Chu Representations of Categories Related to Constructive Mathematics

Home , Constructive set theory

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• In section 2 we present the basic of the Chu construction and the local Chu functor.

• In section 3 we present the global Chu functor that corresponds to the Chu construction.

• In section 4 we present the standard and classical boolean Chu representation of Top and the induced boolean Chu representation of the category of information systems Inf.

• In section 5 we present the constructive normal Chu representation of the category of Bishop spaces Bis. This representation of Bis is the constructive analogue of the aforementioned Chu representation of Top. The notion of a Bishop space is Bishop’s constructive, function-theoretic alternative to the classical, set-based notion of a topological space (see [20]-[22] and [25]-[29]).

• In section 6 and 7 we give the Chu representation of the category PpXq of subsets of a set X and of the category PKJpXq of complemented subsets of X, where X is a set equipped with an equality “X and an inequality ‰X , respectively. All set-theoretic notions mentioned here are within our reconstruction BST of Bishop’s set theory found in [6] and [8] (see [23] and, especially, [24]).

• In section 8 we introduce the generalised Chu category over a ccc C and an endofunctor Γ on C.

• In section 9 we deﬁne the generalised global Chu functor that corresponds to the generalised Chu construction.

• With the help of the generalised Chu construction we provide a generalised Chu representation of the categories of predicates Pred and of complemented predicates Pred‰ in sections 10 and 11, respectively.

• In section 12 we introduce the antiparallel Grothendieck construction over a product category and a contravariant Set-valued functor on it, which has the Chu construction as a special case, in case C is a ccc.

For all notions and results from category theory that are used here without explanation or proof we refer to [17], [4] and [31].

2 The Chu construction over a ccc C

Unless otherwise stated, throughout this paper C, D, E are ccc and γ P C0, δ P D0 are object of C and D, respectively. To show that the Chu construction in Deﬁnition 2.1 is category, one uses the fact the 1 1 product ˆ: C ˆ C Ñ C is a bifunctor (i.e., a functor). Moreover, if f : a Ñ a and g : b Ñ b in C1, then 1 1 1 1 2 1 1 2 f ˆ g : a ˆ b Ñ a ˆ b , such that 1a ˆ 1b “ 1aˆb, and if f : a Ñ a and g : b Ñ b in C1, then

pf 1 ˆ g1q ˝ pf ˆ gq “ pf 1 ˝ fq ˆ pg1 ˝ gq. (1)

1 2 1 If a “ a “ a and f “ f “ 1a, by equation (1) we get

1 1 1 p1a ˆ g q ˝ p1a ˆ gq “ p1a ˝ 1aq ˆ pg ˝ gq“ 1a ˆ pg ˝ gq. (2)

1 2 1 Similarly, if b “ b “ b and g “ g “ 1b, by equation (1) we get

1 1 1 pf ˆ 1bq ˝ pf ˆ 1bq “ pf ˝ fq ˆ p1b ˝ 1bq “ pf ˝ fqˆ 1b. (3)

If a, c, d, j P C0, φ: a Ñ c and θ : j Ñ d P C1, then

pφ ˆ 1dq ˝ p1a ˆ θq “ p1c ˆ θq ˝ pφ ˆ 1jq (4) 1A cartesian closed category C is a csms where its tensor product of C is its product and the tensor-unit is the terminal object of C. The category RelpSetq with objects sets and morhisms relations R Ď X ˆ Y is a csms that is not a ccc.

2 1aˆθ aˆj aˆd

φˆ1j φˆ1d

cˆj cˆd 1cˆθ

p1q p1c ˆ θq ˝ pφ ˆ 1jq “ p1c ˝ φq ˆ pθ ˝ 1jq “ φ ˆ θ

“ pφ ˝ 1aq ˆ p1d ˝ θq p1q “ pφ ˆ 1dq ˝ p1a ˆ θq.

Deﬁnition 2.1 (The Chu construction over a ccc C and some γ P C0). The Chu category ChupC, γq over C and γ has objects Chu spaces i.e., triplets pa, f, xq, with a, x P C0 and f : a ˆ x Ñ γ P C1. A morphism φ: pa, f, xq Ñ pb, g, yq in ChupC, γq, or a Chu transform, is a pair φ “ φ`, φ´ , where ` ´ φ : a Ñ b and φ : y Ñ x are in C1 such that the following diagram commutes ` ˘

´ 1aˆφ aˆy aˆx

` φ ˆ1y f

bˆy γ. g

` ´ ` ` ´ ´ If θ “ θ ,θ : pb, g, yq Ñ pc, h, zq, then θ ˝ φ “ θ ˝ φ , φ ˝ θ . Moreover, 1pa,f,xq “ p1a, 1xq. ` ˘ ` ˘ If C is bicomplete (complete and cocomplete), then ChupC, γq is also bicomplete (see [18], p. 41. The following result is standard (see also [1], p. 712).

Proposition 2.2 (The local Chu functor). The rule ChuC : C Ñ Cat, deﬁned by

C Chu0 pγq“ ChupC, γq, C Chu1 pu: γ Ñ δq“ u˚ : ChupC, γqÑ ChupC, δq,

pu˚q0pa, f, bq “ pa, u ˝ f, bq,

f u a ˆ b γ δ

` ´ ` ´ pu˚q1 φ , φ “ φ , φ , ` ˘ ` ˘ is a functor. Moreover, if u is a monomorphism, then u˚ is a full embedding.

Let Set be the ccc of sets and functions in Bishop’s sense2. If pA,f,Bq and pC,g,Dq are Chu spaces in ChupSet, Xq, for some given set X, and if pφ`, φ´q: pA,f,Bq Ñ pC,g,Dq, then the commutativity of the rectangle

´ idAˆφ A ˆ D A ˆ B

` φ ˆidD f

C ˆ Dg X

2One could have considered some other constructive approach to set theory, like Aczel’s constructive set theory in [3]. Most of the results presented here hold also for sets in a classical sense.

3 is written as f a, φ´pdq “ g φ`paq, d , for every a P A and d P D. In the next two deﬁnitions we follow [30] and` [13], respectively.˘ ` ˘

Deﬁnition 2.3. A Chu space pA,f,Bq in ChupSet, Xq is called separable, if f : A Ñ pB Ñ Xq, where

fpaq pbq“ fpa, bq, p “ ‰ for every a P A and b P B, is an injection. Ap Chu space pA,f,Bq in ChupSet, Xq is called extensional, if f : B Ñ pA Ñ Xq, where fpbq paq“ fpa, bq, q “ ‰ for every b P B and a P A, is an injection.q If pA,f,Bq is both separable and extensional, it is called biextensional. If B Ă XA and f : A ˆ B Ñ X is deﬁned by fpa, bq “ bpaq, then pA,f,Bq is called a normal Chu space. The Chu spaces in ChupSet, 2q are called Boolean.

