Types are weak ω-groupoids

Transcription

1 Proc London Mah Soc (3) 102 (2011) C 2010 London Mahemaical Sociey doi:101112/plm/pdq026 Type are weak ω-groupoid Benno van den Berg and Richard Garner Abrac We define a noion of weak ω-caegory inernal o a model of Marin-Löf ype heory, and prove ha each ype bear a canonical weak ω-caegory rucure obained from he ower of ieraed ideniy ype over ha ype We how ha he ω-caegorie ariing in hi way are in fac ω-groupoid 1 Inroducion I ha long been underood ha here i a cloe connecion beween algebraic opology and higher-dimenional caegory heory [9] More recenly, i ha become apparen ha boh are in urn relaed o he inenional ype heory of Marin-Löf [17] While aemp o make hi link precie have only borne frui in he pa few year [1, 6, 8, 19], he baic idea dae back o an obervaion made by Hofmann and Sreicher [10] Recall ha in Marin-Löf ype heory, we may conruc from a ype A and elemen a, b A, a new ype Id(a, b) whoe elemen are o be hough of a proof ha a and b are propoiionally equal Hofmann and Sreicher oberve ha he ype-heoreic funcion 1 Id(a, a), Id(b, c) Id(a, b) Id(a, c) and Id(a, b) Id(b, a) expreing he reflexiviy, raniiviy and ymmery of propoiional equaliy allow u o view he ype A a a groupoid (a caegory whoe every morphim i inverible) wherein objec are elemen a A and morphim a b are elemen p Id(a, b) However, a i made clear in [10], hi i no he end of he ory The groupoid axiom for A hold only up o propoiional equaliy ; which i o ay ha, for example, he aociaiviy diagram Id(c, d) Id(b, c) Id(a, b) Id(c, d) Id(a, c) Id(b, d) Id(a, b) Id(a, d) doe no commue on he noe, bu only up o uiable erm α p,q,r Id(r (q p), (r q) p) (p Id(a, b),q Id(b, c),r Id(c, d)) Thu, if we wih o view A a an hone groupoid, we mu fir quoien ou he e of elemen p Id(a, b) by propoiional equaliy A more familiar inance of he ame phenomenon occur in conrucing he fundamenal groupoid of a pace, where we mu idenify pah up o homoopy, and hi ugge he following analogy: ha ype are like opological pace, and propoiional equaliy i like he homoopy relaion Uing he machinery of abrac homoopy heory, hi analogy ha been given a precie form in [1], which conruc ype-heoreic rucure from homoopy-heoreic one, and in [6], which doe he convere Received 1 December 2008; revied 14 June 2010; publihed online 12 Ocober Mahemaic Subjec Claificaion 3B15, 18D05 (primary); 18D50 (econdary) The econd-named auhor alo acknowledge he uppor of a Reearch Fellowhip of S John College, Cambridge and a Marie Curie Inra-European Fellowhip, Projec No

2 TYPES ARE WEAK ω-groupoids 371 The connecion wih algebraic opology in urn ugge he one wih higher-dimenional caegory heory A more ophiicaed conrucion of he fundamenal groupoid of a pace (uggeed in [9] and made rigorou in [2]) doe no quoien ou pah by he homoopy relaion; bu inead incorporae hee homoopie, and all higher homoopie beween hem, ino an infinie-dimenional caegorical rucure known a a weak ω-groupoid, whoe variou ideniie, compoiion and invere aify coherence law, no ricly, bu up o all higher homoopie Thi lead u o ak wheher he conrucion of he ype-heoreic fundamenal groupoid admi a imilar refinemen, which conruc a weak ω-groupoid from a ype by conidering no ju elemen of he ype, and proof of heir equaliy, bu alo proof of equaliy beween uch proof, and o on The principal aim of hi paper i o how hi o be he cae In order o give he proof, we mu fir chooe an appropriae noion of weak ω-groupoid o work wih; and ince, in he lieraure, weak ω-groupoid are udied in he broader conex of weak ω-caegorie, which are weak ω-groupoid wihou he invere, hi i anamoun o chooing an appropriae noion of weak ω-caegory There are a number of definiion o pick from, and hee differ from each oher boh in heir general approach and in he deail; ee [13] for an overview Of hee, i i he definiion of Baanin [2] which mache he ype heory mo cloely, for he following wo reaon Firly, i baic cellular daa are globular: which i o ay ha an n-cell α : x y can only exi beween a pair of parallel (n 1)-cell x, y : f g A correponding propery hold for proof of equaliy in ype heory: o know ha α Id(x, y), we mu fir know ha x and y inhabi he ame ype Id(f,g) Secondly, Baanin definiion i algebraic, which i o ay ha compoiion operaion are explicily pecified, raher han merely aered o exi Thi accord wih he conrucivi noion, cenral o he piri of inenional ype heory, ha o know omehing o exi i nohing le han o be provided wih a wine o ha fac On hee ground, i i Baanin definiion which we will adop here; or raher, a mild reformulaion of hi definiion given by Leiner [13] The paper i arranged a follow In Secion 2, we recall Baanin heory of weak ω- caegorie, he appropriae pecializaion o weak ω-groupoid, and he neceary background from inenional ype heory Then in Secion 3 we give he proof of our main reul We begin in Subecion 31 wih an explicily ype-heoreic, bu informal, accoun When we come o make hi precie, i urn ou o be convenien o iolae ju hoe caegorical properie of he ype heory ha make he proof go hrough, and hen o work in an axiomaic eing auming only hee We decribe hi eing in Subecion 32, and hen in Subecion 33 and 34 ue i o give a formal proof ha every ype i a weak ω-groupoid 2 Preparaory maerial In hi ecion, we review he maerial neceary for our main reul; firly, from higher caegory heory, and econdly, from Marin-Löf ype heory 21 Weak ω-caegorie and weak ω-groupoid A menioned in Secion 1, he mo appropriae definiion of weak ω-caegory for our purpoe i ha of [2], which decribe hem a globular e equipped wih algebraic rucure A globular e i a diagram of e and funcion X 0 X 1 X 2 X 3 aifying he globulariy equaion = and = We refer o elemen x X n a n-cell of X, and wrie hem a x : x x In hi erminology, he globulariy equaion expre ha any (n + 2)-cell f g mu mediae beween (n + 1)-cell f and g which are parallel,

