E E B B. over U is a full subcategory of the fiber of C,D over U. Given [B, B ], and h=θ over V, the Cartesian arrow M=f

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1 ppendix : ibrational teory o L euivalence E E onsider ibrations P, P, and te category Fib[,] o all maps 2 =(, ):@ :: P { P 2 o ibrations; Fib[,] is a ull subcategory o [,] ; see [3]. Fib[,] is te total category o a ibration denoted Fib, ; its basecategory is te unctorcategory [, ], and te iber over as objects all te as in () wit te ixed U=, and arrows as in, deined in [3]; te iber o Fib, over U is a ull subcategory o te iber o, over U. Given (:U@V) [, ], and N:@ =θ over V, te artesian arrow = is obtained by te stipulation tat or all, X X, is a artesian arrow over :U()@V() ; te deinition o on arrows is te obvious one; see also below. Te act tat so deined is a map o ibrations is sown by te diagram: θ YG Y X Nθ g UG U g V. Here, θ :X@Y is a artesian arrow over :@ ; te issue is to sow tat θ is artesian (over U ). Te deinition o on arrows makes θ an arrow over U making te upper uadrangle commute (uniue suc θ exists by Y being artesian). s a composite o artesian arrows, (Nθ )v X is artesian; as a let actor o te last, θ is artesian. In wat ollows, te base categories, will ave inite limits. Fiblex, is te 33

2 subibration o Fib, wit basecategory Lex(, ), a ull subcategory o [, ], wit ibers uncanged rom Fib,. Next, assume tat and are U ibrations. We ave te preibration U,, wit base category Lex(, ), and total category U (,). Te iber over U Lex(, ) is te ull subcategory o te iber o Fiblex, over U wit objects te maps o U ibrations :@. U, is not a ibration; owever, or certain maps :U@V, (N) calculated in Fiblex, does belong to U,, as we proceed to point out (rom wic it will o course ollow tat over suc, artesian arrows do exist in U, ). ssume tat is a U ibration, wit Q = rr( ). Let us call rr( ) surjective i t = t.i is surjective, ten or any Y, Y = (t U Y) = t UY = Y (were te second euality is Frobenius reciprocity). It is clear tat a pullback o a surjective arrow is surjective, and te composite o two surjective arrows is surjective. It is also clear tat i r is surjective, ten so is. Let us call a commutative suare g O O a b g a uasipullback i te canonical arrow =P is surjective. Using te stated properties o surjective maps, we easily see tat i in te uasipullback ('), g is surjective, ten so is g. onsider two adjoining suares and @ O O O O O 2 @ 34

3 (2) Te "composite" o two uasipullbacks is again a uasipullback: i bot and 2 are uasipullbacks, ten so is 3. Te veriication uses bot te pullback and composition properties o surjective arrows noted above. (3) In ("), i 3 is a uasipullback, 2 is a pullback, and commmutes, ten is a uasipullback. (3') I in te commutative O _ O _ O O l te two uadrangles and are pullbacks, and te suare is a uasipullback, ten is a uasipullback too. Tis ollows rom (2) and (3). (3") I in ("), 3 is a uasipullback, and is surjective, ten 2 is a uasipullback. To see tis, let P= or 2, and R= or 3. We ave te commutative diagram 35

4 O O O W ) O wit two pullbacks as indicated. Since is surjective, so is RP. Te assumption gives tat R is surjective. Now, te composite P is surjective, and so is its let actor P, wic is wat we want. (4) Te eckevalley condition or olds (not just wit pullback suares, but also) wit uasipullback suares. Indeed, consider te diagram g s O r O F a PG b s p g, and calculate: a X = a X = r X = r X = b g s p s p s g X ; te tird euality is te "uasipullback" property, te last ordinary. Let us continue to assume tat is a "ull" U ibration ( Q contains all arrows), let be an arbitrary U ibration, (:@). We call a map (:U@V) Lex(, ) very surjective wit respect to i te suare UG U g V is a uasipullback. (Te concept o "very surjective" is relative to te ibration, altoug it does not depend on te ibration except or its basecategory.) 36

5 (5) I is very surjective wit respect to an arrow,tensoitiswitrespectto any pullback o ;i is very surjective wit respect to a pair composable arrows, ten so it is wit respect to teir composite. Tis ollows by (3) and (2). We say tat is very surjective i it is very surjective wit respect to every Q ;by(5),it is enoug to reuire te condition or a "generating set" o 's. (6) Te composite o very surjective arrows (in Lex(, ) ) is very surjective; te pullback o a very surjective arrow is very surjective. Tis ollows by using (2) and (3). op Let be a simple category, = on() ; Lex(, ) can be identiied wit Fun(, ) ; tis is te kind o basecategory or te ibrations we are interested in. In 4, we made two dierent coices or te class Q o uantiiable arrows in. Te coice or te purposes o te main body o 5 is Q ; tis, in te version tat is closed under composition, is simply te class o epimorpisms o. Wen we make te coice o = Q or Q, we get as te very surjective maps in te sense o tis section te ones we called normal ones in 5; we leave it to te reader to veriy tis. (6') Let (:U@V) Fun(, ) be very surjective (wit respect to Q ). For every inite context X over, :U[X]@V[X] is surjective. For any, [X] :U()@V() is surjective. Te irst assertion is sown by induction on te cardinality o X.I X is o positive size, we can write X as Y {x} suc tat Y is a context too. y te paragrap ater (4) in 4, or = x, we ave a pusout diagram 37

