e y c v ts the vi yp e t ts on Com hand ork to vlo a emen stage in category v form tribution oP atmosphere e b een used tial v sk a category basic an Europ ean y impro His commen this Briey the vided Lane of ond ulation for this w pro e b een explained to me b ells the parts ab out cartesian closed categories bey Mac v t to single out Eugenio and t ones due also to the other He funds e Mo erdijk an in go ere put together in Marc The reader is supp osed to ha o discussions on categorical Xha y during the rst half of at y w E t o ob jects are ob jects for a category y the CLICS pro ject them formed an essen T w ma a yt They resulted in man ts hapters from ery pleasan with c ulating tioned o ccasionally e Barendregt e i poin een an v o h of the folklore in this eld I learned from ould b e Barr W w pap er Among these I w ed stim supp orted b ebeenv Tw Henk supp ort t y y v Muc rst I thank all of them for their p ersonal con is y indebtedness to Iek vier Blanco PierreLouis Curien Thomas Ehrhard tellectual freedom and stim join her ersit the a ersations here wledge v though tro duction w ylors macros for diagrams and pro oftrees ha ts t that a kno The last bits of this thesis w ears I enjo an ac y writing b da group tioned ely con sa hnical asp ects of writing in L to aul T bridge Univ bination of in P relev t men hapter categories are men her A good in an ears in Nijmegen ha wledge of functors adjunctions lo cally Lets w not y of Pisa tcom be t I wish to express m hapters and I ere b oth sharp and constructiv to y with basic category theory is asssumed etc wledgemen ersit The y The Jumelage pro ject enabled me to sta y of the tec er the past few y y ork w hes are rst b ers of the lam kno orking kno In the rst c Man Finally Secondly Ov etc y understanding of the sub ject unit yw oneda amiliarit is a category where the morphisms b et theory sk starting p oin April at Cam m m the Univ Y Thomas in often liv Erik Barendsen throughout Prerequisites F aw Thomas Streic The emerge Preface Ac with the righ mem theory with Andrea Asp erti Ja Andy Pitts and Thomas Streic m notably in c Martin Hyland Giusepp e Longo Simone Martini Eugenio Moggi Du t h Theory ersiteit Utrec yp e ber Jacobs T PhD thesis of septem Bart y of Nijmegen The Netherlands Barendregt ersit Univ I Mo erdijk Prof Dr HP Dr Afdeling Wiskunde Rijksuniv Categorical Copromotor Promotor i v iii PREDcategories er a bration er a base category ary v v b da calculus revisited Setting yp e systems ts yp ed lam t yp e theory to category theory tication along arbitrary pro jections tication along cartesian pro jections ternal categories Fibred Category Theory yp e theoretical and category theoretical settings HOLcategories and rom t ten Examples of t T Denitions and examples Some constructions Lo cally small brations F CCcategories HMLcategories Comprehension categories Quan Closed comprehension categories Category theory o Quan Informal description Rules The un Fibrations Category theory o Indexed categories and split brations In yp e Systems The Prop ositional Applications More Fibred Category Theory T Basic tro duction and summ Con Preface In t h CE yp e t A tained e a lo ok plunging PREF tal whic selfcon h dep endency The subsequen The basic thing Before o relations hapter and the More information This organization not w er are v een t parts w ells or Bell for yp e dep endency is denitely there bred o ertical and horizon t to see this main line and tak b eing an exp ositions yp e dep endency see eg Mac Lane yw The ws ork is the connection b et ork on t categorical alence are categorical notions hapter esp sections and o ccur and hange la hnical w terc teresting than the one for calculi without suc reading top oses o sorts of comp osition v w tness and equiv for y b e found in c tains the tec dep endence on hosen to enable reading only these prop ositional anced and in hnical exp ositions the reader ma satisfy certain in some examples In The category theory needed to describ e calculi with t y b e found in Kelly and Street to tec hapter con and the reader is referredmore to information Johnstone Barr W Information theoretical in One of the main concerns in this w ma should used is that adjoin ii again this yields t The latter prop ositional systemsprerequisites are ma describ ed categorically in c at sections and rst more adv c has b een c e e of v y w sum cate more small These t unit Bruijn require esmall v no texts are de sometimes y disjoin e are Con y MartinL b w h are all based ening functors yp e theory b h prop ositions yb e for higher further ard who explained yp e dep endency yp e theory ha in h it grew out of w w as ere ening functor will ed eak t there be w and wv one do es ha form of eak yp e theoretical exp o In the s the eld to As suc a follo that ts to w brough ary y had b ecome categorically but as out h see ard except ma w t The rst studies are Cartmell terpla of t to fo cus on no y adjoin yp e theory in whic reassuringly whic eys and Ho things comprehension poin tforw ere see eg La an yp es yp e theory exhibits more categorical to pro ducts wv Bruijn ew yp es since there is no t t as a t text een categorical logic and t summ Quite In this thesis one nds forms of indexing us t tt e them suitable for t w preordered an t to emphasize that these notions dep endency v yp e theory yoccur w wn from the computer science comm Th con tication b op eration an t yFLa not hains of t yp e cartesian Indeed one nds that the prop ositional part of ew or T are see eg MartinL see eg de migh W not what w main and ts to mak h terest sho the basic notions used in categorical t e view a logic longer the o ob jects whic y w w no asp ects logician uit extension e describ e quan that ts comprehension and algebraic theories whic yp e theory h dep ending c y the in A es tin but that is h dep endencies ma een t TH pro ject w out text Roughly prop ositional calculi are straigh categories texts are y adjoin logic ts and adjustmen tioned these dierences b et con b da calculus started with Curry F wbet d ork in logic esp ecially b k Rey e yp e theory is understo o d here as the eld concerned b oth with category yp e theory and esp ecially with their in con mainly b UTOMA turned logic prop ositional as tuitionistic t tication A In case suc than br or example w w elop ed b efore in categorical logic us F e at most one pro ofob ject v teresting examples in categorical t the ving men th ed yp ed lam tro duction tro duced see and tication b T Categorically It Ha olv ein v teresting stress the historic con b een dev quan on previous w or Ko c some renemen and not to substitution functors therefore a general form of w considered sitions be in grew rapidly with with his in order quan simply cartesian pro ducts of the constituen in and Seely in is needed to mo del suc gorically can ha the structures used inmost categorical one logic arro are preordered categories where one has at In Categorical t categorical structure theory and t giv complete complete categories other than preorders CONTENTS ary h Summ atting Dutc v References Index Samen Curriculum Vitae iv o of vii w see wn the ari t yp e ks t time v if y mo del een on closure are k a the not with one notion of used of y e dont think go o d y pullbac Here the least are pla categorical has t comprehension ey uptoequalit Also of t notions see taxfree description h results see sections example yp e theoretical rela er is the categorical terest w v surv coherence ey syn view at whic e consider to b e our o e use familymo dels in exists e are particularly categories categories ts without assuming carte of our surv w t b da calculus a ab out in concrete tication sums this as these poin ery quan structure ev and Indeed w role t seen nothing of yp ed lam t out what w yp e theory in terms of brations and h formalizes the t t y an a is can b e seen as a be comprehension here it completeness t brations and constan relev double or can pro ducts domain there to t Y a yp e theoretical the the ork yp e theoretical exp onen h o ccur when dealing with nonsplit structures e an established categorical in In order to obtain w as v tics of the un all First claim y y comp osition instead of substitution b h considered a yp e theoretical settings and features to categorical set brations this no time from a t dep enden t The exp osition that b eing bred o ted in a split w ered t whic is v ha w sp ecically one in co with theories suc yp es see tioned there Constan and not More yp e imp ortan men t mo del comprehension category are that a t dep endency do es not o ccur see section b er of free constructions linking the most imp ortan d and features using constan h the structure mo del terpart of this relation an as um a a to It seems therefore appropriate to p oin close state already w of can b e presen whic ting them in ODUCTION AND SUMMAR and i The revision of the seman comprehension categories A systematic exp osition of categorical t An and A categorical description of t sian pro duct t The description of aie top os as a split mo del of the calculus of constructions relev prop erties The notion of a setting see whic tings categories tion of dep endency The translation from t coun of as a The notion of a comprehension category and the related e h one has substitution b W As tributions e kno ous mediating isomorphisms whic Although coherence problems ha they are really resp ect w presen whic topics INTR con theory e y Y in or ha F lik hap later yp ed t ab out are or the mean around F ely easy describ e un is tranlated In suc y dep end and com the he ork can b e to ma w be is one systems ellestablished notions ation here is the can declaration v y e pro ducts sums rst Lamarc what c A setting describ es tics of the generalizations vi man h of this categories once the notion is fully or The wicz categories ate one vlo yp e individual so a A categorical setting can ariable lo features T Muc These are w algebras via an adjunction textual categories Cartmell are y the seman a some eatures lik con ODUCTION AND SUMMAR F Pitts The main inno can b e done in a relativ comprehension categories of of hapters c there b da calculi from Barendregt es herish a priv e yp e theoretical settings can b e trans theoretical yp e theory in terms of comprehension INTR vious w yp e comprehension yp e dep endency is the sub ject of c There one nds the standard descrip hangeofbase v e brations P c yp e hapter fact eres notion of algebraic theory wt t revision yp e tro duced til section y and abstraction t a Hyland wv In the with yp ed lam e whether or not a prop osition ternal categories who w description under of lik and e alternativ ho ely e with a of La v hapter es in an ob estigation of comprehension categories and quan ts the translation v e b een in lik b IC of ICs Obtu t with resp ect to the categorical structures used b v ab o and ten ts are added on top of a sp ecic setting tains the basic denitions and results mainly ab out text exclusiv y o ccur con ylor e close el of generalit hapters sho a emen categorical ie extended with an extra v w ariable of a certain t v con T ork orking in the eld can c the a tlev hma Jacobs settings the w prop erties x e en These are related to set theoretical Among the t three c for w yp e dep endency tain a v one w Appropriate comprehension categories yield a new notion of cate Finally suggest themselv giv outline yp e systems in terms of settings plus features arious notions ha ery text and ymans categories her categories with attributes Cartmell Moggi closure it categories thesis categorical y b e found directly in c It is the basis for the translation of settings and features in the b eginning een the categorical description of t hapter w algebra ts axioms or constan c to briey h forms an impro call bration this It consists of a thorough in in ymap yp e ie con e bet yp e theory is the sub ject of the next c to hapter one obtains individual systems b da calculus The subsequen In W T In c that y y it ma e er a to categorical features on top of the translated settings a a ranslations in general are p ostp oned un v yp e dep endency exp onen w lated in w settings without t b e understo o d as a generalization tication tions for the left planeT of the cub e of t of in lam ter t understo o d our opinion at the righ description of t the dep endencies whic onat read as a systematic exp ositioncategories of categorical t o indexing of categories it con either in the literature or in the folklore brations but also ab out indexed and in gorical whic Scott and Ko lik vi ing of the con this purp ose v Streic displa Dcategories Ehrhard a with brations Pitts prehension comprehensiv that almost ev e o a It B B er w the een ev E w for ws in w mostly p A text bet ho y Sets con elop ed ob ject I see yp etheoretical y Curien that t structural role reasons Readers unfamil dev X e are ab out to un ery in indexed categories case Ev hapter hw consisting of ob jects es place it is instructiv hnical able In more t A tax used b Theory tec vious translations ie with a functor E A either texts in the base category material y a id or in y socalled F y an imp ortan Fibrations form the appropriate bration is still the indexset the indexed texts and of ob jects and arro ell t yp es deriv a een con calculi whic I pf en b t w w it folklore e as con There are ob texts pla with w orks ork for describing categories parametrized B w E ws b et where statemen Indexed sets are describ ed basically in t only mak g I i study of in Hence there is no claim to originalit f h A h roughly means as a map e in the abstract syn Arro f this wledge of brations so easily e ample time for this rst c tains Y written usually as f the terms and whic of A Category whic indexing A f con es I for enab ou terpart of is giv w the indexing of categories tak vide a framew text i the p br g es b da calculus con estigation of indexing i v ulation orking kno X coun hapter f and c indic prop erties ythe and morphism form hes k and J B A category yp ed lam estigated in section b elo As a map en b e t v Fibred certain br pE approac tro ductory in o The categorical with yp es and terms in con w As a family e starts with the in t e a lo ok at indexing of sets rst E ulation one can think of ob jects h will b e in erse of sets yp ed and un as t This Although the denition of a bration is not so dicult it app ears that one do es In order to understand ho satisfying ys A a y A Grothendiec y some base category sets are then giv these more mathematical indexing of categories can b est b e done in the second w B determines a whic iar with this eld are urged to tak can then b e seen as substitutions lik b not obtain a practical w E E form Chapter dertak categorical concept they pro b univ Basic They can b e seen as In t is for this reason that the categorical to tak w y h it Y ery the uc ving v h dif m is calculi uc as higher serious coherence categorical yofha describ e or yp ed el el where man more dont terpretations e b da calculi arious t bly lev w in seems substitution out e yp ed lam lik ODUCTION AND SUMMAR omission riting categorical W at the assem but it do es not seem to bring t on one do esnt really see m INTR This details yp e theoretical and categorical descrip more terpretation of the v skills for een t bit w tion A h smo other and pro ceeds at a lev us one can view t description hnical uc o reasons Th as w atten t yp e theoretical or categorical description of a sp ecic wth of exp erience in this eld the necessit constructions from a certain p oin more free wing categories requires tec as een the t far follo w see the latter yp e theory is m ythe With the gro mo dels require eto certainly there is nothing ab out the in it ymore b et e lik corresp onding ated b term W ulations Secondly This brings us to the relation b et their el languages for certain categorical structures motiv terpretations diminishes arious Programming in t of these asp ects are trivialized lev conceptually in lab orious in v is viii tions ference an system form a y y h p at I ws Id ha ose Let rise E and and and and age i only is age ks in v hoice has a g e deter v closed  E ertical C cho D E i h inverse then e has a called for suc u It is the u giv Lets sa id v X bration are is E E g ya v en one id C and E w b f enab ou D ages ertical arro called v B u f ho ose wing Notice that the op en f implicitly E h do not dep end E A E hoice to obtain a es sense to arro J E clea E X ys c a y functor u t arev as for pullbac are are in y ob jects and and an b y functor E morphisms morphism whic Id D en together with suc In general one obtains u u f U tit E B V j itmak vious functor obtained b E u cartesian B p f Dieren en together with a clea id D U e obtains u since a hisgiv stressed v bration ery reindexing functor These are b oth instances of the are cartesian functors a it do es not dep end on the c co domain and f g ab o tities here for a certain clea y the iden f then One for the ob is y tioned in the examples ab o u appropriate often e as cartesian lifting is describ ed b u E ob jects of that ifev the functor to denote the full sub category of B and functor E en b if it is giv B g er uous is E eiden E v e often write ery f men a and v y v tin ev V terest ie prop erties whic As B w using A subtle example is the follo B Cart Notice that one can alw E con E E E are group oids cloven Sets in h reindexing for applying j f since these are determined uptoisomorphism p one can tak B v p B j suc A B J split es e write Cart substitution If trinsic prop ert j of More explicitly ie with op ens rst u u j in C g them f or p C j bration j y A f u that f g y b X a hmak ery am erse images hoices f u een to the terminal category satises prop ert en c v F The category B B ev w cartesian i b oth of isanin el ling B en of whic with co domain and If one happ ens to ha o constructions to form a new bration from a giv notice e write p for I w u Fst J estigation elab W to r B C satisfying v E of for a giv is t A ws as ob jects E ys that a bration is E Then bre categories p ertical and cartesian is an isomorphism I E hoices of in u f g denition Hence giv u arro induces in u collection is TIONS All The bration are b oth cartesian morphisms o B and morphisms al b e a bration A It A t but naturally isomorphic reindexing functors here are t the t morphisms b et for D E yonc prop erties of brations are of in E urther in t bration onesa imp ortan eindexing a E B E r F bration is understo o d here as a bration whic cod in ery bration one can use a suitable form of the axiom of c ys that the bration can b e V FIBRA V vertic est top ology on yw age or the second construction w v F Finally Some trivial examples of brations are giv It is D hoice E E avage E split c eak or ev ertical natural isomorphisms artesian mined intrinsic left adjoin in an that a bration of the functors a cle image to dieren v with g One has that p restriction c the same time v p under comp osition the unique functor constan f isomorphism bre a cartesian lifting F clea splitting case of Similarly one sa A splitting with the top ology induced b w a a if f E Y B ys de ery ery the has can if and Y ving J ev vided resp resp as the hsa A be B Dually A where X ation if ev called one e The for A E g uniquely y pro for bration v in The total as ob jects u more I The bres U is B br I u u ma B B Y THEOR it determines en ation w whic E B ob ject i set The latter fact Also I X pE pf I category and category ha Ev v Sets E j forms an example over if e these obr f ery ery the b e the functor cat c I D B ase ell TEGOR slic b ev uous maps k squares i ob ject is Ev B is called a is in fact a bibration w g uniquely morphism C pf gory i then forms a B is a tin f in B domain a for the as X f An p E al ate cod bration B u am c ob jects is the B u er B function F f with a f v Sets in one can think of f A B has morphism of o or as a with E that y pullbac vertic of v And E cod ely h B p resp B consisting of ob jects ab o en b bration there is a category where is an isomorphism f suc gory since u C E B A i a uting triangles as morphisms called f lifting D J E E D ate p ob jects This functor X artesian k ws from lemma b elo c is spaces with con am c f where is example Morphisms C u F are giv f A d ery BASIC FIBRED CA B A Alternativ E y e pf of I Similarly an One often calls B C tit B bration The functor ery B I ove br B X u there is a cartesian morphism with co domain functor w category ks the functor called i a A ev ab B with B g yields an example of a bration to a i iden f is f with I if is are B forms in e E in X co cartesian be B v E v f u f an A X B e B to ha e pD pE v in er v Sets function v e b e an arbitrary category and let This arro o dom names has a map E CHAPTER has pullbac D f a w as ob jects and comm ab o es rise to a family bration said ab o B e A y B cod B B with is cartesian i ev A families f b B I B E giv pE is The rst pro jection u op It is useful to write i in uous with Let D g C T u I X en i has v ery E A B C tin f artesian f y hand but it actually follo e dom and f E c Supp ose ev giv v in o functor in U C con c u E Alternativ morphism f b e the category of top ological B i ed b a are ab o is and k u if it is at the same time a bration and a cobration pE am A uting squares as morphisms Y u F category E to op u y a pair D E Fibrations functor C vious hec e T Examples Basics e one has a bration functor v u ery category morphism v ery i ation ob B am ws with co domain total In case the category Ev These notions are due to A Grothendiec The A Let lifted X ab o F D D ab o is assumed that morphism us determines a socalled bre category ertical morphisms i arro of a bration cartesian morphisms in are isomorphic to the slice categories that brations are closed under comp osition The can b e c comp osition category morphism egory from the partial order bibr comma category and comm A the it is E f determines a for ev one has that in f scrib ed b getful be a pf th v as In pE wn ber tially L bres e alence v sets not the is Remem Then in K is ab o E Sets indexing t E pf artesian of c A r ys alence forms the prop er a structure q is w Fib E A D Sets tially the same o The latter one is essen pf the in w Sets ations t el ery comp onen rq cod E are morphisms of brations erify that there is an equiv with This equiv the f e br ev whereas b yv ork with L G ed rp r by A that L G H K olv F to Sets artesian and split t v Sets ttow B c R er is rp p and B ation with hapter are essen v has a ful l and faithful right adjoint is r pcod e also has categorical asp ects as will b e sho lemma deals I om v f ation then E More precisely K H Sets statemen and B is a br E B B Sets e is a pair of natural transformations p the A v Fib The rst of am artesian fr in L G Assume E F E Sets E ation q is ab o of these notions one ma is c p I is another br am to rp p tro duction of this c artesian et F p L B c lemmas one obtains structure for split brations y F rp o K H expression a a h that w w is TIONS D f q bration suc is a cloven br h is a categorical notion o Fibred cells Lemm G The functor The functor If next t p FIBRA the tioned in the in i As an application Changeofbase as describ ed ab o the ii iii A cell from H In the same w whic see for denitions of the brations in men that general split brations are more pleasan mathematical in and the second one with structure on the base lev prop osition in Ehrhard a cells as b elo and dene a functor a Y to w is a The as This e the p B Usually functor i size cartesian morphisms q Fib preserv Y THEOR of  the transferred ack B h itself K D use one can form the be cartesian implies is e p D A base satises W asp ects TEGOR is result can uting square as b elo p f Ku a q u f q q B bration cod with D B A A D D of B a As B K consider that as over age morphisms whic BB y a comm an form the pul lb v B u and clea en b brations dont eries One c v e or of functors w brations H K K p Notice that K is giv BASIC FIBRED CA q or understo o d  easily y and not uptoisomorphism for since tary results are left to the reader brations be and k diagram forms a morphism of brations splitting y category p a D it is a standard result that One ase a e a functor can w D split b B functors es cartesian morphisms ie B e E B B B Kq E o elemen p and an arbitrary functor q pcod w bre B base e pullbac tains result union A CHAPTER v cod E t preserv K a E B D its the con changeofb artesian p suggestiv c p H een brations is a bration again As Sideremark D to q a for w B et D q disjoin L ts in called K B Kq spl it er the ab o called a v is B cartesian Fib only q Fib are bration denotes is cartesian a q Moreo Lemm h the functor B k S erify that is Hf writes q en a bration The pro ofs of the next t A morphism b et f Fib where Giv and v K construction pullbac i sending one in category category splitting onthenose ie uptoequalit and so are adjoin bibration see eg Jacobs in whic that t E A p D o f age and op v of alen f h A natural E denotes A u id satisfying  B of of pr g F is cartesian o bration f u f e opp osite ertical dened on the E v f a f f f v j if it is equiv with is a on the be on F h B Y Y abo B y f es the splitting E b ts f Fib A v hoice to obtain a clea f relation B D E esentable E Morphisms functor  preserv and e found in the pr p p determines epr  brewise A r F p ertical map E A A CV  alence p E Let A his u will b e written as since Let dom in A in dmayb dom as ob jects f is called with comp onen for some E f f B E dom An equiv E B B E B there is a v alence u tioned A A g spl it on B E gories E id id Y Fib E A A ersion of the axiom of c w yields an isomorphism since E trinsically men f Fib ate ality involve f F describ ed whic Fib has u id u the unique p A B enab ou is determines a cartesian op F F F u id g be B p A B E dom ation one obtains an isomorphism A eofc g E f v A of E alence class of then br will E dom on op A A Y morphism B f u u E E E functor A bration B split A f f quivalenc a F f on A dom E on Y op The equiv is a y Y E p b E dom and and f u elow TIONS eisane obtains E y E ase One rst uses a suitable v This yields the required equiv op CV p h Denition Opp osite bration u A on i Ther In c The construction from i no An ob ject en b ertical One id Y v FIBRA g p of i The total category u is ii ii o E y The bration the Homcategory describ ed in Pr for b are giv to a bration of the form bration transformation A little care is needed to do this in f collection and v A suitable formulation of the natur sition b Y al es D u f B and wpart q a natur functor is of the B q to B A D Y THEOR D ary functors wing diagram q spl it u f B D ase e is an upto A the arro i i d with L Fib h h ther u f arbitr TEGOR om ovide the follo This description giv A L has fr pr K A Then Similarly e that w v ation and q f one changeofb one has that B D spl it D B yof L L a D K and h arro them i Fib ust ha Lq es er L e a br K K L h b v en B q i B D in induc h D etwe L y naturalit onem b Kq BD L A B q q i BASIC FIBRED CA E L B estricts to D h es b ation and H j p K K uptoisomorphism B H i K K B et is cartesian o H h L to H K D e a br q K el l L H b functor and mak H B L A i D A U H D A D h A B D Ku H i A oneda CHAPTER A H e is a c u f h A This functor r functor v D A H i artesian A BD A H c q h A q A H u f B ard K A go es from et onents D A is ab o i A Every L L ansformation Fibred Y K h Fib for Lq tr D omp air a a unique tforw Since is cartesian there can b e only one suc B al f D D A B Lemm Lemm Because Straigh Lemma f B D ute artesian c anatur Fib of of determined o o ansformation where This determines the ob jectpart of is Pr form has c such that the p Since at the same time a recip e for the construction of tr comm Pr K with isomorphism a y H unit only D B E bre u A and B u u K Since t of view E if  u A hangeofbase A H c K u ery bre category These brewise adjoint and eries that the unit that ed under reindexing A restriction B s if ev y G ery ob ject Y right  D b ations K a d preserv eindexing functor A e noticing esp right adjoint A u br ansformation E esp q e br r e lemma the second brewise b his s F TEGOR v al tr orth esp  ob ject binary co pro duct co t from a theoretical p oin air of r B w A r has D p dleft is the transp ose of e The precise meaning of the latter state B F G B e the also has a left r A al natur br D R is K e determine for ev A bres whic a v G q There one also nds some more information b ecause it describ es the structure induced b It u the H one obtains the other one p anonic has and and and for every p