Definition 2.4. If C is a category and γ P C0, the affine category AffpC, γq over C and γ has objects pairs pa, F q, where a P C0 and F Ď C1pa, γq “ Hompa, γq, and a morphism h: pa, F q Ñ pb, Gq in AffpC, γq is a morphism h: a Ñ b in C1 such that g ˝ h P F , for every g P G.

Next we ﬁx some basic terminology.

Deﬁnition 2.5. Let C, D be categories and F : C Ñ D a functor. F is an embedding, if it is injective on objects and faithful, and its is a representation, if it is a full embedding. If D is a Chu category and F is a representation, we call F a Chu representation. We call a Chu representation F strict, if F is injective on arrows. We call a Chu representation boolean pnormalq, if F0paq is a Boolean pnormalq Chu space, for every a P C0.

a All Chu representations included in this paper are going to be strict. If C is a ccc, let evγ,a : aˆγ Ñ a γ in C1 such that for every f : a ˆ b Ñ γ there is a unique f : b Ñ γ with f “ evγ,a ˝ 1a ˆ f . The next result is also standard, and its proof is constructive. The normal Chu representation` ˘ of Set p p through ESet,2 into ChupSet, 2q is classically the “same” to the boolean Chu representation of Set into ChupSet, 2q in section 4, which relies though, on the classical treatment of negation.

Proposition 2.6 (Chu representation of a ccc). The functor EC,γ : C Ñ ChupC, γq, deﬁned by

C,γ a E0 paq“ a, evγ,a, γ , ` ˘ C,γ ´ a b E1 pf : a Ñ bq “ pf,f q: a, evγ,a, γ Ñ b, evγ,b, γ , ´ b a ` ˘ ` ˘ f “ h: γ Ñ γ , h “ evγ,b ˝ f ˆ 1γb , ` ˘ p evγ,a aˆγa γ

evγ,b

b 1aˆh bˆγ p fˆ1γb

aˆγb is a strict Chu representation of C into ChupC, γq.

3 The global Chu functor

If a functor F : C Ñ D preserves products (i.e., binary product diagrams), then for every a, b P C0 there is a unique morphism Fab : F0paqˆ F0pbqÑ F0pa ˆ bq, which is an isomorphism

4 pra prb a a ˆ b b

F1ppraq F1pprbq F0paq F0pa ˆ bq F0pbq pr pr F0paq Fab F0pbq

F0paqˆ F0pbq.

1 1 1 1 For every a, a , b, b P C0 and every f : a Ñ a , g : b Ñ b in C1 the following rectangle commutes

F1pfˆgq F0paˆbq F0pa1ˆb1q

Fab Fa1b1

F0paqˆF0pbq F0pa1qˆF0pb1q. F1pfqˆF1pgq

If G: D Ñ E also preserves products and pGcdqc,dPD0 are the canonical isomorphisms Gcd : G0pcqˆ G0pdqÑ G0pc ˆ dq, then G ˝ F also preserves products and for every a, b P C0 we have that

pG ˝ F qab “ G1pFabq ˝ GF0paqF0pbq

G0pF0paqqˆG0pF0pbqq prG0pF0paqq

GF0paqF0pbq

pr G0pF0paqq pG˝F qab G0pF0paqˆF0pbqq

G1pFabq

G0pF0paqq G0pF0paˆbqq G0pF0pbqq G1pF1ppraqq G1pF1pprbqq

C The canonical isomorphisms of the identity functor Id on C is the family p1aˆbqa,bPC0 .

Lemma 3.1. Let F : C Ñ D be a product-preserving functor with pFabqa,bPC0 the canonical isomorphisms of F , and let φ: F0pγqÑ δ in D1. The rule F˚ : ChupC, γqÑ ChupD, δq, deﬁned by

pF˚q0pa, f, bq“ F0paq, φ ˝ F1pfq ˝ Fab, F0pbq ` ˘ Fab F1pfq φ F0paqˆ F0pbq F0pa ˆ bq F0pγq δ

` ´ pF˚q1 φ , φ : F0paq, φ ˝ F1pfq ˝ Fab, F0pbq Ñ F0pcq, φ ˝ F1pgq ˝ Fcd, F0pdq , ` ˘ ` ` ´ ˘` ` ´ ˘ pF˚q1 φ , φ “ F1pφ q, F1pφ q , where φ`, φ´ : pa, f, bq Ñ pc, g, dq , is` a functor.˘ ` ˘ ` ˘ ˘ ` ´ Proof. To show that F˚ is well-deﬁned, we show that pF˚q0 φ , φ : F0paq, φ ˝ F1pfq ˝ Fab, F0pbq Ñ F0pcq, φ ˝ F1pgq ˝ Fcd, F0pdq i.e., the following diagram commutes` ˘ ` ˘ ` ˘ ´ 1F0paqˆF1pφ q F0paqˆF0pdq F0paqˆF0pbq

` F1pφ qˆ1F0pdq φ˝F1pfq˝Fab

F0pcqˆF0pdq δ. φ˝F1pgq˝Fcd

5 By the commutativity of the following diagrams we have that

´ 1aˆφ aˆd aˆb

` φ ˆ1d f

cˆd γ g

` ´ F1pφ ˆ1dq F1p1aˆφ q F0paˆdq F0pcˆdq F0paˆdq F0paˆbq

Fad Fcd Fad Fab

F0paqˆF0pdq F0pcqˆF0pdq F0paqˆF0pdq F0paqˆF0pbq ` ´ F1pφ qˆF1p1dq F1p1aqˆF1pφ q

´ ´ φ ˝ F1pfq ˝ Fab ˝ r1F0paq ˆ F1pφ qs “ φ ˝ F1pfq ˝ F1p1a ˆ φ q ˝ Fad ` “ φ ˝ F1pgq ˝ F1pφ ˆ 1dq ˝ Fad ` “ φ ˝ F1pgq ˝ Fcd ˝ rF1pφ qˆ F1p1dqs ` “ φ ˝ F1pgq ˝ Fcd ˝ rF1pφ qˆ 1F0pdqs.

The preservation of the units and compositions by F˚ are immediate to show.

