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

Acquisitions and Acquisitions et Bibliographie Sewices seMces bibliographiques

395 Wellington Street 395. nie Wellington Ottawa ON Ki A ON4 ûtiawaON K 1 A W Canada Canada

The author has granted a non- exclusive licence aliowing the National Library of Canada to reproduce, loan, dism%ute or sell copies of this thesis in microform, paper or electronic formats.

L'auteur a accordé une licence non exclusive permettant a la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/film, de reproduction sur papier ou sur format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts ffom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

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.

~ G F C F'WGFC - VFGFC

and

G'FC

cornmute. Then

and so

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

is commutative in C/B. then one has the following commutative diagram in cat(C):

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?

References

[II J. Adimek, H. Hu, and W. Thden? On pure morphzsrns in accessible categon'es.

Journal of Pure and Applied Algebra 107(1996), 1-8.

[2] J. Adamek and J. Rosickf, Locally Presentable and Accessible Categories: Cam-

bridge University Press, 1994.

[3] SI. Barr and C. Wells: Toposes, Triples and Theon'es. Springer-Verlag. 1985.

[4] hl. Barr and C. Wells' Category Theory /or Cornputing Science. the Electronic

Supplement. 1995

[SI J. Bénabou and J. Roubaud. Monades et descent. C. R. Acad. Sc. X O ( i S T O ) ,

96-98.

[6] F . Borceuq Handbook O/ Categoncul Algebra. Cambridge University Press.

2994.

[7] R. Borger, Makaking factorizatakiom cornpositive, Comment. Math. Cniv. Carolinae

32(1991): 749-759.

[a] R. Borger and W. Tholen, Strong. regular. and deme generators. Cahiers

Topologie Géom. Différentielle Catégoriques

[SI A. Carboni. G. Janelidze, and A. R. Magid. A note on the Galois correspondence

for commutative rings, J . of Algebra 183(1996)! 266-272.

108

[IO] M. Demazure, Topologies et Faisceaux, Exposé IV, in: Schémas e n Groupes I

(SGA3)? Lecture Notes in Mathematics 151. Springer (Berlin), 1970. 159-250.

[Il] P. Gabriel and F. Ulrner, Lokal prientierbare tategorien? Lecture Notes in Math-

ematics 221, Springer (Berlin), 1971.

[12] J. Giraud, Illéthode de la descent, Bull. Soc. Math. France lfernoire 2 (1964).

[13] A. Grothendieck, Technique de descente et théoremes d 'm's tence en géometrie

algébrique. 1. Génerdztés. Descente par morphismes fidélement plats. Séminaire

Bourbaki 290: 1959.

[I-L] .A. Grothendieck. Catégories fibrées et descente. Exposé LI in: Recétements

Etdes et Groupe Fondamental (SG.4 1): Lecture Sot es in Slathematics "4.

Springer (Berlin), 1971, 115-194.

[ lz ] H. Herrlich and G E . Strecker, Category Theo y. an Introduction. Allyn and

Bacon (Boston), 1973.

[16] G. Janelidze, The fundamental theorem of Galois theory. SIath. CSSR Sbornik

64(1989): no.?. 359-371.

[l ï] G. Janelidze, Precategories and Galois theory, Lecture Sotes in SIath.. no. 1488:

Springer (Berl in) . 1991. 157-173.

[18] G. Janelidze and W. Tholen, Facets of descent, I? Appl. Categorical Structures

3(l994)? 1-37.

[19] G. Janelidze and W. Tholen, Facets of descent, II? hppl. Categorical Structures

5(1997), 339-248-

[20] G. Janelidze and W. Tholen, Extended Galois theo y and dissonant rnorphisms,

Journal of Pure and Applied Algebra 113 (1999): 231-253.

[21j .A. Joyal and M. Tierney, An eztension of the Galois t h e o q of Grothendieck.

Mem. Amer. Math. Soc., no. 309 (1984).

[22] GAI. Kellv. - . Monornorphisms, epimorphzsms and pullbacks. J . Austral. Math.

soc. A9 (1969). 124142.

[23] Ivan Le Creurer. Descent of Interna2 Categories, Université Catholique de Lou-

vain, 1999.

[24] S. Mac Lane, Categories for the Workzng i\~athemutician. Springer. the Second

Edition: 1998.

[25] B. 'r~iesablishvili, Pure morphzsms of commutative rings are efective descent

morphisms /or modules - a new proof. preprint. 2000.

[26] J. Reiterman, 'if. Sobral, and W. Tholen. Composites of efective descent mapst

Cahiers Topologie Géom. Différentielle Catégoriques 34(1993). 193-207.

[27] J. Reiterman and W. Tholen, Efective descent maps of topological spaces,

T O P O ~ O ~ Y .Appl. O7(1994), 53-67

[28] J.J. Rotrnan, An Introduction to Homological illgebra, Academic Press, 1979.

[29] bI. Sobral and W. Tholen, E'ectiue descent morphzsms and eflective equiualence

relations, CMS Conference Proceedings: vo1.13 (.OIS, Providence. R.I.. 1992):

421-433.

1301 B. Stenstrom: Rings of Quotients, Springer-Verlag, 1975.

[31] W. Tholen, Relative Bzldzerlegungen und Algebraische h'ategorien. Ph.D. Dis-

sertation, 4Iünster. 1974.