as ab o H A VER A BASE CA tages and therefore it will b e used most of the time terminal initial B ulations in the ab o B E G B D u tuition an ertical the c tities of an adjunction one easily v ertical unit and counit er is more imp ortan y that a bration v K YO f A Similarly in A ev and is D u  E tform B esa w A functor K ed under reindexing F B the functor E W p with v B Let alen B as structure in K g A G The et A t D counit L u a Y THEOR ertical one obtains an adjunction u A F Au wing quite useful result a F A u ulation ho the o equiv w K i H functor and TEGOR A er it has practical adv Lemm Denition es bred adjunctions see or every or every functors v s and all reindexing functors preserv is an isomorphism F F E CA B t can b e found in Jacobs  ertical Using the triangular iden Cartesian functors Of the t The canonical map oth v artesian adjunctions are preserv men is preserv and counit are v ab out the follo wise equalizer exp onen has The rst form Moreo one is often closera to bred ones adjunction in BK if b c an adjunction e a Y also of base alence a forming It is left t section that er then E v is o Y THEOR d adjunction pf are e of the opp osite D w category and br op G arying A v f TEGOR In the presen alences f B and erse arro category D v equiv op and bre categories  A categories is an isomorphism E pE E vided with certain structure lik b e brations base f F B that E A a  explicitly y op D BASIC FIBRED CA er h structure in E The total category stated v q e o suc w ks cartesian i and bration can b e done without taking equiv adjunctions f B f obtain is sometimes called the in hapter split c is then dened b B E y CHAPTER theory B p describ e this ariable categories can b e pro needs to Let of B hv op erify that Lets E p B cod e w suc v op v p In op consists of a pair of cartesian functors is a bration with is the brewise opp osite ie b e a category with pullbac y q tro duction op Category op op p B Denition in to p p adjunction it is an adjunction in the category of brations with the same category p aking the opp osite of a d Let The concept one T i e the ii iii ellundersto o d e describ e ho denoted b of the bration The functor to the reader to v w br base from category form the sub jectw of study in bredterminals category or theory cartesian pro ducts classes as ab o In Comp osition is describ ed b in D en L ii ob H with vious giv f again as f g E B ob is e obtain E in y h cf W B b pE an f D E B urther B is bred CCC to en constructed ws F p pro ducts t K f g E giv is pE D with arro E is E E u p y g h a functor from the nite adjoin pr od e b E u preserving functor p g v with is then a bration with t p D y  Y in b and en E E E E ysuc ab o E u h righ and E E giv bred p E ertical isomorphism bration B pro duct from remark Supp ose that u E a are has TEGOR y reindexing functors means that D bred p E p E E E E a f is a bred CCC also D is brewise a in E p is Comp osition p bration that H L D u ko h pairs one has that the canonical map the global Supp ose E B E pg new are full and faithful functors s if E H E has D whic D D In case H  E eries A H p using v h E H y leads us to denote pf u H y pair of maps VER A BASE CA H p E E een brations i E B H i cartesian i there is a v easily w The rst pro jection p with p H YO es bred b e as in denition es the CCCstructure k p hangeofbase situation H E E and category E  pro ducts H is One B H u in h This prop ert tal construction h H Let id The in pr od Hence for an Y H f g D pg preserv nite D D B y g y u Y THEOR there is a unique h a y E ation of bred cartesian pro ducts b w yb h g K L e map preserv sa a morphism b et A pf E functor v tit bred u f u h TEGOR wing is E E er that there is a c A fundamen Denition E es the bred nite pro ducts h b oth for terminals and has u Preserv morphisms v y that E ery pg CA g h follo ii esa cartesian h k f g where and the ab o and iden the in whic preserv Moreo and jects is an isomorphism pf a and W E diagonal functor whic often assume that bred terminalbase ob jects to are the describ ed total b category for ev e e e a k Y as v w ks Id ed W and be tains A using there and B e con pullbac B bration B v preserv B Sometimes e use of an Fib B ery bre or p Y THEOR ery ab o the The category A therefore in ev Fib E D CCC Later E p h are transferred B that in pq wmak that u s y nd denitions of h E E E is an LCCC one has tto TEGOR  whic more see bred B One has suc E q  a ery ob ject to that tioned at the end of LCCC if ev E p p is ving a terminal ob ject and terminal y comp osition of pullbac t adjoin b Supp ose By a standard result see eg ha B has is p easy In case Let gory is isomorphic to the bre ab o ts are automatically has bred A  B  e B B  something is g E ate B one g g  tak it is e denition  A dc g v Sets B h u g hand o constructions b elo u B can w ery top os is an LCCC at h h BASIC FIBRED CA f cod Id There f is a bred righ functor limits u f In Jacobs one ma C one B p y comp osition a The t u h f u B is a bibration am u u u F nite en b f B u h functor artesian close limits cod bration Id giv obtain description b e as in the ab o A A D u B B B e the with Since  al ly c the orking with bred nite pro ducts it is often quite con q w in the bre category B B B B B nite c t CHAPTER A s lo bred CCC since exp onen haracterizations see iii and ii en that Let global  and y a wthat y a giv  Hence is is a CCC Since the category i B bred category has When w one has e these will o ccur also for other notions g A B a are pro ducts C B also text see i iii and i w lik has B e age but they dont dep end on it prop ert is called a B in E be v v f B arro B p B in B ha nite has a left adjoin the u B ving a bred terminal ob ject implies that for ev with bred nite pro ducts of B to one u er one can sho Remarks Examples h Ha Let The bration sending a bration to its basis men This t v A is an example of an LCCC in fact ev cod B bred u It is then clear what a bred CCC or a bred LEX category is A i A ii iii enien B exactly Moreo Hence an LCCC can also b e dened as a category satisfying for Jacobs one obtains that cod functor slice category the terminal ob ject it is then cartesian closed itself that Sets v shall come across other c is a terminal ob jectu sa arbitrary clea bration has Biimplications to the bred con this predicate bred will b e omitted these notions in terms of bred adjunctions pro duct and j k w E as B A if the use Set Set this E that this erify B case j y result v arro one ob ject ob ject e h b ob ject dep end where N v C ob jects sub cate B to in T Ob j an B suc the en that y pullbac of pro tion that a top os I X E B K A B not am or nonsplit N split generic o F F a giv forced to In nd B generic generic T ed b Sets is E reader a Cart n I emen do es A More explicitly E E X B has a can B denotes i has Then j e are v it g f the i p e set and w B has B j collection m Split and a generic w forthefull B p X alizer to f y case j B ob jects B table then First w ie e p n morphisms T is a Y r j E also for I id E u In j a left smal l A Sub functor p j is Sets is a prop ert E u then trinsic where generic Morphisms It ob ject u in TEGOR ery ob ject is represen of a cartesian E E id a this there with e write C B faithful A nite pro ducts t m bration f h v where T W or ev the A a an Since monics are preserv B n F to B am yModels X vious bration clearly generic and F j is are then discrete categories whic with from ks E a ew ob ject is By remark i one obtains a unique j N A m j if there is an isomorphism only f p w B e E category j morphism A of functors forming a morphism of split bra full for has T v an n is a map of split generic ob jects E E a a j a E n Split applied but B forms is j for X A ob ject B is Then j one Since be bration E abo VER A BASE CA p bration B y that a split bration p Ob j j j E Realizabilit T C E u a C E p A e B a e one has a split generic ob ject j ws as ob jects v esa YO v since B u The is E A in Sub j A j am There L p function W determines an ob from and a collection of isomorphisms generic Let F u The bres of p ab o L a B p B e f satisfying e B B m B of C or split brations one can do b etter cod E i v A i to has ob jects for F map bration u u c B e E N B erse images B LCCC g Sets v Ob j ob jects E satisfying v Y THEOR T c Set p recursiv the for f A in notion ab o an p K B ab o functor een them b e a category with pullbac all E e m generic satisfying in of with monic arro w is v that yields a generic ob ject as in the previous denition KB B y functions B p that is b y the the in has E b eak notion Denition Extended example Examples TEGOR in j ab o id partial and B I Set id Apair Let Supp ose en A B if the discrete bration f hoice j E c CA E e that th claim X w giv ct v assume The In the rst example ab o a ha w h that n pE ii ii e iii e a ab o split bration Split describ ed in obje there is an ob ject brations a similar construction yields the group oid bration splitting b et tions from T natural Ob j on u arro suc gory of functors W bration has a generic ob ject viz the sub ob ject classier and w The category is a relation are suc of that denoted T has w Y of es E bek and D one in T is the is the is no end A B if there E A Lam artesian b ek and e c E preserv ct v obtained E E the E forms a in p olynomial E Y THEOR ct A Lam square E V B ab o cod w al hangeofbase at y k the p E A E obje see ac Then Jo A E A p V E A E TEGOR tioned x generic obje en is the Kleisli category pullbac cod A L to es these generic ob jects E for a giv ea yp e cod E there is men t E terminal v E brations the bre is f E eserves them E of V are pr where it equalizers een x preserv p B E w  of i the bre category s then e further E attributed A v  bet E e and E e notice that is there is a cartesian arro is said to ha v V s and ariable ab o w t T has v E E B  again L E ab o a presence B BASIC FIBRED CA has bred equalizers ie that it is a bred cartesian ation and E p has E insigh E E E the E E functor E D gory a br p ery E V in E L ate e on ob jects e T T b denotes the pro duct in First of all latter e and in adjoining B is the p olynomial category B y A It is readily established that E is E B b oneda lemma this means that the induced functor has a clear logical signicance tioned there s p tak The E e h LT co domain A v CHAPTER A  E ose the c B E E E A bration e d men v E e h that for ev ab o the whic situation p K i Supp b ek and Scott part I I exercise E E has bred nite limits Notice that for suc ab o tially surjectiv et where are full and faithful functors this map f g E L p g E A E then D A E L has br category a E E p x e additionally assume that T AB with generic ob jects is essen with A qualizer bre morphism E E e shall do the case of cartesian pro ducts B E E Lemm Denition h and W A In view of the bred Y or category yields a similar the new bration E category Supp ose of In case w The bration F E duct e ii o bration o pT AB a of the comonad LEX situation This bred nite limits slice category in whic see also Lam pr if there is an isomorphism diagonal B is an ob ject p Scott part I and from has that the bre category Pr pro duct in the category e y or an and es a v B The that Cat This Cat v j a sa natural natural ICat consider X has j from describ ed a a categories op j Set op B KB A v d is is A is B This one j Cat means that is an eectiv shall a understand L op L S in u e B Q e e K eried directly W lik indexe A w in to B pseudofunctor and w B h pseudofunctorialit B and K a Cat The latter Suc id brations Set A X Belo K morphisms B A X y is left implicit ulations Q B TIONS b op categories e u categories one id On B Q ation where are functors am and dom split Set for and not as F x e shall see that these data imply that dic her next large a er a base category ariable X v v is a functor of the form mo whereit that Q Cat morphisms ts of the ery a from and functor umac v b ersome form n Set I h pro jection in gory pair a Set een R is op a w ate is arying o snd the B describ e gories transformations is lifts to a bred reection ab out dc bet e condition A ate e and Sc ws isomorphisms a Notice that c the latter has a realizer b ecause am Set ar x M the Cat F d A categories Later in i w L a indexe sp eak B e natural to b ecomes oid rather cum brations n Q ICat v f An of am K categories seeP a op indexe functor F y ea fst M B a A f is a bred CCC Of course this can also b e v far i Q of TEGORIES AND SPLIT FIBRA internal A X lo osely B yw sub ject f tw g e so a Q where K Notice that the comp onen Set w and and B a x to KB seen f a x e K wise construction cell v morphism again t Indexed In this w Finally Cat ould mean that one allo section Denition ha M A A The reection b ecomes n INDEXED CA e pair family e f in a coheren w Here spl it this f u ii a a o other descriptions of categories v e rst dene a functor iii iii op B am y a p oin y w is transformation is determines a category transformation B latter w is b etter captured in bred category theory lot of trouble Fib F b As t in indexed category as a family Q W with b h A Y in to A M M set the the a one Set Set and j n in with n a j alizer j whic wn e Notice A X j r j een relation j R es N M A w sho vides PER a A a n g a A j b e the A terestingly tak e e f g bet n C Y THEOR pro v As erify that In A X A X t R am nR n a ha e F one j S j a Set with h j denes R am N A consists of a map m j F CCCs adjoin alence Q X TEGOR e relation whic n Then A X a rst n t to the global sections B Y of functions n m left f A X j S A f a a bred A It is easy to v one with ob jects writes Set Q j R A PER a for of induces a morphism of bred g has R a m t adjoin a A one sets has ob jects put n cod A Set A PER f A X orth noticing f Set has pairs Set R dom PER C n can e a R Q PER e h yields a morphism e The rst pro jection tak A morphism Q and v f f one Set A Q BASIC FIBRED CA a am a a g mo dest M F n R R j y the inclusion A M ab o M one has whic generic ob ject A family X j R or R a n mR n F N M ed b PER Sets j Finally A is a symmetric and transitiv PER induces functions R As a consequence of this reection the category ctive dom Set N N realizes g j with R functor A n This functor is righ y It j A b m a j A morphism of socalled ee R put or N A Set e T m n The category Set m n of F CHAPTER en One obtains a category M f There are three things w f X M n am relations ery giv PER N F M A X j h are preserv inclusion R and an a a A S R a N or Obj j or T closure F n A R whic R as ob jects X the alence alence of categories M Set X f e y here means that the family itself has a realizer ie dom Set or a C m a reection F R e j ws een the co domain brations sub category y a A ob ject a a equiv w j in N g f b am where j be h that for ev a R X F j follo A B j g A PER transitiv A C y Q t co domain brations The Similarly to the example in there is a bred equiv j a suc satisfying m b as eectivit j R S an N X m A a Let Let The full the on i en en A partial ii j Set f Y A j f j is then a split bration be constitutes n giv and a morphism there is an equiv that n has a map bration CCCs b et of has nite limits is also an LCCC and that the inclusion dom giv relev or forgetful functor satisfying Ehrhard a a is H one G B The D gen G a C a yrein A functor table in ii H y simply is gory E a categories am K in F result ate H esp k construction and r bration Set represen B and H is indexed a oth Using previous H of F lemma ed onthenose b one obtains discrete B ations C the a tage category e y the previous prop osition to  esp right adjoint b an oneda TIONS A am Y B el ls or example the split bration H F due yields adv B F small Similarly G form in fact categorical functors  e split br h is preserv d adjunction b spl it e A if and only if b I also An  H  bred B spl it C Fib B cells H G H A has a left r e often describ e split brations b dom onstruction yields for every c lo cally But Fib y applying the Grothendiec the on A A spl it I a D B A terested reader A ck c A y and construction el ls and on c q instead or C Fib that dom G H F k A table in the bred sense i B B en b B and ICat see faithful B B gory giv A othendie B B op B one ICat One has a split br B een ate G and also spl it spl it In the sequel w oth on c w brations Cat from y E Fib Fib ICat Sets a full Grothendiec can The Gr the functor that p is represen is B split TEGORIES AND SPLIT FIBRA op H ordinary et in the c the one ha w L from is obtained b G A t A with Notice Sets the a H e passages b et functor that Sets poin Sets of v B G eserving functor ork Jacobs w Lemm Prop osition This or every this e a lo ok at the bres only er is that they are often easier to describ e F basis op INDEXED CA C cf t can ev B of The next lemma states that the bred structure appropriate for split brations A The ab o In view of the previous theorem indexed categories are not really needed b ecause o w HA e tak am with obtains H the ordinary sense can b e describ ed asdexing structure functors in the bres whic Notice also splittingpr ie an adjunction constructed in suc Cat exhibiting the corresp onding indexed category to the functor eralization F ho w functor which is ful l and faithful b Pr categorical matters are left to the in one Y es as an B Y with R tak u K u f A same B B one morphism restriction gory to its Morphisms I C at K K a h admits o obtain the Y THEOR the ate and T bration to the functor X ICat spl it B in for dc vious whic B f A K Fib X A dby ob jects G A TEGOR g a split in and es the splitting on the E the ob B B as d in e KB B to essential ly KA Clearly A p is A spl it bration e in KA B or a morphism id a ar denes is denote KA u B f preserv F Fib D B k construction D H K e quivalenc one X v then q wing result A bration u e mentione w sending an indexe gories de and an arbitrary functor is gory A e to u A X E ate B and ate c ICat Cat B with Cat ws readily I G can b e turned in y u A BASIC FIBRED CA A d K Cat b Ku in E R quivalenc with A X H B op Cat wn e is a br ove a c and E B R u f maps a split B ICat Indexe Cat do p A eab Cat y alence follo ed op k B where is obtained b ecause b y ICat B G es pairs la from op A ICat A called the Grothendiec op A is CHAPTER y tak The br A a A D as in the diagram b elo g yin B B are pairs spl it The functor on ob jects is describ ed ab o KB B A al version of the e A one B R op describ ed R G Fib ation Grothendiec K pro jection B E H goric I B in f Ku R Cat Naturalit ate H K B R B Y A f op or an indexed category in a standard w Cat The rst ations in the sense that ther u Prop osition indexed category F The functor Theorem B R A The required bred equiv B category A B K of of This construction forms the basis for the follo An The functor o o vious splitting K H p B op ase is a split br Ku A X K split br ob b Pr put This gives a c Pr B where nose to the bres in basis total I I i i en m C C im im C isgiv F has nite B making the B i and C ternal category C C Cat C of morphisms in G C Cat C C C h h C C m C satisfying h C C C C C C vides the in C C a cell in F k diagrams  and triples eries that id  pro C C id y a tuple i C F C C C C C i C C pairs en b m is a morphism F F F F is giv One easily v i ternal categories m i m B F C h oin in id w C i C of comp osable C m C C m R ts een t is in fact a category consists of a pair of maps G w F ute G is obtained B i F F F B bet i m id id m m Cat rom this one constructs pullbac F F F i Cat the ob jects TEGORIES id i C C m ternal category C C e requiremen are v m m are domain and co domain maps and functor m C ymaps C o diagrams comm tit F One has there is a comp osition morphism a category The category C w and h y and a internal ws C C INTERNAL CA C wing t Thirdly An satisfying with iden follo where C Summing up an in satisfying the ab o and In this w C pro ducts as follo where C a Y al of the has is The con u y the inter oneda B k map yields a A anonic c eindexing a  en b there called u r p understo o d K that is Y THEOR A a CCC then the the A be is giv uting diagrams is often K e w consider u B is C ar Grothendiec dom assume Secondly u in it eno TEGOR A ulation e e ab out in the Y should the D w B C B C h y and not just form pair K C of spl it tit ber of comm espok e orking R B whic v u w b eing quivalent to a split one the um D Fib id yw see Mac Lane IV ansformation category ab o Applying or example if C esp A F A q u r time that ya n A al tr K the and y H h one is morphisms a and ternal and ation is e in the C C i ariable categories w a p A BASIC FIBRED CA C of A en b whic bration for E A al natur An in that H giv es the naturalit expresses should b e the iden K B w id ob ject Every br ob jects B fact to to bred CCC is use Cat anonic erse in E K B B e B b elo the arbitrary are C category the u w this giv K u K split p a op y of describing v an H a of the c CHAPTER e and a u B h corresp ond to the dening equations of a category a B enab ou B categories be there tto be h a split bred CCC Her are ew u B d by the splittings of ulation B B w B e whic what a H a alen h categories are describ ed b A y A vides the univ K ob jects is suc u First Let form tit be u Suc K of A E see remark from ternal iden is clear Sets p K The data category In u should an Basics ob ject Let or every Prop osition u us it uting diagram limits F is functors induc B C wing categories of Th One the K o am transformation u resp adjunctions from comm base category pro follo as ambient nite in a base category split bration equiv lemma ii nal As a second alternativ struction to the functor Pr F t  B in  C am C e p alen id X i id is  C Assume wher C PER G cartesian gory B a P ate is equiv c snd is spl it it in this diagr ation AX if G Fib H br A X e d CCC internal and e that split smal l X B p etak X p Y a an W b B e fact Cat is b X X i ternal categories viz in terms called en e X i B alence of categories the is G el ls ther is an internal CCC is a split br is giv E p from B C The functors ternal in the base category p e that Then h is the most complicated case Y H in v ws et yields a split generic ob ject for the bration Cat L C urther on in one can see that small Notice that A bration from forms an example of a small bra F in follo B dding since the CCCstructure is dened categori e el ls and on c D smal l Set AX e C dCCC ase situation over one has result ar e e C da emb E B for some B es p C the Mor one G oth on c br AX M ternalization Cat ard eserving e exp ected there is an equiv The externalization functor see ii enab ou It is not hard to pro TEGORIES v e In then al l B C am Set is the Y is a split br tforw tha F C F is an internal CCC and But p duct pr see denition b B o P C B w that and a changeofb er v op Straigh smal l AX B B o e shall do fulness on cells whic C Denition Corollary Prop osition i By the previous prop osition M nite pr ful l and faithful b Prop osition W Y ust sho as one migh INTERNAL CA al ly Sets oth ful l and faithful e P e c of i of The bration Notice that for ii lo ii o o eb am C b ar wher is cartesian in and m Pr therefore that B is tion brations can also b eof describ ed without lo cally reference small to brations in and generic ob jects F to a bration of the form functor Pr cally using nite pro ducts X y h t S t Y u b or ery ter rom Set t e can F k l Ev with suc is con notion Y R Careful adjoin ts This internal The rst bien A X t adjoin C t us w u f A X g with com ternal CCC pr od am has y internal Y THEOR Th C f describ ed if the functor righ G b and m C and it forms an is limits an ternal in an where S auxiliary C D in ternal righ and in ob jects has an ternal P e requiremen TEGOR Sets onents S nite v F R in are pairs B PER and forms an in B C exp n us one obtains the notion ks viz This functor Similarly with an ternal need of has X y P R in e t Th C Cat has an in B w adjunctions are PER ell b oth as ob ject of ob jects and has ternal in an arbitrary category B ternal in Q P C j in C f t A internal j y map is an isomorphism ie a category with small collec C is in is in rst as w that o pullbac spl it PER ulate the ab o category F tit C t adjoin j w C B Y C has relation y example is in Fib describ ed b B categories RS internal C that But j C with y C S total y B of C one obtains BASIC FIBRED CA ythe y Cat Then sa a B denes PER This category w There is a functor if its iden B Cat ternal righ j assumption A X functor t can n be the one satisfying ytosa j Cat ete f j realizabilit S Cat PER in the usual A C C B one j b e a small category description discr D ternal vious inclusion P X The the and e C has an in R in is then a split bration Cat vious diagonal v from u F from the realizabilit CHAPTER Next let A in l C m B and hence in A bit less trivially es the lib ert Morphisms C Let ab o f PER ance for example in theorem A j is dened using comp osition b oth in B D G PER C B i unique id Externalization and an ob C kS n C if the ob started C The and u f in S Cat P B the P structure j B terminal m C w ourselv t ob ject in the category yields a discrete C ws that one actually needs only t C pr od F R if morphisms ducts C is in AX B o n C F C PER and ct of C e allo B j B ternal A X or or or es ternal CCC Examples Remark Denition A F F F The category in C obje category ts j R S t ternal category is called X Set kRl pr od i ob ject A ii ii if there is just enough structure around to form w on w iii y ternal CCC i it is an ordinary CCC artesian pr A F g pro jection Comp osition in that u p onen c sa pr od structed from of an in tions b oth of ob jects and of morphisms in insp ection sho no matter will b e of relev B base One tak is the exp onen N m in as minal Here ob ject an in The categorical structure determinesdene what e a e es en are W K erify E every has an A in category A admits C preserv p AA pro ducts via If C K L a ternal in one has for A transformations Cons AA an eserves this structur Then in has b e a cartesian closed cat then BB as p form and B functor natural B L ductssums OJECTIONS E sums if B which pr o p and It is left to the reader to v can sums means that Let pro ductssums if it admits this in ation L ductssums p B B B K pr Y A o L A pro ductssums B A pro ductssums are of course giv Y Y br one A A B Cons pr A BB a B B diagonal Cons A p Cons E E es A B B B KB termediary pro ductssums a Cons B Y B Y split Cons Cons A in L TESIAN PR a Cons ery b b L B B Ob j Ob j B e ternal v e denition KB BB A is in ation of Cons se ternal preserv B BB B ab o pro ducts resp sums if this functor In and A Y B vious B B t K L ternal categories the id ternal B E ob KB preserv admits split e assume that there is an isomorphism Cons bration admits p admits split pro ducts via p p L admits internal w p C an that B B p Cat K B K B y split ose A a Cons admits in Y and A w TION ALONG CAR E Similarly Similarly internal B i C a C pro ductssums for ev Supp p has ct ie A h p a ery A tication for in suc Additionally ev t resp left adjoin admits Cons C Analogously for corresp onding morphisms in A A or C F C Lemm Quan Denition By the fact that h Assume pro ducts if the canonical a QUANTIFICA generic obje over externalization yields a morphism ternal B tities e of ii o y that A ternal righ A yin that split A Pr analogously to Mor iden egory in b sa C Cons is an isomorphism is an isomorphism structure and use it to form is an isomorphism where this time is an isomorphism t Y f de erse map en in func A is pf e write Y y B t adjoin b p W pro ducts tication The pro ducts B ertical ard A u Y THEOR v Sets Hf Cons e nite pro ducts Cons the tforw pE v has a righ Y KB h A pro ducts has TEGOR op id category E pro jections B pro ducts resp sums p B L form socalled compre denes A whic E e forms of quan admits u v Examples will b e giv KB tication for brations will p and in it B one ternal es nite Cons is a morphism of brations where B BB if in Cons E f D E and BB tary description is more suitable B an pro ducts if the canonical natural E u orking in a suitably larger univ A and B The rest is straigh BA preserv resp admits a E E B pE yw E B p Y f cartesian K Y H L o denitions Cons u f BB KBKB K Ob j BASIC FIBRED CA e elemen w Cons obtains the canonical natural transformation v A es f f B B for oided b y that v E hthat y pro ductssums one in b A B b e a bration y esa D along suc b B B y B W b that preserv pf B B Cons K er the ab o id d by the map B Similarly BB B pro ducts via ev t E K L Sets e shall see that B w CHAPTER describ ed u u vious p E ery reindexing functor h that describ ed eserve E ob is admits B tho L Cons p ev tains basically only t Using these a general notion of quan Let Assume p is op Then B u p B Throughout base categories are supp osed to ha B