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BULL. AUSTRAL. MATH. SOC. 18-02, I8AI5, I8CI0, I8E20 VOL. 7 ( 1 9 7 2 ) , 1-76. Aspects of topoi Peter Freyd After a review of the work of Lawvere and Tierney, i t i s shown that every topos may be exactly embedded in a product of topoi each with 1 as a generator, and near-exactly embedded in a power of the category of sets . derived. Several metatheorems are then Natural numbers objects are shown to be characterized by exactness properties, which yield the fact that some topoi can not be exactly embedded in powers of the category of s e t s , indeed that the "arithmetic" arising from a topos dominates the exactness theory. Finally, several, necessarily non-elementary, conditions are shown to imply exact embedding in powers of the category of sets. The development of elementary topoi by Lawvere and Tierney strikes this writer as the most important event in the history of categorical algebra since i t s creation. The theory of abelian categories served as the "right" generalization for the category of abelian groups. for - no less - the category of sets. For each the motivating examples were categories of sheaves, abelian-valued sheaves for the set-valued sheaves for the second. So topoi serve first, But topoi are far richer than abelian categories (surely foreshadowed by the fact that abelian-valued sheaves are just the abelian-group objects in the category of set-valued sheaves). Whereas abelian categories, nice as they are, appear in various contexts only with the best of luck, topoi appear at the very foundation of mathematics. The theory of topoi provides a method to "algebraicize" much Received 29 December 1971- Prepared from a series of lectures delivered at the Conference on Categorical Algebra, University of New South Wales, 13-17 December 1971. The author was in Australia as a Fulbright Fellow. I 2 Peter Freyd of mathematics. In this work, we explore the exact embedding theory of topoi. the subject is richer than that for abelian categories. Again It is not the case that every small topos can be exactly embedded in a power of the category of sets or even a product of ultra-powers (Theorem 5-23). It is the case that every topos can be exactly embedded in a product of well-pointed topoi (Theorem 3.23) ("well-pointed" means "l is a generator"), and near-exactly embedded in a power of the category of sets (Theorem 3.2U) ("near-exact" means "preserves finite limits, epimorphisms, and coproducts" ). We thus gain several metatheorems concerning the exactness theory of topoi (Metatheorem 3-3l). The obstructions to the existence of exact functors lie in the "arithmetic" of topoi (Proposition 5-33, Theorem 5.52). No set of elementary conditions can imply exact embeddability into a power of the category of sets (Corollary 5.15), hut rather simple, albeit non-elementary, conditions do allow such (§5.6). The easiest to state: a countably complete topos may be exactly embedded in a power of the category of sets. The most impressive use of the metatheorems is that certain exactness conditions imply that something is a natural numbers object (Theorem 5.i+3)A consequence is that a topos has a natural numbers object iff it has an object A such that 1 + A - A (Theorem 5.UU). We begin with a review of the work of Lawvere and Tierney (through Corollary 2.63). All the definitions and theorems are theirs, though some of the proofs are new. Sections 4.1 and 5.1 and Propositions 5-21, 5-22 are surely also theirs. It is easy to underestimate their work: it is not Just that they proved these things, it's that they dared believe them provable. 1. Cartesian closed categories A cartesian closed category is a finitely bicomplete category such that for every pair of objects representable. A, B the set-valued functor (-M, B) is This is the non-elementary (in the technical sense of "elementary") definition. We shall throughout this work tend to give first Aspects of topoi 3 a definition in terms of the representability of some functor and then show how such can be replaced with an elementary definition - usually, indeed, with an essentially algebraic definition. The translation from representability to elementary is the Mac Lane-notion of "universal element". For cartesian-closedness (forgive, oh Muse, but "closure" is just not right) we obtain the following: A A For every A, B there is an object Er and a map a x A -»• B (the "evaluation map") such that for any X x A -*• B there exists a unique f : X -*• a such that • B . A Holding A fixed, f : a -»• U U becomes a functor on B : given / : B •*• C , i s the unique map such that B 4 x A ^ X 1 > (^ x A »• C = S 4 x A >• B -$-+ C . A Holding B fixed, B^ becomes a contravariant functor on A : given g : A' -*• A , £r : a •* a / is the unique map such that XA, One can easily verify that that is, e is a bifunctor. Recall that a group may be defined either as a semigroup satisfying a couple of elementary conditions or as a model of a purely algebraic theory (usually with three operators: multiplication, unit, and inversion). It is important that groups may be defined either way: .there are times when groups are best viewed as special kinds of semigroups, and there are times when they thrive as models of an algebraic theory. So it is with lattices, 4 Peter Freyd and every elaboration of the notion of lattice; and so it is with cartesian-closed categories. To begin with, a category may be viewed as a model of a two-sorted partial algebraic theory. The two sorts are "maps" and "objects". We are given two unary operators from maps to objects, "domain" and "codomain", and one coming back, "identity map"; and we are given a partial binary operator from maps to maps, the domain of which is given by an equation in the previous operators. Partial algebraic theories for which such is the case, namely those such that the partial operators may be ordered and the domain of each is given by equations on the previous, are better than just partial algebraic theories. We shall call them essentially algebraic. A critical feature of essentially algebraic theories is that their models are closed, in the nicest way, under direct limits. Finite bicompleteness becomes algebraic. called the "terminator") is a constant, operation, The terminal object (better 1 , together with a uninary t , from objects to maps, such that domain ( t ) = A , codomain(t.) = 1 , h-h • B -±+ 1 = tA . The equations that stipulate domain and codomain are conventionally absorbed in the notation, thus: t{A) = A -*• 1 . For binary products we have four binary operations, one from objects to objects denoted /I x A^ , two from objects to maps denoted one from maps to maps denoted and one unary operation from objects to maps: A •A x A . The Aspects of topoi equations: A x A —^ ] A = 1A , i = 1, 2 , A x B 2 = ^ - 2 * yl xyl ±1 2S _B x B^ . For equalizers we have partial binary operators from maps, one to objects, one to maps. The domain of each operator is given by the equations domain(/) = domain(g) , codomain(/) = codomain(g) . We may denote the pairs f,g in the domain by object-valued operator is denoted denoted A ? '9+ B . The E{f, g) , the map-valued operator is E(f, g) -* A . We have a third partial operator from triples maps. (h, f, g) of maps to The equations that define the domain are: codomain(?i) = domain(/) = domain(g) , codomain(/) = codomain(g') , hf = hg . Given X • A I'$ > B the value of this operator is denoted X -^+ E{f, g) . The equations: £•(/, g) + A -£+ B = £ ( / , g) •* A - 2 - B , X-^ X JU E{f, g) . E(f, g) * A = h , X ( X - W . g ) * ) . E(f, g) . We can make cartesian-closedness essentially algebraic by taking two binary operators: one 'from objects to objects denoted a , one from A objects to maps denoted Zr x A -*• B ; and a quaternary partial operator from quadruples (X, A, B, f) such that domain(/) = X x A , codomain(/) = S , valued as a map X -*->• a . The equations: Peter Freyd B = / , S 4 = (X, A, B, X Such is the nearest approximation to the previous elementary definitions. It is usually more convenient to ask that a is a bifunctor covariant in the lower variable, contravariant in the upper, equipped with natural transformations e A , B •• ** * A + B e > A,B : such that x A The existence of elementary definitions should not, in itself, oblige us to give elementary proofs. The great technical tour de force in Godel's incompleteness proof, namely that primitive recursive functions (a second-order notion if there ever was one) are all first-order definable (indeed Gode I-recursive) does not oblige us, but allows us to stop worrying about primitive recursive functions. Certainly it is worth knowing when things are elementary - we shall use the elementary nature of topoi (for example, Corollary 5.15, Theorem 5-23) and their essentially algebraic nature (Theorem 3.21). Pirst-order logic is surely an artifice, albeit one of the most important inventions in human thought. thinks in a first-order language. But none of us The predicates of natural dialectics are order-insensitive (one moment's individuals are another's equivalenceclass) and our appreciation of mathematics depends on our ability to interpret the words of mathematics. The interpretation itself is not first-order. The reduction of a subject to an elementary one - in other than the formal method of set theory - usually marks a great event in mathematics. The elementary axioms of topoi are a testament to the ingenuity and insight of human genius. I will refuse to belittle this triumph of mind over matter by taking it as evidence that mind is matter. Aspects of Topoi PROPOSITION 1.11 for cartesian-closed categories. The following natural maps are isomorphisms: 0 =0 xA , (A*C) + (B*C) - (A+B) x C , 1 =1 , (A*B)C - AC x BC , AB+C ~-AB*AC A -A ABXC Proof. , , « (^ S ) C . Each is a result of the existence of an adjoint. have a right-adjoint coproducts. Dually For -*A to i A\ [- j , it must be cocontinuous, hence it preserves (- ) , having a left-adjoint, preserves limits. A~ is adjoint to itself on the right, hence carries colimits to limits. Elementary proofs would go like this: unique 0 x A •* X coterminator. because there exists unique X 0 •+ X there exists , hence 0 x A is a D Caution: 0 PROPOSITION then A = 0 . Proof. for each need not be 0 . (indeed 0 = 1 .) 1.12 for cartesian-closed categories. If A •* o exists The existence of A -*• 0 yields a map A •*• A x 0 , and we obtain A+A*O->-A I? 0 As always, 0 -»• i4 ->• 0 = 1_ . = l. . A O A degenerate category is one with just one object and map. finite bicompleteness implies non-empty.) (Note that 8 Peter Freyd PROPOSITION 1.13. there exists 1 -*• 0 . PROPOSITION A aartesian-closed category is degenerate iff D 1.14. For any small A the category of set-valued Aop contravariant Proof. denoted functors S H Aop J. S is cartesian-closed. has a generating set, the representable functors (herein For any T , -xT : S Aop Aop -*• S is easily seen to preserve colimits because such are constructabie "object-wise" and preserves colimits. Hence for any X T, S , {- T, S) -*TA : S •* S carries colimits into limits and the special adjoint functor theorem says that it is representable. We can, of course, construct fir more directly. s{A) of - (H.XT, A - \H., S\ A K ) S (A) . object-wise: n : H S) and we could use the latter as the definition The evaluation map given x T •* S s x T -*• 5 and an element x £ T{A) , define S ->• (S * T) , let e*{x) € (5 x T)T(.A) as the transformation is such that is easily constructed < !"|, x > i (ST x T) (A ) , that is, a transformation For the co-evaluation map n We know that r\. (l.) = x . e{r\, x) = ^ ( l . , ^) • x € S{A) r\*l: and define HA*T-»-SxT where All the equations are directly verifiable. D H Note that if 5 U) A is a monoid representations of , where If by has finite products, then is the set of transformations from If S A M M M , S and £T A° P T 5 to A = S(Ax-) , and hence S(A*-) . may be viewed as the discrete is the set of homomorphisms from M x T hi is used to denote the "regular" representation. is a group, then a homomorphism f{l, x) , x € T , and given any function /"(a, x) = ag(a~ x ) . Thus 5 f : M x y •+ 5 g : T •* S is determined we may define is the set of all functions from T to to Aspects of topoi S S . ag •* S is the function - (ag)(x) = a\g(oT x)\ • The forgetful functor preserves exponentiation. 1.2 Heyting algebras. A Heyting algebra is a cartesian-closed category in which for every A, B , (A, B) \j {B, A) has at most one element. The latter condition says, of course, that we are dealing with a partially ordered set. finite bicompleteness says that it is a lattice with 0 and 1 . The The cartesian-closedness says that there is an operation on the objects such that (-4, BC) + 0 We switch notation: and y is denoted operation" as lattice with (AxC, B) * 0 . the objects are lower-case existence of a map from x iff x y x x v y , the "Heyting x = x A y : O v x = x , x = x , xAj/ x 5 y , the product of We recall that the following equations give us a defined as l A x = x , A x, y, z, ... , the is stated with x A y , the coproduct as x -*• y . x - y to x v x = x , = j/Ax, x v y = y vx , x A (j/As) = (a;Ay) A z , x v (yvz) = (x</y) v z , x Hon-equationally, z Ax 5 # . with x x •+ y A {yvx) = x = (xAy) v x . is characterized by That is, x -* y is dominated by PROPOSITION z 2 (x •* y) y . 