3 372 BENNO VAN DEN BERG AND RICHARD GARNER in he ene of having he ame ource and arge Globular e alo have a coinducive characerizaion: o give a globular e X i o give a e ob X of objec, ogeher wih for each x, y ob X, a globular e X(x, y) The algebraic rucure required o make a globular e ino a weak ω-caegory i encoded by any one of a cerain cla of monad on he caegory of globular e: hoe ariing from normalized, conracible, globular operad Informally, uch monad are obained by deforming he monad T whoe algebra are ric ω-caegorie To make hi precie, we mu fir recall ome deail concerning ric ω-caegorie If V i any caegory wih finie produc, hen one can peak of caegorie enriched in V, and of V-enriched funcor beween hem [12] The caegory V-Ca of mall V-caegorie i hen ielf a caegory wih finie produc, o ha we can ierae he proce; and when we do o aring from V = 1, we obain he equence 1, Se, Ca, 2-Ca,,whoenh erm i he caegory of mall ric (n 1)-caegorie Now, becaue any finie-produc preerving funcor V Winduce a finie-produc preerving funcor V-Ca W-Ca, we obain, by ieraion on he unique funcor Se 1, a chain n-ca 2-Ca Ca Se 1; and ω-ca, he caegory of mall ric ω-caegorie, i he limi of hi equence Unfolding hi definiion, we find ha a ric ω-caegory i given by, fir, an underlying globular e; nex, operaion of ideniy and compoiion: hu for each n-cell x, an(n + 1)-cell id x : x x, and for each pair of n-cell f and g haring a k-cell boundary (for k<n), a compoie n-cell g k f; and finally, axiom expreing ha any wo way of compoing a diagram of n-cell uing he above operaion yield he ame reul There i an eviden forgeful funcor U : ω-ca GSe, where GSe denoe he caegory of globular e; and i i hown in [14, Appendix B] ha hi ha a lef adjoin and i finiarily monadic The correponding monad T on he caegory of globular e may be decribed a follow Fir we give an inducive characerizaion of T 1, i value a he globular e wih one cell in every dimenion We have ha: (i) (T 1) 0 = { }; (ii) (T 1) n+1 = { (π 1,,π k ) k N,π 1,,π k (T 1) n } The ource and arge map, :(T 1) n+1 (T 1) n coincide and we follow [14] in wriing for he common value Thi oo ha an inducive decripion a follow: (i) (π) = for π (T 1) 1 ; (ii) (π 1,,π k )=( (π 1 ),, (π k )) oherwie We regard elemen of (T 1) n a indexing poible hape for paing diagram of n-cell For example, (( ), (, )) (T 1) 2 correpond o he hape (1) We formalize hi correpondence by aociaing o each elemen π (T 1) n a globular e ˆπ which i he hape indexed by π (i) If π =, hen ˆπ i he globular e wih ob ˆπ = { } and ˆπ(, ) = (ii) If π =(π 1,,π k ), hen ˆπ i he globular e wih ob ˆπ = {0,,k}, ˆπ(i 1,i)= π i (for 1 i k), and ˆπ(i, j) = oherwie By a furher inducion, we define ource and arge embedding, : π ˆπ (i) For π (T 1) 1, he map, : ˆ ˆπ end he unique objec of ˆ o he malle and large elemen of ob ˆπ, repecively (ii) Oherwie, for π =(π 1,,π k ) he morphim and are he ideniy on objec and map π(i 1,i)ino π(i 1,i)via, : π i π i

4 TYPES ARE WEAK ω-groupoids 373 Taken ogeher, he globular e T 1, he globular e ˆπ, and he map and, compleely deermine he funcor T ; hi by virue of i being familially repreenable in he ene of [15, Definiion C31] (hough ee alo [4]) Explicily, TX i he globular e whoe cell are paing diagram labelled wih cell of X: (TX) n = GSe(ˆπ, X), π (T 1) n and whoe ource and arge map are induced in an obviou way by he map and The uni and muliplicaion of he monad T are careian naural ranformaion, which i o ay ha all of heir nauraliy quare are pullback; from which i follow ha hee are in urn deermined by heir componen η 1 :1 T 1andμ 1 : TT1 T 1 The former map aociae o he unique n-cell of 1 he paing diagram ι n :=(( )) (T 1) n, while he laer end a ypical elemen (π (T 1) n,φ:ˆπ T 1) of (TT1) n o he elemen φ π (T 1) n obained by ubiuing ino π he paing diagram ha φ indexe (ee [14, Secion 42] for a picorial accoun of hi proce) A globular operad can now be defined raher uccincly: i i a monad P on GSe equipped wih a careian monad morphim ρ : P T The careianne of ρ implie ha he funcor par of P i deermined by i componen a 1 ogeher wih he map ρ 1 : P 1 T 1, and i will be convenien o have a decripion of P in hee erm Given π (T 1) n, we wrie P π for he e of hoe θ (P 1) n ha are mapped o π by ρ 1, and wrie, : P π P π for he correponding rericion of he ource and arge map of P 1 The value of P a an arbirary globular e X i now given (up o iomorphim) by (PX) n = P π GSe(ˆπ, X), (2) π (T 1) n wih he ource and arge map deermined in he obviou way Thu, if we hink of a T -algebra rucure on X a providing a unique way of compoing each X-labelled paing diagram of hape π, hen a P -algebra rucure provide a e of poible way of compoing uch diagram, indexed by he elemen of P π I follow from he careianne of ρ ha he uni and he muliplicaion of P are hemelve careian naural ranformaion, and hence deermined by heir componen η 1 :1 P 1and μ 1 : PP1 P 1 The former end he unique n-cell of 1 o an elemen ι n P ιn,whichwe hink of a he rivial compoiion operaion of dimenion n; while he laer aign o he elemen (π (T 1) n,θ P π,ψ:ˆπ P 1) of (PP1) n an elemen θ ψ P φ π (where φ i he compoie ρ 1 ψ :ˆπ T 1), which we hink of a he compoiion operaion obained by ubiuing ino θ he collecion of operaion indexed by ψ No every globular operad embodie a enible heory of weak ω-caegorie (ince, for example, he ideniy monad on GSe i a globular operad), bu [2] provide wo condiion which ogeher diinguih hoe which do: normalizaion and conracibiliy Normalizaion i raighforward: i aer ha he monad P i bijecive on objec in he ene ha (PX) 0 = X0, naurally in X; or equivalenly, ha he e P i a ingleon The econd condiion i a lile more uble A globular operad P i aid o be conracible if: (a) given π (T 1) 1 and θ 1,θ 2 P, here exi an elemen φ P π wih (φ) =θ 1 and (φ) =θ 2 ; (b) given π (T 1) n (for n>1) and θ 1,θ 2 P π aifying (θ 1 )=(θ 2 )and(θ 1 )=(θ 2 ), here exi an elemen φ P π uch ha (φ) =θ 1 and (φ) =θ 2

5 374 BENNO VAN DEN BERG AND RICHARD GARNER Conracibiliy expree a globular operad ha enough way of compoing o yield a heory of weak ω-caegorie In homoopy-heoreic erm, a conracible globular operad i a deformaion of he monad T ; an idea which can be made precie uing he language of weak facorizaion yem (ee [7]) Definiion 21 A weak ω-caegory i an algebra for a conracible, normalized, globular operad: more formally, i i a pair (P, X), where P i a conracible, normalized, globular operad and X i an algebra for i Remark 22 Some conideraion mu be paid o he exac force of he erm conracible, which ha been ued in differen way by differen auhor; our uage accord wih ha given in [15, Definiion 913] In paricular, he reader hould carefully diinguih beween he propery of being conracible decribed above, and he correponding rucure of being equipped wih a conracion We now urn from he definiion of weak ω-caegory o ha of weak ω-groupoid For hi we will require he coinducive noion of equivalence in a weak ω-caegory Definiion 23 Le (P, X) beaweakω-caegory An equivalence x y beween parallel n-cell x, y i given by he following: (i) n + 1-cell f : x y and g : y x; (ii) equivalence η : g f id x and ɛ: f g id y We ay ha an (n + 1)-cell f : x y i weakly inverible if i paricipae in an equivalence (f,g,η,ɛ) In order for hi definiion o make ene, we mu deermine wha i mean by he expreion id x, id y, g f and f g appearing in i We do hi a follow Fir, for each n 1, we define he paing diagram 0 n and 2 n (T 1) n o be given by 0 n :=(()) }{{} and 2 n :=((, ) ) }{{} n ime n ime Nex, if P i a normalized, conracible, globular operad, hen we define a yem of compoiion for P o be a choice, for each n 1, of operaion i n P 0n and m n P 2n Noe ha he conracibiliy of P enure ha i will poe a lea one yem of compoiion Finally, if we are given a yem of compoiion and a P -algebra X, hen we define he funcion id ( ) : X n 1 X n and : X n+1 X n+1 X n o be he inerpreaion of he operaion i n and m n, repecively Thi allow u o give meaning o he undefined expreion appearing in Definiion 23 Definiion 24 A weak ω-caegory (P, X) iaweak ω-groupoid if every cell of X i weakly inverible wih repec o every yem of compoiion on P I will be convenien o give a more elemenary reformulaion of he noion of weak ω- groupoid due o Cheng [5] Thi i given in erm of dual Iff : x y i an n-cell (for n 1) in a weak ω-caegory, hen a dual for f i an n-cell f : y x ogeher wih (n + 1)-cell