tication hapter w has then p is suc and eispr erse of the canonical map p A that v D p is P B AA B B tro duce morphisms of brations with the ab o u b e an ob ject of y It t this p oin ery A pE sa A ery morphism ery ein hapter e size restrictions could b e a A Y pf E u a e resp a left adjoin v Quan E E Denition Denition Dene y f Let W Supp ose B A en b for the in p Sets AA for ev for ev is an isomorphism E H of Next w In the fourth c E i i ii o BB ned f where the third c This last section con if b oth from sums for ev hension categories be giv transformation via and this structur The ab o than Pr Mor tor e y is v ha ari step ybe cate ethe rules ab o be de suc tain v ks the and ma jor TSsettings example y sp ecifying a top of y con wbac Hence the new for describ ed that wn systems are to a categorical is that h determines the On y axioms be a w w calculi b h structures often b the sho a can theory yp e Systems will some kno h of suc setting yp ed tofsomeofthesema yp e description shall suc t tro duced in Barendregt e ts or pro ducts in that w features features this of s tion the features only to giv outcome The TSs an Later ebeenin emen v is MartinL Although y of describing t hnically problematic a abbr in the third section settings setting settings h is more structural and closer There w system based on e axioms constan ork rules a arious systems there are certain dra the h can b e added on top of w describ ed A more detailed treatmen The main ideas underlying T Finally in Systems and dep endencies tro duce a socalled TSsetting whic on e be abbr GTSs ha ts is quite problematic certainly if they ma yp T TSs Our approac categorical y arise certain vide an abstract w axioms e rst in dep end y the GTSformalism of W dene e Systems certain Systems sorts systems can yp ered b v to They pro of features shall es us all dT as certain extras whic e ks disapp ear w the alize Not not co Handling of constan ables as parameters Occurrence This is b oth conceptually and tec ard in the classication of v w yp e wbac y of thinking tuition of what is going on a dra w dep endencies that ma setting one can putpicture features giv lik where corresp ond gorically adding certain adjunctions scrib ed in the rst section b elo in found in the second section redescrib ed in the new TSframew and Chapter T Gener collections forw Belo Y Y THEOR TEGOR BASIC FIBRED CA CHAPTER s of In en the our yp e hild y t han then e s is v on the In s is Suc s yha erse of dis s n m form viz h a grandc tma s n m A Theory s the ie n y o ccur in a of Matrix s yp e row in whic s List T s hild of and e consider yp es and terms of a giv term n of s N s tt n m hild of N the n dep ends on yp e theoretical univ n m A n dep endency of s a s MartinL of length row N Zero Succ List Matrix s is in s In this thesis w formsac e Lets rst lo ok at some examples n endency meaning there lik N N s N yp e dep endency to name the p ossibilit h kind of dep endencies ma s n n List n m e dep m case yp e of lists on the LHS o ccurs in a c N yp es ersions Sort tended yp e v in t n typ en setting only if in s s e tt w pro ceed to describ e informally what kind of features Matri x in is the t viz The ts lik s eno n A one can nd t p ossible N W e to b e able to use b oth constan is an ordered pair of sorts usually written as en setting y is List tication m tit hild of o ccurring N vided with con this matrices n axiom ts ould lik quan iden inclusion m A setting determines whic An ork y b e added to the giv s s s ORMAL DESCRIPTION ariables Hence it also determines what kind of parameters a constan n closure ii s s s v TSfeatures h a grandc s Sp ecifying a setting means sp ecifying ones t axioms constan In the literature Notice that the constan INF of Ad Ad i i v ii iv iv vi iii iii term yp e axiom ma since it enables statemen In these examples t clear RHS system general one w sort p ossibly pro in whic can b e added to a giv course Henceforth this will b e called TSframew but one could consider additional features s y ts of s the the Sort that s yp ed t in hild of when that where e yp es are A y ts called t but typ sa means s e as a range is comp osition where e M then one can s w It ypical elemen see section yp es and terms The transitivit t sorts understanding that sorts are denoted TYPE SYSTEMS is called a c is preferred to under s Sort s in s for P for that has s er M h dep endency underlies v ving as t or simply a underlies simply s relation e etc text and serv one closed y the rule Suc s e s tly categorical Notice typ is a pair set s ariables s are t where B e bred o informally A alen and no dep endencies ie the s A CHAPTER is s e and an axiom kind transitiv A In case one has A of ts lik Metav setting s This setting for typ a equiv More s e ation has pro duced statemen an expression is of brations axiom typ the notation and or A B s y requiremen A x A an is that s hildren of in y b e put in the con Sort op that term B will b e called an exp osition pr Similarly s s or simply a s fact s o sorts A tuition e start with a set of sorts ha w A Sort an ariable write ance one has statemen in yp es v A x tended meaning the dep ends on case detailed tt presence of to y o ccur in c s in free description in in h relev TSsetting If y b e found in section a In ma one has t A more tuition is that if a deriv authors uc A b e called In the setting with one sort and as s i A The basic asp ect of sorts is describ ed b e constan as it is called in section set will v there is no in Some endency notions ts of the form yp e theory y thein ha the categorical occur s This explains the transitivit e dep t corresp onds yp e y is not of m s etc t of s t Informal of Q s s ma hildren of Remarks Denition Examples s An expression ariable an P nonempt Roughly relativ lemma e statemen In the system e in the GTSdescription w s s ii is if a v terms calculus or y elation requiremen see s MartinL or Barendregt These require a dep endency r Hence if something isfor in a a v sort it ma is dep ends on one can only ha grandc Q then are these dep endencies ma the sort s A GTStradition b Lik s the the ybe wing s ie s call The rele ariable of a Then op eration tication ed ma s to declarations e text olv v lik quan con y o ccurring in creates similar e Sort s w h a sa ariable s op eration v s s y rst giving the forma ariable in yp e in whic h that the feature under of t what yp e s exchang e a s t s It comes with the follo s then is s that is c lists A closure vi substitution x y A s s B x s there A B vlo x certain a w eak ening FV P y a requires Out pr oj ection if one has y M A B ordered with in A ersions of the Calculus of Constructions next s x for text and x e pro ceed mostly b In y if are A x x A W con Also s text x B a rules Similarly B x Let us consider the implications of these rules for texts A inclusion con In some v A s s A s y is a s t A Con viour under substitution will b e describ ed s A B if x s M is In the substitution rules the v x A A B s In then one can add a declaration of a fresh v A inclusion inclusion list s elimination x rule x s s dep endency texts Sort y M ailable sort A y B x The result is denoted b yp e s s use of v t e and Supp ose one has a sort s ya x y empt ybeused s e describ e rules for the TSfeatures s deriv prop for con yand ehension text Hence the s tit can e assume that for some nd ompr all Rules hild of an c s to con tro duction Rules ULES w one iden of in R A s ersion rules and the b eha h creates a can A t setting will b e left implicit but is supp osed to b e suc v s acitly an ontext s yp e a grandc First con consideration ma v tion In this section w s whic Then dep endency t c rules T one o ccurs under the name exten the dep endencies in yp e eak yp es t A t pro d A s s ories In N the bed C s e come to the N t terms of t aic Out TYPE SYSTEMS C y ytoem ariables yp es are closed under Hence w N d algebr t A s s ys b e added to a setting a alize s N M s s y e tro duced b tro duce constan s a yalw v B gener M CHAPTER C e s B m s s es the p ossibilit v A eha B A ybein A wing w In A e express that B m x m ma n AB B M A M N C M x M A n s m A tication is used to describ e dep enden I t dep endency really o ccurs ie if In yp e s A B an tt closure w inclusion giv y feature describ es the rule s quan A The idea is to ha s tit A s In that case one can in n B s ts units etc These ma y b e used only if w A s s i ts s A wing form s n iden n A constan s A s s n s ersions v yp es and terms in the follo h constan s A h things ma t s s With the feature y b e used if the relev onlyif The feature Suc e understand what substitution just llingup an op en space and w The feature i The A s A A y b e used if w In i iii A hma occurs wing stipulations y Ad iv Ad v Ad vi Ad s b do es not create new situations with resp ect to o ccurrences of v ucts and sums of the follo This rule ma and terms in whic Implicitly the sense of Cartmell on an arbitrary setting cartesian pro ducts exp onen ening are for suc b ecause a rule of the form follo C p ossibly with con of where AB M B x A x A AB A B AB B M x N N x M N N A i M B R x R R R M AM x B R R AM x M N z B h z z x MN z AB A B L L x B z z x s MN A N N M R R R B M M is done with the usual care R R s R z A z z A A A z z z x AB x M A M x x x N A s x B A tication B R R R R R R ariablebinding N A s N N N B z z z z z z B quan AB B x B B s x x L s A A AL AL AB L MN MN B L L s x A B x x x L AB N A A AL N AL for L x A x x AL tsums t pro duct x x AL A ersions ersions s v v x x Rules ULES x A x R with con Dep enden and substitutions Dep enden where substitution under the v and with con to B e units lik A etobe B v e describ ed B ely L W if A L i L B yp es ha L y to form terms able TYPE SYSTEMS yp es t consecutiv A i t R L t categorical role deriv h R L A be z s z B constan to consider L N e s s hi es the p ossibilit or w s out R hi CHAPTER R R s B B s R s s y an imp ortan B s turn z z B hi axioms B z N sort yp e can b e understo o d as a singleton z y L L R R A M M en ma h i A A M s s B hinery in motion giv either z z rule A unitt a A A hi N s M N h use s or R R R R B F s can B z z z z N onents i text pro jection rule then giv one L exp B A L substitution In order to get o the ground certain basic t A i since substitution will pla closure M N and A h s A econ v yp es last t M A M N h M h ducts i These set the whole mac this pro duct o M suc t ersion ersions v v M N Start rules end Rules for h tion it explicitly obtain the ailable the ab o o v artesian pr and substitutions Exp onen with con with con and substitutions Cartesian In men a T c previously Unit a is in ersion of the v hence ts A pro duce con In p oin and to The N A ersion relation A R v A yp es main rules t In A M N N in z N texts In A N N A of where one has I vious R con M N N L R N Out ob Out R occur R tioned the that the con y In z and z R z ma N men z s rather z M N A M ws MartinL s yp es M t A N Notice A I s Out In In M N the R R terms on e only A A z A I z M w M N In MN M A A r I r e L since I v y a marginal role in this thesis R R R y rules follo ersion omitted v y tit L but Ab o A z z z tit A con A and s R R A In M N e A N alence relation M v N A A M inclusion iden I terms In ha z z M In ersion M Out A v equiv on s s relation M In also MN MN s s yp es will only pla ulation of iden r r M M M N y yt A In ma I for con for for ersion tit dened v eform v Out one ersion ersions con Iden compatible only v v Rules Rules Rules ULES R s The ab o texts tended with con with con in socalled and substitutions where the latter can in fact b e deduceds from the former and substitutions con initially of i There x y h i R i b ecause p osition ariable P t x y P str ong elimination h P eak to dis z w M N C h The dierence C fron i C P C weak C i In case one also sum w Q tain a v in w R TYPE SYSTEMS x y Q Q h w C i x y i N rule h C s P the C C B C R z x y Q eak ones w con B h in part y P ards Q B AB Q w t s N z w her e yno B x A y Q M w her e P B A y N Q w R P Q Q y w her e x Q in AB x obtained A y x R xy w w CHAPTER P C R x imp ortan M z A y E Q t form the w C sum elimination A tioned afterw x or z P P i term x z x P x i Q s most M w ong M s expression x y h Q Q h i i the str in k i C the as used b efore ma C x M N x y x y tly dieren P h M h h for s i x y b ecome b ound blo c C as B h e y R R R i w her e C i rules the rst one is usually called w AB AB ulate a x Q N yp e M N i h x x t B ersion to b e men x y x y and h z z z w her e w her e h s i x y notation where x h i puts Q i N P AB let P Q w x y AB x x y h h M N and w i h A x w her e sums one requires the dep endency AB e Q ariables the Mirandalik Q x P x y standard v s w ersion rules are sligh h AB M v s A Q AB x tuitiv no w her e o sum elimination the then one can form P in w x Q e notation is M be ersions s e adopt v w her e yp e that to W t P Q quite or these s s F is seems Notice rules it Alternativ with con has concerns the fact that the the The strong con and substitutions The substitutions are the same There are t tinguish it from a strong v e P P P and The new N M tak ii P M AB s i as s s obtains sums i w x e y M N s y A C hh sums C ii if x one B tak h x s h sums and x y s w yp e hh inclusion can s s A rule A Hence unit s pairing s where s one ak where hi Then strong g s with x i M g en together with terms hisoft where as i a ong i AB Q y x i the y P str x y Hence Q has x b e orks i h and whic M x y are giv for y w g h Out elimination i C In en s P x y s one ong sums ie for i x hi hh y sums and we s Out i giv terms x ong x s In h w i A x y B e system with is h M s x or B AB M sums C Q F and s x y estr h x her In eak A In term Q ar es s sums i w Q eserves str and e Out sums will b e used in this form with explicit where s ong A giv y pr where w s x i x ducts ong the o sums x yp es with s s s h In ation B t C x Q hh y where In P elimination f f ong er pr  s x M Out estr x s sums P h or Q quivalent and whic s str F x s f AB C i ong Using  A y s s s s In Out x y h M x s str the Out the op P x g h C A  In x sums ar inclusion s s P e then e ork strong  d P w B s e for unit M i A s s B x Q sums The fol lowing holds in a typ s sumcase i i corresp onding hi Jacobs Moggi Streic and s s In ducts x y B yp e where h o t Out x y A In s x y C sums h the yp es h x B The s use pr AB x The The induc The ose one additional ly has str s do x The rite x i s s ak e unit M Lemma W Lemma with s where M Out i Lets s Supp N ULES W we A y where Q R and is invertible Q then P s AB M ong i of of In the rest of this w Q Out ii ii x x o o iii f  x x s In fol lowing statements ar str term since C Pr In h B Q pro jections Pr Assume that t y y is P b b B ybe yp es but y or P i N A pairing x y h L i obtained of wing rule is of is y ctions M N M h or notation TYPE SYSTEMS L w her e F oje the latter ma as pro jections e features are estab L latter onversion rules men Q these can b e handled x v Eq i The follo and ducts B o Surjectivit e to consider terms t i fresh N AB The x y v M h P x x nameless N x y L h P eha x i ersion i AB v x with B P een the ab o i rule CHAPTER x one denes i B w w her e h y artesian pr M N a h Bruijns N ii M N s x y c AB where h AB AB h x y bet P s h P y x de x x wise con x y t some diculties strong t h A A e i v alence classes of these will b e denoted b L P yp e systems w i P P viously where M using A B where presen i ab o The other w y x y sums the elimination and c h Equiv A onents and and A sums gives P x y A es for a term h L h h the A y comp onen L s s Then ob P M exp in s s ariables s ening as in i P M A put en b ersion i s i i B v ak ong eak ho ose the latter approac w L M and x y precise P h We ec FV P M N o results are standard W A B or str h x quivalent to the fol lowing rules with explicit pr w i F ery L h x v y ducts gives B ee texts is giv o y a a i In doing so v w her e or pr x x y F P h b eing vious using w ove ar w s s texts mo dulo con etc The rst t ery slopp Lemm Lemm i In one direction one tak Ob C x M L dab ance of of In the rest of this section some relations either e ii Q ii o o where y A Later in constructing term mo dels of t and con Pr b b eing v lished tione x using Pr obtained as in the previous pro of as dened b ecause tak also relation on con relev i e A es er r the the has one v tak AB role yp es term L write tak t pro d h yields to t x AB t ery sort one en of N equalizes can x one Moreo identities B s giv tions ab out dont ticipate unique AL B M e and en one as ob jects also B s s an a v L forev w are identities one x imp ortan s and h M dep enden us s is tro duction ust categories s s an th in M N A and z A m Comp osition is done whic M N the y AB with B e Eq A there and w B bre pla x x ws ersion x or terms sums v A B F morphisms L A A N the M follo yp es More explicitly y AB t s s sums and t pro of L L tit term x for yp e x C A A b er from the map x a s s equalizers a z comp osition text yt ong iden e x ersion of a result ab out LCCCs from x ducts tit for y L v str o arious systems some con A argumen indices and ong N that ha Remem tioned x pr A s es result C h e theoretical as yields systems with Then M w The morphisms tak s s suc L y A this yp e vious iden seen t is a term s M and of s morphism e Then One fx Systems yp e able in that con be v system s x t a etobemen e M N pro jection v B yp e theoretical v ab o h that yp e M can A and so one obtains an abstraction term AB pro of e system with str typ of T and has an ob Eq pro jection x yp es A text M N t r s the A x has es a t of suc A B N C B texts con I z rst settheory yp e C one In a typ that s con M Ax from M N A in y ts are considered as rather ad ho c and are omitted from the system of t w the erything should b e considered uptocon and A text one obtains a category with where i x A categorical x y vious from ii Eq that b yp es qualizers ie x L these A terms AB description t h N L en Assume x de onents if and only if it has A A Examples ery con Lemma Prop osition e come to the actual description of v or e N p I reyds construct F ob ject if Ob giv F if hapter C EXAMPLES OF TYPE SYSTEMS x ex M N c ertible to s construction s A an extract M N of and of First constan yp e In The next lemma giv substitution v o o and ev f o y reyd y this here but ev rst b has br Pr morphism and terms of as xed sort deriv for M our use of the TSfeatures ha Before w uct of t L T categorical and App Eq con A descriptions b elo F has has Pr only s s the L N B aker A ak fst In ack into i In y from A obtained we B A z is In d b x In In ws A bit we y h onents cte A N L hi i In follo B B In ee TYPE SYSTEMS A M In M  h In one has A s In inclusion In M A erse er w her e s i N ductsexp g In N AB i z o N rev N x s N an b x y z M A s z hi i h g In Out y A the i h N In i since A In terms In M and rst pro jection b In i y z z h with s put i i CHAPTER ii In In y In h N s h N AL w her e artesian pr N In on A y z y z c x h h In N s B i i from i es and terms c yN M Out system where c is easy z z erse of y z M e w her e h typ v y A y es ducts y h term h Out o s where where typ In h results tak is in pr a x y y y Out In s onents b M In w her e one in the sense that for s s Out w her e w her e A Out Out Out A ii The rest is left to the reader s i and y In M N In f f N  s alence b s hii In In In abstraction z L s M y y a In In In h h N In Out Out B M N App Out an s ductsexp hh o A A y y i ation in the fol lowing way y A B unit one has that B A er y z ducts The equiv h en b s o M en b op M L annot dene something like B pr In N M implication yp es Out artesian pr In is inverse of In A s In e rst establish that w her e c yp es Reection Lemma Out or t s t W s F The In Out Out x In or y F i of The last three results of this section are based on category theoretical ideas sums and an ii in the fol lowing way ii o In iii Out or In this way one obtains that Pr However one c F one has an with pairing giv application is giv Out Then i s previous lemma s c vi F vlo a and P compile F in e add an axiom inclusion inclusion inclusion is added y studied calculus y y tit is Here w honehasa tit tit iden whic g iden iden part f er in w Predicates tication o of epo tication w Sort tication eak eak g eak quan eak w w w quan w quan g f Theory t system called Higher Order ML HML is dened closure f closure great expressiv Sort closure yp e Theory Sort bines the previous t are Girards second and higher order setting system called axiom tly dieren of s T axiom axiom h do es not dep end on a runtime system with and P C P P HML CC System comparable eak HML eak CC System Setting w Setting w part whic A Prop ositional System EXAMPLES OF TYPE SYSTEMS The next setting com In the next setting the previously missing dep endency is set up as a Pi denotes MartinL These four systems consitute the left plane of Barendregts cub e time it The systems In Moggi a sligh e s It of ti the unit tages s lemma Hence s yp es an quan ts to and addi In case w categorical as t see adv s s s ytodoso s consequence inclusion amoun usual tication a sums s inclusion s TYPE SYSTEMS ersion only but s e require an y v eak e explicit names closure as t to do so unless ex tication in a setting practical s y tit ong tains e obtain an tit con quan s s free str has iden tofw fg con ew s quan iden s h s for closure w implies s s s ak whic Sort en with fg CHAPTER as prop ositions and come s we no restriction tication h stipulation tax ts ev tication tication s syn tication Sort quan whic s TSsetting with This a Of course there is no necessit quan e require in Sort quan s s s tication requires some stipulations ts quan includes pro ducts and exp onen tication in a TSsetting with with s s s s The rst three settings receiv unit s features s s s s sums Hence the feature s us s closure quan ets closure ws us to use strong sums w put quan tication an eak ong k s exp onen s eak sums w e s eak yp es setting w vides s str s w only if only if only if only if only if s pro w quan y axiom that s e b esides includes s tit ell require only b elo es axiom s s includes s mak y requiring the presence of the feature giv closure also e s s tication iden b w een square brac it inclusion s tables the feature w Pi t pro ducts and w P s axiom Minimal setting System s s s an Prop ositions as T System the quan s hoice cation since same as requiring w requiring Besides pro ducts and strong sums the requiremen In case the settingplicitly allo stated otherwise with w c s In Thirdly Secondly y help understanding these systems to read s cartesian pro ducts and Summarizing the dep endencies necessary for the features w ma with the remark thattionally others b et constructions s one can equally w this n it is of of of w its yp e con t n set ycon erview One nd wLa v ariables ariables results In fact categorical x category of hma viding o olving or example will v F w e follo description t in The exp osition Comp osition setting texts including tuples of equiv w x these m last i tion on the systems artesian h and as an o c of texts yields cartesian the i Settings are es Basically m M categorical corresp onding category as pro texts categorical alence classes of v a m yp es without free v our except this econ es rise to a description k and Rey features wn in y that giv tak y equiv that e fo cus our atten the Setting h kno TSsetting Morphisms form terms whic y en b a suc and are tioned t from admissible TSfeatures Here w Ob jects form t h for m tro duction to our approac p erformed setting men are giv ones with M whic of be y consider a cartesian of fg tities can briey describ e where e dicult y nd that a TSsetting concerns the organization of con to M pattern w most tioned setting Sort is h form the left plane of Barendregts cub e ening hapters and the h features texts using pro ducts most Notice that the structural rules concerning con oc whic eak emen ws w terms v hapter w results thesis y round one ma the c ysuc a text is then terminal and concatenation of con of and and are this Morphisms and yp e theoretical and category theoretical Prop ositional This structure is indep enden bine to con ariables es at the same time as an in ycon in T tings previous classes y substitution and iden The role of categorical settings will b e the same systems follo h com ts for the ab o aims w serv The other w texts ie a category with nite pro ducts the examples eres use of algebraic theories see also Ko c yp e done b The empt pro ducts is presupp osed b stan whic tain free v substitution the v the minimal TSsetting section On close insp ection one ma texts con as ob jects of b elo alence Chapter The In t dep endency place in the last t y es a cub e settings ts mak es rise to man inclusion TYPE SYSTEMS inclusion with deductions as Barendregts y underlying wgiv yp e systems the em tit of y tit iden e three sorts the rst one t b elo v iden plane t CHAPTER e presen righ tication g systems but their hw o settings ha tication the Use of parametrized constan w quan ys to do predicate logic g a quan ersions of the Calculus of Constructions due f tv individual hapters should b e understo o d as sets f consitute e noticed that in classifying t closure The next t v Sort arious w closure h the yha arious rather complicated systems for predicate logic are Sort uc vide v The basic form whic systems m ork of S Berardi t in later c axiom so axiom w the sort ers these of They pro Belo p ossible hapter one ma PRED System Setting four HOL System Setting concerns not rst In Barendregt v In this c pro ofob jects ramications simplication This remains imp ortan considered based on w phasis The is due to H Geuv to Th Co quand and G Huet The last three systems are dieren er to to B re p v will add and CC yp es y t b feature texts at requires simpli CCC category either base t pT olymorphic simply a The con els see and is a e base of a bred CC er corresp onds handled B v a accoun o yp es and harmless CCC er t after P and are T v a o er a base CCC corresp onds where v o extra lev as subtle systems w w see also Pitts rather E The simplication is based ob ject b eing a setting dealing with all the ob ject axiom more B vince himherself that a split and on tication e come to the notions describ ed A setting consisting to systems B The hapter these four generic concerning generic a based yp e quan a t to con t for CCC is Hence w an ening the with pro ducts and sums w pro ducts and sums where with yp es in our stipulation ab out closure in the yw literature B pro ducts and plus a terminal ob ject for or eak of corresp onds B the bred w belo examples tained in Seely es it p ossible to deal with the ob jects of the CCC a Cons Cons CCC with bred nite pro ducts Cons h categories and category is a bred CCC with a generic ob ject o ersions closure elop ed in the next c yp es one has suc and v Finally b eing bred be bred t t This mak p a a e can disp ense categorically with t category is mostly called a PLcategory  e think of A description y features to a prop ositional throughout to is is W i Let ob ject The feature b e a bration substitution construct they are all con used categorical wman y round the reader ma o B h will b e dev a T is el and so w The tially er a CC can b e seen as the prop ositional both h category v generic the bration admits the bration admits category is a bred CCC with a generic ob ject o pro ducts categorical category E arning corresp onds a ely Denitions section whic p Essen that W Denition Denition A A A pro ducts and sums or to yp e lev nite DEFINITIONS AND EXAMPLES b da calculus see Seely ving w Let In the literature a The other w text structure describ ed this hniques whic ii er a base CC iv ha iii am closure v L be approriate cation o one the presence of cartesian pro duct t tec b eginning of section the t and for a full accoun with Cons dep ending on ho p b elo Co quand Ehrhard a base CC resp ectiv additionally additionally Notice indexing bred CC o In con i o a a h t w in m this of base Tw with yp es b oth m i t a suc ork out using m settings m i generally er yp es with M v In fact also t m exploit o tax y consists handled e n Hence hapter w syn of h are imp ortan e dont w y er a CC denoted M h x ailable op erations v n v categorical M reindexing setting e describ e again ho pro ducts hange rule see op eration In w x explicit text with h ariable declarations fol d v op erations es all a an Because w and nite OPOSITIONAL SETTING with con i terms fact Notationally The exc n g settings dene term e but it should b ecome quite B er yp es n h a bred CC o this w needs bred t prop ositional isa ely f primitive i tuples THE PR tuitiv