1.21 for Heyting algebras. 0 = 0 A X, (xAt/) v (xAz) = x A {yvz) , x -" 1 = 1 , x iff is the largest element whose intersection -»• (t/Az) = ( x •*• y) A (x •*• z) , 0 -»• x = 1 , (xvy) •*• z = (x •* z) A {y •* z) , x = 1 ->• x , •* z = x •* (y •*• z) . 10 Peter Freyd Proof. Translate Proposition 1.11. PROPOSITION 1.22 • for Heyting algebras. We can characterize x •*• y with the following equations: x •* x = 1 , y A (x -»• y) = y , x •* (y A z) = (x •* y) Proof. z A x = s A Given z £ (x •* y) , that is, (x ->• y) Ax = zAxAy<y we note first that the fact that . is, x •*• (z Ax) z £ x ->• z . z = z A ( X •*• y) , then Conversely, given f(u) = x •* u implies that it preserves order, and hand A (x •+ z) . x-+(aAx)±x-*-y. = (x -*• x) A (x •* s) = x -»• z Hence that s £ ~l x iff x . z A X = 0 , that is , ~1 x A complement of x v y = 1 . "1 x : x is an element In a Heyting algebra, if because PROPOSITION xAy 1.23 On the other and z A (x •+ z) = z , that 3£x-*-3 = x-*-(xAs)£x->-j/. We define the negation of an element, denoted disjoint from z A X £ y , preserves intersections = 0=>ys~\x y x n ~\ x , as x -»• 0 . Note is the largest element such that x A y = 0 and has a complement it must be and for Heyting algebras, x £ ~l ~l x ; if x s y then Proof. The f i r s t two statements are immediate. (~l x) £ ~1 ~l (~l x) by the f i r s t statement, ~~\ ("1 ~\ x) £ ~~\ x by the second applied to the f i r s t . D A boolean algebra i s a Heyting algebra which s a t i s f i e s the further equation x = ~l ~1 x . In a boolean algebra, negation i s thus an order-reversing involution and De Morgan's laws are easy consequences: ~\(xAy)=~\xv~\y . (Note t h a t in any Heyting algebra Hence xv~lx=~l~l (xv~lx)=~l(~lxAx)=~IO = l and every element Aspects of topoi has a complement. Conversely if for all l l l = l l l A I I x , x v "I x = 1 then ( l V 1 l ) = O ~1 X A x ) V (~l ~1 x A ~1 x ) = ~ l ~ l x A x It is easy to verify that in a boolean algebra = X . I + J ="li v J . A complete lattice has a Heyting algebra structure, by the adjoint functor theorem, iff if for any set f(n) = x A n {y .} . preserves all conjunctions, that is, , x A V y . = V [x A y .) . a complete distributive lattice. Such is usually called The lattice of open sets in a topological space is therefore a Heyting algebra. Consider the unit interval, and let: V ~l X = I- >' ' , x ~1 ~1 x = x , 1 (x v 1 x) = 0 , •"i ~i (x v n x) = i , (x v "1 x) t 1 ~1 (x v 1 x ) . In any space, negation yields the interior of the set-theoretic complement. Double negation yields the interior of the closure. Another ready example of a complete Heyting algebra is the lattice of left-ideals in any monoid. ordered set with 0 and For a non-complete example, take any linearly 1 but otherwise not complete and define (l (x *y) if x S y , = [y if y < x . In any Heyting algebra, define z 2 (x «-»• y) which meets iff x and x •*-*• y = (x •*• y) A (y •+ x ) . z A x = 2 A y , that is, x •<->• y j/ in the same way. Then is the largest element We can reverse things to obtain the symmetric definition of Heyting algebras, namely the operations and equations of a lattice together with a binary operator satisfying: l-t->-x = x = x-i->-l ( l i s a unit), 12 Peter Freyd x •*-*• x = 1 (a; is a n inverse of x ) , (a; «-»• y) A z = (x A z -<->• y A z) A z Then (almost distributive). (x •*->• y) A x = (x *-*• x A y) A x = (l «-+• y ) A x = y Ax (x-<->-y)Ay=xAy. Hence if 3 < (a: -*-»• i/) . Similarly then z A x = s A (a; •*-•• i/) Ax = zAyAx and z that i s A y = s A (x •*-*• y) z A x = z A y Conversely, i f A S = ( x A 3 - < - » - y A 3 ) A S= 1 A 3 = 3 , z £ (a; •<-*• j/) . x •*• y as a; •*-*• x A y and v e r i f y z S (x -»• jy) 3 Ax S y . +-»- i s a symmetric b i n a r y operation with a u n i t and i n v e r s e s . not a s s o c i a t i v e i n g e n e r a l , for A s s o c i a t i v i t y implie s t h a t "1 x = x •*-*• 0 ++ i f we can find and It is ~1 ~~\ x = (x •*-+ 0) *-*• 0 . "l~la: = a;-t-+ (0-^-»-0) = x •*-*• 1 = x . Conversely, i n boolean a l g e b r a s , comment: •*-»• i s a s s o c i a t i v e . An orthogonal i s a loop o p e r a t i o n only in a boolean a l g e b r a , for given y such t h a t Given a congruence set , , We may t h e n d e f i n e iff = z A x A y z A x = z A y , then (a: « - » - ! / ) that i s , Ay F = {x\x = 1} = y •*->• 0 = x then x a; = ~ l j / = ~ i n ~ l j / = " l ~ l x on a Heyting a l g e b r a i t i s easy t o see t h a t t h e has t h e p r o p e r t i e s : 1 €-F , i f f ' l v i / f F , We call such a set a filter. We can recover the congruence from iff (x -»-»• y) € F because if x = y (x *-+ y) £ F ; if (x •*->• y) = 1 Moreover: then then F :x = y {x •>-*• x) = (x •*-*• y) and • Aspects of topoi PROPOSITION relation = Proof. suppose 1.24 for Heyting algebras. defined by = 13 x = y iff If F (x •*-*• y) 6 f is clearly reflexive and symmetric. (x *-*- y), (y <->• 3) € F . For transitivity (This is the only use of the fact that F And for that inequality we need only show {x *-* y) A (y *-+ z) A x = (x •*->• y) A {y *-* z) A s For is a congruence. It suffices to show that (x «-• y) A (y *->• z) 5 (x •<-*• 3) . is closed under intersection.) is a filter then the xSy=>xAz=yAz , an easy matter. i t s u f f i c e s t o show (x •«-*• y) 5 (x A 2) +-* (y A 3) , t h a t i s , (x «-»• y ) A (x A z) an easy matter. = (x •<-»• !/) A (1/ A a ) For x = y = * x V 2 = y v z , w e m u s t sho w (x •*-+ y) A (x V z) = (x *-*• y) A (y v z) , which because of distributivity is again an easy matter. Finally for x = y =* (x -<->• 3) = (y -*-»• 3) it suffices to show x *->• y < (y *-+ z) *-* (x *-*• z) , that is, (a; *->• 1/) A (a: «-»• 2 ) Using the third defining equation of = (a; <-^ j/) A (1/ •<-»• 2 ) . «-»• , the left-hand side is (x -<-»• y) A [ (x +•+ y) A x +->• (x *->• y) A 2 ] and the right-hand side is (x •«->• !/) A [(x -<-»• j/) A y f-*- (x *-+ y) A 3] , clearly equal since (x-M-t/)Ax=(x-(-*-j/)Aj/. D 1.3 Adjoint functors arising from cartesian-closed categories. PROPOSITION 1.31. Let A be a cartesian-closed category and A' c A a full reflective subcategory, /? : A -»• A' the reflector. Then R preserves products iff for all B (. A' , S 4 is in A' . Proof. B € A1 Suppose , that for any R(A x- C) - RA *• RC , all C U , A, C . (RC', s") = (C, fl4) . We wish to show for (We are invoking the "Kelly view" of full reflective subcategories, namely that all those objects letting that Y = X . X such that We obtain from X •* RX ->• X = 1 . RX •*• X •*• RX = 1 . From (RY, X) ~ (Y, X) . (RX, X) •++ {X, X) {RX, X) «• {X, X) A' consists of This can be seen by a map RX •* X such we obtain The Ke I I y view is that every full reflective subcategory I4 Peter Freyd is definable by stipulating a class of maps such that C and then looking at all X (/, X) is an isomorphism for all / € C .) To return: {RC, BA) = (RC*A, B) = (R(RC*A), B) = (RCxRA, B) = (fl(C*4), s) = (C*A, B) = (c, B 4 ) . Conversely, if for all B € A' , a € A 1 , then we wish to show that is, as defined in A' , a product of RA, RC . ) It suffices, then, to show that is, for all B (. A' , A1 (RA*RC, -) = A'(R{A*C), -) , that (RA*RC, B) - [R(A*C) , B) . But [R(A*C), B) - (.A*C, B) and , B) == (iJ4, / ^ = (4, / ^ = (4x/?C, B) = (i?C, S 4 ) For any category objects are A A and B£A , A/B denotes the category whose A-maps of the form A •*• B , and whose maps are triangles •* A' \ / • B £ D : A/B •*• A o an isomorphism. denotes the forgetful functor. Note that A/1 -»• A is The naive construction of colimits in A/B works, that is, given colimits in A . The naive construction of equalizers in A/B works; and and it is the purest of tautologies that the A/B-product of A •*• B A' •* B is their PROPOSITION A-pullback. Note that the terminator of A/B is 1.32. Z D : A/B + A preserves and reflects colimits, D equalisers, pullbacks and monomorphisms. If D A has finite products, we can define XB : A -+ A/B by A t-+ [A*B -£•+ B) . PROPOSITION 1.33. ZD is the left-adjoint of XB . D D The next proposition says for right-adjoint. A cartesian-closed, that XB has a F i r s t , note the necessity of cartesian-closedness: if XB Aspects of topoi has a right-adjoint 15 IIg , then (-xfl, A) = lzB(xBl-)), /) = (XB(-), xB(it)) - [-, nfl(xfl(i4))] , that is, given IIg we can construct PROPOSITION 1.34. A as II (xB(4)) . For a artesian-closed A , xB : A -* A/B has a right-adjoint. Proof. where Given 1 •* B [A — £ + fi) f A/S corresponds to B define >• B . IIB(4 - ^ * fl) by the pullback For any C £ A we obtain a pullback in the category of sets (c, n U - B)) (c, i) ->- (c, 4B) [c, sT) . Three of these sets are naturally equivalent to other things and we obtain a pullback (C, IIR(/1 •+ B)) I 1 where the bottom map sends as a subset of 1 to •* {C*B, A) |(\(C*B, f) • (C*B, B) , C x B —2-> B . Viewing (C, U (A •* B)} (C*B, A) , we note that i t can be described as {j : C x B + < | C i j -2-»- 4 - £ * B = C * B -&-*• B} , which i s precisely the description of A/B(C x B-^- B, A-£+ B) = A/B{*B{C), A -£+ B) . D 16 Peter Freyd We note here, in anticipation of the next chapter's development, that for any / , B1—^-* B2 E _ : A/B we can define •* A/B f : K/B^ •* A/B by composing with can be viewed as an object in A/B / . by pulling back along On the other hand B —•*-»• B , and we could consider : (A/B 2 )/(B 1 -B 2 ) - A/B2 , \."]_ But '2 '"2^ '•I 2^ ' (A/Bp) / (B^ ->• B_) i s isomorphic t o A/B , and the isomorphism reveals Z C_ B D )i as (V 2' £,, , ^ xfBX n •+ BA 2 adjoint, each / , iff 1.4 2^ ' as f^ . Therefore j has a r i g h t A/B is cartesian-closed, each B . Modal operators in Heyting algebras. In any partially-ordered set, viewed as a category, the full reflective subcategories are in one-to-one correspondence with the order-preserving inflationary idempotents, that i s , the functions / that i s r such fix) < fiy) , x s fix) , fifix)) = fix) . Clearly, the reflector of a full reflective subcategory is such. Conversely, given such Given / then y (. Im(/) then Image(/) is reflective as follows: x 5 y •» fix) 2 fiy) = y and fix) 5 y => x 5 fix) 5 y . As a corollary of Proposition 1.31 we obtain PROPOSITION denoted x 1.41. Let H be a Heyting algebra with a unary operator such that x = x , x s x ior x A x = x )j Aspects of topoi x 5 y =» x 5 y V (x A y = x A y ) x,j/ We shall say that i// x [or 17 i = i A (x v j/) ) . V (x = x •=» (j/ + x ) = (y -*• x ) j . •*•>& is a closure operator if x =x, x 2x (or x A x = x ), x A i/ = x A y . Note that the l a s t equation implies x 5 y =» x 5 z/ . By the above remarks, the image of a closure operator i s a reflective sub-Heyting-algebra with the property that PROPOSITION operator. 1.42 for Hey ting algebras. In particular Proof. x = x =* (t/ ->- x) = (y •*• x) 1 ~l x is a closure 1 "1 ( x A j / ) = n i A l l i / . We have a l r e a d y noted i n P r o p o s i t i o n 1.23 t h a t ~ l ~ l ~ l ~ l x = ~l~la; . s From x S i / ^ n n x s n i y . x<y '~\y'S~\x then y-*-x = ~\~\(y+x). 3 = ~1 x . Then x = ~1 ~1 x yields x 2 ~I "I x and we e a s i l y o b t a i n By P r o p o s i t i o n l . U l i t s u f f i c e s x = ~1 ~l x . that t o show t h a t i f For purposes of c l a r i t y , l e t x = ~l z . We u s e only t h i s last e q u a t i o n from now on. x = ~l 3 = (2 -»• 0 ) , y -* x = y -* (z -»• 0) = (y A Z) -* 0 = ~l (2/ A 3) , 1 1 ( j + X) = "1 1 1 ( j A «) = n (j A 8 ) = i/ + * D (we have proved that the defining equation of a Heyting algebra implies the equation ~l ~1 (x Aj/) = 1 1 x A ~\ ~\ y , There is, therefore, an entirely equational proof. My attempts to find one (essentially by translating the proof herein) have yielded only the most unbelievably long expressions I've ever seen.) The image of double negation is easily seen to be a boolean algebra. 1.5 Scattered comments. Besides topoi and Heyting algebras important examples of cartesian- closed categories are the categories of small categories and various modifications of categories of topological spaces, which modifications 18 Peter Freyd exist precisely to gain cartesian-closedness. The chief such modifications are fe-spaces and Spanier's quasi-topological spaces. For any category as elements and skeleton of A A < B A' . P define the pre-ordered set iff A U , B) i- 0 . Define A' with P(A) A-objects to be the is a reflection from the category of categories to the category of posets. P clearly preserves products and hence the category of posets is cartesian-closed. Moreover, if x preserves Let , + M is cartesian-closed then be a monoid, Heyting algebra point, A P[^) P\S ) S is, and A •+ P(A) its category of representations. changes wildly depending on = P(S) = {0, 1} . For M a group, be described in terms of sets of subgroups of numbers P(A) and exponentiation. p(x) M . For p[sP) The M a single is a set (and can M ). For M the natural is huge but does satisfy a transfinite descending chain condition, a fact which requires a long proof and can be found - together with a description of P[S ) - in a paper by (of all people) me ([2], 225-229), in a section entitled "When does petty imply lucid?". For M almost anything else, P(i ) fails the transfinite chain condition. 