6 TYPES ARE WEAK ω-groupoids 375 η :id x f f and ɛ : f f id y, ubjec o no axiom Again, hi definiion i o be inerpreed wih repec o ome given yem of compoiion Propoiion 25 A weak ω-caegory i a weak ω-groupoid if and only if, wih repec o every yem of compoiion, every cell ha a dual Proof By coinducion 22 Marin-Löf ype heory By inenional Marin-Löf ype heory, we mean he logical calculu e ou in [17, ParII] We now ummarize hi calculu I ha four baic form of judgemen: A ype ( A i a ype ); a A ( a i an elemen of he ype A ); A = B ype ( A and B are definiionally equal ype ); and a = b A ( a and b are definiionally equal elemen of he ype A ) Thee judgemen may be made eiher aboluely, or relaive o a conex Γ of aumpion, in which cae we wrie hem a (Γ) A ype, (Γ) a A, (Γ) A = B ype and (Γ) a = b A, repecively Here, a conex i a li Γ = (x 1 A 1,x 2 A 2,, x n A n 1 ), wherein each A i i a ype relaive o he conex (x 1 A 1,, x i 1 A i 1 ) There are now ome raher naural requiremen for well-formed judgemen: in order o aer ha a A, we mu fir know ha A ype; o aer ha A = B ype, we mu fir know ha A ype and B ype; ando on We pecify inenional Marin-Löf ype heory a a collecion of inference rule over hee form of judgemen Firly we have he equaliy rule, which aer ha he wo judgemen form A = B ype and a = b A are congruence wih repec o all he oher operaion of he heory; hen we have he rucural rule, which deal wih weakening, conracion, exchange and ubiuion; and finally, he logical rule, which pecify he ype-former of our heory, ogeher wih heir inroducion, eliminaion and compuaion rule For he purpoe of hi paper, we require only he rule for he ideniy ype, which we li in Table 1 We commi he uual abue of noaion in leaving implici an ambien conex Γ common o he premie and concluion of each rule, and omiing he rule expreing abiliy under ubiuion in hi ambien conex Le u remark alo ha in he rule Id -elim and Id -comp, we allow he ype C over which eliminaion i occurring o depend upon an addiional conexual parameer Δ Were we o add Π-ype (dependen produc) o our calculu, hen hee rule would be equivalen o he uual ideniy ype rule However, in he abence of Π-ype, hi exra parameer i eenial o derive all bu he mo baic properie of he ideniy ype Table 1 Ideniy ype rule A ype a, b A Id-form; Id A (a, b) ype Ideniy ype A ype a A r(a) Id A (a, a) Id-inro (x, y A, p Id A (x, y), Δ(x, y, p)) C(x, y, p) ype (x A, Δ(x, x, r(x))) d(x) C(x, x, r(x)) a, b A p Id A (a, b) (Δ(a, b, p)) J d (a, b, p) C(a, b, p) (x, y A, p Id A (x, y), Δ(x, y, p)) C(x, y, p) ype (x A, Δ(x, x, r(x))) d(x) C(x, x, r(x)) a A Id-comp (Δ(a, a, r(a))) J d (a, a, r(a)) = d(a) C(a, a, r(a)) Id-elim

7 376 BENNO VAN DEN BERG AND RICHARD GARNER We now eablih ome furher noaional convenion Where i improve clariy we may omi bracke in funcion applicaion, wriing hgf x in place of h(g(f(x))), for example We may drop he ubcrip A in an ideniy ype Id A (a, b) where no confuion eem likely o occur Given a, b A, we may ay ha a and b are propoiionally equal o indicae ha he ype Id(a, b) i inhabied We hall alo make ue of vecor noaion in he yle of [3] Given a conex Γ = (x 1 A 1,,x n A n ), we may abbreviae a erie of judgemen: a 1 A 1, a 2 A 2 (a 1 ),,a n A n (a 1,,a n 1 ), a a Γ, where a := (a 1,,a n ) We may alo ue hi noaion o abbreviae equence of hypoheical elemen; o, for example, we may pecify a dependen ype in conex Γ a (x Γ) A(x) ype We will alo make ue of he noion of elecope defined in [3] Given Γ a conex a before, hi allow u o abbreviae he erie of judgemen (x Γ) B 1 (x) ype, (x Γ, y 1 B 1 ) B 2 (x, y 1 ) ype, (x Γ, y 1 B 1,, y m 1 B m 1 ) B m (x, y 1,y m 1 ) ype a (x Γ) Δ(x) cx, where Δ(x) := (y 1 B 1 (x), y 2 B 2 (x, y 1 ), ) We ay ha Δ i a conex dependen upon Γ Given uch a dependen conex, we may abbreviae he erie of judgemen (x Γ) f 1 (x) B 1 (x), (x Γ) f m (x) B m (x, f 1 (x),,f m 1 (x)) a (x Γ) f(x) Δ(x), and ay ha f i a dependen elemen of Δ We can imilarly aign a meaning o he judgemen (x Γ) Δ(x) =Θ(x) cx and (x Γ) f(x) =g(x) Δ(x), expreing he definiional equaliy of wo dependen conex, and he definiional equaliy of wo dependen elemen of a dependen conex Le u now recall ome baic fac abou caegorical model of ype heory For a more deailed reamen he reader could refer o [11, 18], for example If T i a dependenly yped calculu admiing each of he rule decribed above, hen we may conruc from i a caegory C T known a he claifying caegory of T I objec are conex Γ, Δ,, in T, conidered modulo definiional equaliy (o we idenify Γ and Δ whenever Γ = Δ cx i derivable); and i map Γ Δareconex morphim, which are judgemen (x Γ) f(x) Δ conidered modulo definiional equaliy The ideniy map on Γ i given by (x Γ) x Γ, while compoiion i given by ubiuion of erm Now, for any judgemen (x Γ) A(x) ype of T, here i a diinguihed conex morphim (x Γ, y A(x)) (x Γ) which end (x, y) ox We call morphim of C T of hi form baic dependen projecion By a dependen projecion, we mean any compoie of zero or more baic dependen projecion An imporan propery of dependen projecion i ha hey are able under pullback, in he ene ha, for every (x Γ) A(x) ype and conex morphim f :Δ Γ, we may how he quare (w Δ, y A(f(w))) (x Γ, y A(x)) p Δ f p Γ,