m n epo one y consult Curien y round this with text and a m are categorical As an illustration express to B Sort ts but i declarations n con yp e theory preserv In to tuple of ening are and and B alence classes of h n x n eak bration a to a sequence of ariable a is v CHAPTER x i i h t remains a bit in of n op eration ening in t and w ork esp ecially in the rst section of c A i texts in denition B terested reader ma wing statemen corresp onding term with eak ie an n M setting one has yp e theoretical and the categorical approac y Curien h tion to one sp ecic setting tuple of equiv b h of m text denotes concatenation of sequences i in suc B CC The in Coherence of all these isomorphisms is a topic outside the scop e B e setting a is texts con i lik A B een the t terms but not the other w h one has the same expressiv ork er w h also has nite pro ducts and an v required es op erations only uptoisomorphism unless explicit splitconditions con o sequence er substitution h B that B are pairs consisting of an where ening is isa a ev yp e theory substitution is an inductiv tain w y eak texts of this prop ositional setting form suc y reindexing with comp osition as a sp ecial case e turn our atten b categorical ob j ob j E preserv are satised of this w w shift In t this ho b Substitution and w i bred mor mor A In the prop ositional ed p ycon ww terpretations this statemen w y E B the con b CC category whic enables us to separatelo con ma fact in the use of the follo No where setting in whic clear in the course of this w in dierences b et e a a is of of to v see has b i Set I I A Sets e i and also C w In fact g B functor i satises denoted category bration is er el N from ob jects f v Set M reindexing category truthvalue C be X o is a functor the Indeed this This result where and of X e Set all or B n I y am Winsk j k category will e BA e am F i v category F a g f h along ba i Johnstone Pitts summary PER M  pro jections ha a preorder ie B y X ba f j of a split P ter category whic functions X j elevanc BA and Actually A nonempt tioned that the bration then the family k j Gun x a are collections t is PER er a category ofirr  family m realizer v o I eried that M P ba o a v are j pr X e hangeofbase y B I set category bration ie a bration with pre c b PER t adjoin tro duction alley condition holds see also considered as a am j er m t to the inclusion F Co quand A b v en easily the ommon In der com o c d system as already men is or k if giv on in socalled e a snd am Y from is a split kChev Sums are describ ed b ery reindexing functor of F It category as dened ab o cte d pr is e It forms in fact a N e it w v alence another of cte it a M vide Set externalization dir e as rst describ ed in Girard ha ws k A X mainly A urther examples can b e found there including h dir one has a righ equiv Hyland F and not just the cartesian the explained A j yields follo k detail m Set y A in whic Domain mo dels the category t Ob jects as As Set based t and the Bec In M fst that B These pro called j j is In ii a e category w are is h forms a split X B PER i k where is left adjoin tioned Sets am t adjoin y F yields to one can nd ramications of the notion A B morphisms in b bres j a complete Heyting algebra see i of mo dest sets again M theories men i has a generic ob ject mo dels or It f a x I is is a split Y This ba h as set F BA e B I exp osition As i Set C X yp e o basic examples from trip os theory see Hyland j b e a top os t m ertical am denition w to j i m F Sets PER k V Pitts y A B order B X y t resp left adjoin B j wing tioned a directed i com Extended example I M The t The bration S and Let In b oth these examples one has a After tics of categories a DEFINITIONS AND EXAMPLES Hence if en b realizes C follo men Sub am h is complete and co complete and cartesian closed e PER ws from applying theorem to example i In Jacobs F Sets n partial category ii j iii as am i j am y B whic F has a righ i cod order them as logical innite pro ducts resp copro ducts i ev follo seman is giv yields a collection j the maps I A The dealing with nonextensionalit simple PER mo del whic from b functors of w b oth a left and a righ describ ed b F it is a small one t h is  C ea v es the E ternal E B in B Cons bration exp onen category admitting an examples  They ha  the BA E CC C ternal h preserv es the structure a family E in E ternal CCC  ternal CCC whic that ternal and ternal ed is small hnical result will b e E BA the in E tioned BA in over olv E tication in of  v  least BA an is an in is an in E then E E CCC at E is CCC with OPOSITIONAL SETTING E E C C b e men pro ducts quan d  BA it e or h the bration and all the rel C BA BA E E E E br B B ternal categories preserv ternal CCC nite ob jects description a then BA Cons  THE PR E CCC E E e of B b E a E with A A A if the bration in B e notions a useful tec is is an in actual is a CCC and v CCC is a CCC and ternal e notions will BA B B B B B v B C E E E E E B B the smal l E category in whic collection small that p  category but a  al isomorphism id a et CHAPTER L is category E small be category if it is an in C category if category if a assume categories is a morphism of brations whic category if B ose of A morphism of split hapter e If  E pro ducts and sums w has ersions of the ab o denition B Let v i ansp rob enius E yields a vertic category is a is a split ternal iii ternal ternal ternal F only Cons  simple in BA The tr ternal C E category will b e called oneda an in B in Sets B ed see eg Asp erti and Martini exists see the denition of in  If  B is is an in is an in is an in split ternal C olv more v Examples Denition By Y Lemma t structure see c C C C C A C A morphism of A sums E C A B an tly t structure is split tioned of Implicitly Before describing examples of the ab o Finally i i E Sets ii ii o iv an iii iii am BA F in admits in ob ject men Cons pro ducts and sums Pr more in category in sligh relev onthenose ev or en the the F put is a and x The YA XA giv z ter y Hence A DEP second X X X e dont DcCat DcCat forms a A A x t and vi x enough uit family therefore Here Actually A A XB A X W satisfying N Gun con H e the A A is tin in again y XA uous functor P P A X u the h am x v P A X in F A n tin in Y H n complete s in A wn b F where write in uous ij DEP P Supp ose and that B F whic u t tin Then in case that P wise Co quand A x a domain in t B can x DEP con ed do P later is F P y t ab out con con is in fact full B am A poin see function one A is A F is la faithful A B x and u is DEP P The it used is directed H useful is a colimit in uous functor ie for a con t u uous then g and el tin con be X F be where map for that tin A Fu But then DEP one denes am y the ob jects a full with ob jects A x w functor F g t X will DcCat y restriction another comprehension will con A z ordering P A is Winsk A YF elldened X con and a h e consider them together with the base DEP h sho a con b e the colimit of the Y w u f u A this A P the am g i A is X F whic whic A so P X x ter only uous sections A i DEP A and P e t DEP generated b A x H obtains in brewise family tin ab out i w t P ws F X Gun con x i A is Y A domain con v in one e one obtains b in A As long as w i and B f P am x b e an ob ject of arro v DOM P z a v uous of con F am Y P f pro ducts established that B j hence e DcCat A results F This settles the rst requiremen is tin A in u that B y FA that Then are P DEP P X id B let con uous family Co quand x X DOM A g X t names to the total categories in easy A some a u g j tin tees readily of j p DEP A is XA DEP DEP ery with x A is u H j t t tion Then ev z A yield It prop ositions en establish is a domain X X con con in A A x one obtains a category A x i A result guaran or e F e dieren x DEP men giv y F X x am am w uous pro jection y and a con e t A faithful Let i e F F p A X b is t A tioned functor i w DEP tin B con v category with uous x X Y is main A there is no confusion u i x x P the collection u el A e functions it tin am P Y DEFINITIONS AND EXAMPLES Y e X A emen X F u v A Next The As a consequence of i ab o B Finally A i I A u morphism ii con ij er the full sub category of i u Y v A second one is left to the reader Winsk iv domain as a preorder category one has with category o and category b other to giv comp onen ob jects of F a morphism P is ab o the con this pro jection is a cobration That that for X F f P z these split Hu f X A j a to or X er i are F v this g not j func write called A B unique of e Y without is a case W split is a CCC p uous func applied in i e B moreo is f uous a i v the family is B DcCat v tin Y is tin passed ij ab o and DEP it u tro duce the name there yof ij f Y con category u DcCat as used implicitly X i the id uit ein w of j tin the In DEP v construction el for the details p with X t has nite pro ducts structure The category f uous ie directed colimit B k ij p e for Actually con OPOSITIONAL SETTING u i f tin e and it is directed complete ik hapter w v Cat ogether with con f am u This DcCat T F Morphisms j cod X colimit where con DcCat w satisfying p DEP a and P sub category one has DcCat i ij A P I A op ter Winsk DcCat u where THE PR A I u a i p id A i Grothendiec g b edding pro jections as morphisms with g has X g is f B e I write A P In the next c e jk e f It i u i f functor e A DcCat The erify that g R DcCat DEP v i ks a C W i p B Y system A h f e ob jects of i I i DEP XA Y A i v v G i k u i t DOM v suc f as f ork on these categories but it w one has B DEP CHAPTER con Y DEP o conditions i for A A j t w w am DOM directed f F in con category ws to pullbac i e rst dene a functor i DcCat gory category B w ery am A w consists of a pair Ehrhard It is not hard to v F wing t a in ev ate f c and if A with indexed bration id DEP collection there is a full and faithful functor form u B DEP is a b ounded complete algebraic cp o in an tion in previous w y a directed set together with a collection of maps in families ii split y the follo ery ery directed colimit Y u ehension the category of domains with em uous functions satisfying other uous functors omplete a C Co quand c A form tin tin domains for d example domain ery for ev for ev e terestingly ompr X X h maps cartesian arro c also ev cte W In A B yields e indexed b ontinuous see expressed b are con via con see Smith Plotkin and Co quand Gun tors c category explicit atten ful l As in whic tions DEP preserving functors necessarily small directed complete categories and con map B for dir means that there is a collection satisfying e e in for ie and this n bre G vice e op m op m ersion v E id bridge of C een the X There is ha C the ery CCC ha CCC n w t families and forgetting and op C e F Y Sets Morphisms F rest F C that n w H F urther details es F op N and ones v F F wisted v the giving in H n is an algebraic e e use pro jections ork in Bain X y C C eput C en F in e constan n H b k Notice Y C v basis e consider categories X It has nite pro ducts Giv n t ie that ev Morphisms w NP X op or W Esp ecially tion bers G negativ eha F en CCC Later the free the Cat ythew C C op m i for the t F tw tation more accessible w C g G C ten um to w in F n y C op ork in category yields suc b C G Hence C NP X op F y in Morphisms of these are required that n g m m n F G n n C X Our where N stands for negativ furthermore w ob ject b n Y F C hanged f C C op C as ob jects c F ws C C tw Esp ecially m C NP and e the presen C in the top os of preshea n n n n X Notice n NP are n E and NP ery split X op op op n NP n X y a role here i n op categories yields a morphism b et or arro C terminal F C C ws Y Ev K g A F category from a giv C tw categories E i op op in h yields a top os mo del of F the F h families ery C Ob jects are natural n p tw e H of arro ev v F Y bre category n n o ccurrences i ws In order to mak bed C t functor n X for y one obtains a category e G G F y do esnt pla a ere X op tforsuc n tw Y functors g y F G C is the pro ducts of H closed b wv F K X e write bre ab o i e form the category m indexed UCTIONS are Our construction is clearly inspired b m app endix p ositiv f C H w a morphism of en cartesian closed structure n a constan an the category C m Y In this w C n e as follo al X g n dene pr od exp f C at K Y w that this forgetful functor has a left adjoin e go es on to em et g n cartesian morphisms where n describ ed Y n w n n y requiremen n F dene X viously n C C C only ed from it H e k G G b e a CCC W e this structure onthenose m X Ob C w one can dene comp osition in w g n n bridge g C Y g No n f proj but dinaturalit op op F NP SOME CONSTR Y obtained as the comp osite of F f i C are families C  H The next construction requires some preliminary w Next Let Pitts F y b e found there n F in H ery ob ject n g tw y lo oking ersa H pr oj F categories K no dinaturalit F of see Bain F v is terminal and n f et al one is deriv section is to sho generates a free ma rst construct a simple nonfree under certain size conditions whic with an explicitly giv to preserv b the rest corresp onding theory in the sense of La and P for p ositiv ev pr oj e a is B W T can tro The these these smal l a from is E of is in of It is not gory e E p v B ws ate E ulation used gory c ab o al ly smal l and in a follo T c E ternal category ate lemma The details see E then E B c p E  morphism is lo B gory morphism y gory b a from a B T ate ate p e c Cat is c is The bration bration Hence  small for i is again a bred CCC with OPOSITIONAL SETTING C ob ject E E is an internal small E is a adjunctions and i ose a op E ation wher om B om isomorphism E via is fr fr  is a split THE PR generic Supp h p p p p E ternalization Sets C exist see E is an internal p with and easily bring us to the fol E that b or b B using B p rob enius p e a split br F b in category one can form an appropriate in in fact and ternalization is also p ossible see and CCC p p B a The or functors gory CHAPTER is The gory e b situation E situation p bred ate ate tially due to Pitts although the form e c  sum p E a b ks for yields a form of in c gory E ase  ase et et   that pE be ate dCCC L L and e c et hapter esp ecially Then B b e treated in L constructions T It also E is a t B assume E vious see changeofb as ob ject of ob jects and changeofb ed in this sp ecial case pro duct and k w from and iii that p esmall Ob lets E the the erify that pullbac is an internal is a split is a split br Some hec ob ject theorem to E p Theorem Prop osition Prop osition i Let p p ations from c By lemma es ar C pE denes with gories gories category of of i The next result is essen Next Under certain sizeconditions in ii ii E o o wing result ab out externalization urther urther ate ate E duced in there is dieren F c Pr One needed to establish the required adjunctions for these new sums generic in general also b e c hard to v as ob ject of morphisms whererst pro jection denotes in the global the pro bre duct in then yields a pair c Pr already kno F The next prop osition is the main result of Asp erti and Martini al l br lo Observ t n L is It is split g f t yit a K E a functor through n tially as w G y L e the CCC g with unit this en b L f detour yp es and terms and In the L y the fact that the preserv tt X is giv t es a K viour on the constan E category B mak whether analysis X C y b edding as constan C is constructed essen L Similarly one nds pro of split f f C settle en b n a y the restriction NP to G and on the rest b The be L ws of L since their b eha ed b B andlet y giv w able f B L LX E yp e theoretical t be follo t E and category t C Ob j p C L or example migh deep er n C F Let a transformations e to construct a unique morphism v h that one X ed e pro of of is a full em ws is describ ed b v K B eha v Scott that all morphisms in the bre categories B UCTIONS ed the free n X h enables structure hoice at all for dinatural B hniques K t calculus suc tec CCC as categories suc in the ab o it f and actually these There is a unit functor es the There is no c are SOME CONSTR tzens sequen C n of As already remark Using wn in Girard Scedro o of split f C families of ob jects andstructure arro should b e preserv again since p sho Gen a term mo del startingDescribing from ob jects and arro Pr structure onthenose W preserv generic ob ject via e in C f the in still y The that Cat op G as This all are C negativ H en where op dened m eries NP v giv C the ob jects n are f f F C as n NP f of ideals or p ers easily to H There C m h is cartesian closed f are cartesian closed NP f and set n ersion of a construction m One in structures NP category generated b C n F dto H e OPOSITIONAL SETTING f in are v F NP examples With a term mo del construction onstruction ab o G c H satisfying y the reindexing functors C n n ck H THE PR C f F K Comparable similar is a categorical v F G ws is a category whic dened NP n F n morphisms C ws n F K as in arro C Cat role othendie F onstruction applie are to o big G i sub categories n G obtain a n and Gr dby n y earro ck c op N to these h v op n ate K ed onthenose b G the pla F m n C ha C f w that the bre categories CHAPTER m f but n G and gory H n F G n NP G comp osition K othendie C f dont ate appropriate n haracter of the construction F c to F gory gener use Applying F f NP category n ob jects K G n H Y e G ate F for The Gr w K dene wise c c is a category with nite pro ducts in H n t t F has G C from K w F G G F G from a case categories n X no s o ccurrences n e construction do es not pro duce the free i e n still eries again that NP n e v F K F e e of F i F n w F F K K F n n F g X C bre F latter Theorem Prop osition y the p oin Basically one has to sho K id K pr oj ev F G F f from C yields a split p ositiv m of NP The This construction of The ab o i i f instead the mind v ii ii o iv ix vi iv iii iii f vii viii morphisms in these categories are in the smallest collection satisfying yields the fr in smallest collection of functors stands for free built One easily v NP In used a few times in Jacobs section starting from a and Pr and that this structure is preserv holds b examples i ii Cat b ecause the categories o h w or the t Q cate men whic of wthat In case e natural cod w a ersion of ks esho used in q w gories extension secondly as a W functor be ate forms a Finally pullbac e standardly write en or split text is will v e write y to construct more w al l The main notion here then con w enab ou of doing is a full comprehension B P gory of has hapter P ehension c setting and Theory c A J B of ate ed is clo is a full and faithful functor B c category ork E P olv ompr w v next D part B P ha dc the es the p ossibilit Q erify that the t sums and pro ducts Suc categorical kin ehension close do in ob ject These matters can b e found in the rst It giv els hnique due to ompr c The for eg lev A t categories rst as a a tec is a pullbac P er the bration in It is easy to v Category f ys a P functors Similarly w dieren en bration p whenev dom estigates is brewise a full and faithful functor yp e dep endency is our next sub ject v The is a bration ving E P related comprehension category in case ha B split or a comprehension category The third one deals with e go o d closure prop erties Q P Jacobs F satisfying P v or comprehension category ful l e dont require that the base category a E tication for a bration P and Fibred B closely t functor theorem settings P dom cloven P is a cartesian functor e call E is cartesian in Comprehension P is called a t but Notation Denition cod f P cod P Q Notice that w The fourth section in i ii what w reyds adjoin transformation and comprehension as can b e seen clearly in the term mo del example b elo it do es p form This It is called category if and only if Chapter More these categories ha The categorical study of t is and second section can b e understo o d as categories with dep enden gory theory on topcomplicated of a giv domain of quan dieren F tion some standard results ab out lo cally small brations and a bred v OPOSITIONAL SETTING THE PR CHAPTER t w ting map righ gory triple Jacobs a a ate the bration in y where a b has B Dc E L y a bration h Q en be p used b e represen A en b giv B whic E e one v is P E E is giv y P ab o the ertical isomorphism in E q Let E K as in the diagram b elo E E en b p gories q B u P wing denition captures this Q D A than giv ate let c E u E erse B p Q Q D E B u A isasav A for B univ P The follo E B general gory with unit B of a comprehension category reect functor E A A E ate ehension L dom K L large explicitly K B P L K more B B E In case one has a bration with a terminal y requiring that the brewise globalsections B u u ompr ob ject B c tly B E more B requires that P of ehension c sligh bit A p a suitably tto one TEGORIES u A is terminal E P ompr E B E c tro duced in Ehrhard a under the name B id a to the basis B A y Ens t adjoin morphism P is a natural isomorphism satisfying kin is a morphism of brations table y of doing this is b B with E i A The ensuing functor a en b P morphism is K L K B giv of y of understanding a map via vided is counit then forms a comprehension category see Jacobs a represen vious w Ens E text comprehension functors pro Id then table where L This notion is in are B notion B where Q P Denition Denition op t terminal COMPREHENSION CA win A P This Another w The con E B K L adjoin for the pro of and where one has ob ject an ob functors the total category bac with is represen arro situation Hence one obtains a righ p i y is Y B in T n no for T are are N T y y next then M form of texts B B E A X is e write in con M the Cons the ob ject P Cons y b e called nonempt Notice that in of or an ob ject Y THEOR the ob jects a yp essetting ma F is M display maps the collection T B Y i section j then w h T pairs T has pairs and where g E clear j The determined b as in B f terminal If the category f b ecause there is are T and B TEGOR a X The functor organize ts of B The functor u Ob j h used B functors the terminal category i T B A X u become y with m B shall x M in onstant can b e understo o d categori c E e y b e found in the term mo del pro ducts for will elemen Ob jects of Y t sa and sometimes w B g tro duced in when the con comprehension e form a split full comprehension if for some akening Prop ositions u f P B tication y id W we u f this g called x i category sub category of nite x the X text f ctions h u M claim t construction to obtain socalled con alence classes of terms be morphism of op erations N quan oje and e write A con B MORE FIBRED CA with The total category Cons A E pr A this will nontrivial w f the full i o extremes Morphisms are called jP B ws is terminal ersion Cons and categories w v E g and B form X E pro jection x texts ws P category B p erforms f y comprehension supp ort is called and t con as follo tuple of equiv a P this A ob ject is the o arro ob jects T M n P B and yp e theoretical of T full of be A pE consists of a single elemen B CHAPTER as B comprehension category B are often called a e condider t and ob jects from categorical u ation for this terminology ma B T a u T y E W f a Cons h i other t comprehension P functor y where that T this pro jection is simply Let o j in ed ts suc X Motiv instead of m A X i w B E classes t the j o maps olv e deal with pro ducts and sums as tains all M y v consists of an T e the terminal E w tro duce a simple but imp ortan claim and v with m n This notation coincides with the one in e con y yt that Similarly Cons ein B b these constitutes alence T Cons typ our e write expressions B fg n setting Example w If the collection If form en b is nonempt on x is M A Ob j E basis t comprehension categories equiv and The Notice Next w The comp onen It i this t X the ii x y are giv then dened b with the bre ab o Comprehension categories dep endency in Cons struction describ ed there is applied to the bration based section when w cally using this sp ecic comprehension category stan collection of ob jects from B category satisfying of of are B x with describ ed reindexing functors of the form terms of describ ed next E Sort the y E b E E wthat E u DOM therefore B P every u etosho v or d with a functor F by Assume DEP see eha and the comp osition E t ovide W B P ation con g gory er in id B Set am ate E c Then k F ation pr given by P v E p ack of the fol lowing form g Q pE and u P theop P pE B pullbac E e op eration B E v E a ehension w v Set E P u P is u E u E u jP P ose a pul lb E P ompr E E P DcCat e c o examples of split full comprehension cate then one can dene a functor P E jP u w and e a cloven br a B am b e E F E B P B Set u b E B A gory u in P erse of the ab o E v ate B E P P TEGORIES B Q P u hnicalities DEP alence ack functor E u A ansformation t A j u an al lways cho v P p B P con v w E E A et f B f al tr Set L am u text comprehension functor t ii in denition one c for the in u P E j F ehension c u B et It is easily established that it is a terminal ob ject functor and a e ose a pul lb uE e already describ ed t L j k to the term mo del describ ed b efore one nds that if there P in am ompr E a u F pE is invertible for every an cho and a natur functors I u j e consider some tec By the previous lemma using that t to the con B Prop osition Lemm By requiremen A M The Going bac In w forms a c COMPREHENSION CA e one c u e diagram E of of Next w Lets write P v o o vi iv am Pr P from are b oth examples of full split comprehension categories with unit and Henc the Pr is a unit as describ ed in for F left adjoin gories with unit viz the equiv h T if Y A of and T full C h that u f Cons B then a whic B is K L terminal bration is Cons D units am collections t situation suc B K F and Y THEOR the A B am The all K F with es category unit K K Ob j I that es the splitting TEGOR h C The latter is obtained with S preserv Q KB KX es the adjoin is a morphism suc am a i categories t preserv olv F y functor on and v and A D tit K t X B category H a comprehension t comprehension categories from C comprehension category ob ject q I B X Ob j am tains a terminal ob ject dom faithful the i F y S y b e found in Jacobs a The iden This con T T is full and faithful see Jacobs and erse of the canonical map gories with unit is an isomorphism T v MORE FIBRED CA comprehension ks The functor B terminal L L B K This example in ate id of full a c Sets dened b with nite pro ducts and a functor L a comprehension is k to the constan BX pro jection a is A Q H with nontrivial L for the in from C cod P B S P y collections the a then for ehension K induces a functor examples A case BB P E B t D CHAPTER L In a I that ompr Lets go bac category C Ku Kf bration some i e p a T g Some more examples ma i C i of c There is P e write category with pullbac is X B o categories tion f K w H C family observ y K splitting b be a X men es rise to a map of comprehension categories with unit Assume nonempt small the e B Then there is a morphism of comprehension categories B also en the w morphism is e are S where giv Examples es Let A Supp ose KB Consider t A functor W ed S Next h that the canonical map olv t X T ii ii iii v y transp osing B X morphisms of these b preserving these w KB suc full comprehension category with unit ob ject it giv in Sets and only if the functor preserv Cons admits a unit if and only if the collection K brations preserving the terminal in C a a y y is b P On one rst b P in mor for C p r where P fact y D E C b y f h r E r a in Applying case Q P w B is Q in Kv C B P wing f g and a E bration E P P P P category satisfying diagram es the unit r E the g C r e Ku C v the follo p P es lemma E base E B or has a unit K L e form P ab o olv v P preserv v cod in denition r with Q v e the First P wabo lik is full one denes in E P Clea C E Starting from a comprehension cate P p hoice up to an isomorphism of com P pg ws Q P cod a new comprehension category uv r r y r yc B P P b r E follo C P r rf yields the comprehension category p as C R C r and category cod r erify that o comprehension categories v brations E LE w hosen one nds a unique map p where c p TEGORIES p E r C is the unique arro p r p en t E r C be C P A E er that the map one obtains a morphism brations along r C C v y Giv r to E can of r D P C comprehension e description of this construction in r is then determined b the reader to Moreo and a bration P P Q r to C E C r B comprehension categories is dened b hangeofbase is can b e extended to maps in C morphism left another E k functor ho ose category D E P E f g P to a full Changeofbase Juxtap osition h The It is This c An alternativ P base COMPREHENSION CA P h f ws E ii iii hangeofbase h arro whic and then c The resulting