(p) can be viewed as the category of Eilenberg-Moore coalgebra category
of p. i p! and p is codescent (effective codescent) with respect to P if and only if p.
is precomonadic (cornonadic) for each p : B -t E in K.
For example, if P is basic cofibration or if P O P satisfies the Beck-Chevalleu con-
dition for each p : E -t B in Kap' then P satisfies the above hypotheses.
106
Clearly, if the cofibration P : E + K satisfies (4): al1 results in this section
remain true with respect to the cofibration P.
4.3.6. Conjecture. I j K is a locally A-presentable category (A-accessible category
&th pushouts) and B zs a fized object of K. then EB is closed under A-directed
colirnits. In particular, X-pure morphisms in K are efect ive codescent morphisms.
By the process we used in 4.3.5, Ive see that the above conjecture may depend
on an answer to the folloming:
Question. Let K be a locully A-presentable category (locally finitely presentable
category or A-accessible catego y m'th pushoutsj. and let T be a A-accessible comonad
ooer K . Ts KT a locally A-presentable ca tegoy (locally finitely presentable category
or A-accessible category)? I j yes. mhat do A-presentable objects of IC* look like?
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