2. The fundamentals of topoi Let A be any category with pullbacks. to be the set of subobjects of We can make Sub A Given (so assume that A f A A SubU) is well-powered). into a contravariant functor by pulling back. A topo8 is a cartesian-closed category for which representable; define Sub is that is, there is an object ft and a natural equivalence (-, ft) •*• Sub . Recall that any n : (-, A) •*• T is determined by knowing D. (l-) € TA , hence there exists fl1 >* ft such that if we define n : (A, ft) ->• S u b U ) isomorphism; by r\(f) = (sub(/))(ft* >* ft) then that is, for every A' >* A A' ->• Q' is an there exists a unique A' such that there exists n such that A -»• ft •* A 4- + ft' •*• ft is a pullback. The Aspects of topoi 19 representability of Sub is thus revealed as an elementary condition. we can do better. Note that in particular for any A there exists a unique A -*•ftsuch that there exists A -*•ft'such that + + ft' • ft Given any A •* ft' then for 4-»-ft=i4-»-ftl->-ft pullback. But is a it is the case A -±>A that . + ft' t is a pullback. • ft The uniqueness condition then says that (A, ft') has precisely one element. In other words ft' is a terminator. Thus we could define a topos as a cartesian-closed category together with an object ft and a map unique A -»• ft such that A' •*• 1 because 1 1 • ft such that for any A' >* A A' + 1 •A + -r° is a pullback. is a terminator.) there exists (We needn't quantify Of course, this elementary condition impli.es well-poweredness. fl° P PROPOSITION 2.11. For any small A , S Proof. is a topos. Aop We showed in Proposition l.lU that S is cartesian-closed. For the ft-condition we again assume the result to discover the proof. ft exists, then an U{A) = fa^, ft) = Subfa.). A sub functor of A-crible, alternatively described as a collection of maps such that B->-i4«C=>B'->-B->-/l€C. Defining SI (A) H^ C If is called into A as the set of 4-cribles, we make ft into a contravariant functor, again by "pulling back": given B •* A' such that Any A' -*• A and an 4-crible C define 1 •* ft is a choice for each A T' c T as the set of maps of an element in SiA . Define t : 1 •+ ft to correspond to the maximal crible, each 4-crible is the set of all maps into Given C B •* A' •* A i C . define A . (The maximal A ). T •* Si by sending x i TA to the 4-crible of 2" a l l maps B -*-* A such that i s a pullback. (T/)(x) €T' (B) and verify that • T + + 1-^ft The uniqueness of T -*• ft can be directly verified. • 20 Peter Freyd PROPOSITION Proof. 2.12 for topoi. Every monomorphism is an equalizer. Given A' »• A A' + A + 4- be a pullback. let 1 Then 4' is the •* Q e q u a l i z e r of A * 1 — ^ A •+ fl a n d 4 * 1 - ^ 1 -»- 0 . B u t 4 X 1 = 4 . COROLLARY D 2.13 for topoi. If A •* B is both mono and epi, it is iso. Proof. It can be the equalizer only of f, f : B •* C . O In any finitely bicomplete category we define the "regular image" of 4 —'-*• B as the equalizer of the cokernel-pair, that is, the equalizer of B x, y : B -*• C where f\ Ix is a pushout. It is a routine exercise to •• C B y verify that the regular image is the smallest regular subobject through which f factors ("regular" is equal to "appears as an equalizer"). Because all subobjects are regular in a topos, the regular image is the smallest subobject allowing a factorization of the image of / , denoted PROPOSITION 4 — £ + B . We shall call it Im(/) 2.14 for topoi. A ->• lm(f) is epi. If A -*->• C *- B = / then there exists unique Im(/) = C sueh that Given B + •D + C there exists unique Im(/) -»• Jm(g) such that 9 A •* I m ( / ) •* B 4C Proof. By construction, 4- Im(/) = B first statement it suffices to show minimality of Im(/) does Just that. + iff / is epic. Hence, for the Im(4 •*• Im(/)) = Im(/) ; but the Aspects of topoi 21 The other two statements are immediate from the characterization of Im(/) as the smallest subobject through which with Corollary 2.13. Sub(l) / factorizes, together • is a Heyting algebra. It is clearly a lattice (because images allow us to construct unions of subobjects) and for any [A, if) - (.A*B, U) monomorphism. Uc 1 , has at most one element, hence the map The subobjects of if •*• 1 - is a 1 , in fact, form a full reflective subcategory (the reflector constructed by taking images: X i—> Jm(X -*• l) ) . Aop The subobjects of collections of 1 in collection and there exists if A may be seen to correspond to 5 •+ A then B A is in the is in the collection. is strongly connected (for example, a monoid) then two subobjects it always must. subobjects of than S A-objects with the property that if 1 Aop S in If A is 1 •-»••-*••••->•• Hence has only the then the form the linearly ordered set one bit longer A . Returning to ft in 5 Aop , suppose A is a monoid M . We may view Aop S as the left-representations of M x X •*• X such that left-ideals in division: group then M . I** = x , The action of ct'A = {&|Ba £ A } . £2 = 1 + 1 ; M , that is, sets a(6*x) = (aS)\r . M X together with. Then fi is the set of on ft is not multiplication but 1 — • ft is the unit ideal. If M is a and, for monoids, conversely. 2.2 The representability of partial maps. A partial map from A Formally, we consider pairs to B is a map from a subobject of such that to < A' ** A, A' •* B > , define <4' >*• A, A' •*• B > B (A" «• A, A" * B) A' •*• A" A if there exists an isomorphism B . 22 Peter Freyd and define Par(i4, B) as the set of equivalence classes. Fixing B, Par(-, B) is a contravariant functor, by pulling back. A relation from of such A to B is a subobject of A x B . Calling the set Rel(A, B) and fixing B , Rel(-, S) becomes a contravariant functor, again by pulling back. Rel(-, B) is representable, namely by B si . Every map from a transformation to B can be viewed as a partial-map and we obtain (-, B) -»• Par(-, B) . Every partial-map yields a relation, transformation A A' >* A x B < A' »• A, A' •* B) (its "graph") and we obtain a Par(-, B) •*• Rel(-, B) . Both transformations are monic. The transformation (-, B) •* Rel(-, B) must come from a monomorphism B -*• SI , the singleton map. B -*• SI may be computed to correspond to B —^ B x B -»• £J where 1 4- * We shall show that PROPOSITION there exists such that + C Proof. Par(-, B) is representable. First: 2.21 for topoi (unique existentiation). Given Q>* A 4- is a pullback. > n such that Q-±+ Q + + C—• A is a pullbaok and for any is a pullback, there exists C •* A X •*• A X -*• Q •*• A = X •*• A . »• A Define (-, A) * Rel(-, C) by sending X -*• A to the pullback R •* X 4C 4- (H to be viewed as a subobject of X x C ). This transformation is •* A C ( r*A QP C \ i n d u c e d b y a map A •*• Si Define ( w h i c h c a n b e c o m p u t e d a s A •+• Si •• SI J . Q>+ A (X,Q) * (X,A) Q by t h e p u l l b a c k 44- . For any X , 44is a C - SlC pullback. Viewing X' iff in pullback 4- C {X, Q) (X,C) - as a subset of {X, A) (x,SiC) we see that / € (X, Q) •* X 4- , - A X' c X * C as a r e l a t i o n from X •+ C , Aspects of topoi describes a map from X to 23 C , that is, X' -*• X is an isomorphism. •+• A C Given a relation from A to B described by D + let Q >*• A be B as described above. ParU, B) . We obtain a partial-map, This operation, above lemma. Moreover Rel(-, B) •+ Par(-, B) splitting of g .) g . ft is the identity. Rel(-, B) -*• Par(-, B) •+ Rel(-, B) " > ft , and we define ft •* B , B-^ft [B -*• ft can be defined as the equalizer of (-, ftB) •* (-, B) , Clearly in is natural by the Par(-, B) •* Rel(-, B) •* Par(-, B) The idempotent transformation from an idempotent (Q>+A>Q-+C-*-B) (-, B) "• (-, ftB) splits must come as a 1 and (-, g) and: PROPOSITION 2.22 for topoi. Par(-, B) is representable. ft —**-• ft corresponds to a map which corresponds to a subobject of B ' — * B x ft where B -*• B s PROPOSITION exists unique B * ft which can be computed to be 2.23. For any partial A -*• B n g (l B ) (. Par(B, B) . fact that n is unique A •*• B map (A'>+A,A' A' * A such that + + The transformation Let In a telling sense 1 •+ ft , to wit: B Proof. B * ft ->• ft (not the evaluation map) is the singleton map. is a generalization of is a -*B> pullback. r| : (-, B) •*• Par(-, B) < B' * B, B' •*• B) represent is an equivalence says that for any such that there is is determined by 1 B (l g ) • A' •* B' exists unique A •* B is a pullback and = A->-B. such that A-*B'-*-B=A-*-B. If A •*• B' B' -*• B isomorphism, then we would obtain a contradiction. Note that Given a map Sub (4) = Par (A, l) B •*• B' and we can define there ->• A + In particular, for any such that there exists Then the (A' >* A, A' -*• B > B' A'-*-B'+B there ->• B A' pullback and D + is a •* B A -*• B such that + B' there + >- B were other than an • 1 = ft . Par(-, B) •* Par(-, B' ) by 24 Peter Freyd composition and obtain a map B •*• B' . Alternatively, from the above B proposition, there exists a unique map B •* B' such that ->• B + 4- is a B' •* B' pullback, and B is revealed as a covariant functor, B •* B as a natural transformation. 2.3. The fundamental theorem of topoi A logical-morphism of topoi is a functor that preserves finite limits, colimits, £1 , and exponentiation. THEOREM 2.31 for topoi. For any topos T and B e T , 1 IB is a topos. For f back along bi-continuous Proof. : B1 •*• B2 the functor f , has a left-adjoint and a logical f : T/Bx •* T./B2 defined I - and a right-adjoint by II- j pulling / is morphism. We noted at the end of 1.3 that the second sentence follows from the f i r s t . The fi condition i s easy: S u b T / B U ->• B) = Sub T (4) = T[ZB(A •* B), 0.) = T/B(A •* B, U*B ->• B) . -F Given T/B . A —*-»• B , Let B -*• IT A that n (A-*-B) a C - -»- B we wish t o construct in T correspond t o —=-"•—>• J 4 X B + B P -*• B i s Given i " B (C •*• B) , •*• CT . ,g be a pullback. ~A B -* ET as follows: X —»• B , T/B(X •* B, P •* B) -* T{X, P) • T(X, B) i s a pullback, by definition of T/B . in k : A*B ->• S , t h e unique map such . P is a pullback and let (C -»• B) By definition of P ->• B , Then Aspects of Topoi 25 U , P) •+ (x, C4) - Par^x/l, C) + 4- + 4 (X, B) •* (X, S ) = Par(Zx4, B ) is a pullback. Then T/B(X •+ B, P -+ B) -* ParT(X>^4, C) 1 is a pullback where X x A -hxl> • ParT(X*A, 1 •+ Par(J><4, B) B x A -^+ S . B) corresponds to The element in Par(,X*A, B) is therefore the result of pulling back: • ->• X*A + Vk B -* B Q and that is the same as the pullback of X •*• B and 4 + J T-partial maps from X ; x that is, A to C •* X + -I- , that is, the product, in A + B T/B{X •* B, P •* B) T/B is the set of such that when composed with C •*• B yield just what they should. That J preserves exponentiation is reducible, as discussed at the end of 1.3, to seeing that wish to compare XS : T •* T/B [-, xB(c^) XBfC 4 )) * B, [D - B, XB(C)*B{-A)) and = \tB(D preserves exponentiation. We (-, * B), C4) = {D, C 4 ) = (D*A, C) , = ((£» + B) x XBU), XB(O) = fsg((£» •* B) * XB(4)), c| . I t suffices to show D * A - £ g ((0 * B) x xB{A)) . P (D -*• B) x xB(i4) is the pullback + D AW verified as p+ •*• • +p . D -»• B Hence £ „ ( ( £ •* B) -\p which can be directly B x xB(4)) = y5 x z) . 26 Peter Freyd preserves limits and colimits because it has both a left and right adjoint. D COROLLARY 2.32 for topoi. Fullbacks of epimorphisms are epimorphisms. Proof. T/B . Given A -£•*• B , j (C •*-*• B) is the pullback. and terminal objects, COROLLARY C ->-»• B , we view Because / (C •* B) •* f (B ->• B) 2.33 for topoi. Given C •** B j preserves epimorphisms is an epimorphism. A -£•+B + C A, + A = ~< as an object in D there exists A I B+C Proof. apply f View B -»• S + C to obtain Given a filter A F and C •+ B + C ->• A , A ->• A as objects in in T/A . T/B+C D (as defined in 1.2) on the Heyting algebra we obtain a Serre-class of maps in the topos, namely those maps such that there exists isomorphism. U £ F , U € F . (U c l) so that T -»• T/£/ T/F(A, B) = lim(/5xU, B) . T/F T/F T/U + T/V Sub-j-.p(l) = (Sub-j-(l))/F . to an which T/U , for are logical morphlsms and that topoi are essentially algebraic, easily believed to be a topos. f The most insightful is to take the direct limit of the topoi Using the fact that all the induced maps Sub(l) f : A -*• B sends The result of inverting all such maps is a topos may be also constructed by way to construct and V c V T/F is Finally, one must note that Strangely enough, we shall not use this construction. R Given relations define the composition ->- A + B i?2 * B , + C R ° i?2 as R let •* R1 + -IR2+ B be a pullback and Im(if •+ A*C) . PROPOSITION 2.34 for topoi. Composition of relations associative. is Aspects of topoi Proof. Give n X' ->• B' Y' 4- + , + X ->- B Y * X •* A*B , Y •*• A*B and 27 B' •* B let B' 4- be pullbacks. Corollary 2.32 says that if -*• B Im(X •* A*B) = Im(y •+A*B) then ImU' ->4><B') = Im(Y' •* A*B' ) . That fact, together with the associativity of pullbacks provides the proof. Let R 6 * nh " ffl + 4- + A i? 5 •* i? 2 - B + + i? 3 -• C be such that a l l squares are pullbacks. = (if ° R ) ° i?3 . I m ^ -*• A*C) = Ft. ° R~ and hence Equally Im(flg •+ AXD) = R o [R ° R ) . D Given a topos T we obtain a category of relations Rel(T) and an embedding T •*• Rel(T) . In particula r Rel i s a bifunctor from T to s e t s , contravariant on the f i r s t variable, covariant on the second. Given / : A •* B the transformation Rel(-, / ) : Rel(-, A) ->• R e l ( - , 5) yields a transformation to il (-, if) •* (-, il ) which must be induced by a map from t o be called 3- . The transformation ( - , - ) -»• R e l ( - , - ) if is natural and we obtain A •+ if n iv B •+ i? Rel(4, B) is of course, a Heyting algebra. Heyting-algebra-valued functor, but unions but not intersections. order. 