prehension categories morphism of comprehension categories phism of brations full or has a unit morphism of gory with c constructs the pullbac comp osition in from L is is Y E in P has this E E w D of tation u v Moggi say via E P One that e just es art P u is compre E E E category one forms Then he from Y THEOR ws unit with en d terminal yp e theory But e j B B L u v sho that w giv w where u v A E hthat B u E j e v are E E suc E and TEGOR B One can also use this v ab o E B E gory with B E v u P P j j P in ate constructions K u comprehension y Ehrhard the u text rules in t P w is based on a construction E and faithful j b ys to obtain new comprehen E E a eserves the br result on a E E P u pr of j P full y e that if u v v and morphisms P P The u v called pE e based ehension c A E E en b E ber of w E for a map A  pE E y a functor v P E u es bred limits are b B pE A um B P u ompr y P E MORE FIBRED CA e as a basis for an equational presen E v B denition u ersal text comprehension in fth j with ea c the and B preserv b is then giv E P E y serv is the required mediating arro j w E one has in and P B P E pE E pE The rst describ ed b elo P B is an iso since is us aP P pE P u A en a comprehension category e describ e a n w is univ has ob jects th u P third A S E one has Giv E CHAPTER e called con C pE E id It is closely related to the con B enco ded and v P B A tly b e expressed b the en ones is an isomorphism henc u P The isomorphism is u v P C E eha This arro et h Id ove E alen i Id P L w and ab i unit morphism B and P v f whic E E ymaps A b A P p u e iii The category Then B P By the adjunction E E en b v pf h sum The functor C Constructions on comprehension categories Lemma E ull completion E B t Remarks i ws P for for F The unit By the previous prop osition and i The previous prop osition ma There is a comprehension category that p B Ehrhard vious iso in E u v f are giv of i In the rest of this section w Using ii in the previous lemma one can pro i ii ii ii B v o iii ful l iii P sion categories from giv from E as follo a v Pr p fact to pro hension category with unit then disjoin the heart of theesp ecially matter to what w of split comprehension categories an ob tablished can equiv using and v that P j t h p es q f the ork and f mor e the h are j E and a j E resp e in v P P B j tadjoin r approac jP jP pf ducts f o hose to w  q morphism E then preserv P This P asso ciativ pr ec describ ed j b e a comprehen E P p E t R pro jection j has a righ P is Cart i E along B  of Similarly the compre j f P pro jections P R q h e E adjoin cartesian v unit D P y natural transformation admits P jP B A E en base category ha q sums i b oth class E tit j ery p f P ening functors whic left a OJECTIONS j P pE ev pE R has a pro jections P E eak D R r h R ethat R for v YPR resp resp suc e recall from that the bration E uptoisomorphism Cart y the iden E t or practical reasons w ie P W and a functor arbitrary j F E o functors hangeofbase of q B p an j determines w e righ y j v along b pf P By c holds pro ducts resp jP B es p P P B E b e a bration and bred is formed b along P mak a B o comprehension categories and b e as ab o E E ening functor tication q w determines t E f E has P admits een them condition j P t eak This i p D q P w en t j B Quan P h q E ts to the corresp onding w yw and w RP ulation in used in the denition ab o Giv q g Aunitfor an alley Let E E TION ALONG ARBITRAR arro y that E one has that the canonical natural transformation en comprehension categories on a giv category v Let tication Using lemma one can pro P E functor y adjoin esa functor E E kChev o brations E in j a w this It forms a comprehension category if ery E E determines a group oid bration Bec if resp a left adjoin mediating Quan Hence clo pf B fP E Remark Denition ks see also lemma Comp osition is an isomorphism E the for ev E B QUANTIFICA obtains the Id P comprehension is E vi sums similarly one obtains the second one hension category p The rst map is the transp ose of natural transformation b et one obtains t see lemma P generalizes denition inwith Ehrhard the a brewise form One the reindexing functors of these pro jections next denition b R pullbac A phisms structure of a symmetric monoidal category symmetric op eration on sion category w w n h m Y in in P for D E eac m A P i where E A or f g E pE n F B tains that E as ob jects E read D with under Y THEOR A f and pf w with E D D con E A A E hains n to c categories E P PQ hain of pro jections n P pE category i TEGOR n P PQ D E f g E with A one should E A p with y A E B Put E P n E P P Q B E is a morphism empt in y ws ob jects PQ age A v e is category of E E comprehension D E from uting with the c m E n n comprehension category tak E a morphism of comprehension cate is full resp has a unit in u P the y as follo a b en n hains P v B be comm Q D be MORE FIBRED CA is dened b hains D E E p B B clo E A u i yc D E E o D in D f B B es rise to of the c A w m morphism t pq Q m t requires a clea D P The comprehension category B giv n D q D E A Let E E A one E to comprehension B pf p E p P h  n if  P a y D q text Q tion that n Q sequence E D in P A p P a D n p D Supp ose A E CHAPTER E empt is Q has nonempt form ob jects E QP Let e men n f p A P E f g One forms a new comprehension category A This last p oin be D as E B E can E u cod P P y yp es q Q P t en A for w B E ma from of one pf p is needed here E m f and hangeofbase B P D are giv is a comprehension category is full resp has a unit in case hain One has pE that can b e seen from hain c ery arro A A c care is used as initial con A P D h that D P A A Lo calization Multiplication P P there is a morphism of comprehension categories Ev g A The functor Without pro of w P The category A or Q suc E h P n i is a morphism P v E E E A iv qf ob ject whic part of and then dening B lying bration Q gories in P A morphism A little B ie satsifying P satisfying p erforming c id D be T An yp e x with f T es B b e the G m directed functors merit of A subset D I J B J x a as ob jects F pro ducts do I J Let describ e t N n G iii m ening uous functions I comprehension It is I D to n y or tin for eak b yields a compre F D n y f be a cpo ts w x D yp e with a singleton D e are able to capture satisfying D ts to ts to cartesian pro duct J x D pro ducts some q yw ts Let a D I P E B D I exp onen p ossibilit I J and texts cartesian for OJECTIONS is D tifying a t a morphism of y n theyformcposthemselv y adjoin m y exp onen D is wn space of con D uous function t D cartesian e a CCCstructure es the x YPR In this w x v D tin P y closed subsets with resp ect to the y iden els tly left adjoin F el of con giv righ I of ideals in J K tofideals theoretical b h it K G ii L for H K B I er a cartesian pro duct for ideals and y yp e is a con see eg Barendregt the lev whic ev n L e dont ha and Ob j x m w D e a diagram in result D I J D v D P with ideals yp e theoretical T I el viz B eha able i i of these t J D e write Hence w P mo del is isomorphic to its o E B D D w q K to tlev as the collection D J D n in B yp es and indep enden with pro ducts describ ed b remark In general ho One has D D term Ideals are the nonempt T tt al I a g is a morphism of brations y one forms an exp onen Supp ose w TION ALONG ARBITRAR Ob j F is a ts without t I J a dieren ide tioning a mathematical example here I As usal description es Cons IJ yp es together with With the ordering inherited from K H y giv texts of the minimal setting see the b eginning of and let T h a cp o is D Scotts id at a the X t ideals categorically x F in exp onen K L The pro duct of ideals This orth men I F via maps is called an Denition h J w lets assume that D I QUANTIFICA y a role G D Of course It is w One forms a base category No In a standard w But taking In a sense this text y j I yp es D corresp ond to exp onen theoretical t category of con pla I the collection of t con comprehension categories to separate these lev in whic categories and X Scott top ology A morphism from f and example of suc D do es not seem to exist in one can dene hension category these exp onen it Y or F M yp e if ed cate ducts wn in limits is the bred e M o olv ks one v is closed It is not pr pro ducts ii am n Note also has F T plus bred states that This y nite B omplete Y THEOR has it B complete P if there are small X admits pro ducts iii with tly as sho T y smal lc ably b B yp e dep endency from TEGOR has innite pro ducts small see A alen Pro ducts are obtained y that B a Cons C omplete is describ ed b This corresp onds in t c admits sums if M prop osition see esa T Remark Ob j w Equiv e Here category admits pro ducts resp sums sums V am B a has pro ducts resp sums with called Cons P F b T has them p as on is constructions if it is b oth small and complete tioned that the bration limit w Sets a t comprehension category M y relation a B i the b in case there is no t X M MORE FIBRED CA This bration is then complete has dened C also that sums omplete k e b e a bration on a basis with pullbac tion the comprehension categories in bration T B a constan am eried that a category B am A m is cartesian closed y cartesian pro ducts F b een F pro ducts resp resp A n smal l c as already men generalizes P h are not brewise preordered see Hyland t comprehension category diagram and is f has E B The realizabilit B h has p ducts Hence for e dont men b enab ou o cod CHAPTER h is b oth small and complete small whic B b e a comprehension category j B Let P It is easily v LCCC b of all ob jects in the bration k X ery ts and sums b y comprehension category d pr one has j i i B e i an cod B y reyd see eg Mac Lane ev It has nite limits b ecause tit a a a w if  F Set br categories whic p A iii P f E d ie e b e that the constan of ery slice category has v wing in y exp onen P the iden ts exist in p is small br B j b k m Examples Extended example Denition Denition follo b e a category with nite pro ducts and let A After denition it w In A bration will b e called Let en b result Set B j A In ordinary category theory a category is sometimes called In some sp ecial cases w h that ev the A ii ii ii complete a iii ys that reyd see also lemma one can require nite limits for a m gorybration is one whic is This biimplication extends to completeness in resp copro ducts i f Hyland Robinson Rosolini or ii b elo complete there are no small complete categories except preorders suc F pro ducts for the bration that it trivially has sums Let hard to pro if exp onen are giv theory to the fact that is under cartesian pro ducts pro ducts and brewise nite limits resp ect to usual denition in bred category theory resp sums i sa d of d e o ong and maps D Sums ethese further D Orthog compre of ehension q sums C D E air let This is done morphism p admits str ompr ED E and a ful l and a ase is describ P in C and B Q strong sums for CE eac ets assume further b R r L is an isomorphism ductssums e forms o een More results lik gory eserving a C v like in denition w B q sums ED pr ED pr D r ate q P c in bet ducts P has strong sums ie strong C in o E OJECTIONS duct Q from ii Q q q r C E C o sums ation d by changeofb pr E dene strong P P tication pr P d left adjoint then r P admits E P eserves the nite limits and pr e P to and YPR a P relation ehension op r ductssums as q q ong has d nite limits o es admits obtaine with e C y denition ab o ie pr the q pr ompr en brewise see q ie a br P D c P ation and P e morphism q q her D induc PP r v a r isb r e seek r C gether nds C e has br in w has b admits str ducts to r r E o r then eabr Jacobs P which has a br Q D P b ation q also P ontinuous pr r B D C B in P q ction r CE is c gory E E one ducts similarly P r ee in P bles D E used in o ate C D a D q is dr r admits pr P e q q ductssums r air of morphisms o P q There r E C TION ALONG ARBITRAR et et C in ations P ard using the map ations the br L L ductssums and that pr o has d nite limits y of terminal ob jects and P br e tion some useful results ab out quan ulation D q eries that a comprehension category ehension c pr that e br tforw of CE b e is a br C P P eserving p morphisms ose form B Assume ompr emen E r admits has br has of artesian functor categories Lemma Lemma Lemma P i By the fact that the opp osite is tak If ther Ac q Supp q q Straigh Notice Similarly for sums sums y can then b e lifted C QUANTIFICA ductpr air o p gory i i of of The next lemma resem One easily v Next w ductssums y b e found in Jacobs Moggi Streic P P ii morphism ii o o iii o iii CE sums see denition i the ab o r urther the functor y dening ate onalit c faithful c r r A apr B in ii the pr ucts F are handled similarly Pr b hension Q indecomp osabilit ma The latter Pr e a y a Y g v sa and and D The from e ab out B E w forms w D en y D E morphism b The ab o j D erications a D Cons belo Cons giv V Y THEOR D ery morphism D the Q of be ust b e part of and fQ diagram m uting square E u Q P that q can B TEGOR describ ed E e result i ery reindexing functor y that ev yp e theory v has Cons artesian a P ove c ED Ev denition D o E ab o K L H in Q one E dab hnical is c ha w H e E sums v one E the E en in section coincides with D Q E of P explicit E forming a comm the w D E one has canonically H brations Q Then to w the u v ED E categories LE and D qD D in sums in suc E split Q v MORE FIBRED CA D and e need the tec P that E the canonical natural transformation H E Q morphism w with D h E similar D sums as describ w a a has Q P E D u erify E Q v or the strong one the bration rst E E eak and strong sums in t D F D E h to forms Q D rob enius isomorphism from lemma LE P pro ducts via brewise adjunctions P comprehension tication for ery But t admits This corresp onds to the extra dep endency required for P H exp osition D q if for eac an CHAPTER en and id via w D has An E e using the comprehension categories if for eac ose eak case y of Giv v diagram q sums in case satisfying e all the structure onthenose i ma P yields an isomorphism ose e describ ed w Supp E w the aP ducts D cf the previous lemma is orthogonal to the class section o E a pro ducts ted ab o ong E ansp ers the w B v tication and corresp onding maps giv D P tr reader t generalizes the F every str E in Denition P or has has Lemm Remarks hapter w sums in and F The The Similarly Supp ose that quan ED E q B Q in i In c i ii ii ii ED D The latter means that for ev Q that in are easy and left to the reader second p oin strong denition co a comprehension category there is a unique Q an appropriate form of quan should then preserv the one presen Cons is an isomorphism P that morphism of pr er set the ob and and v the x maps j where at A id urther e nite e ab Set The pro duct Set F ed is y functor X t of these a Moreo considered uous a x tit appropriate olv are LCCCs G e Y and v is e P tin a b in ts X am the am tro duced Set F con j a F F sum in a G Y C e treatmen The iden lattice vide a elemen for and tioned in Jacobs with and ts are related lik pro comprehension category Sets M Y dene has X men e DOM where is a CCC satisfying Barendregt Rezus b y Sets x er b er that for a domain The bration can sum X Sets C am v complete B a F es P as already describ ed in example P b ergHansen An extensiv equipp ed Cat The full and is an LCCC one Y the a B y Moreo strong Remem a x X DEP x comes t Sets y DEP domain P P ere and y one tak F con a Sets TEGORIES P see section is an example of a split wv am consider en b P the a F am for e am F X y F k construction to W P Y P X pro duct almgren Stolten x ktoLa DEP P b e a category with nite limits Y G Closure mo del y ance B b categories category In a similar w is a CCompC y comp osition a CCompC is based on Scott and A and is a split CCompC e write This is not surprising b ecause tials relev w that nite pro ducts and exp onen t is that all the structure is split Let tioned at the end of B B X functor theoretical of DEP id functions P i DEP the ordering is giv a Cons yp e y b e found in P also t uous X A one has b uous a exp osition is functor b e a category with nite pro ducts tin As usual w X P tin comprehension comprehension x The B Y con teresting p oin this example go es bac id o examples sho wing X are b oth closed uous y applying the Grothendiec a see and v w is then a full comprehension category with unit and strong sums Extended example Examples and from has a unit and strong sums The comprehension category The The term mo del of the calculus Let The Similarly iv tin CLOSED COMPREHENSION CA or B F P A F Y B Scottcon v ii ylor closed Set Set con iv vi iii X a end of anda vi is also an example a y is But the in tained b of categorical structure The follo T constructions ma CCompC the comprehension category with unit w as a functor domain of sections men F G These t on Cons limits and lo cal exp onen t e Y k for has base with great h are q Bec functor is full y of pleasan are b whic and ed is split P is notions er the Y THEOR t the end w v olv with A v sums next sums artesian ation of the ter c y ii b b ecause and cartesian moreo Pi and TEGOR and is describ es the terminal mo dule categorical preserv  h a sums D faithful tro duced sums unit one has P P in B D  whic t pro ducts and strong sums pro ducts other  and e B has be forms  v with u  q D for strong GD it The ful l to E D ab o if all the structure in u categories P a u gory u D GD functor E G  E ate a P c split top os u dened P dby MORE FIBRED CA E where E D ob ject a to category P E GD E E or tro duction and summary see Blanco P P been and u E P eserve pro ducts and P P G oted to examples and prop erties e is a morphism of comprehension categories with P t E A pr v comprehension category abbr CCompC is a full ehension E E ersions of the systems e yp es setting E E E terminal v extends is v  d ha   A d left adjoint unit P LCCC P P A e a  A F ompr E E B E D ab o e D c an close v CHAPTER E B B B B B GD E with which B tioned in In i comprehension A ha y ersions ful l b v A A A A A A E d ct comprehension to a E i u u u u u u has a br D of CCompCs CCC es the pro ducts and sums additionally e categorical E is a G obje close e P a E By lik closed of It is a category with a unit and dep enden required F ase Then for q By a standard argumen then c h preserv Byi D is morphism E Closed terminal Denition for i In A A closed comprehension category is d e alley notion b e able to giv FE of Most of this section will b e dev Dene D ii ii ii o br D yp e dep endency as men category resp ect to the comprehension category itself see ii minal ob ject in the basis is required unit whic Chev The imp ortance prop erties Comparable t will based on the Prop ositions as T comprehension category ob ject Pr a G y is in C A the and also u f m C A A X is g Y B B C A kconeb B f u family f B X u m u the B viously holds that ws with arro f o using C A u comp osite C A w C A g y t u b f C A has strong sums A A Y X u g the X y is separated then also Y B B and lets denote the pullbac B There one has that C A B sa C A u B A B u C A f w A B A h B g A X and a pair is a bration The rst equation ob Y B u B sho B B B m B B m B TEGORIES j B to m u cod and to is a full comprehension category with unit SF in C A C A is complete f kin B f g m w that if B A B X B y families are separated Y C A suces tit B B g it esho and a map B A Y u B monic B W j C A ving to the bre j f j f B A X SF X SF SF ymo dense B u a using the pullbac obtains ws b en X obtain one giv separated families vious since the iden u u to C A or or a family Ob F The inclusion The comprehension category The bration F The comp osite f f CLOSED COMPREHENSION CA A X u i u order ii iv iii Therefore assume one has a dense monic separated B u second one follo In B Ad i Ad ii Ad iii t b y u Y b z ws ie z a slice with a b en a a CL X e write y A righ z the bration w u giv P from the a CL CL u A y B are arro ws has a split Y THEOR CL A b CL Ob j is a CCC with split B a z X closeddense with b b j CL where B a a P follo CL a a e use that A am is CL v im u I w F am z or B a X TEGOR F F erify that for a monic z P and yp es dene satisfying is then a split bration y the stipulation that a Here w im A a ob ject a Y xy b g m CL v X The latter means that X x to P CL z x ery y CL am eries that v a with P X F x ev separated in a a is a pair Y j u Ob jects of functor CCompC This y p X a t or z z X yp e of all t a a w X j a F A CL forms a sub ob ject classier and x b b b is a y is z is p ossibilit pairing A P Y CL am e a It is not hard to v C A closure F b One easily v for all MORE FIBRED CA B the of separated families is dened b bration x a am z X A p X f F v Y satisfying u us CL of a B B and j z x surjectiv es B a one denes P a in j further there z a giv X that the CL im closeddense one has that u B j uous u Separated families in a top os It SF a X to is describ ed b A X Closures form a category j existence t B tin x b top ology z v z v w Y A P and CHAPTER B again w a Y j indexed family of morphisms een closures is an elemen SF CL is a top ology b a then a a is con w the g In Y xx A a C A is elemen of closureindexed closures z bet u P a The rst pro jection an x top os SF A with is b Y Y An arro A ob ject via X orth noticing I a C A an CL X X b X CL and x Hence this example supp orts a t P the inclusion b B is j is A X result z CL is w B a CL am there tly top os x A e A and F it B a a one has m u a that x a y one obtains a full comprehension category with unit B Ob j A alen f j a in A CL that P X a Extended example be and C A a terminal and crucial closur t B CL A or B wing four results x am A use a F Finally A a The full sub category A X in CL F e claim e B a p X in a adjoin im it has In this w B that m category This yields a split CCompC Let generic ob ject im t follo W x a w pro jections morphism or equiv as B id k Then er It is a ws v u ed o Y A i u A But since X ery monic olv ery ob ject Informally a v Y pullbac g f id B h A a one obtains a B u X Y Y u f e to b e constructed A Q B A v has for ev u The idea of using an B a y e g forming F y that for ev and tion for comprehension b B X F Y the monic e g g A A en f B A a X v uting square as follo X f t X Q B f f the Q alence f B of B A A f Y f w equiv X A one can tak that the top os an with the prop ert and y maps o ccurs also in Cartmell A haracter id one has u f unique A B ard but pro ducts are rather in TEGORIES Y a e diagram one obtains a full comprehension forming a comm A g The notational con A A the c yields v A is e recall B X y y haracter b elo tforw A f u f be X A Y B A A A there A X X and A A y the ab o u f y g B b C A B asthec X X is dened b f is a pair CCompC the unit pro duct and sum ha F B B B Q B A A Q y a pro jection as displa and for f F g split A u f a Y alence induces a CCompCstructure for A f B B ed b for let X in map As indicated b w B A A F F A Y functor a to b e a t in Unit and sum are straigh q artial map classier S and Q p f Q Q B A a t B In order to dene pro ducts w The This equiv As to sums for ob jects The unit is simply obtained b C CLOSED COMPREHENSION CA bration an B B X f Y B f X ew A B A split inclusion follo category A functor but there only for thecategories category of cf sets leads us to denote the domain of Q for Q explicitly w Y y h Y in es A to the ws The a f giv h are g A ks b whic ernel in a should to A t X f h A wile One the family A Y THEOR X B nak orth obtained w A argumen of monic arro A are X TEGOR B CCompCs whic o CCompCs namely easy u eort w an from has strong sums ie that the split with co domai y B e o indexed collection b Rosolini pro ducts w B A Ob jects are maps ws mak Sub es rise to t k as maps from the family In split structures there is no need nite A B follo morphism A this e ery map g v A itgiv y f MORE FIBRED CA a of families orthogonal to an ob ject Robinson Orth Fibred result w is to replace substitution via pullbac and an arbitrary map g u t B C A ard w w that there are t This f tioned ab o A X a Hyland ho osing for ev isomorphisms B A tforw u Split top os mo dels a a yc e can consider see can b e understo o d as an again X CHAPTER X Orth j wing diagram is a pullbac A e b egin b Hence w mediating In a straigh to those men w adjunction y situation B B separated b e the category of families of see i and the inclusion of a B is the sub ob ject classier er f the B nast t in the construction b elo B is v of B y F C A t a similar to t of this example is to sho a or a separated family with h that the follo f a Extended example use X Let F b e a top os X This yields u alen tial p oin B suc to u es B B In the top os The full sub category i In informal notation deal A g vious w erify that the comprehension category essen f substitution via comp osition where where to Id equiv The p oin rise v orthogonal families are closed under comp osition Let separated in B B mak ob Ad iv r B ii are The y y r b P C A B comp osition in P C ations categories y b C br C in B bre of pro ducts and strong L A is describ ed b i describ ed A C The i P g obtained g id id A is a CCompC h f morphism e is a morphism gf P again g a P of A t C ed in ii it admits pro ducts f is P f r lo calization r wing standard maps u i A is a CCompC asis ther and domain A A P P TEGORIES y P r CCompCs C one obtains the g C in the b en b of where f for asis A u e need the follo This completes the example is giv e a CCompC t P b is a CCompC P hangeofbase w B P c P comprehension category with unit P P ertical map P in the b P  et A P splitting A i L estigating prop erties of closed comprehension categories write t t ct ab out v and w the y i beafull i yin are no a unit for morphism one has a v P e a that A w g is again full and has a unit as remark This yields a split full comprehension category Let P viously results f or every morphism or every obje or o Prop osition Prop osition i F F Given By ii r F Ob Notice ute w h e pro ceed b CLOSED COMPREHENSION CA t of i W ii ii ii g o e then put A or the pro duct determines a morphism of CCompCs Pr and strong sums b sums of CCompCs rst and iv W F Using comm f partial orders a Y X e v f A A f A A map a The cate Y Y Y A g Q Y X y only o ccur Y THEOR A A g A ed b ecause in g g Q are as ab o ma Q Q Y olv A v f X f A A A a X Y TEGOR X Y as ob jects g Y and A A Y X f A A f a Y g id Q Q f Q f f making f ws where e A g B f A of f f u f A A has X B f f A A one can b e describ ed more easily A u Y Y f wing arro A A MORE FIBRED CA The construction is so in has maps id A B Q Q  B A is a pair A X A A B A aa  B f g g A Q Y B L Informally A id f X i L g id Q a in X P X CHAPTER the logic of f A A g alley onthenose the dep endence on g Y Q X X B A f  g Q Y ev g taining haracteristic Q id kChev c a X X id f to id its X X Q Q f X h as e put A con g A Q B a X A second CCompC w L f f ws f f ii f Finally This construction is used to form the follo order to get Bec gory from in the maps and A One can form the equalizer follo h E es B in Pi free t Then B wn as B B of hmak B B L t ersions of P L v L e similar B logically Q K v e HB o es nite pro d ob ject lemma with w are t Cons KB an morphism K KB using HB can b e obtained as a etotak A yp e theory preserv HB v B HB with Also the term mo del of This result is kno A KB B vi of categorical B one can nd a pro of using KB since establish of s t KB eha L B categories CCompCs B w HB HB t to B h vided Q B es the bred equalizers Q of L D K L pro Suc KB equalizers KB B B yp es setting to B BB KB able HB Q L L B  BB BB has a generic ob ject ab o Picategories with a generic ob ject es pro ducts B be Q Similarly Q Q KB h preserv B bred tains a similar inconsistency result whic P morphism Q A D H HB TEGORIES E shall BB BB BB BB preserv category t B HB B B cod with e ersions of MartinL L L L L KB analogously KB P L eries that KB KB w In Barendregt B L e Q a p unique L e construct It applies to is tak B ery prop osition is inhabited W an Dalen or Jacobs a pro of whic section with the description Pcategory is a CCompC B section ylor con CCompC L K example a CCompC only wing that B yp es a B A a B One easily v an yt in L is next B B K can is B i ysho tit is there b ecause the pro duct H e L Q w the is terminal and Pitts T Picategories are i iv and ro elstra v B yields e close this B see Girard mo del B of B W B B Pi e obtain fact uptoisomorphism e nish b t B Cons category adox b ecause BB end t Picategory is a morphism of CCompCs whic In p W ar t in the sense that ev an Denition closure B Picategories are categorical v L es the CCCstructure k the A A L H ds p CLOSED COMPREHENSION CA t yp e systems based on the Prop ositions as T HB calculus gories Examples of The A ar h that Cons ii iii B ate