8 TYPES ARE WEAK ω-groupoids 377 wherein he uppermo arrow end (w, y) o(fw,y), o be a pullback in C T Le u now recall from [6] a econd cla of map in C T which will play an imporan role in hi paper A conex morphim f :Γ Δ i aid o be an injecive equivalence if i validae ype-heoreic rule: (y Δ) Λ(y) cx (x Γ) d(x) Λ(f(x)) b Δ E d (b) Λ(b) and (y Δ) Λ(y) cx (x Γ) d(x) Λ(f(x)) a Γ E d (f(a)) = d(a) Λ(f(a)) The name i moivaed by he groupoid model of ype heory, wherein he injecive equivalence are preciely he injecive groupoid equivalence Inuiively, a morphim f :Γ Δ i an injecive equivalence ju when every (dependen) funcion ou of Δ i deermined, up o propoiional equaliy, by i rericion o Γ The leading example of an injecive equivalence i given by he conex morphim A (x, y A, p Id(x, y)) ending x o (x, x, rx) Tha hi map i an injecive equivalence i preciely he conen of he Id-eliminaion and compuaion rule Diagramaically, a map f i an injecive equivalence if, for every commuaive quare of he form h Γ Δ d (Δ, Λ) p Δ wih p a dependen projecion, we may find a diagonal filler E d :Δ (Δ, Λ) making boh induced riangle commue By he abiliy of dependen projecion under pullback, hi i equivalen wih he propery ha we hould be able o find filler for all commuaive quare of he form d Γ (Φ, Λ) h Δ k p (3) Φ again wih p a dependen projecion See [6, Secion 5] for an elemenary characerizaion of he cla of injecive equivalence 31 An overview of he proof 3 The main reul We are now ready o begin he proof of our main reul, which ay ha if T i a dependenly yped calculu admiing each of he rule decribed in Subecion 22, hen each ype A herein give rie o a weak ω-groupoid whoe objec are elemen of A, and whoe higher cell are elemen of he ieraed ideniy ype on A In fac, we will be able o prove he ronger reul ha A provide he ype of objec for a weak ω-groupoid which i, in a uiable ene, inernal o T A explained in Secion 1, we will give our proof wice: once informally, uing a ype-heoreic language, and once formally, uing an axiomaic caegorical framework which capure ju hoe apec of he ype heory ha allow he proof o go hrough In hi ecion, we give he informal proof We hall concenrae in he fir inance on conrucing a weak ω-caegory, and defer he queion of wheher or no i i a weak ω-groupoid unil he formal proof

9 378 BENNO VAN DEN BERG AND RICHARD GARNER We begin by defining wha we mean by a weak ω-caegory inernal o a ype heory T More pecifically, given ome globular operad P, we define a noion of P -algebra inernal o T The underlying daa for uch a P -algebra i a globular conex (Δ) Γ T; which i a equence of judgemen a follow: (Δ) Γ 0 cx, (Δ,x,y Γ 0 )Γ 1 (x, y) cx, (Δ,x,y Γ 0,p,q Γ 1 (x, y)) Γ 2 (x, y, p, q) cx, Like globular e, globular conex alo have a coinducive characerizaion: o give a globular conex (Δ) Γ i o give a conex (Δ) Γ 0 and a globular conex (Δ, x,y :Γ 0 )Γ +1 (x, y) In order o define he operaion making a globular conex Γ (where henceforh we implify he noaion by omiing he preconex Δ) ino a P -algebra, we fir define for each paing diagram π (T 1) n he conex Γ π coniing of π-indexed elemen of Γ Thi i done by inducion on π: (i) if π =, hen Γ π :=Γ 0 ; (ii) if π =(π 1,,π k ), hen Γ π i he conex (x 0,,x k Γ 0,y 1 Γ +1 (x 0,x 1 ) π1,,y k Γ +1 (x k 1,x k ) π k ) For example, if π i he paing diagram (1), hen he conex Γ π i given by (x 0,x 1,x 2 Γ 0,, Γ 1 (x 0,x 1 ),α Γ 2 (x 0,x 1,,), u,v,w Γ 1 (x 1,x 2 ), β Γ 2 (x 1,x 2,u,v), γ Γ 2 (x 1,x 2,v,w)) while if π (T 1) n i he elemen ι n =(( ) ), hen Γ ιn i he conex (x 0,y 0 Γ 0,x 1,y 1 Γ 1 (x 0,y 0 ),, x n (x 0,y 0,,x n 1,y n 1 )) indexing he oaliy of he n-cell of Γ Now o give a P -algebra rucure on he globular conex Γ will be o give, for every π (T 1) n and θ P π, a conex morphim [θ] :Γ π Γ ιn inerpreing he operaion θ, ubjec o he following axiom Firly, he inerpreaion hould be compaible wih ource and arge, which i o ay ha diagram of he form Γ π [θ] Γ ιn Γ π [θ] Γ ιn Γ π [θ] Γ ι n 1 and Γ π [θ] Γ ι n 1 hould commue; here, :Γ π Γ π are ource and arge projecion, repecively, defined by a furher raighforward inducion over π Secondly, he rivial paing operaion hould have a rivial inerpreaion; which i o ay ha [ι n ]=id Γ ιn :Γ ιn Γ ιn

10 TYPES ARE WEAK ω-groupoids 379 Thirdly, he inerpreaion of a compoie [θ ψ] hould be given by he compoie of [θ] wih [ψ], in he ene ha he following diagram commue: Γ π φ [ψ] Γ π [θ] [θ ψ] Γ ιn Thi i no ye enirely formal, becaue we have no indicaed how he map [ψ] :Γ π φ Γ π hould be defined Inuiively, i i he morphim ha applie imulaneouly he inerpreaion of he operaion indexed by ψ :ˆπ P 1; bu i i no immediaely clear how o make hi precie We hall do o in Subecion 33 below, uing Michael Baanin machinery of monoidal globular caegorie [2] A general reul from hi heory allow u o aociae o he globular conex Γ a paricular globular operad [Γ, Γ], he endomorphim operad of Γ, which i uch ha we may define P -algebra rucure on Γ o be globular operad morphim P [Γ, Γ] Thi operad [Γ, Γ] ha a operaion of hape π, all erially commuaive diagram Γ π f n Γ ιn Γ π f n 1 Γ ιn 1 g n 1 f 0 Γ Γ g 0 (4) of conex morphim The ource and arge funcion [Γ, Γ] π [Γ, Γ] π end uch a diagram o i ubdiagram headed by f n 1 and g n 1, repecively; he ideniy operaion ι n [Γ, Γ] ιn ha each f i and g i given by an ideniy map; whil decribing ubiuion of operaion in [Γ, Γ] i preciely he problem ha we encounered above, and which Baanin machinery olve I i eay o ee ha a map of globular operad P [Γ, Γ] encode exacly he rucure of an inernal P -algebra keched above We may now give a precie aemen of he main reul Given a ype heory T admiing he rule of Subecion 22 and a ype A T, we conruc a normalized, conracible, globular operad P uch ha he globular conex A given by A cx, (x, y A) Id A (x, y) cx, (x, y A, p, q Id A (x, y)) Id IdA(x,y)(p, q) cx, admi an inernal P -algebra rucure Now, i i raighforward o find an operad for which A i an algebra, namely, he endomorphim operad [A,A], wih algebra rucure given by he ideniy morphim [A,A] [A,A], bu hi doe no help u, ince here i no reaon o expec hi operad o be eiher normalized or conracible However, i come raher cloe o being conracible, in a ene ha we hall now explain For [A,A] o be conracible would be for