If η : F ñ G, we cannot define a natural transformation η˚ : F˚ ñ G˚ i.e., we cannot show ˆ that F ÞÑ F˚ is a functor on the category Fun pC, Dq of product-preserving functors from C to D. What we showed though, in the previous lemma is that the pair pF, φq generated the functor F˚ : ChupC, γqÑ ChupD, δq. Next we describe an instance of the (generalised) covariant Grothendieck construction that defines the category with respect to which pF, φq ÞÑ F˚ becomes a functor. Definition 3.2 (A covariant Grothendieck construction). Let ccCat be the category of cartesian closed categories with morphisms the product preserving functors3. The Grothendieck category

Groth ccCat, IdccCat ` ˘ over ccCat and the covariant identity functor IdccCat : ccCat Ñ Cat has objects pairs pC, γq, where C is a cartesian closed category and γ P Ob ccCat “ C . A morphism pF, φq: pC, γq Ñ pD, δq is a Id0 pCq 0 C D ccCat product-preserving functor F : Ñ and a morphism φ: Id1 pF q 0pγqÑ δ i.e., φ: F0pγqÑ δ. If C pG, θq: pD, δq Ñ pE, εq, then pG, θq ˝ pF, φq“ G ˝ F,θ ˝ G“1pφq . Moreover,‰ 1pC,γq “ Id , 1γ . ` ˘ ` ˘ Theorem 3.3 (The global Chu functor). The rule Chu: Groth ccCat, IdccCat Ñ Cat, deﬁned by ` ˘ Chu0pC, γq“ ChupC, γq,

Chu1 F, φq: pC, γq Ñ pD, δq : ChupC, γqÑ ChupD, δq, ` ˘ Chu1 F, φq“ F˚, ` where F˚ is deﬁned in Lemma 3.1, is a functor. Moreover, if F : C Ñ D is a full embedding and φ is a monomorphism, then F˚ is a full embedding of ChupC, γq into ChupD, δq.

Proof. By Lemma 3.1 Chu1pF, φq is well-deﬁned. Clearly,

Chu 1 Chu IdC, 1 IdC 1 . 1p pC,γqq“ 1 γ “ ˚ “ ChupC,γq ` ˘ “ ‰ 3One could have considered the cartesian closed functors i.e., the functors preserving the whole structure of a cartesian closed category, as morphisms of ccCat.

6 If pG, θq: pD, δq Ñ pE, εq, we show that pG ˝ F q˚ “ G˚ ˝ F˚. By deﬁnition pG, θq ˝ pF, φq“ G ˝ F,θ ˝

G1pφq , and by the equality shown for the canonical isomorphisms rpG ˝ F qabsa,bPC0 we get ` ˘ pG ˝ F q˚ 0pa, f, bq“ G0pF0paqq,θ ˝ G1pφq ˝ G1pF1pfqq ˝ pG ˝ F qab, G0pF0pbqq “ ‰ ` ˘ “ G0pF0paqq,θ ˝ G1pφq ˝ G1pF1pfqq ˝ G1pFabq ˝ GF0paqF0pbq, G0pF0pbqq ` ˘ “ G0pF0paqq,θ ˝ G1 φ ˝ F1pfq ˝ Fab ˝ GF0paqF0pbq, G0pF0pbqq ` “ ‰ ˘ “ pG˚q0 F0paq, φ ˝ F1pfq ˝ Fab, F0pbq

“ pG˚q0`pF˚q0pa, f, bq . ˘ ` ´ ` ˘ ` ´ The equality rpG ˝ F q˚s1pφ , φ q “ pG˚q1 pF˚q1pφ , φ q follows immediately. Let F : C Ñ D be a 1 full embedding and φ a monomorphism. The` equality F˘0paq, φ ˝ F1pfq ˝ Fab, F0pbq “ F0pa q, φ ˝ 1 1 1 1 F1pf q ˝ Fa1b1 , F0pb q implies a “ a , b “ b , and as φ is` a monomorphism and Fab an˘ isomorphism,` 1 1 hence an epimorphism,˘ we get F1pfq “ F1pf q, hence f “ f . The fact that F˚ is faithful and full follows immediately.

The local Chu functor is a special case of the global one. Namely, C C Chu1pId , u: γ Ñ δq“ u˚ “ Chu1 puq: ChupC, γqÑ ChupC, δq.

If F : C Ñ D, a left F -coalgebra is a triplet γ P C0, δ P D0, φ: F0pγq Ñ δ . If G: D Ñ C, a right G-coalgebra is a triplet γ P C0, δ P D0, φ: `γ Ñ G0pδq . If D “ C, a right˘F -coalgebra of the form γ P C0, γ P D0, φ: γ Ñ F`0pγq is traditionally called an˘F -coalgebra. The relation between Chu spaces` and coalgebras is studied by˘ Abramsky in [2].

4 Boolean Chu representations

The following Chu representation is standard. Recall that the category Top of topological spaces is not cartesian closed, and hence we cannot use Proposition 2.6 to represent it. Proposition 4.1 (Chu representation of Top). The functor ETop : Top Ñ ChupSet, 2q, deﬁned by Top E0 pX, T q “ pX, PX,T , T q,

PX,T : X ˆ T Ñ 2, 1 , x P G P px, Gq“ X,T " 0 , x R G, Top cnt Top ´ E1 f : pX, T q ÝÑ pY,Sq “ f, E1 pfq : pX, PX,T , T q Ñ pY, PY,S ,Sq, ` ´1 Top˘ ` ´ “ ‰ ˘ ´1 f “ E1 pfq : S Ñ T,U ÞÑ f pUq, is a strict Chu representation of Top“ into Chu‰ pSet, 2q.

Notice that although the proof of the previous proof is constructive, the definition of PX,T is classical. One can show classically that the Chu space pX, PX,T , T q is separable if and only if the topology T is T0. Clearly, pX, PX,T , T q is always extensional. The special properties of a topology T on a set X play no role in the above definitions i.e., this representation applies to more general categories. E.g., a classical Chu representation ESet : Set Ñ ChupSet, 2q is defined similarly by Set E0 pXq “ pX,fX , PpXqq,

fX : X ˆ PpXqÑ 2, 1 , x P A fX px, Aq“ " 0 , x R A, Set ´1 E1 f : X Ñ Y “ f,f . ` ˘ ` ˘ If we consider the full embedding ∆: Set Ñ Top, where ∆0pXq “ pX, PpXqq and ∆1pf : X Ñ Y q“ f, the following triangle commutes

7 ETop Top ChupSet, 2q.

∆ ESet Set

For all notions mentioned next we refer to [32], chapter 6. Recall that the Scott topology is Hausdorﬀ, only in a trivial case, and hence it is not completely regular.