2.4 Rel(4, -) Rel(-, B) is a is not. 3,. preserves We will use the fact that i t preserves The propositional calculus of a topos PROPOSITION 2.41 for topoi. For every B , the subobjeats of B form a Heyting algebra and the operations of same are preserved by pulling 28 ' Peter Freyd back. Proof. T./B . The subobjects of B are the subobjects of the terminator in D We can t h e r e f o r e view Sub a s a H e y t i n g - a l g e b r a - v a l u e d N e c e s s a r i l y , t h e o b j e c t which r e p r e s e n t s T . £2x£2 That i s , t h e r e e x i s t maps • £2 , £2x£2 1 • £2 , functor. Sub must be a Heyting a l g e b r a i n 1 —£+• £2 , £2x£2 -^+ £2 , • £2 which s a t i s f y t h e e q u a t i o ns of a Heyting a l g e b r a (t = X, f = 0) , which maps y i e l d t h e Heyting a l g e b r a s t r u c t u r e on each Sub(S) . f 1 —-1-*- £2 can be computed as the unique map such that i s a pullback; pullback. £2x£2 >• £2 as the map such that + 0— + 1 +/ +n i s a Let IX t] v £2 = Im £2+£2 Then + +u i s a pullback. unique map such that + F i n a l l y , define ++-»• i s a pullback. ftxfi >• £2 t o be the One may directly verify the equations of the symmetric definition of Heyting algebras for these maps so defined. Note that for each B , are two Heyting algebras for sets, n which lives in f2 B : becomes a Heyting algebra in Sub(S) T . which lives in the category of T . A boolean topos i s one in which There Q is boolean. £2 • £2 i s the Aspects of topoi fl +~1 is a pullback. unique map such that + fl Note that every • n — • fl = 1 . boolean, and given B' >* B 0 + -• B' t is a pullback, B" •+ B that B'+B" •* B Sub(S) 29 Hence is epi. is boolean if is thus forced to be there exists a complement B'+B" -*• B T B" >*• B such that In any topos, such implies is an iso, just by showing that B'+B" -*• B is mono, an easy matter using Corollary 2.33. Products of boolean topoi are easily seen to be boolean. PROPOSITION 2.42 for topoi. isomorphism. (1 T ia boolean iff iff 1 + 1 -±LU fl ie an 1 + 1 -iiX* fl -Lt Proof. Clearly i f 1 + 1 works as fl then fl other than the twist map on 1 + 1 , and ~1 ~I = 1 . Conversely for T boolean, the complement of and the remarks above yield 1 + 1 = fl . Q PROPOSITION Proof. If c the form S , 1 +1 . Aop 2.43. S A i s a groupoid then G a group. Conversely, suppose is, there is no is boolean iff B •*• A •+ fl can be none 1 —• SI is 1 —*-• fl A is a groupoid. Aop S i s a product of categories of c We observed earlier that fl in S is A •*• B such that in A does not have a left inverse, that B •* A •*• B = 1_ . A •*• B generates a D 5-crible, neither empty nor everything; that is fl(fi) has more than 2 elements. But i f 1 + 1 = fl then fl(S) = (l+l)(B) = 1 + 1 . D By a closure operation on B we mean a closure operation as defined in 1.1* on Sub(S) , that is an intersection-preserving inflationary idempotent Sub(S) •*• Sub(B) . A global closure operation is a choice for each B of a closure operation that makes Sub •+ Sub natural. Necessarily such must be induced by a map j : fl •* fl , and j itself is a 30 Peter Freyd closure operation in the internal sense, that is, j2 = j , £2 <1,.7> luJ _ —*• £2*£2 n • £2 = 1 £2x£2-^£2 and JXJ+ ftxft ftx +j . Rather mysteriously: 1- ft n PROPOSITION Proof. Given applying 2.44 for topoi. First a less algebraic proof: 0 :fi•*ftconsider (B,ft)v ft = 1 -^+ ft then for 1+ -r n+ >;/ '•» (B,ft). If a pullback, the result of (B, j ) yields B 1 ->- B ft' * ft and t h e r e e x i s t s i s a pullback and 1 ->• ft' such t h a t (B, j) More directly, l e t < 1 , j > i s monic, exists 1 •* y such t h a t is. that is + + (1, j > be a pullback. If Because 1 —>- ft - ^ * ft = 1 — • fi then t h e r e 1 •*• U •* £2 = 1 1—1_ isomorphism; B' -+~B' + 1 is inflationary. + U •*• 1 1 •* ft1 ->• ft = 1 —*• ft , hence »• £2 and tf * 1 i s an i s a pullback. Hence ft +<1, j> + •*• ft1 Aspects of topoi is a pullback and the uniqueness condition on As shown in Proposition 1.1+2, Another example arises as follows: Sub(B) -* Sub(B) by is a monomorphism.) SI let • SI V c 1 (B1 •* B) >-+ (B'u(B*U)} . 31 SI implies the result. D is a closure operation. and define for each (Notice that B , B*U •* B*l •* B Clearly this operation is inflationary and idempotent. To see that it is intersection-preserving consider B', B" c B . Then 11 (B'u(B*£/)) n (B"u(Bx£/)) = (B'nB ) u [fl' n{B*V)) u (CBxt/)ng") u (Bxtf) , by distributivity. The middle two terms are contained in B x t/ , hence we obtain (B'u(Bxy)) n (B"U(BX£/)) = (s'nB") u (flxy) . Bx[/ -* B + + i s a pullback. U +1 The operation is natural because 2.5. Injective objects 2.51 for topoi. a is injective. PROPOSITION Proof. A' let Given and A + 8 let A' • • A 4- be a pullback. Then <• B + 4- b e a p u l l b a c k . COROLLARY Proof. epic. A >* B Then verify that A->-B + n = A + Q . O 2.52 for topoi. ftC ie injeotive. Given A »• B we wish to show that But (B, QC) - [A, Sf) I? I? , n) [B, SI ) + (At SI ) is 32 and Peter Freyd A*C -*• B*C is monic. D Using the singleton map B -*• ft we obtain: PROPOSITION injective. 2.52 for topoi. Every object may be embedded in an D COROLLARY 2.53 for topoi. Pushouts of monomorphisms are monomorphisms. Proof. Given a pushout A >* B + + choose C *+ E , E injective. There C •* D exists A yB + + , h e n c e t h e r e exists C •*• monic. C-<-D->-E =C>*E a n d C •*• D i s E • PROPOSITION 2.54 for topoi. Snb(D) •* Sub(S) 4+ Snb(C) •* Sub(A) Proof. Just use the distributivity, COROLLARY intersections show that A H- B $ ¥ is a pullback, then C>+ D is a pushout. for Heyting algebras. Proof. If X A («/ v z ) = (x A y) v (x A z) D 2.55 for topoi. If E is injeative then (-, E) carries (that is, monomorphic pullbaaks) into pushouts. If E (-, Cl ) is injective, then -p E **• il splits and it suffices to carries monomorphic pullbacks into pushouts. But (-. fi ) - (-*£, ft) and -*£ preserves (as in any category, any E ) monomorphic pullbacks, and Proposition 2.51* says that desired. (-, ft) is as • PROPOSITION 2.56 for topoi. Given A H- 8 \ 1 , it is a pullback iff for C >* D {D, E) •* (C, E) injective objects Proof. E3 The family + t (S, E) -* (A, E) {(-, E)} , E is a pushout. injective, is collectively faithful Aspects of topoi 33 by Proposition 2.52 and hence reflects isomorphisms. Any family that reflects isomorphisms reflects the limits it preserves (any category). • 2.6. Sheaves Let that i s j : ft •* ft be a closure operation as defined in the l a s t section, j =j and j i s a homomorphism with regard t o ftxft -°->- ft and 1 —• ft . Given B' » B , we will write (B, j) to Sub(B) . We will say that B' = B . B' H- B i s B' j-closed We say that A i s j-separated (B, A) •* (B' , A) i s monic. for the result of applying if B' = B' if for a l l j-dense We say that A i s a j-eheaf if for a l l ./-dense (B, i4) •+ (B' , -4) i s an isomorphism. and j-dense if B' >+• B , B' H- B , A functor i s exact i f i t preserves a l l f i n i t e limits and colimits. THEOREM 2.61. The fundamental theorem of sheaves. The full Bubaategoviea of j-eeparated objeote and j-sheaves are reflective and eaah ie cartesian-closed. The full subcategory of exact. Proof. We fix j j-sheaves is a topos and its reflector and drop the prefix B' then so is B'*C -*• B*C is equivalent to because if "j-" . If is B' >* B i s dense • B + +3 is a pullback, then density B -&+ft-2-+ft= B •* 1 -^* ft , and B' *C •* BXC +p B 1 is a pullback. Hence if A - ft is separated (sheaf) and B' >* B dense then 34 Peter Freyd ( S , A°) - (B 1 , AC) \i I? (5XC, / I ) •+ ( S ' x C , 4 ) , and t h e h o r i z o n t a l maps a r e mono ( i s o ) , and A i s separated (sheaf). Let Sep. be t h e f u l l subcategory of separated objects and 3 the full subcategory of sheaves. Sh. be 3 We have just seen that Sep. and Sh . are cartesian-closed (except 3 3 possibly for finite completeness) and hence by Proposition 1.31, when we know that Sep. and Sh . are reflective, we will know that the reflector 3 3 preserves products. Define Q . •*ftas the equalizer of 1 3 and j . Because j is i d e m p o t e n t , there exists ft •*ft. such that ft •*ft. •*ft= j , 3 3 ft . -<• ft ->•ft. = 1 . ft. i s i n j e c t i v e . 3 3 3 LEMMA 2.611. Proof. Let of ft . yields 0 B> ft. 3 is a B' >* B be dense, and and we have Then B 2.612. ft.. and B 5 1 n B' = B COROLLARY B' •+ Q. 3 given. The i n j e c t i v i t y B'—• B \ J . We need only the uniqueness condition. ft. 3 -»- B -£-+ft. = B' •* B -2-+ pullbacks. sheaf. Let 4- 4-/ and 4- are both closed. n B1 = B1 , For any A, But B B^ n B' = B2 tfi 3 is a sheaf. Suppose 4-0 b e n B' = B^ n B' and f = g • Q D The definition of Sep. and S h . easily says that both are closed 3 3 under limits, Sep . under subobjects. 3 The idempotence of j says that for B' C B , B' is dense in B' Note that the closed subobjects of B are in natural correspondence . Aspects of topoi with 35 (B,fi.) . 3 LEMMA 2.613. Proof. If A A is separated iff A — • A*A is a closed subobject. is separated, let A c A*A be the closure of A . A -A There is at most one I •* Ax-A A •-- J . hence Thus =J —i*A Conversely, suppose /I B' •+• fl -i-»- A = B' •* B -2-+ A . S as the pullback • + S"1 C E C B ; thus LEMMA 2.614. _ A p. _ po •* A A , and hence A c Eq(p , P2J = A . •• A • .4*4 is closed. Let B' >* B The equalizer of /, ^ be dense and can be constructed B ^f>9^ . hence B = B and / = g . A is separated iff E1 is closed. But B' c E and D A •+ $/ = A -* fi4 -J2—•• fi!4 . Proof, j yields a transformation j : Rel(-, A) -*• Rel(-, A) alternatively described as taking closures of subobjects of -X/l . The equation of the lemma is equivalent t o : (-, A) ->- Rel(-, A) = (-, A) - Rel(-, A) -i+ Rel(-, A) which may be tested on 1. € (A, A) . The left side yields the right side yields the closure of A proof. O LEMMA 2.615. For any A the image of reflection of A into Sep • . 3 Let and A A •* A be the image of A -*• U •* U . Sep. 3 A A •* fir -*—+ fir is the if •*+fi4.+ j / and A i s a subobject of ft4. . J 3 u. € Sep. 3 3 • A*A ; and the last lemma yields the A Proof. A j factors as Lemma 2 . 6 l l said that i s clearly closed under subobjects, hence A t Sep. 3 36 Peter Freyd If B i. Sep • then by the last lemma B •+ B i s an isomorphism. 0 f : A -*• B , S € Sep • we need only show that the outer rectangle of Given A •* fT4 -2—*- if A commutes to obtain -*• A \ ^ 4- . Equivalently we consider B ( - , A) •* Rel(-, A) ( - , B) -+ Rel(-, B) The right-hand square does not commute. clockwise we obtain «U Rel(-, A) T* Rel(-, B) Given show t h a t when R c X*A graph of Im(R • But the fact for inverse Im(i? -»• X*B) c Im(i? -»• X*B) and because direct images preserve order we also have But i f and chasing Im(i? -*• X*B) and in the other direction Inverse, not d i r e c t , images preserve closures. images yields that R c: X*A Im(R •* X*B) c Im(/f -»• X*B) . R i s the graph of a map then Im(i? -»• X*B) = Im(R •*• X*B) . i s the graph of g : X •* A then X " > A —'-> S I t suffices t o Im(i? •* X*B) i s the and the last lemma says precisely that graphs of maps i n t o separated objects are closed. For the r e f l e c t i v i t y of • Sh . i t suffices to show i t reflective in J SeV. . LEMMA 2.616. Proof. Given If A' » A is closed, A € Sh. then A' € Sh. . 0 0 B' B >* B dense and B' •+ A , l e t •*• B + A' + commute and •+ A B" * B + A' +. a pullback. Then B" is closed, B' c B" and hence B" = B . D •* A ' We'll say that a separated object is absolutely closed if whenever i t appears as a subobject i t appears as a closed subobject. Aspects of topoi LEMMA 2.617. 37 Sheaves are absolutely closed and separated. Absolutely closed objects are sheaves. Proof. exists Given A •*• A A >* B , A € Sh. , let A >* B be the closure. J A _ such that + yB , that is, A c A c A . Conversely, if 4 fl €. Sh . 3 3 3 a sheaf. B and if A A is separated then it appears as a subobject of is absolutely closed, then by the last lemma, LEMMA 2.618. fi . satisfies the LEMMA 2.619. G-iuew is the reflection of Proof. For any Let A >* B , A A •*• B C € Sh. , 3 There in 0.-condition for B (A, C) = {A, C) A D a sheaf, then the closure of A in because Lemma 2.6l6 says that A -*• A is dense. A € Sh. . 3 • can be embedded in a sheaf (for example, u. ) we obtain that every separated 3 A has a reflection in Composing the two reflections, the reflection of an arbitrary closure of the image of is Sh. . 3 be the closure. Because every separated Sh. . A Sh. . 3 A is the A •+• U. . 