11 380 BENNO VAN DEN BERG AND RICHARD GARNER u o ak ha, for every erially commuaive diagram A π A ιn A π f n 1 A ιn 1 g n 1 A f 0 A g 0 (5) of conex morphim, we could find a map A π A ιn compleing i o a diagram like (4) Le u conider in paricular he cae where π i he paing diagram of (1) Here, o give he daa of (5) i o give he judgemen (x A) f 0 (x) A, (x A) g 0 (x) A, (x, y, z A, p Id(x, y),q Id(y, z)) f 1 (x, y, z, p, q) Id(f 0 (x),g 0 (z)), (x, y, z A, p Id(x, y),q Id(y, z)) g 1 (x, y, z, p, q) Id(f 0 (x),g 0 (z)), (6) while o give i compleion f 2 : A π A ι2 would be o give a judgemen (x, y, z A,, Id(x, y), α Id(, ), u,v,w Id(y, z), β Id(u, v), γ Id(v, w)) f 2 (x, y, z,,, α, u, v, w, β, γ) Id(f 1 (x, y, z,, u),g 1 (x, y, z,, w)) We migh aemp o obain uch a judgemen by repeaed applicaion of he ideniy ype eliminaion rule Indeed, by Id-eliminaion on α, i uffice o conider he cae where = and α =r(); and by Id-eliminaion on γ and β, i uffice o conider he cae where u = v = w and γ = β =r(u) Thu i uffice o find a erm (x, y, z A, Id(x, y), u Id(y, z)) f 2(x, y, z,, u) Id(f 1 (x, y, z,, u),g 1 (x, y, z,, u)) Bu now by Id-eliminaion on and on u, i uffice o conider he cae where x = y = z and = u =r(x); o ha i even uffice o find a erm (x A) f 2 (x) Id(f 1 (x, x, x, rx, rx),g 1 (x, x, x, rx, rx)) (7) Ye here we encouner he problem ha f 1 and g 1, being arbirarily defined, need no agree a (x, x, x, rx, rx), o ha here i in general no reaon for a erm like (7) oexihowever,here i a raighforward way of removing hi obrucion: we reric aenion o hoe operaion of hape π ha, when applied o a erm coniing olely of reflexiviy proof, yield anoher reflexiviy proof We may formalize hi a follow For each π (T 1) n, we define, by inducion on π, a poining r π : A A π uch ha: (i) if π =, hen r :=id:a A; (ii) if π =(π 1,,π k ), hen r π i he conex morphim (x A) (x,,x,r }{{} π1 (rx),,r πk (rx)) A π k ime

12 TYPES ARE WEAK ω-groupoids 381 In our example, if he judgemen in (6) commued wih he A-poining, hen we would have ha f 1 (x, x, x, rx, rx) =g 1 (x, x, x, rx, rx) =r(x) Id(x, x), o ha in (7) we could define (x A) f 2 (x) := r(rx) Id(rx, rx) and in hi way obain by repeaed Id-eliminaion he deired compleion f 2 : A π A ι2 Moivaed by hi, we define he uboperad P [A,A] o have a i operaion of hape π hoe diagram of he form (4) in which each f i and g i commue wih he A-poining ju defined Again, i i inuiively clear ha hi define a uboperad, which i o ay ha he operaion wih hi propery are cloed under ideniie and ubiuion, bu o prove hi require a econd excurion ino he heory of monoidal globular caegorie, one which for he purpoe of he preen ecion, we omi However, we claim furher ha P i boh normalized and conracible Thi will hen prove our main reul, ince he globular conex A i a P -algebra, a wineed by he map of globular operad P [A,A], o ha we will have hown he globular conex A o be an algebra for a normalized, conracible, globular operad P, and hence a weak ω-caegory Now, o how P normalized i rivial, ince i operaion of hape are hoe conex morphim A A ha commue wih he poining id A : A A, and here i of coure only one uch On he oher hand, we ee ha i i conracible hrough a generalizaion of he argumen given in he example above The only par requiring ome hough i how o decribe generically he proce of repeaedly applying Id-eliminaion The key o doing hi i o prove by inducion on π ha each of he poining r π : A A π i an injecive equivalence in he ene defined in Subecion 22 The injecive equivalence rucure now encode he proce of repeaed Id-eliminaion Uing hi, we may how P conracible a follow Suppoe ha we are given a diagram like (5), where each f i and g i commue wih he A-poining We le BA ιn denoe he conex obained from A ιn by removing i final variable, and le p : A ιn BA ιn denoe he correponding dependen projecion Then we have a commuaive quare A r ιn A ι n p A π BA ι n, r π where he lower arrow i obained by applying fir he projecion A π BA π, and hen he map f n 1 and g n 1 Commuaiviy obain by virue of he fac ha f n 1 and g n 1 commue wih he poining; and o, becaue r π i an injecive equivalence and p i a dependen projecion, we can find a diagonal filler, which will be he required map f n : A π A ιn 32 An axiomaic framework We now wih o make rigorou he above proof; and a we have already menioned, we hall do o no in an explicily ype-heoreic manner, bu raher wihin an axiomaic caegorical framework In hi ecion, we decribe hi framework and give he inended ype-heoreic inerpreaion Definiion 31 A caegory C i an ideniy ype caegory if i come equipped wih wo clae of map I, P mor C aifying he following axiom: (i) Empy: C ha a erminal objec 1, and for all A C, he unique map A 1iaP-map; (ii) Compoiion: he clae of P-map and I-map conain he ideniie and are cloed under compoiion; (iii) Sabiliy: pullback of P-map along arbirary map exi, and are again P-map; (iv) Frobeniu: he pullback of an I-map along a P-map i an I-map;

13 382 BENNO VAN DEN BERG AND RICHARD GARNER (v) Orhogonaliy: for every commuaive quare f A C i p B g D (8) wih i I and p P, we can find a diagonal filler j : B C uch ha ji = f and pj = g; (vi) Ideniie: for every P-map p : C D, he diagonal map Δ : C C D C ha a facorizaion where r I and e P Δ=C r e Id(C) C D C We make wo remark concerning hi definiion Firly, by (Empy) and (Sabiliy), any ideniy ype caegory will have finie produc, and produc projecion will be P-map Secondly, in order o verify (Orhogonaliy), i uffice, by (Sabiliy), o do o only in hoe cae where he map along he boom of (8) i an ideniy Propoiion 32 Le T be a dependen ype heory admiing each of he inference rule decribed in Subecion 22 Then he claifying caegory C T i an ideniy ype caegory, where we ake P o be he cla of dependen projecion and I o be he cla of injecive equivalence Proof The empy conex ( ) provide a erminal objec of C T (Compoiion) i immediae from he definiion (Sabiliy) correpond o he poibiliy of performing ype-heoreic ubiuion (Frobeniu) i hown o hold in [6, Propoiion 14]; i i a caegorical correlae of he fac ha we allow an exra conexual parameer Δ in he aemen of he Id-eliminaion rule (Orhogonaliy) hold by he very definiion of injecive equivalence, ogeher wih he remark made above Finally, (Ideniie) ay omehing more han ha ideniy ype exi; i ay ha ideniy conex exi: which i o ay ha, for every dependen conex (Δ) Γ cx, we may find a conex (Δ, x,y Γ) Id Γ (x, y) cx uch ha he conexual analogue of he ideniy ype rule are validaed Tha hi i poible i proved in [8, Propoiion 331] We alo require wo abiliy properie of ideniy ype caegorie Propoiion 33 Le C be an ideniy ype caegory and le X C Then he colice caegory X/C i alo an ideniy ype caegory, where we ake he cla of I-map and P-map o coni of hoe morphim ha become I-map and P-map, repecively, upon applicaion of he forgeful funcor X/C C Propoiion 34 Le C be an ideniy ype caegory and le X C Then he caegory C X, whoe objec are P-map A X and whoe morphim are commuaive riangle, i alo an ideniy ype caegory, where we define he clae of I-map and P-map in a manner analogou o ha of Propoiion 33