Deﬁnition 4.2. Let Inf be the category of information systems pX, ConX , $X q together with morphisms r : pX, ConX , $X q Ñ pY, ConY , $Y q the approximable mappings i.e., appropriate relations r Ď

ConX ˆ Y . If s: pY, ConY , $Y q Ñ pZ, ConZ , $Z q, the composition s ˝ r is deﬁned by

Aps ˝ rqz :ô DBPConB ArB & Bsz . ` ˘

Moreover, 1pX,ConX ,$X q “ $X . Let |X| be the set of ideals of pX, ConX , $X q and SX the Scott topology on |X| that has the sets OA “ tJ P |X| | A Ď Ju, where A P ConX , as a base.

To show that $X pX, ConX , $X q Ñ pX, ConX , $X q we use the deﬁnition of an information system.

To show that 1pX,ConX ,$X q “ $X we use the deﬁnition of composition of approximable mappings. Proposition 4.3 (Chu represenation of Inf). The functor S : Inf Ñ Top, where

S0pX, ConX , $X q“ |X|,SX , ` ˘ S1 r : pX, ConX , $X q Ñ pY, ConY , $Y q “ |r|: |X| Ñ |Y |, ` ˘1 |r|pJq“ y P Y | DJ1ĎfinJ J ry , ` ˘( is a full embedding of Inf into Top. Consequently, ETop ˝ S : Inf Ñ ChupSet, 2q is a a strict Chu representation of Inf into ChupSet, 2q.

Proof. First we show that |$X |“ id|X|. If J P |X|, then

1 |$X |pJq“ x P X | DJ1ĎfinJ J $X x . ` ˘( 1 1 fin If x P|$X |pJq, then J $X x, for some J Ď J, hence x P J “ J. If x P J, then txu $X x, and hence x P|$X |pJq. The equality |r ˝ s| “ |r| ˝ |s| is straightforward to show. S is full, as if f : |X| Ñ |Y |, then f “ |rf |, where Arf y :ô y P f A . S is injective on arrows; if |r| “ |s|, then r “ r|r| “ r|s| “ s. To show that S is injective on` objects,˘ we suppose that |X|,SX “ |Y |,SX and we show that pX, ConX , $X q “ pY, ConY , $Y q. If x P X, then txu P |Y |,` hence t˘xu Ď` Y , and˘ consequently x P Y . Similarly, we get Y Ď X. If A P ConX , then

X A “ tx P X | A $X xu P |Y |.

fin X As A Ď A P |Y |, we get A P ConY . Similarly, we get ConY Ď ConX . If A $X x, then

Y A “ ty P Y | A $Y yu P |Y | “ |X|.

Y Y Hence, there is I P |X| such that I “ A . As A Ďfin I and I is deductively closed, we get a P A i.e.,

A $Y a. Similarly, we get $Y Ď $X .

As the category Inf is cartesian closed, then, according to Proposition 2.6, there is a normal Chu representation of Inf, which avoids classical reasoning.

8 5 Normal Chu representations

We have seen already the normal Chu representation of Set through ESet,2 into ChupSet, 2q. Next we present the normal Chu representation of the category of Bishop spaces. The notion of Bishop space is a constructive, function-theoretic alternative to the set-based notion of topological space, which was introduced by Bishop in [6], revived by Bridges in [9] and elaborated by the author in [20]-[22] and [25]-[29]. For the sake of completeness we give next all necessary deﬁnitions related to the proof of a strict Chu representation of the category of Bishop spaces.

Definition 5.1. If X is a set and R is the set of real numbers, we denote by FpXq the set of functions from X to R, by F˚pXq the bounded elements of FpXq, and by ConstpXq the subset of FpXq of all constant functions on X. If a P R, we denote by aX the constant function on X with value a. We denote by N` the set of non-zero natural numbers. A function φ : R Ñ R is called Bishop continuous, ` ` ` or simply continuous, if for every n P N there is a function ωφ,n : R Ñ R , ǫ ÞÑ ωφ,npǫq, which is called a modulus of continuity of φ on rń,ns, such that the following condition is satisfied

@x,yPr´n,nsp|x ´ y|ă ωφ,npǫq ñ |φpxq´ φpyq| ď ǫq, for every ǫ ą 0 and every n P N`. We denote by BicpRq the set of continuous functions from R to R, which is equipped with the pointwise equality inherited from FpRq.

Deﬁnition 5.2. If X is a set, f, g P FpXq, ǫ ą 0, and Φ Ď FpXq, let

UpX; g,f,ǫq :ô@xPX |gpxq´ fpxq| ď ǫ , ` ˘ UpX;Φ,fq :ô@ǫą0DgPΦ Upg,f,ǫq . ` ˘ If the set X is clear from the context, we write simply Upf,g,ǫq and UpΦ,fq, respectively. We denote by Φ˚ the bounded elements of Φ, and its uniform closure Φ is deﬁned by

Φ :“ tf P FpXq | UpΦ,fqu.

A Bishop topology on X is a certain subset of FpXq. As the Bishop topologies considered here are 4 F FpXq F all extensional subsets of pXq, we do not mention the embedding iF : F ãÑ pXq, which is given in all cases by the identity map-rule. The uniform closure Φ of Φ is an extensional subset of FpXq.

Deﬁnition 5.3. A Bishop space is a pair F :“ pX, F q, where F is an extensional subset of FpXq, which is called a Bishop topology, or a topology of functions on X, that satisﬁes the following conditions: X pBS1q If a P R, then a P F . pBS2q If f, g P F , then f ` g P F . pBS3q If f P F and φ P BicpRq, then φ ˝ f P F

f X R F Q φ ˝ f φ P BicpRq

R. pBS4q F “ F .

If F :“ pX, F q is a Bishop space, then F ˚ :“ pX, F ˚q is the Bishop space of bounded elements of F . The constant functions ConstpXq is the trivial topology on X, while FpXq is the discrete topology on X. Clearly, if F is a topology on X, then ConstpXqĎ F Ď FpXq, and the set of its bounded elements

4 If X is a set and P is an extensional property on X i.e., P pxq & x “X y ñ P pyq, the extensional subset XP of X is defined by separation, XP “ tx P X | P pxqu, its equality is inherited by that of X and the embedding of XP into X is defined by the identity rule (see [24], Definition 2.2.3).

9 F ˚ is also a topology on X. It is straightforward to see that the pair R :“ pR, BicpRqq is a Bishop space, which we call the Bishop space of reals. If X is a metric space, the set CppXq of all weakly continuous functions of type X Ñ R, as it is deﬁned in [8], p.76, is the set of pointwise continuous ones. It is easy to see that the pair WpXq “ pX,CppXqq is Bishop space. Bishop calls CppXq the weak topology on X, but here we avoid this term, since in [20] we use this term for the Bishop topology that corresponds to the weak topology of open sets, and we call CppXq the pointwise topology on X. If X is a compact metric space, the set CupXq of all uniformly continuous functions of type X Ñ R is a topology, called by Bishop the uniform topology on X. We call UpXq “ pX,CupXqq the uniform space. If X is a locally compact metric space, the set BicpXq of Bishop continuous functions from X to R i.e., uniformly continuous on every5 bounded subset of X, is a Bishop topology on X.