3 We saw at the beginning of the proof that Sh . is closed under 3 exponentiation. Lemma 2.618 says that Sh . has an fi , and we have just 3 seen that it is reflective, hence finitely cocomplete. Thus Sh . is a 3 w topos. Note that for E injective in Sh . , that E >* il. retracts and 3 3 is injective in the ambient category Since the reflection T . H : T -* Sh . preserves products it suffices for 3 exactness to show that it preserves equalizers. any category, the equalizer of /, g Given /, g : A •+ B in may be constructed as the pullback of 38 Peter Freyd hence i t suffices t o show t h a t R : 7" •* Sh . 3 preserves monomorphic pullbacks. A y- B ? I in I , it suffices according to Proposition C w- D Given a pullback 2.56 to show that for any injective {RV, E) ->• (RC, E) + + is a pushout. (RB, E) •* {RA, E) {D, E) + (C, E) + + and since (S, E) •* (A, E) the finish. (A' But that diagram is isomorphic to is injective in T , Corollary 2.55 provides D Consider 3 E E £ Sh . , it is the case that 3 S* . Si = (3 •*• 2) . Let i / c i t h e c l o s u r e o p e r a t o r t h a t sends •* B) c (A -*• B) . Then t h e (A •* 1) . fl . = (2 •*• 1) . 3 reflection of ft Recall that is ft •Q (0 -»• l ) c ( l -»• 1) and (A' + B' ) c {A ->• B) to j - s h e a v e s a r e of t h e form The r e f l e c t i o n (3 -»• l ) be not of (A •* B) is U -»• l ) . The Q. . 3 is a closure operator. PROPOSITION 2.62 fortopoi. Sh-,-, is boolean. Proof. 1 + 1 —*—>• fl is easily seen to be dense, hence becomes an isomorphism in Sh-, -. . COROLLARY co-continuous Proof. • 2.63. Every non-degenerate topos has an exact functor to a non-degenerate boolean topos. 0 is a 1 1 -sheaf. D Following the remarks at the end of Section 2.k, let U c 1 and consider B' Q - ^ + ft t h e closure operator so t h a t for = B' u (B*U) . B' c B , T •+ Sh. sends 0 •+ V tc an isomorphism, 3 that i s , U is sent to the zero-object. Sh. is degenerate iff U = 1 . 3 COROLLARY exists The r e f l e c t o r 2.64 f o r t o p o i . a boolean topos Given x, y : 1 + B , 6 , exact co-continuous F •* 8 x f y , there that separates Aspects of topoi Proof. Let U c 1 39 be the equalizer of x, y . described above, T •*• Sh . sends x, J/ : 1 -»• B 3 Now apply Corollary 2.63. O as equalizer. THEOREM 2.65 f,g:A-*B3 Then for (the plentitude of boolean topoi). f ? g , there exists a boolean topos ao-continuous T -*• 8 that separates j as .to a pair of maps with • 0 For every 8 exact 3 f, g . Every small topos can be exactly embedded in a boolean topos. BM Proof. T - T/A sends f, g to \ / A*A - 2 ^ 1 * BxA \ , ^/ A , A is the terminator for T/A 5 as follows: define J OO t e a partially ordered set. for + A and T/A •* 8 Hence we can apply Corollary 2.6k to obtain Let A . We define as desired. j : ft + ft in ^ P 5 x (. B , Q(x) = {A c H \ u < v £ A "* u £ A ** u < xi ; to be the set of all with the property that then this simply says V' eH\y eA> u € H such that there exists (v 2 s) •» (v 2 u)J . u 5 IM1 .) j (If U4' A' c A exists, is easily seen to be idempotent and inflationary. It is natural and it preserves intersections iff distributive, for example, a Heyting algebra. (l, ft.) is its completion. 3 (l, ft.) = H 3 For B Hence for H For B B a Heyting algebra a complete Heyting algebra and ft .(a:) = {u i. B \ u < *} . 3 That is, 3 (A) = UA . x (l,ft.) 3 Sh . is the classically defined category 3 the lattice of non-empty open sets in a space is the lattice of all open sets, is very X , of sheaves. The definition of a Grothendieck topology on definition of a closure operator in more general. S A A is almost the op . Closure operators are a bit 40 Peter Freyd 2.7. Insoluble topoi, or how topoi aren't as complete as you'd like In a powerful way, topoi are "internally complete". example, define a map given any Sub(fT) stated B c!T = fl, fl •+ ti which acts as a union operator. we obtain a subobject ->- ( l , ?iA) = Sub (4) I fl One may, for UB c A Hence [via the functions which has a l l of the properly properties of a union. There i s a catch. subobjects of A UB i s not necessarily the least upper bound of the described by ( l , B) c ( l , QA) = SubU) . completeness we often need i s just t h a t ; upper bound for singleton map ( l , B) c Sub(4) . A •+ u (1, A) c Sub (A) (l, A) = ( l , A) . that 1 , UA = A . that i s , given UB f a i l s miserably. B c tf4 , a least For the But the least upper bound of would be the least subobject To say t h a t The type of A = A , all A of A such that A , is equivalent to saying is a generator (that i s , a well-pointed topos as defined in the next s e c t i o n ) . If for every upper bound of A there were such A c A , then UB is the least (l, B) c Sub(j4) . (We shall not use this construction and hence will not prove it. ) The existence of A c A independent of the axioms for topoi. is elementary but Indeed, as we shall show, it is not equivalent to any essentially algebraic axioms. We will call that T solvable if for every (1, 2) - (1, A) is the case that PROPOSITION Proof. and for all Be A such that A c A such (l, B) - (l, A) it 2.71. Not all topoi are solvable. We saw at the end of the last section that every complete (l,fi). Let T-. be such that order-type of the unit interval and let exists there exists A c B. Heyting algebra appears as submodel. A TV, c J (l,fl)is the be a countable elementary In Tp , (l,fl)is a dense ordering and countable, hence there iu c l} , u c \j such that n n n n+1 be such that u n c v al l n • Uu does not exist. n Let V C 1 f Aspects of topoi 41 / n Consider the direct system T^j ] \ (l+^-) induced by the obvious n+l n 1 T{l+U-) •* 1 [(l+U.) and let projection maps T be the direct limit. The essentially algebraic nature of topoi insures that can give a more elementary description of j T is a topos. We as follows: objects: {n, A -+ TT(l-*J / , maps: <n, A A ^ —>• "TTM^)) (l+^-j) t to (m, (n, o (m, from are equivalence classes of pairs < k, f) B where k k f : A x J~f (!-«/ ) ^S x ] i such that T M) X X | 7 (1+y) _^U T T M J - TT d^J i=m+l i=l ^^f T The equivalence relation is generated by The notation is eased by replacing in T . Hence For - rr <k, f> = (k+1, fxl> . Tg/ ] T(l+^-) with its image T/n T/0 = T 2 , UT/n = T , T(A, B) = lim T/n (/I, S) . A i T/n , B i T/0 , TU, k /" 1 S) = lim T/0U x 7 7 (l+y ) , B —• (an easy verification). that i=m+l k ] [(l+U.) If Thus "• i=«+l J T(l, 1+7) = lim T/O(T~T(l+i/-) » 1+^ l —• W=l > • Note k is a coproduct of 2 subobjects of 1 , one of which is 42 42 Peter Peter Freyd Freyd one, one,one oneof ofwhich which is is U. U. ,,the theother other smaller smaller than than U, U, .. Hence Hence the theunion union of of ((k k )) the images of T/0 the images of T/0MM [{l+U.), [{l+U.), 1+V\ 1+V\ is is 11 ++ U. U... Suppose Suppose WW>+ >+1+1/ 1+1/€€TT were were such such that that (l, (l,U) U) -- (l+V) (l+V) .. We We shall shall show show that •»•l+V that there there exists exists W' W'-»l+V with with the thesame same property, property, and andsuch such that that kk 3 V V<£ <£WW .. WW•* •* l+V l+V in in TT appears appears as as AA xxYJ(l+U.) YJ(l+U.) --3-- l+V l+V some some kk .. For For all // cc BB all n ncc Im{g) all nn ,, 1+U 1+U Im{g)cc 1+7 1+7.. Let Let BBcc VV be be such such that that ££ all «« ,, but )) .. Then •»•l+V ffff but 1+5 1+5££ Im( Im( Then 1+5 1+5-»l+V as asaa subobject subobject in in TT is is contained contained in in W W -*• -*• l+V l+V.. DD 3.3. Well-pointed Well-pointed topoi topoi AAtopos topos isis well-pointed well-pointed ifif itit isis non-degenerate non-degenerate and and ifif 11 is is aa generator. generator. PROPOSITION PROPOSITION 3.11 3.11 for forwell-pointed well-pointed topoi. topoi. ( 1( 1, , fl) fl) ==22 ; ; 1 1 ++11 ==ftft; ; ( l( ,l , - -) ) preserves preserves ooproduot8, ooproduot8, epimorphiemSj epimorphiemSj epimorphia epimorphia families, families, and and pushouts of monomorphisms; pushouts of monomorphisms; AAf f 00=*=*AA isis injeotive; injeotive; 44^ ^00, ,l l= => >. .44 isis aa aogeneratOT. aogeneratOT. Proof. Proof. For For 44^^00 there there are areaat t least least two twomaps maps from from AA ttoo Q Q ,, hence hence t ht he er er e e exxi si st st s 11-»•-»•44 . . Let Let UUcc l l . . IfIf UU$$00 then then there there exists exists ll ++U U and and 11 •+ •+ UU ++ 11 = = 1^ 1^ forces forces UU•+ •+ 11 t o t obe beepic. epic. Hence Hence ((l l, , fi) fi) ==22 . . a 11 ++11-*-*nn i si s always always monic. monic. Because Because ((l l, , 1+1) 1+1) a ((ll,, ^^)) and and 11 r er ef lfel ec ct st s isomorphisms, isomorphisms, 11++11--flfl. . For For ( (11, ,4) 4) ++( (11, ,5) 5) ==((11, ,4+5) 4+5) use useCorollary Corollary 2.332.33C C •*•*11 Given Given AA•*-* •*-*BB and and 11-+-+SS l leet t ++ 4-4- be be aa pullback. pullback. By By Corollary Corollary AA-*-* BB Aspects of topoi 2.32, C •* 1 i s epic; thus Cf 0 43 and there exists 1 •+ C yielding 1 4*+ B Given any family can view {A • •* B] {A. -*• B] collectively epimorphic, and 1—£-*• B , we as a collection of objects in T/B which collectively if cover the terminal object. that do the same. Applying Hence for some f^ we obtain a collection of objects i , / (A . •* s) ^ 0 and {fl, 4 .) -*- (l, B)} is collectively epimorphic. A x- B + 4- we use the booleanness to write Given a pushout A •* A+A' + + C * C+A and we see that Given Then B = A + A' •*• D C is a pushout, clearly preserved by A >+ B , A \ 0 , again write B = A+A' •+ A+l •*• A B = A + A' (l, - ) . and choose is a right-inverse for A -*• B . 1 •+ A . 0 PROPOSITION 3.12. If 8 is a boolean topos then it is well-pointed iff for all A (. 8 , A f 0 there exists 1 + A . Proof. f>g> Let f,g: Ac B f t g . Let E e B be the equalizer of the complement of E then there exists 3.2. B + C, 1 •+ B in B . Hence if there exists that distinguishes f,g. 1 -»• A • The plentitude of well-pointed topoi A logical morphiem of topoi is a functor that preserves all the structure. THEOREM 3.21. there exists preserves a well-pointed epimorphic Proof. a topos & and logical 8 and A € 8 , r : B+8 , TA % 0 . A \Q T show: For every boolean topos 8' , logical epimorphic families; boolean topos families. We f i r s t LEMMA 3.211. exists For every snail T : 8 •*• 8' , and for all B $. 8 8 and A i 8 , T(A) f 0 . either T TB - 0 A J= 0 there preserves or there exists 44 Peter Freyd 1 •*• TB . Proof of lemma. object. Well-order the objects of 8 , taking A as first We construct an ordinal sequence of topoi and logical morphisms as follows : If T : 8 •*• 8 a sequence at and property, hand, there exist 5*8 terminate the a . on the other If, has the described ( 1 , TB) = 0 then take such that B to be the first TB \ 0 such and define 8 cc+l = V ™ • If 3 is a limit ordinal, and 8 a is defined for all a < 6, then Bg is the colimit of the 8 's . The essentially algebraic nature of topoi insures that The functor 8 •*• 8 carries C i8 such that Moreover for every Bg is a topos. TB to an object with a map from 1. 8 (l, TC) * 0 , 8 ( 1 , T'C) * 0 and eventually the sequence must terminate. • Now, for the theorem, define a sequence on the finite ordinals by 8 (as defined in the lemma) and 8 That 8 . 8 = lim B 8 •*• a i s logical and is boolean because implies the same in n+l = K ' Proposition 3.12 says that 8 8 is well-pointed. T preserves epimorphic families follows from the fact that colimits of such functors are such functors. COROLLARY in a product families. 1 + 1 - fl in 3.22. D Every small boolean topos can be logically of well-pointed topoi, and the embedding preserves embedded epimorphic Aspects of topoi Proof. iff For exact TA - 0 =» A - 0 . 45 T : 8 •* B' between boolean topoi, T is faithful D Composing with Theorem 2.65 we obtain: THEOREM 3.23. Every small topos can be exactly embedded in a product of well-pointed topoi and the embedding preserves epimorphic families. Composing with THEOREM (l, -) and using Proposition 3.11 we obtain: 3.24. For every small topos T there exists faithful T : T ->• IIS . T preserves all finite limits, coproducts, epimorphisms, epimorphic families, and pushouts of monomorphisms. 