14 TYPES ARE WEAK ω-groupoids 383 The proof are rivial; he only poin of noe i ha, in he econd inance, we could no ake C X o be he full lice caegory C/X, a hen (Empy) would no be aified 33 Inernal weak ω-groupoid In hi ecion, we decribe he noion of weak ω-groupoid inernal o an ideniy ype caegory C We begin by defining inernal P -algebra for a globular operad P Definiion 35 A pre-globular conex in C i a diagram Γ 0 Γ 1 Γ 2 Γ 3 aifying he globulariy equaion = and = A pre-globular conex i a globular conex if, for each n 1, he map (, ) : B n Γ (9) i a P-map, where B n Γ i defined a follow We have B 1 Γ := Γ 0 Γ 0, and have B n+1 Γ given by he pullback B n+1 Γ (,) (,) Bn Γ (10) Oberve ha requiring (9) obeap-map for n = 1 enure he exience of he pullback (10) defining B 2 Γ; which in urn allow u o require ha (9) hould be a P-map for n =2,ando on Once again, we have a coinducive characerizaion of globular conex: o give a globular conex Γ Ci o give an objec Γ 0 ogeher wih a globular conex Γ +1 C Γ0 Γ 0 The fir ep in defining P -algebra rucure on a globular conex Γ i o decribe he objec Γ π of π-indexed elemen of Γ Definiion 36 Le Γ be a globular conex in C and le π (T 1) n We define he objec Γ π Cby he following inducion (i) If π =, hen Γ π :=Γ 0 (ii) If π =(π 1,,π k ), hen we fir form he objec (Γ +1 ) π1,, (Γ +1 ) π k of C Γ0 Γ 0 Thi yield a diagram (Γ +1 ) π1 (Γ +1 ) π k Γ 0 Γ 0 Γ 0 Γ 0 in C Noe ha each and i a P-map o ha hi diagram ha a limi, which we define o be Γ π We define he map, :Γ π Γ π by a furher inducion (i) For π (T 1) 1,wehaveΓ π given by he limi of a diagram Γ 1 Γ 1 Γ 0 Γ 0 Γ 0 Γ 0 ; and o we may ake, :Γ π Γ =Γ 0 o be given by he projecion from hi limi ino he lefmo and righmo copie, repecively, of Γ 0

15 384 BENNO VAN DEN BERG AND RICHARD GARNER (ii) Oherwie, given π =(π 1,,π k ), we fir conruc he map, :(Γ +1 ) πi (Γ +1 ) πi Thee give rie o a diagram (Γ +1 ) π1 (Γ +1 ) π k (Γ +1 ) π1 (Γ +1 ) π k Γ 0 Γ 0 Γ 0 Γ 0 and correpondingly for We now ake, :Γ π Γ π o be he induced map from he limi of he upper ubdiagram (which i Γ π ) o he limi of he lower one (which i Γ π ) Propoiion 37 Le Γ Cbe a globular conex Then here i a globular operad [Γ, Γ] whoe e of operaion of hape π comprie all erially commuaive diagram of he form (4) We will prove hi propoiion uing Michael Baanin heory of monoidal globular caegorie [2] The noion of monoidal globular caegory bear he ame relaionhip o ha of ric ω-caegory a he noion of monoidal caegory doe o ha of monoid; in boh cae, he former noion i obained from he laer by replacing everywhere e wih caegorie, funcion wih funcor, and equaliie wih coheren naural iomorphim Definiion 38 A monoidal globular caegory E i a equence of caegorie and funcor E 0 S E 1 T S T E 2 aifying he globulariy equaion SS = ST and TS = TT, ogeher wih, for each naural number n, an ideniie funcor Z : E n E n+1 and, for each pair of naural number 0 k<n, a compoiion funcor S T E 3 S k : E n k E n E n, T where E n k E n denoe he pullback E n k E n E n S n k E n Ek T n k In addiion, here are given inverible naural ranformaion wineing: (i) Aociaiviy: α n,k : A k (B k C) = (A k B) k C; (ii) Unialiy: λ n : Z n k T n k A k A = A and ρ n : A k Z n k S n k A = A;

16 TYPES ARE WEAK ω-groupoids 385 (iii) Inerchange: χ n,k,l :(A k B) l (C k D) = (A l C) k (B l D) (for k<l) Thee daa are required o aify a number of coherence axiom, which he reader may find in [2, Definiion 23] Ju a monoidal caegorie provide a general environmen wihin which we can peak of monoid, o oo monoidal globular caegorie provide a general environmen wihin which we can peak of algebra for a globular operad The underlying daa for an algebra in hi general eing i given a follow Definiion 39 A globular objec X in a monoidal globular caegory E i a equence of objec X i E i, one for each naural number i, uch ha S(X i+1 )=T(X i+1 )=X i for all i To decribe he addiional rucure required o make a globular objec ino a P -algebra, we employ one of he cenral conrucion of [2] Thi aociae o each globular objec X E an endomorphim operad [X, X]; which allow u o define a P -algebra in E o be a globular objec X ogeher wih a globular operad morphim P [X, X] We now decribe he conrucion of [X, X] Fir oberve ha if E i a monoidal globular caegory, hen o oo i E +1, where (E +1 ) n = E n+1 and he remaining daa i defined in he obviou way Moreover, if X i a globular objec in E, hen X +1 i a globular objec in E +1, where again we define (X +1 ) n = X n+1 Now, given a globular objec X E and a paing diagram π (T 1) n,we define, by inducion on π, an objec X π E n uch ha: (i) if π =, hen X π := X 0 E 0 ; (ii) if π =(π 1,,π k ), hen X π :=(X +1 ) π1 0 0 (X +1 ) π k Propoiion 310 Le E be a monoidal globular caegory and X Ebeaglobular objec Then here i a globular operad [X, X] wih [X, X] π := E n (X π,x n ) for all π (T 1) n Proof Thi i [2, Propoiion 72] We now ue hi reul o prove Propoiion 37 The fir ep i o conruc, from our ideniy ype caegory C, a monoidal globular caegory E(C) Definiion 311 Le G denoe he caegory 0 1 2