A Bishop topology F is a ring and a lattice; since |idR| P BicpRq, where idR is the identity function on R, by BS3 we get that if f P F then |f| P F . By BS2 and BS3, and using the following equalities

pf ` gq2 ´ f 2 ´ g2 f¨g “ P F, 2 f ` g ` |f ´ g| f _ g “ maxtf, gu“ P F, 2 f ` g ´ |f ´ g| f ^ g “ mintf, gu“ P F, 2 we get similarly that if f, g P F , then f¨g,f _ g,f ^ g P F . Turning the definitional clauses of a Bishop topology into inductive rules, Bishop defined in [6], p. 72, the least topology including a given subbase F0. This inductive definition, which is also found in [8], p. 78, is crucial to the definition of new Bishop topologies from given ones.

Deﬁnition 5.4. The category of Bishop spaces Bis is the subcategory of AﬀpSet, Rq with objects pairs pX, F q such that F Ď FpXq is a Bishop topology on X.

Consequently, if F :“ pX, F q and G “ pY, Gq are Bishop spaces, a function h : X Ñ Y is a morphism from F to G in Bis, which is called a Bishop morphism, if @gPGpg ˝ h P F q

h X Y F Q g ˝ h g P G

We denote by MorpF, Gq the set of Bishop morphisms from F to G. As F is an extensional subset of FpXq, MorpF, Gq is an extensional subset of FpX,Y q. Similarly to Top, the category Bis is not cartesian closed. The following Chu-representation of Bishop spaces is completely constructive, and its proof is equally simple to the proof of Proposition 4.1.

Proposition 5.5 (Chu representation of Bis). The functor EBis : Bis Ñ ChupSet, Rq, deﬁned by

Bis E0 pX, F q “ pX, evX,F , F q,

5 As in the case of BicpRq, it seems that this deﬁnition requires quantiﬁcation over the power set of X i.e.,

BicpXqpfqô@BPPpXqpboundedpBqñ f|B is uniformly continuousq.

A bounded subset B of an inhabited metric space X is a triplet pB,x0, Mq, where x0 P X, B Ď X, and M ą 0 is a 1 bound for B Y tx0u. To avoid such a quantiﬁcation, if x0 inhabits X, then for every bounded subset pB,x0 , Mq of

X we have that there is some n P N such that n ą 0 and B Ď rdx0 ď ns “ tx P X | dpx0,xq ď nu. If x P B, then 1 1 1 1 dpx,x0qď dpx,x0 q` dpx0 ,x0qď M ` dpx0 ,x0q, therefore x P rdx0 ď ns, for some n ą M ` dpx0 ,x0q. Hence,

Bic N pXqpfqô@nP pf|rdx0 ďns is uniformly continuousq, since rdx0 ď ns “ tx P dpx0,xqď nu is trivially a bounded subset of X.

10 evX,F : X ˆ F Ñ R,

evX,F px,fq“ fpxq, Bis ˚ E1 h: pX, F q ÝÑ pY, Gq “ h, h : pX, evX,F , F q Ñ pY, evY,G, Gq, ` ˘ ` ˘ h˚ : G Ñ F, h˚pgq“ g ˝ h, is a strict Chu representation of Bis into ChupSet, Rq.

˚ Proof. First we show that h, h : pX, evX,F , F q Ñ pY, evY,G, Gq i.e., the following rectangle commutes ` ˘ ˚ idX ˆh X ˆ G X ˆ F

hˆidG evX,F

Y ˆ G R evY,G

˚ evX,F pidX ˆ h qpx, gq “ evX,F x, g ˝ h ` ˘ “ gphpx`qq ˘

“ evY,G hpxq, g

“ evY,G`ph ˆ id˘Gqpx, gq . ` ˘ It is immediate to show that EBis is a functor, which is injective on objects and arrows. Next we show Bis ` ´ ` ´ that E is full. Let φ , φ : pX, evX,F , F q Ñ pY, evY,G, Gq i.e., φ : X Ñ Y and φ : Y Ñ X such that the following rectangle` commutes˘

´ idX ˆφ X ˆ G X ˆ F

` φ ˆidG evX,F

Y ˆ G R evY,G

´ ´ evX,F pidX ˆ φ qpx, gq “ evX,F x, φ pgq ` ˘ “ φ´pg`q pxqq ˘ “ g“ φ`px‰q ` “ ev`Y,G φ ˘pxq, g ` “ evY,G`pφ ˆ id˘Gqpx, gq . ` ˘ From the resulting equality F Q φ´pgq “ g ˝ φ`, and since g P G is arbitrary, we conclude that φ` P MorpF, Gq. By the same equality we also get φ´ “ φ` ˚, since, if g P G, we have that ` ˘ φ` ˚ gqspxq“ g φ`pxq “ φ´pgq pxqq. “` ˘ ` ` ˘ “ ‰ Bis ` ` ´ Hence, E1 φ q“ φ , φ . ` ` ˘ ˚ Bis ´ In [20] the mapping h “ E1 phq : G Ñ F is the ring homomorphism induced by h P MorpF, Gq. R Let the Chu space pX, evX,F ,“ F q, and‰ by Deﬁnition 2.3 let evX,F : X Ñ pF Ñ q with evX,F pxq “ x.

Consequently, the Chu space pX, evX,F , F q is separable if and only if F separates the points of X: z z p 1 1 x “FpF,Rq x :ô@fPF xpfq“R x pfq 1 `f x R f x ˘ p p ô@fPF pp q“ pp q 1 ô x “X`x . ˘