3.3. Me ta theorems By the universal theory of exactness of a category we mean all true universally quantified sentences using the predicates of composition, finite limits and colimits. By the universal Horn theory of exactness we mean all the universally quantified Horn sentences in the predicates of exactness, that is, sentences of the form A A A^ A ... A A =» A where each A . says either that something commutes, or is a limit, or is a colimit. By theories of near exactness we mean those using the predicates of composition, finite limits, coproducts, epimorphisms, and pushouts of monomorphi sms. As easy corollaries of Theorem 3.21 through Theorem 3.2U we obtain: METATHEOREM 3.31. The universal Horn theory of near exactness true for the category of sets is true for any topos. The universal Horn theory of exactness true for all well-pointed topoi is true for all topoi. The universal Horn theory of topoi true for all well-pointed topoi is true for all boolean topoi. The universal theory of topoi true for all well-pointed topoi is true for all boolean topoi in which (l,fi)= 2 . We will show later that there do exist universal Horn sentences in 46 Peter Freyd exactness predicates true for S but not true for all well-pointed topoi. An equivalence relation on A is a relation E c AM satisfying the usual axioms. COROLLARY 3.32 for topoi. Every equivalence relation is effective; E •* A that is, given E a A*A there exists A •* B such that + A 4- is a •+ B pullbaok. Proof. We get rid of the existential quantifier by defining to be the coequalizer of the two maps from E A •* B to A . We note that the statement is in the universal Horn theory of exactness and it suffices to prove it in well-pointed topoi. E' ->• A Accordingly, let + A 1 -*• E' E . be a pullback. that can not be factored through x, y : 1 •* A in + such that Let A' If E % E' there exists ->• B E •*• E' . Hence there exists 1 — ^ A •* B = 1 - ^ A + B be the complement of but 1 Im(x)' u Im(t/) X ^ ' A*A no 4 and define 1+1+1 by A' •* A -£+ 1+1+1 = A' •+ 1 x U a 1 -^* A -iL* 1+1+1 = ^ E •* A -&-+ 1+1+1 contradiction. • 1+1+1 , 1 — ^ * 1+1+1 . A equalizes and there must be • B \^ / ,a 1+1+1 • COROLLARY 3.33 for topoi. as: 2 1 A -2->- 1+1+1 = Then • 1+1+1 , Every epimorphism is a coequalizer. Proof. We again get rid of the existential quantifier by stating it every epimorphism is the coequalizer of i t s kernel-pair. (The E •* A kernel-pair of A •* B is the pullback + + .) It suffices to prove i t A •* B in well-pointed topoi. Given A •*-*• B let A •*• B' be the coequalizer of Aspects of topoi E -*• A . x' 1 I t s u f f i c e s t o show t h a t B' •* B u' »• B' •+ B = 1 -a—> B' ->• B and find 1 1 —^+ 4 •+• B = x E c 4x4 1 , 1 -&+ i4 -• B' = j / x ' = z/1 . and 1 47 i s monic. Let x , z/ : 1 •* 4 . Then a>y 1 so t h a t » /5M lies in D A «• B COROLLARY 3.34 for topoi. If * 1 is a pullback and C+B •* D C y- D is epi, then it is a pushout. Proof. The statement lies in the universal Horn theory of near exactness and it suffices to prove it in S , an easy matter. • 3.4. Solvable topoi Given a topos T and object if the maps from 1 A i T we'll say that to are jointly epimorphic. A A to be the full subcategory of well-pointed objects. is a well-pointed Define T c T In Section 2.7 we defined a solvable topos , which definition is equivalent to the coreflectivity of 7" PROPOSITION Proof. is that 0 . 3.41 Suppose Let for topoi. A, B € J . T is closed under finite products. We may assume that neither /, g : A*B •+• C , f * g . 1 •+ XXB—•-• C + 1 •+ i4*B -^*" C . /, g , and let 1 -2+ A Because there exists B f 0 be such that We wish to find Let f,g A 1 •+ A*B nor B such : A •* CT correspond to 1 — • 4 - ^ if * 1 • 4 - ^ C^ . 1 -• B , and hence B •* 1 is epi, which yields: By following with the evaluation Cpx-B •* C we obtain B •* A*B -£-*• C * B •* A*B -2-- C . Finally, l e t C . D 1 -*• B be such as to separate these two maps from B t o By a two-valued topos we mean ( l , fi) = 2 . 48 Peter Freyd PROPOSITION 3.42 well-pointed topos. f o r solvable two-valued topoi. Subobjects of T objects are in P T is a T . hence T ->• T P V is exact. Proof. Because T is closed under products and coreflective, exponentiation is effected by exponentiating in Given f : A •* B , collection of B (. T T/B-objects , , consider {l •* B} s ubt ermi nat or s to A only two subobjects, Given in T . A «• B in T there exists either 3.5. T/A . let A' •* B Hence j The applied to that (Thus in any topos, the maps from be the negation of A 1 has as defined A u A' = B , for such implies that T . 1 + 1 It suffices to show that for any l*^->-5 = l + S (i4'ulm(l -»• B)) n A = 0 contradiction. : T/B •* T/A . A € T satisfies the ft condition in 1 -* B f form a jointly epimorphic family. ) Because We wish to show that neither, then and then coreflecting. are such that their maps to the terminator form a jointly epimorphic family. family yields a similar family in T or 1 + A'-+B=1-+B. and the maximality of A' If yields a • Topoi exactly embeddable in well-pointed topoi We have shown t h a t every topos i s exactly embeddable in a product of well-pointed t o p o i , and a residual question presents i t s e l f : which are exactly embeddable in a single well-pointed topos? Because of the elementary nature of the l a t t e r (they are closed under ultra-products) , we are asking which topoi have the universal exactness theory of well-pointed t o p o i . For example: (U H- l ) =» (0 •** U) v {U ->->• l ) . That i s , such topoi must be such that ( l , ft) = 2 . I t suffices to show for each n f X'Bl. B „ f 2'g2. B and A f n* i = 1, 2, . . . , n t h a t there i s a well-pointed topos T' and exact T : T •* T' such that Aspects of topoi (/"„•) * T(3J) » i = 1, 2, . . . , n . and E. c A E. = 4. x ... x A . . x E\ x A. f.,g. • Let E = E I T : 1 •* T , T T[fi) * T{9i) . (Ultra-products again). the equalizer of ... * A u ... u E . If where E + A Let A * A. E. is the equalizer of we know that there exists 71 J 1 49 well-pointed, such that T{E) * T(A) and hence It is a statement in the universal exactness theory of well-pointed topoi that if each E'. t A . then E # A . Hence such is a necessary Lr If condition for exact embeddability into a well-pointed topos. it elementary by noticing that the case for arbitrary case follows from the n = 2 : If (A'*B) u (AXB') = A x B then either A' = A Further reductions can occur by replacing A' of n We can make A or with B' = B . A/A' , the pushout •* A + 1 If (l*B) u (Axi) = A x B Together with (l,fi)= 2 then either A = 1 or B = 1 . this can be seen to be sufficient. Finally, if one considers l/T , that is, the category whose objects ^A are of the form 1 •* A and whose maps are of the form 1 + , define ^ (1 •+ A) v (l •* B) cokernel of as the coproduct in l/T , (l •* A) v (l •* B) •* (l •*• A*B) , B (l •+ A) A (l •+ B) (-A- as the has a right-adjoint, namely exponentiation), then the condition for exact embeddability into a well-pointed topos is that the half-ring of isomorphism types with addition and A Let L n-ary The first order calculus of a topos be a vocabulary of predicates and operators; L as as multiplication is without zero-divisors. 4. objects of v are either pairs predicate, / an w-ary <P, n) or operator. <f, n) that i s , the where we call P an 50 Peter Freyd An interpretation P a Bn subobject of L for each in a topos T is an object (P, n> i I , a map B € j , a J : B™ ->• B for each < / , n> 6 L . Given any derived L n-ary predicate or operator using the vocabulary of and the classical logical connections and quantifiers, we wish to stipulate a subobject of Bn . The definition is recursive. defining maps from expressions in the two n-ary predicates easily define PTQ Given an each P, Q , L-operators are well known. P v Q , P -* Q m-ary, we define Bn . as a subobject in P[x-., •••, x ) and operators is the obvious. Given an 3 where Bn P+ tT -»• B f , ..., f • + if as the pullback + •)- P(f , . . . , f ) Given Sub(s n ) , we can already assigned values in n-ary predicate The rules for n-ary P[x., ... , x ) modeled as P[x , . . . , x ) as the image of P c B P ->• B™ •+ Bn~ , we define . For V we need: n PROPOSITION 4.11 exists a maximal Proof. S'+B We define for topoi. B' c B V is Given such that g l(i)'-»j(). 9 P{x , ... , x j n inverse image is contained in g : A •* B (B' ) c A' . A' c A , there B' is a model of T whose P . B S (that is, no , that is, an element of We shall call such the truth value of the sentence, T B1 as the maximal subobject in unquantified variables) a subobject of Given any set is called • In this manner we obtain for each sentence (1, fl) . and Such t(5) . of sentences, we say that an interpretation of if every sentence in T has truth value 1 • £2 . L Aspects of topoi (Yes, that's why it's called T we say that S^ implies of 5 2 , and denote same by S- in T same by S3. K S~ S Given sentences in T ^ . Note that S^ L S 1= Sp . ^-r S We say that if for all topoi L , S., S^ T , S H=y S S. is a model t(s,) - £ (<S2) » and denote 1 Hj- (s we say that . and a topos S-. strongly implies is equivalent to and sentences S,, Sp if every model of if for every interpretation of Given any language implies "t" .) 51 S •* S ) . semantiaally We denote same by 52 • PROPOSITION 4.12. If S 1 !=„ S 2 Proof. B (. J Let t[s ) ^ t(s 2 ) . model of S Then in . then for all T, be an interpretation of T/i(S ) L we obtain a model of SX^TS2. in T and suppose S that is not a G The definition of semantic implication reduces a host of assertions in intuitionistic logic to exercises in classical logic: PROPOSITION 1=% T 4.13. For fixed L h* 3 obeys Craig's interpolation theorem. If every finite subset of a given has a model in some topos, then so does For each monoid is recursively enumerable. T . D M , we obtain an intuitionistic logic H „ . We suspect that the connection between such logics and classes of monoids will be a fruitful pursuit. Problem: For every topos H" ,. for some monoid 4.2. T is \=j the same partial ordering as M 1 The boolean case If we restrict our interpretation of a language to boolean topoi, we can replace that v, A, 1 P •*• Q and with 3 ~1 P v Q and V with 1 3 1 . The advantage is are all definable using only the predicates of 52 Peter Freyd near-exactness. (p = "I Q i f f P + Q •* EF i s an isomorphism.) More p r e c i s e l y , l e t L = {<P 1 , n>, . . . , <P a , n a >, < / 1 , jn1> Let Jg be the set of interpretations of elements of Jc o L </ & , m^ >} . in a boolean topos are of the form {B, P 1 <= B X, . . . , Pa c B a, f± : B X - S ^ : Bb + B) . PROPOSITION 4 . 2 1 . Given any elementary sentence universally quantified Horn formula auah that for each boolean F{B,P~X, ...,Jb) B and (B, P , .... iff there is an exietentially predicates G{B, P , ..., <B,P±, S F in the predicates D there exists of f, ) i J_ 1 that 8 . The a near-exactness it is the case D .. . , Jb > is a model of S , Also, quantified conjunction of near-exactness f^) with the same property. Proof. There is a tree. Its root is 5 , each branch a wff (well formed formula), each leaf a single variable, each branch point (we'll allow degenerate branch points) marked with either P , . . . , P , / - , . . . , / " , , A, "1 , = or < 3, n > ; and, if a branch point is marked P. expressions then g,, n. branches lead into i t , they are all operator ... , g and the branch leading out is. i P.\g , . .., g I ; if the branch point is marked lead in, they are all operator expressions /. then g.s ..., g m. branches and the branch i leading out is f.\g,, ..., g I ; if the branch point is marked two branches lead in, both are predicates out is P A Q ; P, Q if the branch point is marked in, it^ is a predicate P and the branch out is g = h ; ~l then one branch leads "IP; if it is marked g, h , and the branch finally if the branch point is marked then one branch leads in, it is a predicate then and the branch leading = , two branches lead in, both operator expressions leading out is A P{x , ..., x^) < 3 , n> and the branch Aspects of topoi leading out is Let of F JC a P K and 3 P[x. , . . . , x ) . n be the number of variables in G 53 are defined as follows: S . The quantified variables for each variable in 5 , a map i •S ; •* B g xB for each branch other than a leaf, we introduce a variable if the branch is an operator expression, a variable the branch is a predicate. new variable To each leaf marked x. P •+ B if we make correspond the p. . For each branch point we define a near-exactness predicate as follows: If a branch point is marked P. then let g , ..., g variables corresponding to the incoming branches; and let A be the P •*• b to the outgoing, say that n _ • P. •* B v is a pullback. If a branch point is marked /. , then let g^ , ..., g be the 1 variables corresponding to the incoming branches; outgoing and let A g :B •*• B say that ^- BZ g = ET If a branch point is marked A , then let ^* B . P •* a , variables corresponding to the incoming branches : R •* a and let A to the say that JC R + a is a pullback of _ ^ + . Q •* B^ be the to the outgoing 54 Peter Freyd If a branch point is marked incoming branch; R •* B ~1 then let P •* IT correspond to the is the outgoing, and let A say that _ 4- is a coproduct. If a branch point i s marked = , then l e t correspond to the incoming branches, say t h a t P •*• B P •* a i s an equalizer of If a branch point i s marked t h e incoming branch, Q •*• a g : a -*• B , ~h : ET •*• B t o the outgoing, and l e t A g, H . < 3 , n ) , then l e t P •*• B t o the outgoing, and l e t A correspond to say t h a t is a pullback. Finally, let S •*• a be the variable assigned to the root. F = V [ A / •• (S = / ) ] . G = 2{/\A A (S = / ) ] . COROLLARY boolean 4.22. If a theory has a model in any non-degenerate topos, it has a model in sets. Proof. ?y Theorem 3.23 there always exists a near-exact functor into sets, which functor must preserve COROLLARY Proof. G . 4.23. For any boolean Suppose that an interpretation of 5 D L D B if S^ l=s S2 then S^ £ g 5 2 . S h o S^ but not S1 ^ g S 2 . Then there exists in B/t (s ) that is a model for S^ but not . Let t(S 2 ) be the truth value for Sp , and reflect the interpretation into becomes an iso. Corollary U.22. Sh . for j 3 the closure operator such that We then have a model in Sh . of S • A ~1 S 0 -»• t{S^\ . Then apply 55 Aspects of topoi Of course H ^ is classical logic. In a later section we find necessary and sufficient conditions on 8 so that 5. A pair 1 —^* N , short, if for every NNO's 1 Arithmetic i n topoi N -^-»- K i s a natural x •A N=g coincides with t •A numbers object, t h e r e e x i s t s unique or N -*• A NNO for such t h a t a r e c l e a r l y unique up t o unique isomorphisms. PROPOSITION 5 . 