17 386 BENNO VAN DEN BERG AND RICHARD GARNER The generic n-pan S n i defined o be he colice caegory n/g In low dimenion, we have ha S 0 =, S 1 =, S 2 =, The monoidal globular caegory E(C) i defined by aking E(C) n o be he full ubcaegory of he funcor caegory C Sn on hoe funcor ha end every morphim of S n o a P-map The remaining rucure of E(C) may be found decribed in [2, Definiion 32] A a repreenaive ample, we decribe on objec he funcor Z : E(C) 1 E(C) 2, which i given by f A C g B C 1 C 1 C C C g f g A B; f and he funcor 0 : E(C) 2 0 E(C) 2 E(C) 2, which end he objec ( H m n D g E h A B f k, K u v F q G ) r B C of E(C) 2 0 E(C) 2 o he objec H B K m Bu n Bv D B F E qπ B G 2 hπ 1 fπ 1 π 2 A C of E(C) 2 Noe ha he requiie pullback exi by virue of he requiremen ha every arrow in he above diagram hould be a P-map

18 TYPES ARE WEAK ω-groupoids 387 We nex oberve ha if Γ i a globular conex in C, hen here i an aociaed globular objec X Γ E(C), where (X Γ ) n i he n-pan Γ 0 Γ By a raighforward inducion on π, we may now prove ha, for any π (T 1) n,(x Γ ) π E n i given by he n-pan Γ π Γ π Γ Γ ; Γ π from which i follow ha he hom-e E(C) n ((X Γ ) π, (X Γ ) n ) i preciely he e of commuaive diagram of he form (4) Thi allow u o complee he proof of Propoiion 37: indeed, we may ake he globular operad [Γ, Γ] whoe exience i aered here o be he globular operad [X Γ,X Γ ] obained from an applicaion of Propoiion 310 Definiion 312 Le C be an ideniy ype caegory An inernal P -algebra for a globular operad P i a pair (Γ,f), where Γ i a globular conex in C and f : P [Γ, Γ] a map of globular operad By a weak ω-caegory in C, we mean a riple (P, Γ,f), where P i a normalized, conracible, globular operad and (Γ,f) an inernal algebra for i I remain o exend hi o a definiion of weak ω-groupoid in C To do hi, we exploi he characerizaion of weak ω-groupoid given by Propoiion 25 Definiion 313 Le f : P [Γ, Γ] be a weak ω-caegory in an ideniy ype caegory C Now a choice of dual for Γ, wih repec o ome yem of compoiion (i n,m n )onp,i given by map ( ) :, η : +1, ɛ : +1,

19 388 BENNO VAN DEN BERG AND RICHARD GARNER for each n 1, making he following diagram commue: ( ) (,) (,) 1 1 η +1 η +1 1 [i n] Γn (( ),id) [mn] Γn (11) ɛ +1 η +1 1 [i n] Γn (id,( ) ) [mn] Γn We ay ha (Γ,f)iaweak ω-groupoid if i ha a choice of dual wih repec o every yem of compoiion on P 34 Type are weak ω-groupoid We are now ready o prove our main heorem I follow from a general reul ha how a paricular cla of globular conex o admi a weak ω-groupoid rucure Definiion 314 Le C be an ideniy ype caegory A globular conex Γ i aid o be reflexive if i come equipped wih morphim Γ 0 r 0 Γ 1 r 1 Γ 2 r 2, where each r i i an I-map aifying r i = r i =id Γi Theorem 315 ω-groupoid Every reflexive globular conex (Γ,r i ) admi he rucure of a weak To prove he heorem, we fir exhibi a weak ω-caegory rucure, and hen how hi o be a weak ω-groupoid To obain he ω-caegory rucure, we how he endomorphim operad [Γ, Γ] of Propoiion 37 o admi a normalized, conracible uboperad P, whereupon he incluion of operad P [Γ, Γ] exhibi Γ a a P -algebra, and hence a weak ω-caegory Definiion 316 Le (Γ,r i ) be a reflexive globular conex We define, for each π (T 1) n, a map r π :Γ 0 Γ π by inducion on π Ifπ =, hen we ake r π :=id Γ0 :Γ 0 Γ 0 Oherwie, if π =(π 1,,π k ), hen we fir oberve ha (Γ +1,r +1 ) i a reflexive globular conex in

20 TYPES ARE WEAK ω-groupoids 389 C Γ0 Γ 0, where (r +1 ) n := r n+1 Hence by inducion, we obain, for each 1 i k, map Γ 1 r π i (,) (,) Γ 0 Γ 0 (Γ + ) πi (12) in C Γ0 Γ 0 Thee now give rie o a diagram Γ 0 r π r 0 1 r π r 0 k (Γ +1 ) π1 (Γ +1 ) π k Γ 0 Γ 0 Γ 0 Γ 0, (13) wherein, by a raighforward calculaion, any map from Γ 0 a he op o ome Γ 0 a he boom i an ideniy In paricular, hi mean ha Γ 0, ogeher wih he map ou of i, form a cone over he remainder of he diagram However, Γ π i, by definiion, he limi of hi ubdiagram, and o we induce a map r π :Γ 0 Γ π a required Propoiion 317 Le (Γ,r i ) be a reflexive globular conex in C Then he globular operad [Γ, Γ] ha a uboperad P whoe e of operaion of hape π comprie all erially commuaive diagram of he form (4) in which he f i and g i commue wih he poining r π :Γ 0 Γ π of Definiion 316 Proof Le u wrie Γ o denoe he globular conex id Γ 0 r 0 Γ 0 Γ 1 r 0r 1 Γ 2 in he ideniy ype caegory Γ 0 /C We claim he objec (Γ ) π Γ 0 /C i given by r π :Γ 0 Γ π Oberve ha hi implie he reul, becaue he endomorphim operad [Γ, Γ ] i hen preciely he uboperad P [Γ, Γ] we require We will prove he claim by inducion on π When π =, i i clear So uppoe now ha π =(π 1,,π k ) By he decripion given in Definiion 36, and he inducive hypohei, we ee ha (Γ ) π i given by he unique map Γ 0 Γ π induced by he following cone: Γ 0 r π1 r πk (Γ +1 ) π1 (Γ +1 ) π k Γ 0 Γ 0 Γ 0 Γ 0 Thu, i uffice o how ha hi cone coincide wih (13); which i o how ha, for each 1 i k, wehaver πi = r π i r 0 Now, oberve ha r πi i obained a ((Γ ) +1 ) πi, where (Γ ) +1

