Preface Categorical Typ e Theory Acknowledgements PhD thesis of The rst to be mentioned here is Henk Barendregt He provided an atmosphere with the rightcombination of intellectual freedom and stimulation for this work to emerge The years in Nijmegen havebeenvery pleasant ones due also to the other memb ers of the lamb da group I thank all of them for their p ersonal contribution Bart Jacobs Secondly I wish to express my indebtedness to Ieke Mo erdijk His comments on mywork were b oth sharp and constructive They resulted in many improvements notably in chapters and Over the past few years I enjoyed stimulating discussions on categorical typ e University of Nijmegen The Netherlands theory with Andrea Asp erti Javier Blanco PierreLouis Curien Thomas Ehrhard september Martin Hyland Giusepp e Longo Simone Martini Eugenio Moggi DuskoPavlovic Andy Pitts and Thomas Streicher Among these I want to single out Eugenio and Thomas Streicher writing a joint pap er with them formed an essential stage in my understanding of the sub ject Much of the folklore in this eld I learned from Thomas in often lively conversations a Many of the technical asp ects of writing in L T Xhave b een explained to me by E Erik Barendsen Paul Taylors macros for diagrams and pro oftrees have b een used throughout Finally I want to acknowledge supp ort from two funds of the Europ ean Com munity The Jumelage pro ject enabled me to stay during the rst half of at the University of Pisa The last bits of this thesis were put together in Marchand April at Cambridge University supp orted by the CLICS pro ject Prerequisites Promotor Prof Dr HP Barendregt Familiarity with basic category theory is asssumed The reader is supp osed to have Copromotor Dr I Mo erdijk aworking knowledge of functors adjunctions lo cally cartesian closed categories Afdeling Wiskunde Rijksuniversiteit Utrecht Yoneda etc Lets say that the rst ve chapters in Mac Lane form the starting p oint A good intro duction would b e Barr Wells the parts ab out sketches are not relevant though Two points may go beyond this basic category theory In the rst chapter categories are mentioned o ccasionally Briey a category is a category where the morphisms b etween anytwo ob jects are ob jects for a category i ii PREFACE again this yields two sorts of comp osition vertical and horizontal which should satisfy certain interchange laws see eg Mac Lane The basic thing used is that adjointness and equivalence are categorical notions More information may b e found in Kelly and Street In some examples top oses o ccur The exp ositions there are not selfcontained and the reader is referred to Johnstone Barr Wells or Bell for Contents more information Information for reading One of the main concerns in this work is the connection b etween two relations typ e Preface i theoretical dep endence on and categorical b eing bred over Before plunging into technical exp ositions the reader maywant to see this main line and take a lo ok Intro duction and summary v at sections and rst Basic Fibred Category Theory The category theory needed to describ e calculi with typ e dep endency is denitely Fibrations more advanced and interesting than the one for calculi without such dep endency Category theory over a base category The latter prop ositional systems are describ ed categorically in chapter and the Indexed categories and split brations prerequisites may b e found in chapter esp sections and This organization Internal categories has b een chosen to enable reading only these prop ositional parts The subsequent Quantication along cartesian pro jections chapter contains the technical work on typ e dep endency Typ e Systems Informal description Rules Examples of typ e systems The Prop ositional Setting Typ e theoretical and category theoretical settings Denitions and examples Some constructions More Fibred Category Theory Comprehension categories Quantication along arbitrary pro jections Closed comprehension categories Category theory over a bration Lo cally small brations Applications From typ e theory to category theory CCcategories HMLcategories HOLcategories and PREDcategories The untyp ed lamb da calculus revisited iii iv CONTENTS References Index Samenvatting Dutch Summary Curriculum Vitae Intro duction and summary Categorical typ e theory is understo o d here as the eld concerned b oth with category theory and typ e theory and esp ecially with their interplay As such it grew out of categorical logic Roughlywe view a logic as a typ e theory in which prop ositions can have at most one pro ofob ject Indeed one nds that the prop ositional part of the structures used in categorical logic are preordered categories where one has at most one arrowbetween two ob jects Thus typ e theory exhibits more categorical structure than logic A logician might want to point out that there are no small complete categories other than preorders Quite reassuringly one do es havesmall complete bred categories which are not preordered see and further These giveinteresting examples in categorical typ e theory Having mentioned these dierences b etween categorical logic and typ e theorywe stress the historic continuity the basic notions used in categorical typ e theory have b een develop ed b efore in categorical logic In this thesis one nds forms of indexing quantication by adjoints comprehension and algebraic theories which are all based on previous work in logic esp ecially byFLawvere see eg Lawvere or Ko ck Reyes Wewant to emphasize that these notions require some renements and adjustments to make them suitable for typ e theoretical exp o sitions For example we describ e quantication by adjoints to weakening functors and not to substitution functors therefore a general form of weakening functor will be intro duced see and Typ ed lamb da calculus started with Curry Feys and Howard who considered prop ositional asp ects Typ e dep endency was brought in by de Bruijn with the AUTOMATH pro ject see eg de Bruijn followed by MartinLof with his intuitionistic typ e theory see eg MartinLof In the s the eld grew rapidly mainly by the interest shown from the computer science community Categorically prop ositional calculi are straightforward except mayb e for higher order quantication but that is not what wewant to fo cus on now Contexts are simply cartesian pro ducts of the constituenttyp es since there is no typ e dep endency involved In case such dep endencies mayoccur things b ecome categorically more interesting contexts are no longer cartesian pro ducts but a form of disjoint sum is needed to mo del such dep ending chains of typ es The rst studies are Cartmell and Seely It thus turned out that the main op eration which had to be explained cate gorically was context extension or context comprehension as we sometimes v vi INTRODUCTION AND SUMMARY INTRODUCTION AND SUMMARY vii like to call it given a context and a typ e Typ e what is the mean ing of the context x ie extended with an extra v ariable declaration For As already mentioned this work can be seen as a survey of categorical typ e this purp ose various notions have b een intro duced contextual categories Cartmell theory It seems therefore appropriate to p oint out what we consider to b e our own Streicher categories with attributes Cartmell Moggi contributions displaymap categories Taylor Hyland Pitts Lamarche The notion of a comprehension category and the related results see sections Dcategories Ehrhard a b IC of ICs Obtulowicz categories More sp ecically the double role these categories play one time with brations Pitts comprehensive brations Pavlovic and com as a mo del and one time as a domain of quantication Also the notion of prehension categories Jacobs In fact there are so many notions around a closed comprehension category it can b e seen as a syntaxfree description that almost everyone working in the eld can cherish a private one of a structure with dep endent pro ducts and sums which has go o d closure In this thesis we work exclusively with comprehension categories to describ e prop erties typ e dep endency Among the ab ove alternatives comprehension categories are in our opinion at the rightlevel of generality and abstraction once the notion is fully The notion of a setting see which formalizes the typ e theoretical rela understo o d closure prop erties like under changeofbase or generalizations like tion of dep endency The exp osition that b eing bred over is the categorical over a bration suggest themselves in an obvious way Much of this work can b e counterpart of this relation read as a systematic exp osition of categorical typ e theory in terms of comprehension categories The translation from typ e theoretical settings and features to categorical set We briey outline the contents of the ve chapters The rst one is ab out tings and features using constant brations and constant comprehension indexing of categories it contains the basic denitions and results mainly ab out categories Constant brations