1 1 . If 1 -2+ N - ^ N is an NNO then 1° : 1+N ^ N is an isomorphism. Proof. Define 1 L e t w . : 1 •+ 1+N a n d w_ : N -*• 1+N b e t h e c o p r o j e c t i o n s . e ' : 1+N •> 1+N b y us' •N og = u = CM , u s ' = ew *• N i s an NNO , t h e r e i s a u n i q u e a n d sg = gs' . We c l a i m t h a t g . Since g : N •* 1+N such t h a t i s t h e inverse of : 1+N - N . That of g\A V j =1 follows from the uniqueness clause in the definition NNO applied t o the commutative diagram 8 N 3 1+N 1+N o 8 Peter Freyd 56 That \ \9 = 1 is equivalent to og = u and eg = u . The first we have, and the second comes by applying the uniqueness clause to the commutative diagrams N N 1+N -T+ N 1+N 1+N -jr- 1+N . O 8 PROPOSITION aoequalizer of Proof. if l N is an UNO then N •* 1 is a s and !„ . N •*• 1 to show that if that 5.12. is epic (because there exists N -^-* N -*-+• A = N -^-* A N -£+ A = N •* 1 -»• A . Define 1 •*• N ) and it suffices then there exists 1 ~^* A = 1 - ^ N -£+ A . N 1 -*• A such Then both N and N -2- N A -r* A and the uniqueness condition on KNO's yields N -"*-»• <4 = TV ->- 1 -^ A . We shall show that the exactness conditions of the last two propositions characterize NHO's in topoi. In the last section we observed that near-exactness conditions are equivalent to stating that something is a model for a finite theory. Here we will see that the addition of a single coequalizer condition can yield a categorical (in the old sense) definition. C Aspects of topoi 57 We consider first the category of sets: PROPOSITION coequalizer of 5.13 s for sets. and A -s* A If 1. , then either rl A - N , s = s . is monio and A - Z , A •*• 1 s{a) = a + 1 is a or Yt D Hence, if we add the requirement that s is not epic, we can characterize the natural numbers. Let is an T be a well-pointed topos, NNO in T , then Ts 1 * 2W T *S • TN exact. If is the standard 1 NNO •N in •N S . (Any exact functor from a well-pointed topos to a non-degenerate topos is faithful.) Suppose then that in no natural n does can't exist. there exists 1 —°->- N -^* N —^+ We obtain such ultra-power of T T 1 —^* N . . . -^* N = 1 -^+ A? . pointed topoi.) Then T T' and is faithful. T : T -+T' (l + 1 suppose preserves monomorphisms if it is faithful. 1 - ^ N-^ N is an C o e q m i J , 2"(s) Proposition 5.13. NNO T and = TlCoeqfl^, s) Thus for T : T •*• S . PROPOSITION in 5.14 T T(l) = 1 , 7(1) = 1 , , then is as described in (the scarcity of right-exact functors). S , then for every example, right exact T ), it is the case that T : T -*• S If T is such that C o e q m i f f ) , T(s)\ - r(Coeq(l, s)) , (for Suppose for some View 5 , there We can go one further step: and and define r(l) + T(l) = T(l+l) , 1 •+ TN + TN T(l) + T(l) - 2"(1+1) r(l) k 0 . Hence if a non-principal ultra-power of a non-principal ultra-power of 1 * T(l) For Because non-zero objects in well-pointed topoi are T Proof. T . is a cogenerator for well- injective, 1 * 4 , 3? just simply by taking a non-principal T , non-degenerate 17(1+1) = 1 + 1 . is no such Then S . In fact, we are using very little of the exactness of well-pointed such that for T 4 € T , T = 0 . TA \ 0 . Then because there exists as a functor with values in T' = T •* S/T(l) * S/l . T' S/T(l) , choose may be alternatively 58 Peter Freyd described by the pullbacks colimits as T , and exist. Nor can T . T' (B) •*• T{B) + + . 1 •* r(l) T' (l) = 1 . • T' preserves at least the same Hence by our remarks above, T' can not In particular, Theorem 3- 2U can not be improved to make T exact. Moreover, we have laid to rest the idea that the exact embedding theorem for abelian categories has a nice generalization to "base" categories other than abelian. For: COROLLARY 5.15. No set of elementary conditions true for the category of sets implies exact (even right-exact) embeddability into the category of sets. (Even if you add countability.) Proof. We can take the complete elementary theory of S , and let T be an elementary submodel of a non-principal ultra-power of S and apply Proposition 5.11*O The embedding theorem for abelian categories was motivated by the consequent metatheorem for the universal theory of exactness. The latter can be true without the former. But, alas, not for topoi. We need, f i r s t , a b i t more about NNO's . 5.2. Primitive recursive functions in topoi PROPOSITION for every 5.21 for topoi. If 1 - ^ N -2->- N is an A —^->- B and B —• B there exists unique A Proof. Transfer the problem to solving for NNO , then A*N •+• B such that 59 Aspects of topoi N ——• N D For the standard natural numbers in g : A •*• B , h : Axflx-B -*• B S we know that given there exists unique / : A*N •* B such that f(a, 0) = g(a) , f(a, i/+l) = h{a, y, f(a, y)) . (Usually A is a power of PROPOSITION g : A •* B , 5.22 N , B = N .) for topoi. Let 1 -^* N -2+ N h : A*-N*B -*• B given. be a There exists unique NNO , f : A*N •* B such that A •+ A * l A*N -J-* B = g , A*N Proof. k (Notation: B = for any P±,P2,f X , X ~^+ N B . means X •*• 1 -2-»- N .) be such that k . (5-231) as insured by Proposition 5.22. lt p28, h>px = (lxe)kp1 We will show that and kp~ works. < 1, o ) ^ = 1 , hence First: Let 60 Peter Freyd AxN p works as well as kp. and Proposition 5-22 says, therefore, that Second: kp2s = k(p±, p2s; h>p2 = {l*s)kp2 and < 1 , o)kp2 = o hence lxs . N •- N s and kp2 p2 vrorks as well as fcp and again Proposition 5.22 says t h a t = p2 . F i n a l l y , for the existence of k = < px, p2, /> / , define / - kp. . Then and < 1, o >/ = < 1, o >kp3 = < 1, o, g >p3 = g , (1*8 )f = (l><s)fcp3 = k<pv For the uniqueness, suppose Define 5.22. h)p3 = < P l , p2, f)h . is as described in the proposition. k = <p , p , /> , verify that (5.231) commutes and use Proposition D Note that given f p2s, h need not "depend" on A or N , or B . That is, h : A*B •* B we could define fc1 : <4XWXB •+ B = AxN*B •* A*B >• B and apply the proposition. Thus we can define on any NNO in a topos a, m, e : N'X-N -*• N by Aspects of topoi 61 N = N*N - ^ N - ^ N , = o = o s , tf xtf - ^ + ffxtf -^+ N = N*N In the category of sets exponentiation; a, m, e — •- N*N -^* N . are addition, multiplication and that is, x + 0 = x , x + (t/+l) = (*+i/) + 1 , x • 0 = 0 , x • (j/+l) = (x'jy) + x , x° = 1 , Take any elementary sentence x(i/+l) = ( * » ) • * . 5 in the operators Add to it the six equations above which define o, s, a,tfi,e . a, m, e on N . We saw in Proposition It.21 that there is a universal Horn exactness predicate that says that a given <N, o, 3, a, m, e ) is a model of O formula to include [ : 1 + N - N and 5 . Coeq(l, s) = 1 . Enlarge that The universal quantification of that formula then says that the arithmetic of the satisfies S . THEOREM conditions T NKO Hence, 5.23. FOP any recursively enumerable set of elementary T , true for the category of sets, there exists a model T of and a universal Horn sentence in the predicates of exactness, true for sets but false for Proof. T . Add to T the axioms of a well pointed topos with an NKO . By Godel's Incompleteness Theorem we know that there is an elementary sentence 5 true for standard arithmetic, whose translation, S' , into a universal Horn sentence in the predicates of exactness is not a consequence of T (else number theory would be decidable). Godel's Completeness Theorem implies that there is a model of T u D S1 } . 5.3. D The exact characterization of the natural numbers PROPOSITION 5.31 for topoi. If a topos has NNO then 62 Peter Freyd 1 -2+ N -£+ N is an NNO Coeq(l, s) = 1 . Proof. iff : 1 + N •* N P is iso and D Let 1 -^-* A -£•*• 4 be such that I* is iso i V.*J Coeq(l, t) = 1 . 1 - ^ ff - 1 * if Let be NNO and let N -^* N Q, • A A t commute. We wish to show First, f is epi. theory, namely that imply / is epi. Accordingly let to show that We have a universal Horn sentence in exactness isomorphs and Coeq.(l, a) = Coeq(l, t) = 1 It suffices to prove it in well pointed topoi. A' = Im(/) and A" the complement of t{A") c A" , for such allows us to "split" A' * A , then A' . It suffices t as t\A' + t\A" Coeq(l, t) = Coeq(l.,, t\A') + Coeq(l „, t\A") . A" # 0 suffices to show Let an isomorphism. [° , I* and obtain a splitting If f and Coeq.(l, t) is bigger than 1 . Hence it t(A") c A" . t' = t\A' , 1 -±—+ A' = 1 - ^ N •* A' . Then universal Horn sentence: " I f A' + A" - A r, I5 ; is iso. The a n d I , i I, I.I i s o t h e n V- ) \t) t(A") c A" " is in the predicates of near-exactness and it suffices to prove it in S , an easy matter. Second, / is mono. 1 can show t h a t o+ N Using just that •1 A ~T + and and I are isomorphs we N -£+ A S+ N +t A are pullbacks since such are y* sentences in the universal Horn theory of near-exactness and it suffices to prove them in S . Aspects of topoi Let Q >* A 63 be as described in Proposition 2.21, that is A i s a pullback, and for a l l X -*• A such that X + Q+A=X +A . there exists 1 •+• Q •* A = 1 — ^ + A . + -A A. + + i s a pullback, // there exists + N • A Then b e c a u s e o+ +x N -* A i s a pullback, Because Q -^ Q 4+ is a pullback, there exists a N >• A It *• A Q -&•*• Q t Q -**-•• Q + A = Q •* A 1 Q + . + "~~~~> N that such that Q^+ •A . Because Because 1 • It Q + N + i s a p u l l b a c k we o b t a i n • A >• If i s a NNO we o b t a i n It •*• Q such •• N N -*• Q •*• N = 1 Hence so is / . . That i s Q •*• N i s e p i and Q •*• A i s an i s o m o r p h . D Thus, exact functors between topoi with NNO's preserve NNO's . Hence, THEOREM with 5 . 3 2 . If NNO'e , then Proof. 1*1' the elementary ie exact for 1,1' well-pointed of 1, 1' Combine Propositions it.21 and 5.31. D We can push a bit further. arithmetics topoi coincide. By the existential second-order arithmetic of a topos , we mean the second-order sentences in arithmetic in which a l l second-order quantifiers are existential. that the truth of such a sentence in sentence in the exactness theory of PROPOSITION well-pointed 5.33. Propositions 1+.21 and 5-31 say T is equivalent to an existential T . The existential topos is determined by its Hence, second-order arithmetic of a universal Horn theory of 64 Peter Freyd exactness. 5.4. O Inferring the axiom of infinity PROPOSITION A' 5.41 for topoi. Given 1 -*-A -^+A there exists V- A and xy A' -£-+ A1 i A t. A A -J-"A ,[ : 1 + A' -*• A' is epi. Before proving Proposition 5.1*1 we show its consequences: PROPOSITION and 5.42 (the Peano property). Coeq(a;, t) = 1 t(A' ) c A' Proof. i t is then for every the case that A' c A If [x. : l + A •* A Xt) such, that is iso Im(a;) c A' , A' = A . The sentence i s in the universal theory of exactness and i t suffices to prove i t in well-pointed topoi. By applying Proposition 5-^1 x' t' (x1) to 1 • A' • A' we can assume t h a t I , , I is epi. (x) Let A" a A be the complement of A' . Using just that is i sio \v) and K , I is epi we can show that t{A") c A" in S , hence everywhere. I* ) Thus t splits as t' + t" (t" = t\A") and Coeq(l, t) = Coeq(l^ ( , t'} + Coeq(l^,,s t") . If A" f 0 then is bigger than THEOREM 1 . Proof. 5.11, 5.12. • 5.43 for topoi. : 1 +N + N Coeq(l, t) 1 -s- N-$->• N is an NNO iff and Coeq(l, s ) = 1 . The necessity of the exactness condition was Propositions For the sufficiency, note first that the Peano property above yields the uniqueness conditions, that is, if both /, g : N •* A were such 65 Aspects of topoi that A •A t then the equalizer of f, g would satisfy the hypotheses of Proposition 5.k2 and hence would be all of N . For the existential condition, let epi , maps A •* N , A •* B . Again, the Peano property says 1 —**-»• B •B It suffices to show that A -*• N is epi. A -*• N is iso. We need: 5.431. If [tXJ : 1 + A — • A is epi, \ .) LEMMA be given, and apply [° | : 1 + N ->• N iso, Xs) Coeq(l, s) = 1 j and f f, N —• N s then A ^s mono. Proof. The sentence is in the universal Horn theory of exactness and it suffices to prove it in well pointed topoi. kernel-pair of that is, N' . We need Note that to / , E' = E - A , E' - 0 , equivalently Let E c A*A N' = Im(E' * A •+ N) . N' = 0 . Let N" be the We need E = A , be the complement of N" = N . 1 -^+ N factors through iff g !i factors through A . We may verify that 1 N does, then so does 1 ~^->- N . hypotheses of Proposition 5.U2 and N" That is, N" = N . • N" has a unique lifting //' and that if satisfies the 66 Peter Freyd THEOREM 5.44 for topoi. The following are equivalent: (a) there exists an NNO ; (b) there exists- a monomorphism A >*• A and a map 1 -• 4 such that 0 -• 1 + A + is a pullback; •* A (a) there exists an isomorphism 1 + A -A . Proof. to obtain Clearly (a) =» (c) =» (b).' For (b) =» (a) apply Proposition 5.1*1 A'^-A' 1 ( ,) 0 f , : 1 + A' -* A' hence , c l;••} ) Let epic. It is clear that A' -*• C be a coequalizer of i ,/ • 4 ' -»• C . and t h a t C along +i' is a pullback, is monic, and an isomorphism. 