11 If evX,F : F Ñ pX Ñ Rq with evX,F pxq“ f, then pX, evX,F , F q is always extensional. Clearly, all these proofs concerning the Chu space pX, evX,F , F q are constructive. ~ ~ q As in the case of the classical Chu representation of Top, the Chu representation of Bis does not involve the special properties of a Bishop topology F and it can be applied to other categories too. The functor CTop : Top Ñ ChupSet, Rq deﬁned by Top C0 pX, T q “ pX, evX ,CpXqq,

evX : X ˆ CpXqÑ R,

evX px,fq“ fpxq, Top cnt ˚ C1 h: pX, T q ÝÑ pY,Sq “ h, h : pX, evX ,CpXqq Ñ pY, evY ,CpY qq, ` h˚ : CpY˘qÑ`CpXq˘, h˚pgq“ g ˝ h, is only an embedding of Top into ChupSet, Rq. To show that CTop is full, one needs to show that ` ´ ` if φ , φ : pX, evX ,CpXqq Ñ pY, evY ,CpY qq, then φ P CpX,Y q. What we can show only is that φ´`pgq “ ˘g ˝ φ` P CpXq, for every g P CpY q, something which does not imply, in general, that φ` P CpX,Y q. One can show that φ` P CpX,Y q, if Y is completely regular i.e., a Hausdorﬀ space Y such that every closed set F and a point y R F are separated by an element of CpY q. Let crTop be the full subcategory of completely regular topological spaces. It is not a coincidence that such a result holds (classically), as one can show classically that the canonical topology of open sets induced by some Bishop topology is completely regular. From the point of view of the theory of rings of continuous functions, the restriction to crTop is not a loss of generality, as for every topological space X there is a completely regular space ρX such that the ring CpXq is isomorphic to CpρXq. Actually, crTop is a reﬂective subcategory of Top (see [14] and [35]), as for every topological space pX, T q there is a completely regular space pρX, ρT q and a continuous surjection τX : X Ñ ρX such that for every completely regular space pY,Sq and continuous function f : X Ñ Y there is a unique continuous function ρf : ρX Ñ Y such that the following triangle commutes

τX X ρX f ρf

Y .

Proposition 5.6 (Chu representation of crTop). The functor EcrTop : crTop Ñ ChupSet, Rq, where crTop E0 pX, T q “ pX, evX ,CpXqq,

evX : X ˆ CpXqÑ R,

evX px,fq“ fpxq, crTop cnt ˚ E1 h: pX, T q ÝÑ pY,Sq “ h, h : pX, evX ,CpXqq Ñ pY, evY ,CpY qq, ` h˚ : CpY qÑ˘ C` pXq,˘ h˚pgq“ g ˝ h, is a strict representation of crTop into ChupSet, Rq.

Proof. It suffices to show that φ` P CpX,Y q. A Hausdorff space is completely regular if and only if the family ZpXq “ tζpfq | f P CpXqu, ζpfq “ tx P X | fpxq“ 0u, of zero sets of X is a base for the closed sets of X i.e., every closed set in X is the intersection of a family of zero sets of X (see [12], p. 38). As 1 φ` ´ ζpgq “ tx P X | φ`pxq P ζpfqu ` ˘ ` ˘ “ tx P X | gpφ`pxqq “ 0u “ ζpg ˝ φ`q, 1 and g ˝ φ` P CpXq, we conclude that φ` ´ ζpgq is closed in X, hence φ` is continuous. ` ˘ ` ˘ 12 If pX, T q is a topological space a subset C Ď CpXq determines the topology T , if the weak topology of C i.e., the smallest topology τpCq that turns all elements of C into continuous functions, is equal to T . If pX, T q is Hausdorff, then pX, T q is completely regular if and only if τpCpXqq “ T (see [12], p. 40). By the argument in the proof of Proposition 5.6 one shows (see [12], p. 40) that if C Ď CpY q with τpCq “ S, then a function φ` : pX, T q Ñ pY,Sq is continuous if and only if g ˝ φ` P CpXq, for every g P C. A generalisation of the proof of Proposition 5.5 follows next. Its proof is identical to the proof of Proposition 5.5.

Proposition 5.7 (Chu representation of AffpSet, Xq). If X is a set, the rule pA, F q ÞÑ pA, evA,F , F q defines a strict Chu representation of AffpSet, Xq into ChupSet, Xq.

6 A Chu representation of the category of subsets

Next we present the categorical in spirit notion of subset of a (Bishop) set. X Deﬁnition 6.1. Let pX, “X q be a set. A subset of X is a pair pA, iA q, where pA, “Aq is a set and X X X iA : A ãÑ X is an embedding (i.e., an injection) of A into X. If pA, iA q and pB, iB q are subsets of X, X X then A is a subset of B, in symbols pA, iA q Ď pB, iB q, or simpler A Ď B, if there is f : A Ñ B such that the following diagram commutes

f A B X X iA iB X.

X In this case we also write f : A Ď B. Usually we write A instead of pA, iA q. The totality of the subsets of X is the powerset PpXq of X, and it is equipped with the equality

X X pA, iA q“PpXq pB, iB q :ô A Ď B & B Ď A.

If f : A Ď B and g : B Ď A, we write pf, gq: A “PpXq B. The category PpXq of subsets of X has objects the subsets of X and morphisms functions f : A Ñ B as above.

Since the membership condition for PpXq requires quantiﬁcation over the open-ended totality V0 of predicative sets (see [24], chapter 2), the totality PpXq is a proper class. It is immediate to show that f : A Ď B is an embedding, and that the category PpXq is thin. X Proposition 6.2 (Chu-representation of PpXq). If pX, “X q is a set, the functor E : PpXq Ñ ChupSet, Xq, deﬁned by X X X 1 E0 A, iA “ A, IA , , X 1 ` ˘X ` X ˘ IA : A ˆ Ñ X, IA pa, 0q“ iA paq; a P A, X X X X 1 X 1 E1 f : A, iA Ñ B, iB “ pf, id1q: A, IA , Ñ B,IB , , is a strict Chu representation` ` of P˘ pXq` into Chu˘˘ pSet, Xq.` ˘ ` ˘

X X Proof. If f : A, iA Ñ B, iB , then by the commutativity of the following triangle we get the commutativity of` the following˘ ` rectangle˘

idAˆid1 A ˆ 1 A ˆ 1 A B X X fˆid1 X i i IA A B B ˆ 1 X X X IB

13 X X 1 X 1 X thus E1 pfq: A, IA , Ñ B,IB , . Clearly, E is a functor injective on objects and arrows, hence an embedding.` Moreover,˘ by` the commutativity˘ of the above rectangle we get the commutativity of X 1 X 1 the above triangle. Hence, if pf, id1q: A, IA , Ñ B,IB , in ChupSet, Xq, then f : A Ď B in PpXq, and hence E is full. ` ˘ ` ˘

The category of subsets of X and its Chu representation are generalised to a ccc C as follows.

Deﬁnition 6.3. The category SubpC, γq of subobjects of γ has objects monomorphisms of C with codomain γ and a morphism f : i Ñ j, where i: a ãÑ γ and j : b ãÑ γ is a a morphism f : a Ñ b such that the following triangle commutes

f a b i j γ.

It is immediate to show that f is a monomorphism and that SubpC, γq is thin.