6 P P in on(),wic,wit V=X, U=X, gives rise to O O j j O 2 O j j to wic (3') is applicable. Te suare is a uasipullback (by being very surjective), ence, so is 2. Since by te induction ypotesis, U[Y]@V[Y] is surjective, so is U[X]@V[X]. Te second assertion ollows immediately rom te irst by te uasipullback π U [] P P U ; π note tat U[] =U[X ], etc. ssume now tat and are UV ibrations, a "ull" one. (7) I :U@V is very surjective, and N UV (, ), te = (N) calculated in Fib(,) is in act in UV (, ). 38

7 First o all, using tat or eac g rr( ), g is a morpism o lattices, we immediately see tat preserves te iberwise operations. onsider XG X θ θ g NX UG g N X U g V X = N X = NX = NX = X ; N ere, te irst euality is te deinition o ; te second te uality o N being a morpism o ibrations; te tird being very surjective; and te last again te deinition o. Now, assume in addition tat bot and are UV ibrations, again wit Q = rr( ). I claim tat (8) I :U@V is very surjective, ten N UV (, ) implies tat = (N) UV (, ). Te additional iberwise operation, Heyting implication, is dealt wit as beore. Let U (:@) Q, X ;wewanttosowtat X = X ;tatis,orany Φ, Φ X U (U) Φ U X. Te lettorigt implication is automatic. ssume and consider (U) Φ X, (9) U 39

8 (U) Φ X= UX Φ? (V) ( Φ) NX Φ UG U g V. s indicated, we consider te object Φ over V, and claim tat te ineuality marked? is true. (V) ( Φ) = (U) Φ (0) by te (generalized) property or wit uasipullbacks. (9) implies tat (U) Φ X = U NX NX. () (0) and () imply wat we wanted. Now, rom tis, Φ V NX = N( X),and Φ Φ N( X) =( X) as desired., N UV (, ) are said to be euivalent, N, i tere is a diagram PG m n W N suc tat m, n are artesian in Fiblex(, ), and m n are very surjective. Euivalence is clearly relexive and symmetric; it is transitive too; given QG RG m t n n t p W g W g N P 40

9 wit te relevant properties, one orms te pullback S G r W Q G g R n n t N W in Lex(, ), and deines S as (n") (N ),or n"=n =n r ;let n":s@n be te artesian arrow over n". Ten n being artesian implies tat tere is a (uniue) over suc tat n=n" ; similarly or r over r.since n" is artesian, so are and r.since, r are pullbacks o very surjective arrows, tey are very surjective. We conclude tat m and pr are artesian arrows over very surjective ones, wic proves wat we want. Let us take T=(L, ), te "empty teory" over te SV L, and let =[T],a UV ibration wit basecategory = (on[]) op and class o uantiiable arrows Q=Q. Recall te canonical i:@ induced by Yoneda. od (T) = Str (L),andwe ave te ibration E:od (T)@ as explained in 5. We also ave te ibration = Fiblex,P() : We ave a "orgetul" morpism () :@E ; () is te euivalence J Uvi : ; and () is deined as 2 P was deined in 4 (see (5)) or te special case wen P od () Fiblex[,P()].Itiseasytoveriytat () P() is a morpism o ibrations. We ave te uasiinverse J (2) 4

10 speciied so tat [U]([X]) = U[X] ; we ave te canonical isomorpism j :[U] U U natural in U. () :@E restricts to an euivalence () : od iso iso P() (T), (3) wose uasiinverse is iso [] : od P() () Fiblex[,P()] constructed in 4, wit te canonical isomorpism j :[] natural in. Tese are connected to (2) by [] =[], (j ) =j. Let us deduce ()(b) o 5 rom (8); let's use te notation and ypoteses o 5.()(b). onsider te ollowing diagram in te ibration E : θ ] j N [θ ] j ] U j V []. Te two uadrangles commute, by te naturality o j. It ollows tat [θ ] is artesian over [] onsider te artesian arrow θ :[] over in [].Since () is a morpism o ibrations, (θ ) :([] [] is artesian over te same [] It ollows tat tere is an isomorpism 42

11 ([] over [U]. ut ten, since (3) is ull and aitul, it ollows tat [] [N] =. Hence, [X:ϕ] =([] [N])[X:ϕ] = ([N][X:ϕ]) = (N[X:ϕ]), X were te second euality is te description o artesian arrows in, te last is te deinition o [N] ; and tis is wat was to be proved. X ontinuing in tis manner, we see tat, or, N od (T), L N in te sense o 4 i [][N] in te sense o tis ppendix. 43