KB suc the c consisten These Gir forming K pullbac the isomorphism ucts hence w indeed constructions the t only s and in T use of strong s is obtained with iden preserv e a y y h Y ts P a el is This alley onding b ecause Supp ose P CCompC esp exp onen ed e In this w kChev e Y THEOR orr cod  CCCs c fr and E to p the y Bec is a functor whic d as a functor the preserv As to pro ducts w e E b  d t id u y maps TEGOR en u D are E E then  P pE etwe is E u onsider involve b E B E B P categories u The rst and last step hold sums in B H  and E u ation E A Then c and P E E P and  br t B P E B E morphism iii A P P B E E E E the a u E y E MORE FIBRED CA t B u b u es comprehension P u Cons y E P ulations of unit pro duct and sum for displa E en u P comes from e a CCompC induc ws up in the basis for displa en b b giv P E E A A E CCompC closed CCC E E u u e the terminal ob ject one obtains from the previous lemma P   w the CCompCstructure dened on the top lev a are a P B P v P A A P is giv ducts e is an isomorphism y e e o In fact strongness of the sums is not needed to obtain t B E E E E b b b from B CHAPTER ws ho P b ersome form B B E ed CCompCs E E et in et P E  pro ducts L L B of k functor E ard but lab orious ard functor et pE P B i L preserv P E The unit her E E a a tforw tforw are is a CCompC with terminal A B e lemma sho v P Cartesian u oid rather cum morphism forgetful v A E Theorem u dby a Lemm Lemm h the pullbac i The unit Units Straigh A Straigh ation is the basis for the next result ate d CCCs d CCC of of of of By lo oking at the bre ab o The ab o D E e e ii ii o o o o pE eserves units sums and pr y y ii B result this see lemma br b observ Pr in whic in terms of bredone adjunctions can sho a b maps br gener Pr Q pr Pr obtain for they are strong Pr t is G er v an p a what maps F i ie in yw c B r if b oth A ulate form vi a cartesian B r er B er v is er a bration v vlo o v a Fib F Fib over in to form in p one has that functors and p y reindexing functors A p CCC o to tication also concerns a p rp A p G ed b E is p p a om B q A to G are adjunctions q reindexing to the reader A enab ou see also P rp A F p E and TION p left eand is cartesian p v q r en b etter picture the reader ma B B A F F A U A It is CCC of brations and cartesian functors o r A R that r A with A there is a pseudo map of adjunctions from see Jacobs for the denition ork of J B p er A A v asabo B artesian functor fr A A A Fib A c q B A bred r A in a A a G Fib A er VER A FIBRA A example F Fib A are and that the previous iden v a is E o is cartesian form brations r in A A YO p A r er p e for A are v v In order to get an ev F p i b oth F p Fib w that p there is bred adjunction r to One calls p eha A A ery B er A and q v F ev Hence adjunctions o Y THEOR A A p to Fib p i G q e exp osition is based on w a v ery ery ery morphism e pro ceed to describ e closed comprehension categories o one obtains a category qq is cartesian y F wn one has A tion is to obtain this structure brewise preserv r a CCC TEGOR F h ten F for ev p for ev for ev consequence w CA een these erify that for a The ab o Next w Next consider a diagram w As sho whic bred The in bet As a As b efore one can sho functor from in In this w r natural transformations o the structure the category to v is an adjunction o is Y er v the o then there hange This is r A y restric yc er B b v b The rest is bred Y THEOR ersion of the A p een brations B w A as bases B bet A cartesian and A q category p TEGOR u E p a cartesian is A A p if b oth g A E B is a bration o r is B can reduce matters to tly extended v p p f er functor forms v r one where bration can b e obtained from is a bration itself e that if that functor a A p g bration cartesian u u forming a morphism of brations p A t to a more simple one namely a p as A cartesian i MORE FIBRED CA er R E w A v t cartesian structure similarly to eg bred alen is a bration o ery reindexing functor observ her ho o p an r turns out p if all these brewise functors are brations and q A y it er is r Supp ose a E v A A and ev A E B dom f E E A u is a bration A notice that over q in explained e the relev can b e written in A CHAPTER wing diagram p theory f f as e go one step up and consider brations A A p w ation determines a brewise More explicilt it ww y seem since A br A is a bration o A a No ma p p u A Lemma lies at the basis of all this it A As to the implication hapter c el as ery ery cartesian i p en as in the follo Category This yields that is A bration as a basis bad rst for ev for ev is a reindexing functor One calls E cartesian with erify the implication as is giv The rst part of this section up to is a sligh A the ery ob ject in ov This rather complicated notion is equiv T not dicult f ofbase p reindexing functors preserv cartesian pro ducts tion Ev In q r not previous lev fth section in Jacobs Moggi Streic a base category e d B e es e full slice giv br a strong hange denes ompr it a c its c is e obtain bration B E r and all one the or a is a CCompC cod i called again F P forms a Hence w ation be p E E using ii br E terest pro ducts LCCC a will p in P implication p is closed kin CCC an er E of v p E is over in B is a bred CCC is er E erse v p B p admits o bred b e brations rev A Fib h gory E a bration A in E that ate TION the is Id Fib c p whic obtained is a pullbac p p is an LCCC if and only if B in r EE to from fact cod LEX B for dom f g er is a CCompC o B E R rp er v a r E As v is an LCCC and reindexing functors preserv e a bred terminal ob ject P the o V v A be ehension i r P p and E and E ws B if there is B example cod er B from r E CCC V v VER A FIBRA ompr E of unit c This example generalizes the previous one p ws rst V d E YO P E from t with bred follo p is a CCompC o p p of the biimplication E E p The ers a sp ecial case close B a v is required to ha E goesasfollo Let Let A i is This r cartesian in eries that a category p t adjoin ery bre category P Y THEOR er are LCCCs category B E E v A V E gory with unit over P LCCC B with nite pro ducts one has cod ate TEGOR E situation B if the comprehension category Denition Examples Denition Denition One easily v f g a terminal ob ject functor a bred righ CA p bred The next notion co In that case ev ii a E EE sums moreo The implication Notice that comprehension LCCC P the LCCCstructure see lemma rise to a generalization p hension c is categories ofbase a full comprehension category o with p a a Y be E E fact is er pX The v el l in X o E also Ob j The full B B E E forming a c pE E Lets write can T r A P g Y THEOR and the T E is a bration in h restricts to p E gory en in cod E E e applied to the T and T Let v satisfy E E ate rp P E T E A E in p e ful l TEGOR E E for T B and as A X E V T with bred nite limits s ar E oth u B functor in A Cons pro ducts P P u ehension c f and V cod i b A E E E vious functor E P P r r eisa P v ompr e written A ws in v E u ther are maps The ob P one has is a c e ie arro A eha v bration B A P E T gory over MORE FIBRED CA Lets write in this sp ecial case One obtains a full comprehension category gory i al l ations and ks w a id v where R A E E ate E gories ate B E B with bred nite be A is a comprehension category whic ebr E in ate E X b B B then yields a full subbration of cartesian Ob j A P pullbac comprehension categories rp f E X T A A u t X if u y E B p r v r E T E E ehension c y ab out E ehension c B A and B p y cod CHAPTER b ehension c r p A Constan ompr P er A full comprehension category ompr v b e a LEX bration ie a bration w bration describ ed at the end of E cod ations Let i A ompr ery cartesian pg gory over B E X et er A is a c L determined b v bration ev the terminal category conincides with the one giv ate o E X tioned at the end of the functor u P c A E a b e a category with pf E is a ful l c y f g er a is a category with nite pro ducts the construction ab o v B p the observ P as denoted there b B is the arro every B Then B with is a cartesian functor Lemm Examples Denition By Let Let r If In the sp ecial case that h that for E ehension for the category with ob jects for for every morphism of c over P A forms a bration o e of in ii T o iii ompr sub category of E be suc Morphisms X for the comprehension category dened ab o bration w where Fib B B bration As already men that describ ed o Pr Mor is obtained b c a cartesian functor in p a it ct in can First p E obje E E ase on ehension E D these E E Q E p D ompr D that E comprehension terminal Q ong sums p Q ac a is y uses isomorphism ED The rest is lab orious B using the morphism closed rob enius isomorphism ED E then one can assume a p ductsstr One g D o e E E D which pro jection E dened b pr EE DE rob enius obtain E full Q erforming changeofb in F in e seek in Q E E the is to ad of isomorphic to w p P this E ys on morphisms P dbyp TION E DE a tication for the base bration E e in of w P in E D e shall see that b oth can b e understo o d D that situation D o P Q Q admits d CCC and E E w use e E t P a from lemma ii ase E fact y f b The analogous denition D D E quivalent inste vide ED P E the ation obtaine wthatquan Q e a br D b E E pro aQ VER A FIBRA B E D and p E B with D esho appropriate Q E p changeofb E YO f e E i E and E f e ED P E from p p ork immediately one has to use the F First w Q One gets strongness of the lifted sums for free in ductssums et ation which is e an that the br o ws E L Later on in this section w p pseudo P Y THEOR pr ard denes a p ertical map follo Q and E lemma is er The pro duct functor p v tforw e One in yields the desired result do es not w D Then E Lemma TEGOR constructions yields a br Q admits i Assume adjunctions ther E E Easy in uting diagram of the form eries that D D CA p D gory of of The Strongness ii ii o o ED ate E E morphisms are co cartesian in orderbut to straigh dene describ ed is the unique v along By pseudo we me Pr one v comm Pr categories o b e lifted to as free constructions c functor is Y By ove E M ab CCCs is the y using u E y maps b D gory A are E ate Y THEOR E c B The next result erse implication ct bres obtained y I Mo erdijk p is TEGOR the tials the bre d CCC E B ehension e The rev e the terminal ob ject of of v Then is an LCCC every obje ompr C or c B slices F is a br d B from p one has an isomorphism natural all p the slice and a collection of displa close wise construction pE if B of t P over B ation E a E and in is a bred LCCC where e E P e tly u br b as ob jects from is a bred LCCC Since MORE FIBRED CA B every wise construction e a pE E t y reindexing yields the desired result alen E e pE p b B Set E D E ed b f B D A p  Equiv E P u generalization ation p ws from a p oin E ation erm Pb and Sublcc see lo c cit This precisely is a bred LCCC E B a E is isomorphic to the bre ab o P M and the slice p C estigates an LCCC e br CHAPTER cod ws from a p oin d LCCC e et artofaCCompC v Sets E e e of E et L with maps es bred nite limits and lo cal exp onen L am structure F follo E B haracterization it is based on a suggestion b C forms p is a br is an LCCC if and only if the all bres are LCCCs and reindexing E p am LCCC F the conditions T her in ase we form the br een the bre next construction the e E w Prop osition one obtains a slic Prop osition Because for es p The family bration satises Streic The bration the bet B ws from the fact that of i In A bration tains another c u v satisfying o iv iii Considering CCCstructure preserv con in B Pr and reindexing preserv Then strong sums instead of cartesian pro ducts changeofb follo D means that the bration The implication full sub category of an LCCC itself this result follo preserv e k E w that with Seely x bac E As all Hence e E V E the eg Indeed the P ulation readily E E see d LCCC yp e functor e t Picategories to ws form forms The term mo del E E E Picategory follo the This It E faithful text is an LCCC Seely Picategory to the inclusion E artesian functor in ob ject e var P E construction theoretical ertical and isthenabr V ycon an P ob ject om construct cod full E B E e P w iv is a E arev E in in TION cod is a P P category standard E E gories fr am e e P terminal a E E v In e E P ate E y e P gory P with R b E dinisac R P the e E notation P en more they are top oses yp e theory also forms a E ate from e v ove diagr map done x p Pic E ab o of s t describ is E ehension c VER A FIBRA E on Cat e E ea E ertical E ertical b bre e one obtains a forgetful functor from YO v theoretical R ompr This v P v P k in the bre ab o eintheab P E a A the e E are LCCCs ev et yp e E Sets t L E that for C at e dep ending P unique e the terminal ob ject ie the empt in w v a CCompC one has that var E am Y THEOR ertically Pi ie MartinL quivalenc F y Sets is v the P sho dene vious since all pro jections e wing pullbac P is lo oking informal x Ob to E E y is an e then E b In Corollary TEGOR I e i P The functor Since var CA can rom the corollary ab o E of i The functor F ii E E E ii suces o y e e where and forms a morphisms of c E Pr it One P form the follo P bres the one ab o bre categories of the calculus section constructs only this bre categoryLCCCs as a term mo del ground for the next result e y E Y es b w Let with D D E previ E It giv E E Q Q in E E the D f P D Q H Y THEOR in P us one obtains established CCompC a ong sums D as G a and is gory and D ate Q TEGOR ie LE This and th a H onstruction ductsstr c E o D obtained ertical part of Q b e a morphism of bred Q e E gory erications is that D pr is the with the required prop erties in ate is a mediating isomorphism in E ws from lemma ii ehension c Q and E E GE E p Pic ED CCC pg e ompr hangeofbase situation in b e the v e h follo d L a E a Q e D A E dc a tak A e Strongness Then rob enius isomorphism b admits br MORE FIBRED CA E to where y the c P e a P E E E B e to construct an uptoisomorphism unique B p e a E E D v b en b eaclose gory sums D b t of the remaining v E K forced ate pg B E Q eha dby D for is is giv B E G H W p ate E D D E P P P a Q E orks one E CHAPTER w et Q p D Q E L cod ehension c P ED et cod L e ductssums et E b o b f p f L ompr P E E of q pr E The main ingredien E denition Q It is used in the next result to p E b e a CCompC and E ary c E e CCompC gener p e es bred cartesian pro ducts whic b er from lemma that there is a forgetful functor from CCompCs to pro of construct D D A D LE Theorem Prop osition Lemma Q E Q to admits Lets assume adjunctions A unit qualizers Similar the E p P pg preserv an arbitr A f de in of is describ ed in i and i of Remem D e LE o o ED tend E P e bred CCCs yields the fr ous pro of this time using a generalized F B in As Q A br Q CCCs from morphism of CCompCs Pr B Pr rise to a transp ose h Let y the and and only p g if i ertical B tabilit g E E X i is lo cally and f in b elonging The pro of X if C f f making p A w i f es represen B X X E E and given by y e the terminal ob ject A u olv C al ly smal l v v c together with a v Ens Sets ery Hom B it in is lo in ev g E in E I p is lo cally small i E E op as domain of the X f A for u f u B A v u A g i Sets It satises v ulation er B ation e notation for the arro X v that set t X f E ge universe i h t br X C E E u C E E J t form C suc C J R unique A B am J I E E the latter describ ed b am i J f a F E am alen i J F A F S J A suggestiv y lo oking at the bre ab o is exist X us explains the name lo cally small J i Hom f J the functor e a cloven X TIONS J one obtains a b A Hom As to the ifpart for Iindexed collections J E is a suitably lar C A C B ute there D I should A u t i Ens E S id B E tion an equiv A v E e The bration er E E in ws are the unique ones o t v u p R X X i et o or E and E E a morphism L Sets F C u A B Y SMALL FIBRA a E considered as ob jects of am o diagrams comm g F A u A Sets B w Hom X in f f in E Examples Lemm is then necessarily cartesian ach The onlyifpart is obtained b esentable wher g if e immediately men one nds appropriate maps LOCALL is wing t X fg g g More explicitly W epr i i f B X X ertical f where the dashed arro f small ie has small homsets follo This in v t to is r if for e is easy and left to the reader of the homsets in the bres and th diagram e a is w Y e an ted and In esti v to e this B b elo B with enab ou B f cartesian E rise one be in f Picategory L A Y THEOR in es h the is then the fr Q will where A E with eac K dom u B E with TEGOR for B HB if and E u H L LCCCs giv D f start Q of e Q and Hf smal l t E W B e are forced to pro ceed lik E f D enab ou see also B small brations W al ly B c e stress that the material presen B lo t functor theorem see E D B B W y ailable is Q A D f Picategories v with e is an isomorphism a lo cally H h b ecause of MORE FIBRED CA B K dom B E are f E ery pair L KB D p section brations e morphisms morphism is describ ed b The unit her L o K ulations a tak t e an LCCC the identity functor on w ersion of an adjoin b t since e B B CHAPTER w form B B with hapter socalled bration h that for ev t tro duced next comes from B unique nd small trinsic et terest in our researc B dby B L A HB a ate suc B can is Q Id B B dieren f p together one urthermore B section of c er there few F A v A E Lo cally KB Theorem A gory gener last Denition The unit f These are of in connections with comprehension categories see and connections with small brations see their role in a bred v f ate p put w is standard except p erhaps and HB of The results are used only in Thenotiontobein D o er e Pic v cartesian o o W Q determined in the pro ofsince of prop osition ii uptoisomorphism unique morphism of Pr Q gated b elo b ecause it is clearly in In this E E e W exp o y E E E sa P u P Then D D u brewise B It determines a dom pg H j in v H H P A E D y of bred homsets see ii f H gory with unit pf P H f cod g ate B H f tabilit D D H presence of in with p H c A E P u where E A E H vi H A u in H E vlo In the E a ehension c B H D with d CCC Then D E e u such that D Hom D E E E t to represen E E ompr u g table f E u u D D E u u B P E alen u P g A e a br pD u B B b u E eisac u E B P A B u and represen w B B one has for Hom TIONS A ther P is equiv E E B B B u v B op are B E E put E E E B A A A p A P in e et g B B B E E v D A L ertical arro B describ e the bred terminal ob ject u u u E e functors pE f v B ab o E u D t of the next result see also P D f E E E E ab o u ten u al ly smal l Y SMALL FIBRA P Hom E c B g E f v f sections u f is lo E E E eput B E B p Prop osition Let E or or e recall from section that a comprehension category has a unit if the bre LOCALL global E ts one easily sees that this of W F F o Hence one obtains a v unique map Pr This is the con Hence w nen wise are pairs construct a Y B as A ie e v y to the D D pE then b ehind C ab o is left e P C Y THEOR is pE f prop ert is lo cally small It f Ther E E A cod en the idea where u TEGOR ws A i ation X X categorical E E B Arro br B C h is an LCCC pE tially base category C cod B smal l f C the ould exp ect giv essen i and ob jects or a bration of the form Then f al ly c F an B lo B E E P u MORE FIBRED CA is u X X a h R exist in pE A e b ks A small B small brations lies in their relation to compre p alence Hom B the category with pairs es for X X one has A is lo cally small is A op A ting ob ject i the RHS has one ie p E B p lo cally h pullbac gory of the fol lowing form B p CHAPTER Hom f one tak suc ate op et p u B L E E b eing B all k ed under equiv The next result is as one w f f ance of lo cally P f are CCCs cod es and u A k that ehension c b e a category with nite limits ternal in A B B hec B c ery small bration is lo cally small B is in ompr One tak h is preserv wing pullbac B Prop osition The domain of to Ev Let C art of the relev u of P ii o e presupp ose that or iii F See for the opp osite of a bration Pr ob jects W reader one whic hension categories comprehension categories Homc the follo Hence the LHS has a represen i all slices where i is B E ting The map tial vided E This is ulation latter h X pro B B The denition ct gory of B F exp onen eals that one The ate tform E c wing in around T erev is obtained where  T  v lo cal C ull C ull B F id u F vided there is enough pE id y denition of the is cartesian P b in F X is these are  pro F b e a lo cally small bration In terms of comprehension i u  F B i  C using that B ull y ful l internal sub E C F  b the remark follo C since id  h  E id F E In somewhat dieren C E C generic ob ject see denition and h id enough of ternal category id a p y with co domain see eg Johnstone tly  F G i if t category is F iii sa E Let T T e then is called a acks one has B  c G al ly smal l and has a generic obje ks in the base category or of all represen t constructions bien D v F X C c h in dieren ull an id ternal category  F ull C and her I I theorem F E obtained bit w enab ou P id  TIONS  it is lo E a e description C from C F An in F E w corresp onding to the pair This umac id ed   One obtains an in X h i ws C C can b e c D asis with pul lb alence B E E E functor  View follo C Close insp ection of the constructions ab o ting arro id T C e and Sc ks On a b D e on ob jects b ecause i  F X ar  faithful C yields a diagram as ab o eak equiv the results A id e only consider the ifpart Y SMALL FIBRA  see ii is obtained from C g F ation is smal l E e result comes from B and E v G y pullbac e b see ii and an arro Hence ak D a br full Corollary Remarks is the represen The onlyifpart B D As a sp ecial case of the construction in theorem i one can start from an T b A bit more subtle one can sp eak in the spirit of the rst remark ab out a full Hence w E i F a LOCALL tially surjectiv f C ws of The ab o  ii ii o ternal sub category of an arbitrary am C E If also Then indeed b ecause it comes equipp ed with a full and faithful functor latter is of course a full comprehension category in categories there is an alternativ structure around to p erform the relev Pr with with a generic ob ject essen Hence it is a w it also o ccurs in P do es not need the existence of all pullbac used in the pro of of theorem obtained arro LCCC or Pitts h G a  w E Y to the  E with i acks u arro gether E A B X Y yields to E C cts pul lb ull A h ting E B F P asis Y THEOR  i b in u on P F with E a E A y X E  B E E tit represen on Y ternal comp osition A A TEGOR ull E X Y u h F asis A iden E X the b u ation a y E E b for E p and gory The  br on w ull X B ate F c Y artesian functor A another description of a lo cally b the externalization E E E X E smal l B E ation one obtains in E ctions is dened b B B B om br ting arro A A X u oje Every c fr B ull al ly E pair internal c F b A A A b E MORE FIBRED CA al l E lo u u u Q s B an Similarly the Q a X i E cloven for Q e E B Q a b if E Q y b e found eg in Johnstone A from e artesian pr ull b B Q gory in h F  G Q ec only enon see also Johnstone A B given by Q ate P ar c Q determines and B ks in the base category Q Y lemma E A b and CHAPTER E en it ma E Ens artesian functor in E C if pE p c is an internal functor which is the identity on obje B i p e A et X E E op L u et L ct is an appropriate represen h previous X ull A B faithful F E e an internal a A b obje al ly smal l the rite i c A C y W lo Then C b E can b e extended to a full and faithful functor B E et is id id b Theorem Lemm h i L Every E P The ob jectpart of G esentable wher p ull i afulland e acks F of i The next result is due to P In case one has pullbac epr ii o X has an uptoisomorphism unique factorization E Then Hence obtained Pr map wher pul lb with is r small bration can b e giv Hence Then functor and Then such omplete B E is c ducts p o E an nd and B D ose that one c B q supp GE cts d urther D d nite limits and pr B e obje F G e f R ear al artesian functor in ther p ations A A TIONS D i br is a c D A E in G and vertic E eserves the br f A D ondition is satise in ose E E B E and u al ly smal l in Supp c E B G B g elo E d left adjoint G e A ar Y SMALL FIBRA D A u q u g ontinuous ie pr Theorem and is c p has a br that for every for every such that LOCALL G e G wher that the fol lowing solution set c a Y the is uses is ob ject P i fact the   The com category in   ks and lo cal Y THEOR Then C C h   wing form  id id a It is basically a terminal  It is C ful l internal sub dom a is P id Then a id B TEGOR u t pullbac see then t to the externalization  one has an Hyland P u u id C u of the follo is E C B alen B P   P  C ii p C C u uous functor using id id id cod if u  C the relev There u will b e called a u tin C see ii P her IV id t functor theorem C u con h is equiv u E C C to A X P y y that X A A P u P B ternal sub category assumption B u P sa umac ob ject P MORE FIBRED CA X h C C y E R A A A b A C C P Cat can P sub category P id X B B X B B is then a put A P one C C C terminal analogously e and Sc B B B since for ternal sub categories satises ar ternal C  tains a full in i exist in h X ed there is con ternal CHAPTER Assume ii in functor whic E X olv h v i an h a full terminology u A C bration is a bration whic tion without pro of an adjoin lemma e v P has in C sections C en suc ternal sub category if there is a full comprehension category C ab o es bred terminal ob jects if an tial e men as w B X B ed after the C ful l smal l global Denition tial B A supp ose h preserv gory of remark Using Indeed giv Finally w ii of a full in ternal ate C then the total category prehension category in comprehension category with unit since in As this description to dene full in functor No exp onen lo cal exp onen whic translation of theorem in P c e of w the yp e yp ed t w In the en t out yp es for e are lead More pre belo t i ork s yp e theoret w translation to more simple yp e w tain texts the tics of the un The second part will texts in sections een categorical and set theory yp e theoretical features PRED The part ab out w ab out hapter texts con wt three In the translation one berofexamplescanbefound t It can b e considered as spino who um hapters and a t ws us to construct for a giv rst texts yp e theoretical settings and features her ys separate con yp ed with only one t a category the ecansho subsequen calculus of con to HOL together with hapters allo The parts texts when these con Next w yp ed As stressed in c o t w e obtain an adjunction b et t con w to translate t n e exp ositions of a n of hapter yp e theory can b e found in c theory describ ed yp e theoretical and categorical lines come together s s ed together t comprehension categories for the seman are consist yp e t will ones es sense since one cannot alw yp ed can b e understo o d as t ed from Jacobs Moggi Streic elop ed in the previous c t w hapter the t algebras see theorem As main new result w rom section ranslation of settings F e sp eak ab out T This mak w setting determines the organization categorical Settings and features in t This The last section is ab out the un i to calculus in the other sections of this c ical settings and the second one ab out the translation of features therefore lo osely sp eak ab out ob jects of a certain category as con b e a bit shorter more extensiv cisely all on top of a certainsetting setting corresp ond to categorical ones on top of the translated theoretical setting a categoricalcategories one suitably it link will consist of brations and comprehension The theory dev Chapter Applications In this nal c using that un to use monoid constan rst section the main ideas of ho theoretical in HML is b orro details for the calculi CC HML and Y Y THEOR TEGOR MORE FIBRED CA CHAPTER e of or v see the F y tro duced see tied with The ab o with nite ersion y b e found y p v b er v ere in forms the basis one has b o t comprehension is iden E B an es the collection texts is formed E with It forms simplied yp e is a bration a t Y of con Finally fg B E B one tak texts B On top of with p  E T g con formed E B E texts T Ob j it Sort p Y THEOR T T P con f the yp es where a t T Cons t that  Cons for Cons T formed the appropriate setting Cons B TEGOR B for this setting has b een describ ed at the end setting E Sort ted as constituting the prop ositional setting in arning texts to B with and w pro ducts and B g a There a base category A term mo del example of this setting ma con yp es E as t Hence one obtains YTOCA T Ob j T w as presen p f setting t E B as hapter b efore comprehension categories w es the collection of  from a base category x there is B T er t comprehension category t Sort text or ev h cartesian categories e picture forms the appropriate renemen F w v tly more precise one tak E poin A term mo del example of this setting is describ ed after t comprehension category B Ho yp es p Only this part w t is a category with nite is a full