21 390 BENNO VAN DEN BERG AND RICHARD GARNER i he globular conex Γ 0 r 0 r 0r 1 Γ 1 Γ 2 Γ 0 Γ 0 r 0r 1r 2 Γ 3 in Γ 0 /C Γ0 Γ 0 On he oher hand, by a furher applicaion of he inducive hypohei, r π i obained a he map ((Γ +1 ) ) πi, where (Γ +1 ) i he globular conex i Γ 1 id r 1 Γ 1 Γ 2 Γ 0 Γ 0 r 1r 2 Γ 3 in Γ 1 /C Γ0 Γ 0 Bu he funcor (r 0 )! :Γ 1 /C Γ0 Γ 0 Γ 0 /C Γ0 Γ 0 given by precompoiion wih he map r 0 :Γ 0 Γ 1 of C Γ0 Γ 0 end he laer of hee globular conex o he former; and hu, becaue (r 0 )! preerve limi, i mu alo end ((Γ +1 ) ) πi o ((Γ ) +1 ) πi, which i o ay ha r πi = r π i r 0, a required Thu, for a reflexive globular conex (Γ,r i ), we have now defined he uboperad P [Γ, Γ] required for he proof of Theorem 315 I remain only o how ha P i normalized and conracible To do hi, we need he following reul Propoiion 318 Le (Γ,r i ) be a reflexive globular conex in C Then each of he map r π :Γ 0 Γ π of Definiion 316 i an I-map Proof We proceed by inducion on π When π =, wehaver π an ideniy map, and hence an I-map So uppoe now ha π =(π 1,,π k ), and conider he diagram (13) defining he map r π :Γ 0 Γ π In i, each of he map r π i i an I-map by inducion, and o becaue r 0 i an I-map by aumpion, and I-map are cloed under compoiion, r π i r 0 i alo an I-map Repeaed applicaion of he following lemma now complee he proof Lemma 319 Suppoe ha A i B id A p A i a commuaive diagram in an ideniy ype caegory C Suppoe furher ha i and j are I-map, and p and q are P-map Then he induced map (i, j) :A B A C i alo an I-map j q C

22 TYPES ARE WEAK ω-groupoids 391 Proof We fir form he pullback quare B A C q B p C q p A Now he univeral propery of hi pullback induce a facorizaion of he commuaive quare B id B B jp p C q A a B j B A C q B p A j C p q p A Since he ouer recangle ha ideniie along boh horizonal edge, i i a pullback However, he righ-hand quare i a pullback, and o we deduce ha he lef-hand quare i oo Now p i a P-map by (Sabiliy) and j i an I-map by aumpion, and o by (Frobeniu), j i alo an I-map I follow, by (Compoiion) and he fac ha i i an I-map, ha A i B j B A C i alo an I-map Bu hi map i he induced map (i, j) :A B A C, ince i ha i a i projecion ono B, andjpi = j a i projecion ono C Propoiion 320 Le (Γ,r i ) be a reflexive globular conex in C Then he uboperad P [Γ, Γ] of Propoiion 317 i boh normalized and conracible Proof Noe fir ha P i he e of all map f 0 :Γ 0 Γ 0 for which f 0 id Γ0 =id Γ0 and hence a ingleon, which prove ha P i normalized To how i conracible, we mu how ha, given a erially commuaive diagram of he form Γ π Γ π f n 1 1 g n 1 Γ 0 f 0 Γ 0 g 0 (14)

23 392 BENNO VAN DEN BERG AND RICHARD GARNER wherein each f i and g i commue wih he poining, we can find a map f n :Γ π compleing he diagram (and commuing wih he poining) Fir we noe ha he diagram Γ π f n 1 1 g n 1 1 (,) (,) B n 1 Γ commue, a may be een by pocompoing i wih he wo projecion B n 1 Γ 2,and oberving ha he reulan diagram are commuaive Thu we induce a map k :Γ π B n Γ We now conider he diagram Γ 0 r n 1 r 0 Γn r π Γ π k (,) Bn Γ Tha hi i commuaive once again follow from he fac ha i i o upon pocompoiion wih he wo projecion B n Γ 1 Moreover,r π i an I-map by Propoiion 318, and (, ) i a P-map by he definiion of globular conex, o ha by (Orhogonaliy), we can find a map f n :Γ π making boh induced riangle commue The fac ha he lower riangle commue indicae ha f n render he diagram (14) erially commuaive; while he fac ha he upper riangle commue indicae ha f n commue wih he poining Thu we have hown he operad P [Γ, Γ] o be normalized and conracible, from which i follow ha he incluion P [Γ, Γ] exhibi Γ a a weak ω-caegory I remain o how ha hi weak ω-caegory i a weak ω-groupoid Propoiion 321 Le (Γ,r i ) be a reflexive globular conex in C and le P [Γ, Γ] be he operad defined above Then he incluion P [Γ, Γ] exhibi Γ a a weak ω-groupoid Proof According o Definiion 313, we mu how ha, for any given yem of compoiion (i n,m n )forp, here i a correponding choice of dual for Γ Now, for each n 1we have a commuaive diagram Γ 0 r n 1 r 0 r n 1 r 0 (,) Γn (,) Bn Γ The lef-hand morphim i an I-map, and he righ-hand one i a P-map; and o by (Orhogonaliy) we have a diagonal filler ( ) : Commuaiviy of he lower riangle implie he commuaiviy of he fir diagram in (11) We induce η and ɛ imilarly, by conidering he commuaive quare Γ 0 r n 1 r 0 r nr n 1 r 0 +1 ([i n], [m n] (( ),id)) (,) B n+1 Γ

24 TYPES ARE WEAK ω-groupoids 393 Γ 0 r nr n 1 r 0 +1 r n 1 r 0 ([m n] (id,( ) ), [i n]) (,) B n+1 Γ Again, commuaiviy of he lower riangle enail he commuaiviy of he remaining four diagram in (11) We have hu hown ha every reflexive globular conex in an ideniy ype caegory C bear a rucure of a weak ω-groupoid Noe ha in giving hi proof, we have nowhere ued he axiom (Ideniie) In fac, he only reaon we need i i o how ha, from an objec of C, we can conruc a reflexive globular conex correponding o i ower of ideniy ype Definiion 322 Le C be an ideniy ype caegory and le A C We define a reflexive globular conex A Cby he following inducion For he bae cae, we ake A 0 = A For he inducive ep, uppoe ha we have defined A 0,,A n Then we may form he n-dimenional boundary B n A of A, and by inducion he map (, ) :A n B n A i a P-map So by (Ideniie), we may facorize he diagonal morphim A n A n BnA A n a A n r n+1 Id(An ) en+1 A n BnA A n, wih r n+1 an I-map and e n+1 a P-map We now define A n+1 o be Id(A n ), and, : A n+1 A n o be he compoie of e n+1 wih he wo projecion morphim A n BnA A n A n I remain o how ha he induced map (, ) :A n+1 B n+1 A i a P-map Bu we recall ha B n+1 A wa defined by he pullback diagram (10), o ha B n+1 A = A n BnA A n, and he induced map (, ) i preciely e n+1, which i, by aumpion, a P-map Taking hi definiion ogeher wih Theorem 315, we immediaely obain he following Theorem 323 Le C be an ideniy ype caegory and le A C Then he globular conex A i a weak ω-groupoid in C In paricular, aking C o be he ideniy ype caegory C T aociaed wih ome dependen ype heory T yield he following corollary Theorem 324 Le T be a dependen ype heory admiing each of he rule decribed in Subecion 22 Then, for each ype A of T, he ower of ideniy ype over A i a weak ω-groupoid Acknowledgemen I eem appropriae o ay a few word abou he hiory of hi paper The main reul wa decribed by he fir-named auhor in 2006 in a preenaion a he workhop Ideniy Type Topological and Caegorical Srucure held a Uppala Univeriy The deail of he proof were hen worked ou by boh auhor during a 2008 vii by he firnamed auhor o Uppala; and i wa a hi age ha he axiomaic approach wa inroduced While preparing hi manucrip for publicaion, we become aware ha, independenly, Peer Lumdaine had been conidering he ame queion Hi analyi may be found in [16] Le u remark only ha, where our argumen i caegory-heoreic in naure, ha given by Lumdaine i eenially proof-heoreic We graefully acknowledge he uppor of Uppala Univeriy

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