Proposition 6.4 (Chu representation of SubpC, γq). The functor ESubpC,γq : SubpC, γqÑ ChupC, γq, deﬁned by SubpC,γq E0 i: a ãÑ γ “ a, i ˝ pra, 1 , ` ˘ ` ˘ pra i a ˆ 1 a x

SubpC,γq E1 f : i Ñ j “ pf, 11q: a, i ˝ pra, 1 Ñ b, j ˝ prb, 1 , ` ˘ ` ˘ ` ˘ is a strict Chu representation of SubpC, γq into ChupC, γq.

SubpC,γq Proof. The morphism pra is an iso, hence a mono. To show that E1 pfq: a, i ˝ pra, 1 Ñ b, j ˝ prb, 1 , we show that the following diagram commutes ` ˘ ` ˘ 1aˆ1 aˆ1 aˆ1

fˆ11 i˝pra

bˆ1 γ j˝prb

` i ˝ pra “ pj ˝ fq ˝ pra “ j ˝ pf ˝ praq“ j ˝ rprb ˝ pf ˆ 11qs “ pj ˝ prbq ˝ pf ˆ 11q, as the equality f ˝ pra “ prb ˝ pf ˆ 11q follows from the deﬁnition of f ˆ 11

a ˆ 1

pra pr1

a fˆ11 1

f 11 prb pr1 b b ˆ 1 1.

If a, i ˝ pra, 1 “ b, j ˝ prb, 1 , then a “ b, and i ˝ pra “ j ˝ pa. As pra is a mono, we get i “ j, and` hence ESub˘ pC,γ`q is injective˘ on objects. It is trivially injective on arrows. To show that it is full, ` ´ ´ let pφ , φ q: a, i ˝ pra, 1 Ñ b, j ˝ prb, 1 . Clearly, φ “ 11. By the previous equalities we get ` ` ` i ˝ pra “ pj ˝ φ` q ˝ pra, and˘ since` pra is a mono,˘ j ˝ φ “ i i.e., φ : i Ñ j in SubpC, γq.

14 7 A Chu representation of the category of complemented subsets

Definition 7.1. Let pX, “X q be a set. An inequality on X, or an apartness relation on X, is a relation x ‰X y such that the following conditions are satisfied: pAp1q @x,yPX x “X y & x ‰X y ñK . pAp2q @x,yPX`x ‰X y ñ y ‰X x . ˘ pAp3q @x,yPX`x ‰X y ñ@zPXpz ˘‰X x _ z ‰X yq . ` ˘ X We write pX, “X , ‰X q to denote the equality-inequality structure of a set X. If A, iA is a subset of X, the canonical inequality on A induced by ‰X is defined by ` ˘

1 X X 1 a ‰A a :ô iA paq‰X iA pa q,

1 for every a, a P A. If pY, “Y , ‰Y q is a set with inequality, a function f : X Ñ Y is called strongly 1 1 1 extensional, if fpxq‰Y fpx qñ x ‰X x , for every x,x P X.

Remark 7.2. An inequality relation x ‰X y is extensional on X ˆ X.

1 1 1 1 Proof. If x,y P X such that x ‰ y, and if x ,y P X such that x “X x and y “X y, we show that 1 1 1 1 x ‰ y . By pAp3q we get x ‰ x, which is excluded from pAp1q, or x ‰ y, which has to be the case. Hence, y1 ‰ x1, or y1 ‰ y. Since the last option is excluded similarly, we get y1 ‰ x1, hence x1 ‰ y1.

An inequality on a set X induces a positively deﬁned notion of disjointness of subsets of X.

X X Deﬁnition 7.3. Let pX, “X , ‰X q be a set, and pA, iA q, pB, iB qĎ X. We say that A and B are disjoint with respect to ‰X , in symbols AKJ‰X B, if

X X A KJ B :ô@aPA@bPB iA paq‰X iB pbq . ‰X ` ˘

If ‰X is clear from the context, we only write AKJX B or even AKJB.

Clearly, if AKJB, then A X B is not inhabited. The positive disjointness of subsets of X induces the notion of a complemented subset of X, and the negative notion of the complement of a set is avoided. We use bold letters to denote a complemented subset of a set. A 1 0 1 X Deﬁnition 7.4. A complemented subset of a set pX, “X , ‰X q is a pair :“ pA , A q, where pA , iA1 q 0 X 1 0 A 1 0 A and pA , iA0 q are subsets of X such that A KJA . If Domp q :“ A Y A is the domain of , the indicator function, or characteristic function, of A is the operation χA : DompAq ù 2 deﬁned by

1 , x P A1 χApxq :“ " 0 , x P A0.

Let x P A :ô x P A1 and x R A :ô x P A0. If A, B are complemented subsets of X, let

A Ď B :ô A1 Ď B1 & B0 Ď A0,

KJ A B A B B A Let P pXq be their totality, equipped with the equality “PKJpXq :ô Ď & Ď .

A B 1 1 0 0 1 1 0 Clearly, “PKJpXq ô A “PpXq B & A “PpXq B . Notice that if f1 : A Ď B and f0 : B Ď 0 1 1 1 A , then f1,f0 are strongly extensional functions. E.g., if f1pa1q ‰B1 f1pa1 q, for some a1, a1 P A , 1 X X 1 then from the deﬁnition of the canonical inequality ‰B this means that iB1 f1pa1q ‰X iB1 f1pa1 q . X X 1 1 1 By the extensionality of ‰X we get iA1 pa1q‰ iB1 pa1 q :ô a1 ‰A a1 . ` ˘ ` ˘

KJ Deﬁnition 7.5. If pX, “X , ‰X q is a set, the category P pXq has objects the complemented subsets of 1 1 0 0 X and a morphism f : A Ñ B is a pair f “ pf1,f0q: A Ď B i.e., f1 : A Ď B and f0 : B Ď A . The unit morphism 1A of A is the pair pidA1 , idA0 q, and if g “ pg1, g0q: B Ď C, then g˝f :“ pg1 ˝f1,f0 ˝g0q

15 f1 g1 g0 f0 A1 B1 C1 C0 B0 A0

X X X X X X iA1 iB1 iC1 iC0 iB0 iA0

X X

Clearly, the category PKJpXq is thin. KJ Proposition 7.6 (Chu representation of P pXq). If pX, “X , ‰X q is a set with an inequality, then the functor EX : PKJpXqÑ ChupSet, X ˆ Xq, deﬁned by

X 1 X 0 X 1 X X 0 E0 A , iA1 , A , iA0 “ A , iA1 ˆ iA0 , A , ` ˘ ` ˘ X 1 0 A B 1 0 1 X X 0 1 X X 0 E1 pf ,f q: Ñ “ pf ,f q: A , iA1 ˆ iA0 , A Ñ B , iB1 ˆ iB0 ,B , ` ˘ ` ˘ ` ˘ is a strict Chu representation of PKJpXq into ChupSet, X ˆ Xq.