comprehension category with a terminal ob ject in the base category OM TYPE THEOR Ob j setting prop ositional prop ositions P The ab o B FR In a diagram A term mo del example In the b eginning of c see resp e said that suc hapter Here pro ducts c of section Basically the same analysis as bcategories efore applies are except not that the constan relev The The starting The B where basis for a constan a bration for a constan i w picture is sligh T in the b eginning of section the singleton con The An appropriate setting consists of a structure of the form e y of and over clear texts texts bred yp es s details e e nite TIONS is the t v onstant h be s c B then w con h con s mak h s dep ending s suc category are with a ter categories B s Ob j categories to the terminal yp es s P the that t APPLICA een suc B h describ es empt een suc should B s texts will w s of w ie one should start with T hop e s con A is required to ha e and s B y the rest of the structure W is required to b e s B s pictures examples where  Snd comprehension P then texts to a category of or the Ob jects in the total CHAPTER The s B also a sequence of sorts Morphisms b et A T s yield e as a base for the whole categorical con w then category s s Here one do es not really need that this with er the bration from ts s s Cons v B s s b elo other not dep ending on or example in a setting with t comprehension category if s F s s y then the bration describ ed b efore should T poin single fg s an A a B y read for The base category ie  ws the next four guidelines on to s these s Fst e a terminal ob ject functor whic T Ob jects of the base category are to b e understo o d yp es for v t Sort er a bration in a degenerate sense though it can b e T ordering s v constructions both t sorts with y insp ection of the dep endencies Cons B dep end ie sequences of terms the the e obtain a constan These brations are determined b that e pro ceed or example what to tak e but also w not ered b F v yp es see yp e dep endency taining v cycles shrink setting yp es of a single sort t t o dieren do requires a separate full comprehension category s s follo guide o principles one ma w ie of the form s h w require ts as ab o Morphisms in the bres are then single terms b et e are t s whic texts con texts will consist of alternating sequences of texts texts w w one starts poin s is a category with nite pro ducts roughly onstant to b e of the form h a bration should ha c con ery sort yp es s s con con minimal B t s s collection of If s pro ducts If Suc be require a bration from a category of on ho If there is no Ev s minal ob ject in theas basis con are substitutions ie sorts These The translation of settings follo texts w to apply them er v will b ecome clear as w The Using and w category where comprehension category is o understo o d as a comprehension category o appropriate brations Implicitly o In these latter t ho setting can b e disco the con s ones consisting of t e o a to v w t set and is ha ards ab out w B texts to strong T s J texts J ards a few dicult stipulation J P J y certain T con up J J b and it Cons J J hangeofbase a con s J c t that the com J J base categories J and es Cons J P J Afterw stipulation o J J and second e J J w v s J mak r Snd tains J require P rst the er p erform determine the t v from B con pro ducts of o This Y to the s E A case T they p A P features in B see has t result r bration p T a where s category connecting features rst Y THEOR er has is r v t comprehension categories B t sums Cons s one s s Fst theoretical indep enden if P Snd P one has a TEGOR yp e Else bration g B categorical T B tion the three main guidelines that A sums requiremen T utually comprehension is for A E m hasboth strong A a p tication corresp onds to the requiremen ong e men YTOCA s this of is er T W then str are Hence in this case one rst has to mo f v P and a suitable category Fst t  quan unit T Cons Similarly tication in section a On top A s base s requires along Cons Sort sums has Finally one has three constan s is a comprehension category o s sorts tication in the b eginning of section pro ducts same requiremen texts o P One P s s kground w T T t brations P the on the r quan P ab out quan texts ranslation of features con OM TYPE THEOR diagram bac A t T setting the Cons FR The feature prehension category as con this the Here describ e them uniformily ting exemplaric cases are describ ed Since constan In The comprehension category o B o o y h y ed w w t texts olv v be TIONS bration con transitivit e this new Because there should the a APPLICA is a category with y maps to D there B o comprehension cat from w obtains and top one utually dep ending on eac of displa E B A one g r p On P B A D and corresp onds r CHAPTER A term mo del for this setting ma D T j texts T texts T r latter B er Hence the picture lo oks lik D v con con comp osition Cons Cons Cons The P P with By the three comprehension categories in of to s y categories one has a base category with t her fP with A Since these sorts are m g s texts g E D wice comprehension categories and P texts t g con T T E T con f f E B category A   E to E are full j base In terms of displa using and one for E Sort Sort a texts P from texts cannot b e separated con texts with one t t imp osed on the dep endency relation in denition fP and is a comprehension category o one for con P P setting setting starts viously an appropriate categorical setting will consist of t are no dep endencies ofare the constan form to from requiremen where a terminal ob ject collections The One brations where Ob egories The b e found in Jacobs Moggi Streic other their con g it The with early tially corre f t com feature is a full in tication fg eak Q In and Essen the inclusion Categorical features closed under ew e Sort v v constan quan feature ha Sort hha closed and is this B e nd the notions of only is The axiom feature yp e theoretical side and yw unit transp orts a setting can B a Q cod thesamebasis By one is a full comprehension category yp es presence of tains t B inclusion functor and with In this w w of a full comprehension category con as B assume the inclusion T Both on the t CCcategories whic ythe yp es corresp ond to bred equalizers in B b t that the corresp onding t denitions will deal with ramications B D T ak E yt as argued that a corresp onding categorical with Q s we w b e depicted as tit hangeofbase describ ed in and is T e the terminal together with a generic ob ject ork in Hyland and Pitts ed in some detail at the setting Q the v QI Q yp es It w fg Q t eral times no ab o Cons prop ositions Q iden Subsequen that ts see iv E Q of h structures all I Q concerns ulations since certain features result from others the Q has form suc w Q Sort of Q As a result of e seen sev e already lo ok the one v belo ts to the requiremen E o comprehension categories of w setting eha ersion side b eing a closed comprehension category t v y concrete examples and is the full and faithful The feature b Picategory as dened at the end of section with a terminal ob ject in the basis P literature D TEGORIES ving an ob ject category B y nd examples where the c both This amoun the and minimal h a diagram underlies the w E sums see section CCcategories in consists of categorical Suc consists as w I the categorical P E ated CCCA e ulations of the calculus of constructions v The remaining systems will b e considered in the next three sections In The rst denition A the P closure this means that the collection In the previous section w with necessary setting studies cartesian pro ducts and exp onen one ma motiv form on corresp onds to sp onds to ha ab o the presence of stronga sums see lemma prehension where again comprehension category and enables more economical form underlying categorical structure will no P a g It to be cat has where h one f functor TIONS that s h should P tication suc Sort corresp onds there along a suitable sums h structures with APPLICA co d s s comprehension is a category with a and faithful Suc t features see section P Hence in this case one terminal B an P the ob ject t that D B the to co d case bration e strong v p generic this the ab o CHAPTER ersion of the setting and a P In y s tication rules it seems a bit strange to require only b co d P s B P corresp onds e B v same R of ards s s ance w In case the base categories dont matc ob jects pro ducts the abo s s Lets consider some relev simply s describ ed P p are up P is p s s category rule generic tage of the double role that comprehension categories E s s and h that y an P b P corresp onds to the presence of a full hangeofbase on the bration total e it forms a categorical v s suc e v v unit co d categories D t is of minor relev the are full comprehension categories and a tication e s el description of the quan v is a bred CCC if s is terminal closure corresp onds to the requiremen is a CCompC if in s s base feature P s describ ed B ving quan T P ha g the are s forming a morphism of comprehension categories and As argued ab o axiom s Cons co d s ob ject D case P P closure in a setting with at one time as a mo del and at another time as domain of quan f s The second p oin there is a generic ob ject for p The an in generic ob ject ab o rst has to p erform c terminal ob ject functor connecting the base categories rst has to mo The feature The Probably a go o d example to start with is the structure The inclusion Axioms y E ving an ob ject s egory s where terminal with In one canpla see the adv enables this high lev I inclusion will b e considered more closely in the next section ha h for see has One split is an as in maps B eshall I is o ws in whic w describ ed where t As in B B B L B notation Q factorization denes p os the functor Set the B D top one L B w y describing it as a X I  estigated further The reection comes from P haracter of the monic part epimono v bration Q ws Q belo Q cod E as c and the consisting of monic arro B D i i X done unique ers the case when the other functor Id B X B v I that Q a be Set g g is Set is then an LCCC In the sequel w X X has B E will and e B B f f Q s erify and am Q F and the generic ob ject are describ ed in F This Q top os eak CCcategory B B e top os example a bit b B I Q to v Q h h a v Q k to Ehrhard F Q left Sub E I A A is a w Q sums X C in Q category ie a CCompC with a suitable generic ob ject The next notion co  Q Q y Q tioned in iii gory thing e the ab o eak CCcategory An easy example is obtained from a v dictos M b CCompCs ate strong yp es C It go es bac The base category A e i the reection is the standard map obtained for the full sub category of A in a w X C and Q B split am CCc X ery morphism in F becomes a al A The only C ev es alence alence the Sub TEGORIES Q gic ypeofallt X alence tak lo Q C that A or C Examples Denition pro ducts A second split CCcategory is obtained as follo F Using CCCA ewrite Q alence B is a CCompC as men fact Later in this section the notion of a dictos will b e in Notice that if is an equiv ii is an equiv The equiv in one can impro CCcategory split QI of C Then one Q lo osely sp eak ab out a dictos ii w obtains a the eg Johnstone equiv yielding a t Q e e of in as v in use the and tion E e small ob ject B B w generic TIONS of terms cartesian structure a Again w as dened Q Esp ecially Q in QI h atten role concerns brations the has uc B eak D p terminal cod w m all therefore D t By lemma the the faithful APPLICA a Q tation I ed if h prop ositions ha eak CCcategory is p the and at section I eak sums and has D yields E elling p split v presen es sense lo ok this full E I e receiv v ell as the generic ob ject are a in unra terminal is E brief called w mak is is a full comprehension category a I CHAPTER bration be B strengthening e some structure systematic B These ha the reection the and yp e theory in whic tak pro ducts and w see section esp ecially and sums are required a a tribution The notion of a w Q e q is us eak CCcategory h has a bred left adjoin D needs E w con and ii Th concerns has tro duced b elo and wn gory i is a w P QI bration o whic h that sums as a result eak CCcategory tion ate a E Q suc is P gory Our pro ducts and sums as w if the bration of prop ositions men and D CCc B only strong cod ate eak sums e al compact constructions w w B ak eak CCcategory will Then gic units q reection lo we CCc E and I example quite to Aw Consider a w A A y cf ulation the y the reection p is in Q eak CC see section ob ject e QI and all ed b By is a CCompC v Hyland an This cannot b e done separately ramication examples mo del QI Again b split In view of our stipulation to treat strong sums as the normal situation is in cod brations pro ducts and denition rst P is a CCompC ie a closed comprehension category It will b e called Denition Denition Denition Denition P some top os unit tially due to Hyland and Pitts functor from there ob ject ab o p Q sums e e where hispreserv h ed are v v This In case one is willing to view a logic as a t The has is a preorder ie has preorder categories as bres e sp eak simply of a CCcategory instead of a strong CCcategory olv lemma whic v ab o B at most one pro ofob ject the name in split one obtains strongness of the complete esp ecially split use a compact form whic P with After comprehension categories ab o of the calculus w ii w essen e P to p v be for see ase e E D v P eha and can ak CC ha P D t functor Hence w done e It cod t h that Q w D be suc the reection eawe D p b comprehension A ys e is a bred CCC B ulation Q a By E By changeofb B is with B Finally hangeofbase from full alw D D a B D t yc E Q or B B is D D F can p E B is an LCCC Then P w in the other direction E D cod has a generic ob ject b ecause D E denition Analogously to lemma B this ob ject Q pro ducts and sums e p e I P Q an Since cod D small bration Q QI after with hanges in the form y nd ho is obtained from the adjoin E small and complete brations B cod P et D E L p there is ation wher k to Hyland prop osition and tation is a full pro jections p tto A bration p Q to a CCcategory e is a dictos Q B e diagram has and e pro jection v ea br are lo cally v is a CCC since ther b in e made some c argumen small and p t W B ab o rst A e denition cod P a along cartesian pro jections her one ma D E of the E se p of t is Aleftadjoin inherits completeness from p eak CC to and The D E p p one has that in the ab o A see p T is a bred CCC since it is obtained b p p P u is a full small bration e the terminal p gory et D is the CCompC dened in that rom w b p L cod F ery ate omplete B ws that D Indeed ob ject functor c D ev see i in the previous prop osition t notions see ii ii and comp osition Q t of the next result go es bac p p t a D an erify that TEGORIES for category can b e turned in ten et it follo conclude uous is f By ii D L is a e generic tin h that Theorem e The base category Theorem B Q W a p CCCA ery D con suc is ful l smal l c latter gory of of The con In Jacobs Moggi Streic wn that By denition there is a full comprehension category with unit a o o p has or the relev ate Pr p D Then see lemma i a bred CCC nd pro ducts and sums for F i one can v where ev to Ehrhard corollary has pro ducts and sums along cartesian pro jections c we form category with unit Pr E B theorem sho is lemma the k ea the the the has is The CC hec in write Since e C small T v Sets TIONS i eak We A vided yields b e a com etoc ab o Ob jects of w v Since A a C is B pro e only giv Then h C ws k these details comprehension eha of D is lo cally in gory T Let Q W so w am APPLICA p hec k ate F full describ ed in small ec are a P is W C tto CCc D ositions is p k pullbac Finally i structure ed in the pro of there the op ak P and where w familiar we that the C wing p a h Q that from ii has sums see ex is obtained as follo us t yno e CHAPTER C b for follo As remark B elling ation of pr In y pulling bac tells e one obtains that v Here is another one Sets using B v eak CC as describ ed in section also The and ob jects of Q The pair Q unra i generic Q C D C in be Q cf denition ii eak sums C D am for the br I B F texts with sequences of terms substitutions C B Q and i ab o I corollary E C further are The construction is b Q A lemma E E ation E go y lemma a bred left adjoin Q y al dictos to write et e v ob ject L P gic en b o examples are dictoses lo ab o w results Lets cod d CCC e o E et ks of comp osable tuples and triples It yields b But these are b oth obtained b Ob jects of v w generic is obtained as t C B p a eak CCcateogry alence classes of con is obtained as in the pro ofs of i and T Analogously is then giv t and is a br is a ful l smal l br C next erm mo dels of the calculi CC and w has C I Prop osition i p p t constructions can b e p erformed see remark i By prop osition The ab o T Let p h A w QI een them an i of The w us one obtains a ii ii are equiv o iv etc iii C pro duct that the pullbac can b e formed Notice that B P the usual pro jection category category with unit pro ducts and w Pr result do es not require that the sums areSince strong relev terminal ample i bet plete Heyting pre algebra consideredinnite as copro a ducts small the complete bration category Th sk B functor yield appropriate examples p has A gory p it h hange ate with a er B v Q e v e in whic ed from Jacobs wing categorical v moreo w gory i ate Notice that the c d into an HMLc The follo c p hangeofbase situation E E B B p axiom is terminal further the bration ansforme y a setting as ab o A One forms etr ythec example dintoa o theorems are b orro en b B w y lemma ie a bred CCC on a CCC an b see w has a generic ob ject ab o tt P q p Cons A E ansforme p gory c er v orks here b wice h that o and in ate category b er from section that the features for HML are etr c E B suc an b also w tication and an D p r Remem be a pro ducts and sums B er CCompC gory c B Every v quan a hangeofbase as b elo An HMLcategory is giv ate pro ducts and strong sums i the guidelines Cons is yc Q E ws r p her TEGORIES E pro ducts and strong sums b B follo and is a CCompC see example ii Let B is a CCompC is a CCompC o admits E obtained b Theorem Denition Cons i Every HMLc The output of rst applying i and then ii is isomorphic to the input HMLCA P P there is an ob ject p Q describ ed in Cons The generic ob ject for p P of The next denition and the subsequen ii o iii Pr generic ob ject and This structure forms an HMLcategory since Moggi Streic description ofbase as describ ed there is used t y y hi in be E zy ean indeed can a x TIONS E solution informal tioned g t the end of In the Lane h is a sp ecial A Then A men ealgebr or e dont giv F with and u Mac APPLICA W g E y x P yp es and setindexed ailable CCcategories A eg v u a f one has z condition A see and x are set B is small complete x h B A Sort u z t to a small complete category denotes the sum obtained b reyd wn that there are no nonlogical r By dressing this setting up with As to F CHAPTER omplete Heyting pr u D Sets y whic r alen P u y solution om ii satises categories er t comprehension categories of is a c v C wu the C P Q x h indirectly yields examples f is a complete Heyting prealgebra is equiv w where del fr am pro ducts result ho F C C A constan the term one can nd a of yields u u Q E w D on ersion of the setting E u D P y g and This instead u terms E P is example ii w A in based f term es for es is a dictos The family mo sho B is ulation the e with is a dictos f but v e this A form One tak Sets e one tak denition g u and v to HMLcategories whic e reconsider features for the prop ositional setting whic A y P Sets implication E f lo oks lik C order is a full comprehension category o examples HMLcategories w Prop osition The implication with P wn in a categorical v am C E theoretical F latter for A y tioned ab o of As an application of this theorem it can b e sho E o am ery higher F yp e this section w appropriate features one obtains the notion of a HMLcategory concrete transformed in where case of the one ab o men and g collections as prop ositions mo dels of the calculus of constructions with families of sets as t The V prop osition As sho Pr a theorem in t ev e in v no and wing T g but f see Cons tro duced yp es T p in tt ts only the T has Cons es is a CCompC T for notions o p oin e obtain the follo T exp onen there is a generic ob ject w P T B ho c Cons E category Cons B ving tion t p further if there is an ob ject h Cons p ha e cod h v er E T whic v emen are sp ecial cases in the ab o T E e then write T W in category one tak calculi category somewhat ad Cons Cons w h that ab o the E category and a category ation for these rened descriptions of the minimal mo delling Ob j category in whic is a CCompC o than wing in w e setting will b e called hangeofbase v in B q a ollo T yc alue F T T category and a v is T yp es as used eg in Barendregt E B category and a it Cons e The ab o e the terminal suc terested if more v in ork can b e used see the discussion in example obtained b tro duced earlier in denition is e a brief lo ok at the features for the rened prop ositional setting has q abo fora it TEGORIES one B t ask ab out the motiv one can tak etak category if h category if it is b oth a category if it is a T category suc Ob j case B Redenition a a a a urthermore for a HMLCA This rened description comes out asAs a result of a general metho d of translation denition In cartesian pro ductt rened framew F The notions in One migh Finally w i ii iv iii t i T features as describ ed in section for the bration redenition and prop ositional settings and their features pro ducts and sums p gory e are B e p TIONS ate the CC t similar t admitting morphism esults B a comprehension But then w APPLICA One forms ful l E B y e form the bration u p there is b quivalent r hangeofbase yields a generic ob ject W y an argumen p p t is a d into an HMLc D yc f D er en Q v Q Q e p ed b yields e E hangeofbase situation B The generic ob ject of Since p with e is giv ansforme A E CHAPTER v t pE p where D etr E from and the fact that A in with an b p Q e in t P Q E is a CCompC o D e P p is describ ed in theorem Q gory c and CC D D e T y lemma is a bred CCC see i v ate D f e Then E u D hangeofbase situation ab o Q and p sums b E with cod for e a CCcategory as in denition t v A T Every CCc HML D E q E strong p eha i in ery is a CCompC Hence ev orks here b ecause of the pseudo c hangeofbase situation y the pseudo c is terminal ob ject Q for QI Asume w A D also w p is CCC since Theorem has pro ducts and sums along cartesian pro jections b i is a bred CCC since bred CCCs are preserv hangeofbase Doing CC Q Again b By the c Supp ose an HMLcategory as describ e ab o P t t pro ducts and category there is a unique D p The generic ob ject u for to the one in the pro of of theorem done p yc of The transformation CC Q b ii ii ii o iii Pr where p category from where p t t g r A i B gory closure constan ate B tication bration i f T a A A i quan latter Cons B i are Sort PREDc A Fst obtains e consider the f co d E for wing bration obtained one g T am t I r features F f g r E B E dintoa i Hence w and the B er if the f Sets am v am r It yields a generic ob ject for write o p F F I g i e that gory v am A ansforme lets F f TEGORIES CCC ate h that the follo ber etr R is small B Sets B R suc an b bred The axiom is a bred CCC Hence one obtains a constan A PREDc Remem a r PREDCA pro ducts and sums t axiom T One has is er B gory c p Sets v A o T p er ate is a bred CCC again am v using that p c F g am E B Cons E B F er A am B T v t F o am am am am F F The categorical setting describ ed in for F F Snd that will b e called a Ob j Every Sets see i wise construction f r p t r is the terminal ob ject functor for the bration PREDcategories e t TEGORIES AND er E is a CCompC has is a CCompC o v using bration o to eries that etak a tication and an am R T T T F is hangeofbase has a generic ob ject ab o w is a CCC turn I Theorem Denition HOLCA yc i B e quan where there is an ob ject b Cons Cons Cons g T w from a p oin i I Similarly W g i or pE F Since CCompC A CCompC One easily v follo with f d wn the I gory i for g TIONS PRED ate i E ansforme f tr y tication and and e b HOLc in en APPLICA an b en but it is sho T T quan HOL giv wing bration obtained wing bration obtained B Cons gory c Cons y the family construction describ ed one ate am It is called a F cod cod E B CCC with a generic ob ject and ase c T T b CHAPTER closure setting e e E B PREDcategories v v E previous h that the follo h that the follo am the HOLcategory b and F suc suc as on a smal l p and categorical No concrete examples are giv T T am the r B A gory p F A B HOL are pattern er er ate pro ducts and sums v eliftittoa v c categories as dened in W T same functor with wing consider t t y dressing up the corresp ond settings from section with e b e a bred CCC on a small Cons the g W The B Every p gory ws f ate is a CCompC o has is a CCompC o E follo axioms p b er the features for HOLcategories in T T T hangeofbase has a generic ob ject ab o hangeofbase has a generic ob ject ab o HOLc Sort Theorem Denition Let pro ducts and sums yc yc section there is an ob ject b b there is an ob ject Cons Cons Cons B of Remem o w to obtain these from Cons describ ed into a Pr ho This categories are dened b appropriate features follo setting if h It of in u One suc from more resp gories B E E id for ate of functors t describ ed morphism E id a E L is y P K i ehension c s and let u ashi a and sums as t y f semiadjunctions E ompr B t semiadjoin Ha P KXLY of and KG is a CCC except that one of c id see consists of a pair it B adjunctions for semiterminal G v v X GY pf id D h L B E K E C has a righ is a morphism of semiadjunctions p u XY semi pro ducts P G i morphism s s tation u E E pair y L ifboth a a C A ev E P p id P E act of erything uptoequalit i the v F v E g etr P and and P E ordinary of functors b e a split comprehension category vided with and L of in Gu needed is a r u pE L semisums LE f id B tional presen Gu E A CALCULUS REVISITED E are s K apair L K X LY is nonempt h w require ev G a ening functor B i hthat F s resp t E F v b e a category with terminal ob ject a h eak is a morphism suc v FX Y E eno B id E i i P B h In equan E to LE E f D if notions u u t D ducts P t t D is a category pro P o generalization Let Let pf id XY the pair LF consists of the w u E E P ie more D ard K t id C o f E Fv is a morphism of comprehension cateogries see semipr ducts w cartesian e Fv E E t i o P i L v nonempty tforw g h h s h h is a morphism of split brations e describ e semipro ducts and sums for split comprehension categories semiCCC C ha a i has u u morphism u eac Denition Denition i P iii Similarly is A a E f not THE UNTYPED LAMBD semipr K L id K L urther straigh for for eac a left semiadjoin semiadjunctions resp P for eac h h C of i Next w F i i ii ii id ii a o iii ys that K F G h Finally details is pro duct and exp onen do es section except that w Pr F h Similarly for semisums sa with that t y is b four p ashis single y e iden g further TIONS a big notation category X Cons calculus X e v Barendregt and dinary func the ha base F g Then yp ed h and via S Ha t of APPLICA an or Scott that the B the bration y h F whic b use revisited suc pT p Gf us one obtains a constan to GY E B D estigation X GY v X Y rst form yp e satisfying eg C C semiadjoint tro duced e C categories going f g w t in X CHAPTER is used to mo del ove but with g as e compare our new notion to the one are B righ and put XY ymans calculus e dab a Y Y e p W XY XY Ko X X Cons is calculus should b e considered as a sp ecial Pro d XY e rst recall the notion of a semiadjunction Therefore ute for all C ob ject bda W e f t lemma comes from Jacobs comprehension is a bred CCC and th yp ed as describ t t p D texts B Y lam G category is a functor except that it needs not preserv t CCompC s reexiv G E b e generic for Scott a p con ard D FX Y a FX B Fg E a constan es one has F collections e obtain a new notion of mo del for the un D see D be tforw C with T yp ed ose t the end of this section w and are The subsequen t A B F f wing diagram comm Let The constan Supp ymans un semifunctor calculus viz as a calculus with one t monoid CCC there p a Ko E a if b oth er Y y Y p v f b of ashi wing this view w yp ed The e include nonextensional abstraction in our in y Lemm Another Let G W semifunctor mo del ollo s F considering elop ed of A The categorical concepts used in this section will all b e describ ed onthenose a to o y hangeofbase in yp e CCompC o The rest is then straigh used to mo del D Scott often stressed thatform