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Monadicity, Purity, and Descent Equivalence 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 Bibliothèque nationale 191 of Canada du Canada Acquisitions and Acquisitions et Bibliographie Sewices seMces bibliographiques 395 Wellington Street 395. nie Wellington Ottawa ON Ki A ON4 ûtiawaON K1AW Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence aliowing the exclusive permettant a la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, dism%ute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. 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. Acknowledgements 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 is encourage- ment and financial support. As iveil: 1 am grateful to Professors J.W . Pelletier. G. Janelidze, and 31. Sobral for their encouragement and comments. 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. Table of Contents O. Introduction 1.1. Revielr- of monads and T-algebras 1.2. Sote 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 SRS? 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 efectiue 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 shom 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]. As is well knom. 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 condztzo~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 (Definition 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 dth 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. Fealso uish 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~hzchcan parantee 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 qT TT C i-, are commutative.
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