1 -»• C = 1 N, • 1 + I .i l t 1 •+ C t o obtain : iV and define i4' ->- C1 as an object in View J 1 . , , t' 1 + 4 ' = A' »• N i n remains t r u e in T/l . If Note t h a t T/C . Pullback We maintain t h e coproduct conditions and gain the coequalizer condition. THEOREM 5.45 for topoi. T/C . • ( l , n) = 2 then either there exists an A A : A •* A is NNO or every mono-endo is auto and every epi-endo is auto. Proof. Suppose t h a t U mono and if + A A " f : A >+ A i s not e p i . X + A •*• A is a pullback where A 1 -»• A Then A j corresponds to Aspects of topoi A — • A , then 11 •* 1 can not be iso. 67 Hence 11=0 and we can apply the last theorem. If g : A •**• A A* : / - / . is not mono, we can repeat the argument for D The proof of Theorem 5.1*1 is fairly easy in solvable topoi (see 2.7). We can there take such that A' as the smallest subobject such that t(A' ) c A' and lm{x) c A' . Remarkable enough, even without solvability we can construct A' , not as an intersection but as a union. Proof of Proposition 5.41. We define B e !T to correspond to the family of subobjects of A such that US as the equalizer of the identity map and works. where Define B c ST Px"l : #4->-tf4 = c h a r a c t e r i s t i c map of For any B c fi^ i f f tf4->-l-*-£?4 1 A' c Jm(x) u t(A' ) and and show that 1 -»• fl4 corresponds t o A •* fi t h e •4 . A' c A , the corresponding map 4 ' c Im(ar) u t ( 4 ' ) . such remains the ease after 1 •*• u factors through Moreover: application by any logical morphism. Let C — • B*A be a pullback. C" For any 1 •+• B C c Im(x) u let •* C + + be a pullback. 1 •+ B Then C •*• C •*• A i s mono and t(C') and euch remains the case after application by any logical morphism. 68 Peter Freyd C' •* 1*4 The proof of this is obtained by noticing that + C + is a pullback, •* B*A hence C' is a pullback, where • 1M 1 —£»• Sr = 1 •+ B •* £r . But (/"xl)e = A -&-* Q the C ->• 4 map which corresponds to / . + Ij is a pullback. 1 -0 characteristic map of C' corresponds to a map through B proposition. Im(C -*• A) works as A' as demanded by the The reversal of the above paragraph shows that given any such that C' c Im(cc) u t{C' ) we can find C + •+ C I i s a pullback and 1 -»• B that which factors and C' c Im(x) u t(C) . We wish to show that C' >+ A 1 •+ u Thus the Im(C ->- /I) c Im (c •+ A C" •*• C -*• A = C' «• A . 1 •*• B such that We wish t o show, first, >• A) , equivalently t h a t in the pullback P •*• 1 + C 1+A P -*• C is epic. (This last "equivalently" is a pair of sentences in the universal Horn theory of near-exactness and may be verified in sets.) Suppose P •*• C is not epic. We may transport the entire situation to T/C P and obtain a map 1 -*• TC such that in the pullback 1 not epic. •* TP + + , P -* 1 is •* TC Because all our constructions are preserved by logical morphisms , we may drop the "T" and show that for any 1 -»• C and pullback Aspects of topoi 69 I+C 1+A 1 + C+ A , + 1 is epi. Let + 1 4- be a pullback. •* C + Then B P~ • 1+C 1+A 1 is a pullback. Let 4- 4- -+ C' + A be a pullback.. It suffices to show that P2 + 1+C P + P + 1 Let is epi. 4C + 4A be a pullback. P pullback. P- + 1 is epic. But pullbacks and we have shown that Call that is epic. + 1+C P 4- and Let C' c lm(x) u t(C') A' and 4- it is the case that C 4- Cc -*• A) . C' c A such hence But note that for Im(x) u t(C ) . Hence t(A' ) c A' , yielding we have be a are both + A = C' + C + A is the maximal such subobject. A' -£-* A' 4- AA C' = Im(x) u A' 41+C' Im(C + A) c Im(x) u Im(C + i4 = A' u t(A' ) it is the case that A u t(A' ) c A' 4- + 1+C' A' = Im(C + A) . We remarked above, that for any C' c A' . Thus C 4- P, •+ C' C' c Im(x) u t(C ) and 4- . For > A t. A I m U ) <= A' , yielding 70 Peter Freyd A' -•£-+ A' s 1 X I— Finally, 5.5. A' c Im(z) u t{A' ) K : 1 + A' ->•+ A' . / , g : A •*• B let Rf = Im(A ^ ? f ? > B*B) C flxg and c S X B the kernel-pair of the coequalizer of f, g . $y Corollary 3.32 s fi'r. is the smallest equivalence relation containing Suppose induced by B —*• C is epi. + B +C a pullback. f, g iff E. Then by Corollary 3-33, = E. . Hote that Rf and are defined using only near-exactness. B is B^ Let E-, c B*B be the equivalence relation h , that i s , + B - ^ - C i s a coequalizer of ff, Q One c o e q u a l i z e r f o r a l l Given £•„ directly yields • C ie a coequalizer the smallest equivalence of f , g : A + B iff relation containing relation containing preserves the i? c B'x-B define R . is epic and EV /?„ j In general, given h is Hi? to be the smallest equivalence Then a near-exact functor is exact iff it = operator on binary relations. Given any R c A*C we can define a transformation P •* Q R e l ( - , A) • R e l ( - , C) as follows: for 8 c W l e t + I be a R •* A pullback and send oR : rf4 •+ if . R c B*B define (if Q to Im(P •+ X*-C) . °i? i s natural and there exists R i s the graph of a map / , then H c fixfl as °R = 3 A u R u Im (i? •+• B*B —1-+ B*B) where .) Given T i s the Aspects of topoi twist map, and define k : N -*• H U B*B °R B*B • si be the map such that 71 as above. 1 In a topos with •N — • ft NKO let corresponds to B - ^ + Bx-B a n d N IV k I I r If °R We o b t a i n a r e l a t i o n from We may look a t N to B*B , Q c tfxgxs , Q backwards as a map from =/? to Q In w e l l - p o i n t e d t o p o i , a t l e a s t , we can d e f i n e t o t h e non-empty s u b o b j e c t s of l e a s t e l e m e n t s , and o b t a i n N and d e f i n e =R -*-•• N . A —>• =R + + i s a pullback and for 1- N 1 f (fl ) S' + + + , —>• N o . [U ) •*• N to to properties: >• =ff + 1 a; correspond t o correspond has t h e f o l l o w i n g S —»• 3? 1 Im(Q •+ B*B) = 3? . pullbaaks, • A? xs S' = (5o?) - S . These two properties, entirely in the language of near-exactness except for N itself, characterize PROPOSITION 5.51. If =R . Hence: j1 : T + T 1 i8 a near-exact functor of well-pointed topoi that preserves epimorphic families, then iff it preserves the coequalizer of 1^ and s . T is exact • Moreover given any diagram in a well-pointed topos, we can enlarge it and add 1 -^+ N •N and translate any coequalizer condition into near-exactness conditions on the enlarged diagram. We can know the universal Horn theory of exactness of T elementary theories have models in in which one of the stipulated sorts T if we know which two-sorted is the natural numbers. If T is a well-pointed topos with the axiom of choice, that is every object is projective, then by a Lowenheim-Skolem argument we can reduce the two-sorts to one and obtain the converse of Proposition 5-33: 72 Peter Freyd THEOREM 5.52. The universal Horn theory of exactness of a well-pointed topos with NNO and axiom of choice is determined by its existential second order arithmetic. 5.6. D A standard recovery PROPOSITION 5.61 for topoi. If T is such that for every R c B*B , =/? is the union of the sequence A, R, R , ..., through the standard natural numbers, then T Proof. may be exactly embedded in a power of S . By Theorem 3.2U we know that there is a collectively faithful family of non-exact functors into S , each of which preserves epimorphic families, hence unions. Hence the operation R i—• 3? is preserved and by our remarks in the last section, such functors are exact. COROLLARY 5.62. • Countably complete topoi may be exactly embedded in a power of S . Proof. Whenever A, R, R , ..., W1, ... has a union it is =R . D From Corollary 5-15 we thus obtain: PROPOSITION 5.63. No set of elementary properties true for the category of sets implies exact (even right-exact) embeddability into a product of countably complete topoi. By the standard maps from 1 to D N we mean those of the form 1 -*+ N -*+ N -*+ 11 -*+... -*+ N . We say that N is of standard generation if the standard maps form a collectively epimorphic family. equivalent to THEOREM 5.64. If T embedded in a power of S Proof. In a well-pointed topos, such is (l, N) having only standard maps. is a topos with NNO if then it may be exactly N is of standard generation. Theorem 5.23 says that there is a collectively faithful family of exact functors into well-pointed topoi each of which preserves epimorphic families, hence in each of which has NNO (l, N) is standard. (Each by Proposition 5.U2.) By our remarks in the last section, Hfl is the union of the values of Aspects of topoi 73 x a map N •*• U , thus a union of a sequence over the standard natural numbers, and Proposition 5-6l applies. If T is solvable then N D is well-pointed and standard generation is necessary. COROLLARY topos with WNO 5.65. If and every order arithmetic of T T is a solvable (for example, well-pointed) 1 -»• N is standard then the existential second is standard. D It seems to me that Corollary 5-65 provides something of a semantics for existential second-order arithmetic. 6. Problems Which categories can be exactly embedded in topoi? Which well-pointed topoi can be exactly embedded in well-pointed topoi with NNO ? Which of the latter can be exactly embedded in well-pointed topoi with the axiom of choice? Which of the latter can be exactly embedded in well-pointed topoi with AC and an axiom of replacement? M i tcheI I and Co Ie have independently shown that the latter are isomorphic to categories arising from models of Zermelo-Fraenkel. Hence we are asking for a metatheorem in which not the category of sets (what's that?) but a category of sets is the model. The answers to the last two questions would tell us which existential second order theories of arithmetic are compatible with Z-F . Indeed a good question is whether each first-order arithmetic compatible with the axioms of well-pointed topoi is compatible with Z-F . Using standard techniques transferred to well-pointed topoi we can see that each existential second order sentence in arithmetic is equivalent to one which asserts the existence of a single unary operator satisfies an equation involving x g, + , , - . g that If we know that each such sentence implies (using the axioms of topoi) that g is bounded by a first-order definable operator (for example, recursive) then the existential second order theory is determined by the first order theory. 74 Peter Freyd We should remark, in regard to the second question, that Theorem 5.23 remains true even if we replace the category of sets with the category of finite sets. The idea of the proof is to find a finite elementary theory such that the sentences true for all finite models are not recursively enumerable. An example of such is the theory of ordered partial rings, with enough axioms to insure that the finite models of such are finite intervals of the integers. The theory of diophantine equations is thus a subset of the theory of its finite models. Editor's note (l May 1972). The Editor is very grateful to Mr T.G. Brook and especially also to Professor G.M. Kelly for reading the proofs with care and amending several mathematical errors. Some last-minute (and later) changes are due to the author. Index of definitions Paragraph Page 36 Absolutely closed 2.6 Axiom of choice 5-5 71 Boolean algebra 1.2 10 Boolean topos 2.1* 28 Cartesian-closed 1.1 2 Closed subobjects 2.6 33 Closure operation 2.1* Closure operator l.U 29 17 Complement 1.2 10 Crible 2.1 19 Degenerate category 1.1 7 Dense subobject 2.6 Equivalence relation 3-3 33 1+6 Essentially algebraic 1.1 h Exact functor 2.6 33 Filter 1.2 12 Global closure operation 2.1* Heyting algebra 1.2 29 9 Aspects of topoi 75 Image 2.1 20 l*.l 51 k.l 50 Logical morphism 2.3 2k Model in k.l 50 Implies in T Interpretation T Natural numbers object 5.1 55 Near-exact 3.3 1*5 Negation 1.2 10 NNO • 5.1_ 55 2.2 21 Relation 2.2 22 Semantically implies l*.l 51 Separated object 2.6 33 2.6 33 P a r t i a l map • Sheaf . Singleton map 2.2 22 Solvable topos 2.7 1*0 Standard generation 5-6 72 Strongly implies It. 1 51 1.2 11 Topos Symmetric d e f i n i t i o n of Heyting algebras 2.1 18 Truth value lt.l 50 Two-valued topos 3.1* 1*7 Universal Horn theory 3.3 ^5 Well-pointed object 3. h 1*7 Well-pointed topos 3.1 1*2 References [/] J.C. Cole, "Categories of sets and models of set theory", Aarhus Universitet, Matematisk Institut, Preprint series, 52, 1971[2] Peter Freyd, "Several new concepts: lucid and concordant functors, pre-limits, pre-completeness, the continuous and concordant completions of categories", Category theory, homology theory and their applications III (Battelle Institute Conference, Seattle, Washington, 19-68. Lecture Notes in Mathematics, 99, 196-21*1. Springer-Verlag, Berlin, Heidelberg, New York, 1969). 76 Peter Freyd [3] John W. Gray, "The meeting of the Midwest Category Seminar in Zurich August 2U-30, 1970, Reports of the Midwest Category Seminar V (Lecture Notes in Mathematics, 195, 2U8-255. Springer-Verlag, Berlin, Heidelberg, New York, 1971). L41 F.W. Lawvere, "Quantifiers and sheaves", Aates du Congres International 329-33**. des Mathematiaiens, T.I N i c e , September 1970, (Gauthier-Villars, P a r i s , 1971). [5] F.W. Lawvere and M. Tierney, summarized in [3] above, Reports of the Midwest Category Seminar V (Lectures Notes in Mathematics, 195, 251-251*. Springer-Verlag, Berlin, Heidelberg, New York, 1971- [6] J.-L. Verdier, "Seminaire de geometrie algebrique", fascicule 1, Mimeographed Notes Inst. Bautes Etudes Sai. (1963/6I1). To be published in Theorie des topos et cohomologie Stale des schSmas (Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidelberg, New York), Department of Mathematics, University of Pennsylvania, Phi ladeiphia, Pennsy I vani a, USA.