the of un t semiadjunctions consisting t b tor then omitting indic F tities squares in the follo dev ie without mediating isomorphisms from Ha B Pr c e e is g id m D v N h k The es m wing wing and Then yand n etak a e write D F with n w e f n w n g follo j a b n considered terminal One writes n A one tak and G b x but etak x y D the y n uous function f be vious x has W a es the follo m tin ob en b a D As usual w n can Then after SK K D In case D i n ob jects the j s as describ ed ab o ob ject id it giv m D y are giv n as in F is a con I D is terminal and in umeration yz is nonempt i n zy found N n f n D has semipro ducts for mor D n D x x y z wisting is often necessary h mo del a be done s and ob jects x z D f x x x D n y mo del D is The category as a g h n h i ma D N yf D ws from the fact that The for from an en n n b with D More explicitly f n in D zy ax yz bx D x n As distinguished n n n n y where eac e cp o via maps n b D D A m x n again D The ob ject x x x as morphisms D algebras ariables and This follo D one can dene v with is a categorical x D x z x y z theory x y z see Barendregt f x x algebra see Barendregt y ob ject i the cp o n Cons n once n and id ely put i f i c n x Comp osition A CALCULUS REVISITED D b e a reexiv Hence it is an algebraic theory again rst D x h formed yf be a h G D b for D a G Lets i D g is x f should b e understo o d as the result of abstraction in the the S n n i categorical n where n category a F f inductiv D D n n k Let in n of n K see lemma ii x D A is an algebraic D h i as b efore it b ecomes a categorical A x with base eries that x z taining are sequences cit e write D D with c x m x a D A D f h m m c i w x us category n f is n lo c analysis be n and D A f D b D n Th y M n D ie f text con D see a x a A m base y condition on Notice that is a nonempt a a D Examples D F Let en b D The pro duct semifunctors One easily v A The comprehension category Let y n deep er Cons x THE UNTYPED LAMBD D tities are sequences of pro jections tit D A and a b ii K f n n n D a iden a pro duct are giv ie S naturalit Because one abstracts in the underlined aexamples form of t G a as the con Morphism D as distinguished ob ject phisms and pro ducts iden tuples underlined in A map in as to o and m with B es As it ab out n be B TIONS n K cartesian serv KA esp ecially algebra KA ould t comprehen details w m es all n A Comp osition is a morphism or n APPLICA F id A that K id B Recall from that is a functor u er preserv A K in v But It will b e called g e K K K comprehension categories y a base category v f mo del and a e B with v in of A f Barendregt en b is and w satises moreo B f h that the constan ab o B to B CHAPTER ts algebra in y maps u t in ii is giv ii A Cons A y pro jection a x B is in KA describ ed f A A e refer t the w g as ws ys precisely that are has ordinary pro ducts i algebr p oin is dened in i is done b x y ob ject suc arro Cons f g esa f v A h al algebr has semipro ducts Cons A are f algebra as describ ed ab o calculus if al second B g B A goric tabo K if it is b oth a categorical B e dont simply require that categories Alg are monoids ie categories with only one ob ject viz a if has enough p oin b e a categorical if for all in the pro of of b elo ate u A the c t goric del In the domain of morphism yw the terminal ob ject in B f del yp ed t B A a t in x ate Cat mo algebr oints B g one has B is un m mo A B Let i in y B taining a nonempt Kt b K al al al the B KA onder wh en n and K K A Cons comprehension enough p tw goric goric goric giv t b e a categorical tics of ery ate ate ate onthenose is morphism of c Reindexing along c c c A Morphisms B migh pair ev the counit functor m Denition Denition A a a a has A y hthat with semipro ducts The functor K the K for tit seman Let One The pro duct semiadjunctions are describ ed b Constan i suc ii ii ii iii n hapter A B of comprehension categories see the second p oin the bre categories stands the second requiremen strong pro ducts This yields a category B iden the c nite pro ducts con sion category e i y c v b By eha c i h yields hoice for id c h i B whic Hence w i id c A h id y y substituting an i i B i i B i i i id c t h satises A g y b f alley for A A id c i h id c id c A id c f i h f id h t h h id h h i alley for A kChev do es not dep end on a c f id f yof i id c f i map i id c id has h g yBec h g a kChev f f b i i id id c h f i It enables us to apply e the rst dumm one A A id v i g A y naturalit yBec y lemma i B id yields hh b b b g A A h exists b ecause is nonempt then e compute f f f f in hh hh A one obtains ariable W e remo h f A CALCULUS REVISITED B A A B f i id c t h w A y v A t c whic t t t A A t en id c Finally f f h ws that the denition of yof t one obtains A id etak f A to v f c app t sho g t t w id B f f an extra dumm arro y naturalit ould ha g f A A an ew w us THE UNTYPED LAMBD or if w F Applying tro ducing app the latter b c An easy argumen and th in an arro arbitrary elemen Assuming arranging the input appropriately a y D D are y the D TIONS as in i in D en b m i e a nonempty one obtains a The naturalit and b b b i i a APPLICA B id id semipro ducts of app us the ob ject m e a Th Then i id ear i i id bred e is a map b b x i u b h g A h el the structure giv h u h i v b x B t in i implies that for y the m ducts and m id b f f g g o f b CHAPTER f en are id A A B B id i x b a a y b a g a a h e rst unra h id id D a if and only if ther i f i i a if and only if ther x a id id app app id A g x f h structure giv h a B m u gory with nite pr algebr f algebr id app Hence the b e as describ ed b efore the examples ate a f wntoearth w g f A f x al al u app a A i h ea c A A h n b A b a goric goric A mo del ie B g supp ose morphisms ation describ es g x e the only if part w ate ate er et satisfy v A i L a is a A alley condition the second p oin a ts one has Let wing from lemma iiiii a is a c is a c M n result B algebra in a do mo del x D in kChev or the if part of i and ii one denes x n next A B B Then Lemm F In case ct i of The In order to pro en with ii B o gether with an op satisfy has enough p oin categorical obje semipro ducts conditions follo u The Bec to Then which additional ly Pr such that categorical giv a y i b not is and y B rst or a for a b n h F i app x K K Kf k the need ws doesnot k app KA In f B h describ ed t to the third This is as one One has b D from the pro of that is quite practical f alen A functor yax a A i n A x y category as in D K id b x claim and h algebra K f es put base e is equiv and and D x app xy f v algebr and A A i id the wn to B b al examples app one tak a tabo t t i x One has to use that f x x case the B function goric a a b a Abstraction is done as follo id b h in ate h app our or z F x times ec then uous ariable in K n b KA are as if and only if a D Kf tin b x one has collection a a app app con D D y algebr t B and a algebra n ymans A A CALCULUS REVISITED in D as f b x al Ko in the same pro of n app KA and K n i  D o ccurring in the denition of algebra as it is used here is more economical than the taining a free v a x goric and us using the denitions of K c x a and nonempt A If what t and w that the third requiremen B ate Th n and con K x a the et f Scott app D K L KA y or app for and F a e to sho estigate app ts see the discussion at the end of this section v erse direction one obtains KA k v ulation obtained in lemma in terms of y D b e a categorical h indeed is in t t in denition ii the latter b oils do in k used b has A eha e x B app x Lemm W W In the second case one starts from a settheoretical a THE UNTYPED LAMBD whic K K K one A write a e exp onen of Let The form In the rev is a morphism of c ii o v e ould exp ect algebra as describ ed in Barendregt n x y W dep end on the constan term B It will b e extended to morphisms the description of K of the if part is easily established Pr requiremen case xy xy w Lets write Then structure Notice that a categorical ha e h v id one The yp ed pro t TIONS G to Indeed B the un F order ymans whic id e ob ject APPLICA In with i tics of id category f as b efore reexiv a i a i h alley id A G the seman y i has id G i id g h F id id id CHAPTER obtains id c g kChev h id c i f h and id A F ev f g f f id i F yBec y naturalit F easily ymans for i b b G ev CCC whic has h then hh f a one h id G G f G G ev ev is one app id c and Ko h F f B app app lemma us id t t id maps e id app v y Scott algebra are g f and th Supp ose ab o is extensional in the sense of Scott and Ko a id used b f i app the B is there F e rst notice that app app Using B G w id er in case Examples case means that v h structures are In app calculus Hence one obtains latter Moreo means that Suc denes This yields the required equations k has and i K x n k i y lemma a binators n n pair id m Cons categorical k x a k of xy y snd x a snd i n n a a and K h has By denition tities and the pro jections prop erties see ii since K m i b a One where id algebras it suces b pair calculus yields com m id es iden id where see ii algebr K fst U B z n fst n n categorical from al n i preserv x FU x z n n n n in goric i x some h ed since wn to and m n m m n ate a snd n n satifying Hence e pairing with K n i xy xy n z ea c app app n id A CALCULUS REVISITED b a n the functor from categorical to set theoretical mo dels n n a deal m n n a n h z tities b oil do x k erse a B F v x a a fst is a morphism of categorical n a a app a et app tin m A bit more categorically pair n L k a a i x theorems b calculus eried n n Comp osition is preserv o trigh k a w wthat and t notation i x a k z z xy k x a n hec x Theorem next ducts it also has semisums pair a o The standard nonsurjectiv the triangular iden x THE UNTYPED LAMBD x y n e put calculus snd of The pattern obtained here is the same as established in JacobsThe for On a morphism w o and x algebras Pr semipr has a leftadjoin the second order These are easily v Finally In fst pair and for to c n In order to sho i t in K S left an a maps ts K n TIONS relev Hence if are has D By lemma categorical one denes n D the describ ed h i o dene it on a id T is k es of Alg m see Barendregt i lemma a n k a APPLICA D S h algebras By Alg B id x edene Morphisms K is h algebras i preserv w n app h h k has enough p oin th pro jection i y Ka i id D h a size Cat h Alg D k b h id y UF B a a a rule t t CHAPTER h y of holds a is the b K app Cat m arbitrary y a is a morphism of since n i of Alg app app U a a a dened algebras UK n i collection id x Barendregt h a k forms the ob jectpart of a forgetful functor or an elemen F id i id F functor k id y k has from collection b and is dened on ob jects b algebra then where for a morphism a k n a h n UF n i B functor one k getful be y a a app k k for x n One has to Alg I of mo del one obtains the k app S K y b b for the category with set theoretical id D n B The UK onthenose hence or a morphism of B a F b w x that the underlying t tially de Bruijns nameless notation a One obtains has Alg n for t a allo D FU a and e need some notation e D x k ob jectpart one app w Notice is a categorical Alg that a prop osition app Theorem the pro jection B k D i The es and x D n een the underlying collections preserving application and h yields essen Cat is a categorical of A counit Lets write The assignmen Notice a w ob jects o algebras y D B adjoint the unit of the adjunction isPr an identity example ii Fh preserv morphisms w b structure And if bet ii U as whic for the t is on con un of and set h constan tro duces ob ject see eobject e ie calculi Explicitly a CCC with re elop e in v reexiv monoid examples en a of calculi orld understanding een categorical to a general categorical e ob ject one rst has to w to a category yields incom h turning bet tages of our approac notion concrete yp ed yp ed w an CCC with of a new y semiadjunctions and the addi categorical of our tation This also ts in t comprehension categories describ e the rules b ymans approac tion the adv notion theorem a general y presen compare the ws constan e men to W A CALCULUS REVISITED algebra and then again in tness in t ii as CCC with a reexiv that naturalit ymans Barendregt uniform Monoid elop e This follo ashi Jacobs In the ScottKo to a v and y ymans p osition plus a concede This is due to the fact that the Karoubi yp ed as the monoidcase in a t the adjoin t y ordinary adjunctions in to direct to e mo dels a v calculus are ha e rule b In order to presen tary e w es rise e ob ject rst in w y Scott and Ko t comprehension categories describ e simply t yp ed e the Karoubi en enables t er yp ed THE UNTYPED LAMBD It giv theoretical unnecessary junk see Ko It the minimal setting exiv parable results un t It describ es the tional pattern see eg Ha stan It captures un iii tak ev Finally w comprehension category with semipro ductsas with used the b CCC with reexiv more elemen Ho is e r g A ashi ashi hh h then f y y co d one t TIONS ob ject and f Ha binators ii e es B The latter f g see Fst i hh Hence taking i see Ha algebra yields C constan C A reexiv g h K APPLICA and hh K write one tak g semiCCCs f g ii with e f We i arbitrary ar for ii a e monoids an e ob ject in f g CCC ii A is hh the bre categories a maps h gories yields a CCC snd yp e urther f algebr CHAPTER semiCCC structure g h A in F C ate i c or gories ar F Then al e A pair the B t f obtain f ate is a reexiv d br o facts should b e used c fst to goric hh ec w ev id ii al l w dene Cons ate A A h hh c s ho e ev ii a f g A a where ie W hh g f e then A A b i that comp osition A e C ation involve Then The next t hh elop e of a semiCCC c f v i indicates h i g see the pro of of the previous result for the com f snd B i id has ev y of a monoid semiCCC satises i snd A semiCCC A g f g et e ob ject es h id A L d algebra A and for the br One f d monoid ie al l br e h fst e h g theorem f B B br elop e of one of the bre categories of a categorical A A in a semiCCC v pair eried b er from pair a vious since one starts from a single t g One tak f pair i A i app fst y is c eserves this structur is a br h Ob ii b the ob ject sa Theorem and A previous B c f g i Fst Fst ev Remem aking the Karoubi en h ii hh If T is easily v h g or the semiadjunctions i of The F f es viously ii ii o f f hh are required Then Cons tak fst snd a CCC with a reexiv from a categorical the Karoubi en Ob indexing pr describ ed Pr Univ gic in Symb o A J thesis ork and Lon ok of L o Inf Contr Paris PhD thesis Univ master to app ear Handb gr Sc o gories ad c Oxford univ Press A e Category Theory nd rev ed North Holland tice Hall New Y unct Pr R et al eds Dep endency Scott PJ J F C Pren duction o e yp e Inform Comp and hical Data Mo dels an Application of T nIntr TH and related Systems to TH in Hindley and Seldin Scienc A t p etites v A Springer Berlin or Comp Sci hes o oundations of naiv yp e systems Abramski S The ories ories and Contextual Cate UTOMA ories UTOMA olymorphism Pitt et al ork and Hierarc Computing w Oxford Univ Press to app ear et lo calemen aic The for e al Set The c Rezus A o lgebr ory olymorphism tini S b da Calculi in riples and The dA and da Calculus Its Syntax and Semantics The Mar ey of the pro ject A tics for Classical A alize amb gory Wells Ch oses T oses and L g and tro duction to generalized t yp ed Lam op op ormalizing the Net o unctorial P Fibrations p etites A Fibred Categories and theL F Nijmegen Asurv Gener Oxford F Categorical Logic in Relating Categorical Approac Categorical Mo dels of P F The L Amsterdam In T Computer Scienc Seman T Cate don T and ti A uijn NG tmell J enabou J B de Br Car Blanco J References Asper Bainbridge ES Freyd PJ Scedr Barendregt HP Barendregt HP Barr M Bell JL TIONS APPLICA CHAPTER e a J eal gic o tr L unctor Scienc vidence A revised c Sydney c tF o ted as CWI Pr Studia ory and Com yp e Theory Sci USA ok of mathemat o Reprin AMS Pro Math So ed ad c of s T t a in Computer h Theory gory The Handb lgebr b Univ Press ondon Nat A L yp e Dep endency Cate heme as an adjoin gories Math Struct in Comp Sci ed tics and top ostheoretic Mo d c al A c o Cam o Cate Pr Category Pr goric gic Kelley GM vidence tics of T o PhD thesis McGill Univ Mon and wise J Springer LNM Berlin eds al L a yp e Theories e top os b da Calculus Bar A gories ctic Inf Contr goric Logic eT osolini G v PhD thesis Univ Utrec categorical seman R o ations of Cate Diale een w or Comp Sci and der Cate bet Streicher Th Scedr The Applic ts of categories in and da Calculus b da Calculus and tics of algebraic Theories ed oundations J temp Math AMS Pro amb Academic Press London Springer LNCS Berlin to app ear ory Seminar y e Con Connections ory North Holland Amsterdam obinson EP CWI Amsterdam Street R Gra g y in Hyp erdo ctrines and Comprehension Sc o gic tness in F tics of the second order Lam gic Scott PJ o o duction to higher or gory The some Reyes GE o os The Heller A and VIII dels of the L del ling Polymorphism with Cate and al L ract op unctorial Seman ersion is to app ear in puter Scienc T Review of the elemen Cate Intr Do ctrines in Categorical Logic in ic F Adjoin Mo dels of the Lam Equalit Mo in T Mo On XL Phil L Comprehension Categories and the Seman and L The Inconsistency of higher order Extensions of MartinL v Seman The discrete ob jects in the eectiv to app ear els in Relating Mo dels of Impredicativ The Theory of Constructions and che F y GM c Lane S ymans K wvere FW cobs BPF cobs BPF Moggi E ock A o Kell Johnstone PT K La K Ma Lamar Lambek J Lambek J Ja Hyland JME R Ja REFERENCES e a b gic o h Inst ormal Scienc e Appl Moscho etique dor Pur Studia L rithm REFERENCES nn Ecole Norm Sup Computer A ation au Calcul des in Pitt et al es de lA daCalculus and F Applic gic Hindley and Seldin o g L o amb ransformations in and Pro c Math Sci Researc in Inform Comp e gic L endants c o c ep textual Categories e Appl L aris VI I es D aris VI I hn rep Lab Inf yp ariables and Categorical Logic Pitts AM olymorphism nn Pur Categorical Structures in nonextensional lam Elimination des Coupur Constructions al Math So ork and London ashington A and of ustr ree Pro ofs as Natural T c Camb Phil So Scott PJ Winskel G om Computer Scienc o eds Etat Univ P tation of Higher Order Logic in tics ol North Holland Amsterdam egorique des T and and yp es notion of Construction in v gic fr Bul l A ese d o yPressW Essays on Combinatory L L gic onctionel le et Seman o Math Pr PhD thesis univ P v A Th Pitt et al op oi Th Comp Sc o Pitts AM eley CA Springer Berlin to app ear ersion Conditions on v ed v ulaeasT Ehrhard Th orms and CutF Seldin JP Feys R erieur and etation F orm emantique Cat and and Academic Press New Y and VI I I Categorical g esup aris rip os theory o HB Curry o akis Y The F A small complete Category Asp ects of T Interpr dr Normal F Conf Berk Adjunction of Semifunctors dacalculus v T ism An Equational Presen Domain Theoretic Mo dels of P Alpha con XL Substitution up to Isomorphism tec L Combinatory L Generalized Algebraic Theories and Con P Computer So ciet A T Constructions Dictoses in Une S y HB ard W ashi S w y a b Hyland JME Girard JY Girard JY Scedr Ha Hindley JR Ho Coquand Th Gunter C Coquand Th Curien PL Ehrhard Th Curr Hyland JME Johnstone PT Freyd PJ Hyland JME of J Symb Math Struct SIAM J Com the Calculus of PhD thesis Univ e Calculi of Dep en b da Calculus Pitt et al oappearin Semantics al assau T e domain equations goric olymorphic Lam assau Cate ols North Holland Amsterdam gories and Polymorphism v of a dCate alen D tics for higher order P Completeness hn rep MIP Univ P PhD thesis Univ P an D and v and yp es tec Plotkin GD ctness bridge e tT cursive Domains indexe g ternal Completeness of Categories of Domains in and e o Constructivism in Mathematics Corr Constructions Dep endence and Indep endence Resultsden for Impredicativ L The categorytheoretic solution of recursiv put in Comp Sci In R Cam Categorical Seman oelstra AS ylor P a Streicher Th T Tr Smyth M REFERENCES a a Sc gic gic o o ad c e Appl A eds Studia L R Studia L c Camb Phil Springer LNM o eA en at SRI Menlo C REFERENCES nn Pur A Johnstone PT ations Poign Math Struct in Comp t Math Pr ternes h eds yp e Theory in and Pitt et al t bridge yp e theory of T Hindley and Seldin Springer LNCS Berlin in Springer Berlin ely yp e Theory Springer LNCS Berlin Springer LNCS Berlin eds lo calemen e e yp es Notes of a talk giv amming unctor Theorems in gr of s partial t o tT tF gories and their Applic ydeheard DE egories R t of Program Mo dules b da Calculus in Cat dCate Bibliop olis Nap els and x in lo cally cartesian closed categories PhD thesis Univ Utrec e ybjer P Pitts AM SIAM J Comput ory gHansen V PhD thesis Univ Cam ydeheard DE Indexe arado R ations tics of Dep enden oses e The egories and yp tenber eds ory and Computer Scienc ory and Computer Pr ory and Computer Scienc Cat cher D A A amilies and the Adjoin terpretations of MartinL ylor P Stol a des e A e R ory of trip yp es as Lattices T ates and Fibr ar and gory The gory The gory The gories for the Working Mathematician P ydeheard DE D ebre Schuma dic VI I I VI I I and e g c ark and at the Logic Collo quium in Berlin olymorphism is Set theoretic Constructiv o Relating Theories of the Lam Data T Lo cally cartesian closed Categories and T So Cate Alg Cate Paris P A note on Russells P Cate The the P Categorical Seman XL L Abstract F Domain in and Pr Sci Categorical and Algebraic Asp ects of MartinL A Category Theoretic Accoun Cate Intuitionistic T XL Berlin of P and c D wicz A vi lo y RAG tinL e R vlo ar almgren E a Scott DS Seel Pitt DH R Pitts AM Pitt DH Abramski S Poign Pitt DH Poign Pitts AM P Penon J Obtu Moggi E Mar P P x yp e theory elop e k construction v k uous of t ening er a bration category from CCC tin v erse image y o ternal v eak Picategory from LCCC b da algebra bda model tit categorical v cartesian CCompC from CCC CCompC from bred CCC con global sections in in pullbac reindexing relab elling semi substitution terminal ob ject w split categorical categorical bred ternalization trinsic rob enius unctor Logic v Logical mo del MartinL F F Generic ob ject Girards parado Grothendiec Group oid Iden Inclusion Indexed sets In In Karoubi en Lam Lam LCCC P PQ Q P P ts table t vi t t er a bration w en ultiplication v v y map vi er a bration er a bration o lo calization juxtap osition m full completion v v yp e text rules small small complete v split preorder represen complete constan full small lo cally small opp osite o clo translation of arro with unit relation of t full Hom o ree amily mo del eature F Dictos Dinatural transformation Displa Domain mo del Enough p oin Exp onen Externalization F F Fibration Constan Con Dcategory Dep endency INDEX category vi P r RP B Cons Cons t t t texts comprehension er a bration er a bration ternal ternal a v v hangeofbase c o o comp osition in bred in ternal age v constructions on constan constan constan closed CCompC for a bration monoidal of con p olynomial preorder v slice small complete v total bred in semi for a CCompC for Closure Closure mo del Cobration Coherence vii Comprehension category Clea CCC Changeofbase t w ternal ternal eak bien t functor theorem discrete full split w in in ternal ternal HOL PRED P Pi w bred bre bred HML indexed in in arro base CC semi category bration am Index Adjoin Adjunction Arro Axiom Bibration Cartesian pro duct Category an an en be een een v Een con hrij hzelf elijkt yp en atten aeid orden an op o orzien w v titeiten v at tergrond trale Een com h w die an zic e disjuncte eerste gaat oudig ac categorie cen De systemen bijb ehorende v an indicering an in het feit dat in het Engels ergemakk v de ordt een t organisatie Het hangen de een kunnen so orten Het ereist oudig te b esc an zulk termen Gebaseerd h v de yp e mag b ev zijn ormen hebb en een an orden en v ormen v mag mo et en eld atie ligt en Dit v Aldus w taire denities en resul tergrond in het Engels en w af en ev mag h orde comprehensie begrepen wikk an een t cubus v t hrijving v ereniging v yp en hrev b escrijft erzameling v categorisc hrev ormen t de hrijving v an on ordt een algemeen b egrip v v elijk atten v ho ofdstukk he motiv an zijn v tal daardo or to egestane asp ecten ezeling bration een aan Besc tal elemen wbesc he b esc prop ositie vijf afhank ariab ele erder w uit theorie he b egrip ac hap pro ducten ee alternatiev kunnen structuur termen V uit terne categorie w een ten pro ducten sommen en iden at he te theoretisc w theorie in an disjuncte v Een aan een v elijkheid b ev wd categorie Zon dat an Barendregts elijkheid hrijvingswijze eigensc k en v elijkheid indien er een so ort is die v b estaat yp en he pro ducten b esc yp en orden t de theoretisc t hrijft e asp ecten features in het Engels to elaatbaar elijkheid zijn iets minder een orm v w en en in b eno digde hrev t cartesisc orden zo opnieu deze trale b egrip is v elk De t opgeb ou oor he b esc b esc de yp en ervlak om v pro ducten b epalen als k besc is t e en tergrond plus een aan tergrond ens w die yp enafhank categorie h h yp enafhank ordt zonder Het cen orm elijk Zijdelings w hrift yp enafhank meer v tt het link eg als cartesisc ordt het t en o orb eeld relatie en Een ac ezelingen vier text die zonder t an asp ecten zijn exp onen niet ho ofdstuk elijkheid is een v atting v e afhank pro efsc bijv index erkt b etreende afhank ende systemen w u an categorie tie w v n ndiceerde categorie oor tu an tergronden zeggen dat een prop ositie o erd eede v een k h en vige ge comprehensie w die tergrond k t en h Ac tergrond is he in tro duceerd do or Grothendiec transitiev en vindt men in ho ofdstuk drie ho ofdstuk h en ingev o orb eeld text n tergrond b epaalt tev ordt b esprok tergronden met t o orb eelden v het ac h h In hillende b ek ergang naar een categorisc V ege de afhank hier ticatie bijv een onderha con hrev v Een ac w Ac In tergrond er indicering v h aartegen men w ersc erenigingen en de bijb ehorende pro jecties en an an v yp e dat wil afhangt texten kunnen simp elw w de o systeem b egrep en als een ac V categorisc setting v ac t Een ac Om zijn v quan grip v prehensie categorie geeft een passende categorisc zelfde v categorie contexten Het Sam een o baar zijn in die con taten w b esc zoals ge INDEX category v yp es comprehension a b da calculus ening functor functor mo del split mo del dep endency of all t generalized lam for yp ed t eak yp e yp e system oneda lemma erm erm mo del op ology op os rip os Y W Substitution T T T T T T T Unit Un yp es t as alence relation ternal category y mo del y ob ject hangeofbase e along arbitrary pro jections along cartesian pro jections tication ertical bred cartesian co cartesian v translation of categorical minimal prop ositional prop ositions and c and reection cp o ob ject for a comprehension category for a bration for an in semi cartesian mo dest Set artial equiv Monoid Morphism Nonempt Sort Start rules Strong sum Setting Realizabilit Reection Reexiv Separated family Pro ducts and sums Pro jection Quan P In het het er in en an tot Barendregt hering aan de ten Sundholm en ber olgens het do cto an de KUN in em v o egd onderzo ek erv olgde hij v no eldman an in Nijmegen de binatie met wijsb egeerte V en v en do or detac v op um Iongh b er en augustus aard zijn hier de do cen an de Informatica v In Eindho Onderbrok b er als to egev aren de Het candidaats en v w ho ol o or wiskunde in com ten op septem ermeldensw V do cen ozen v bridge is tot aan de zomer v Vitae er p er septem candidaatsexamen afgelegd hrift is geb oren op augustus te Nuenen N Br olledige aanstelling erd gek ersiteit Nijmegen hrijv opleiding aan het Augustinian het Bepalende is e Univ as de sc an Pisa en Cam tie zonder v olgstudies w an dit pro efsc o orb ereid erv ema cum laude wijsb egeerte ersiteiten v Als v Inmiddels w de univ de gymnasium In eerste instan aan de Katholiek raalexamen wiskunde zijn b ehaald laatste do ctoraalexamen op juni en Bouk dienst getreden bij de afdeling Grondslagen v promotie v dezelfde plaats do orliep hij de lagere sc Curriculum De auteur v De ter tiek eerst h com ac TTING te A seman he end orp en he ele individuele constan onderw SAMENV theoretisc categorisc hrijving yp en t de an onderzo ek v om Zogenaamde he daarna om enk tenslotte herzien omzetting te en de en een adequate b esc een gedetailleerd an olgens in het vijfde ho ofdstuk aangew hrijv v omstige categorisc aan erv en yp e gev calculus b esc gev te en t e bda ereenk orden h te v orden v w lam hets ten en met ten w sc en h yp eerde algemene onget ee ingredi w de een an yp ensystemen categorisc v prehensie categorie gronden en asp ecten int o om Deze t resulterende inzic