X X 1 0 X X 1 0 X 1 X 0 Proof. Let iA1 ˆ iA0 : A ˆ A Ñ X ˆ X where iA1 ˆ iA0 pa , a q “ iA1 pa q, iA0 pa q , for every 1 0 1 0 1 0 A B 1 0 1 X X 0 1 X X 0 pa , a q P A ˆ A . If pf ,f q: Ñ , then pf ,f“ q: A , iA‰ 1 ˆ iA0 , A ` Ñ B , iB1 ˆ i˘B0 ,B is a morphism in ChupSet, X ˆ Xq, as the commutativity of` the following rectangle˘ ` ˘

0 idA1 ˆf A1ˆB0 A1ˆA0

1 X X f id 0 i ˆi ˆ B A1 A0

B1ˆB0 XˆX iX ˆiX B1 B0 follows from the commutativity of the following two triangles

f 1 f 0

A1 B1 B0 A0 X X X X iA1 iB1 iB0 iA0 X X

X X 0 1 0 X X 1 0 0 iA1 ˆ iA0 ˝ 1A1 ˆ f pa , b q“ iA1 ˆ iA0 pa ,f pb qq “` ˘ ` ˘‰ “ X 1 X‰ 0 0 “ iA1 pa q, iA0 pf pb qq ` X 1 1 X 0 ˘ “ iB1 pf pa qq, iB0 pb q ` X X 1 1 ˘0 “ iB1 ˆ iB0 f pa q, b “ X X‰` 1 ˘ 1 0 “ iB1 ˆ iB0 ˝ f ˆ 1B0 pa , b q. “` ˘ ` ˘‰ Clearly, EX is a functor injective on objects and arrows, hence an embedding. It is also full, as the above equalities also show that the commutativity of the above rectangle implies the commutativity of 1 0 1 X X 0 1 X X 0 the above triangles. hence, if pf ,f q: A , iA1 ˆ iA0 , A Ñ B , iB1 ˆ iB0 ,B in ChupSet, X ˆ Xq, then pf 1,f 0q: A Ñ B. ` ˘ ` ˘

Consequently, one can identify PKJpXq with the full subcategory of ChupSet, X ˆ Xq with objects 1 X X 0 X 1 X 0 1 1 0 0 X 1 triplets A , iA1 ˆ iA0 , A , where iA1 : A ãÑ X and iA0 : A ãÑ X such that @a PA @a PA iA1 pa q‰X X 0 iA0 pa q `. Notice that the˘ Chu category ChupSet, X ˆ Xq “captures” the behavior of the morphisms` in PKJpXq˘, but not the positive disjointness of A1, A0, as there are objects pA,f,Bq of ChupSet, X ˆ Xq, with A B; e.g., we may consider the triplet pX, idXˆX , Xq.

16 8 The generalised Chu construction over a ccc C and an endofunctor

In order to Chu-represent categories like the category of predicates Pred and the category of complemented predicates Pred‰, defined in the following two sections, respectively, we generalise the Chu construction. Actually, it is this embedding that shaped the “right” definition of the category Pred‰, as, at first sight, more than one possible options exist.

Deﬁnition 8.1 (The Chu construction over a ccc C and an endofunctor). Let Γ: C Ñ C an endofunctor on C. The Chu category ChupC, Γq over C and Γ has objects quadruples px; a, f, bq, with x, a, b P C0 and f : a ˆ b Ñ Γ0pxq P C1. A morphism φ: px; a, f, bq Ñ py; c, g, dq in ChupC, Γq, or a Chu transform, 0 ` ´ 0 ` ´ is a triplet φ “ φ , φ , φ , where φ : x Ñ y, φ : a Ñ c and φ : d Ñ b are in C1 such that the following diagram` commutes˘

´ 1aˆφ aˆd aˆb

` φ ˆ1d Γ0pxq

0 Γ1pφ q

cˆd Γ0pyq. g

If θ “ θ0,θ`,θ´ : py; c, g, dq Ñ pz; i, h, jq, let θ ˝ φ “ θ0 ˝ φ0,θ` ˝ φ`, φ´ ˝ θ´ ` ˘ ` ˘ ´ 1aˆφ aˆd aˆb

` ´ ´ φ ˆ1d Γ0pxq 1aˆpφ ˝θ q

0 Γ1pφ q

0 0 cˆd Γ0pyq Γ1pθ ˝φ q aˆj g

0 Γ1pθ q

´ ` ` 1cˆθ Γ0pzq pθ ˝φ qˆ1j

cˆj iˆj. ` θ ˆ1j

Moreover, 1px;a,f,bq “ p1x, 1a, 1bq

1aˆ1b aˆb aˆb

1aˆ1b Γ0pxq

Γ1p1xq“1Γ0pxq

aˆb Γ0pxq. f

To show that composition in Chu, pC, Γq is well-deﬁned, we show the commutativity of the above

17 triangle as follows:

0 0 ´ ´ 0 0 ´ ´ Γ1pθ ˝ φ q ˝ f ˝ r1a ˆ pφ ˝ θ qs “ Γ1pθ q ˝ Γ1pφ q ˝ f ˝ r1a ˆ pφ ˝ θ qs

p2q 0 0 ´ ´ “ Γ1pθ q ˝ Γ1pφ q ˝ f ˝ p1a ˆ φ q ˝ p1a ˆ θ q 0 “ ` ´‰ “ Γ1pθ q ˝ g ˝ pφ ˆ 1dq ˝ p1a ˆ θ q

p4q 0 ´ ` “ Γ1pθ q ˝ g ˝ p1c ˆ θ q ˝ pφ ˆ 1jq 0 ´ ` “ Γ1pθ q ˝ g ˝ p1c ˆ θ q ˝ pφ ˆ 1jq ` ` “ “h ˝ pθ ˆ 1jq ˝ pφ ˆ‰ 1jq

p3q“ ` `‰ “ h ˝ pθ ˝ φ qˆ 1j . “ ‰ γ γ Proposition 8.2. Let Γ : C Ñ C the constant endofunctor with value γ i.e., Γ0 paq “ γ, for every γ γ γ a P C0, and Γ1 pfq“ 1γ , for every f P C1. The functor E : ChupC, γqÑ ChupC, Γ q, deﬁned by

γ E0 pa, f, bq “ pγ; a, f, bq,

γ ` ´ ` ` E1 φ , φ : pa, f, bq Ñ pc, g, dq “ 1γ , φ , φ : pγ; a, f, bq Ñ pγ; c, g, dq, `` ˘ ˘ ` ˘ is an embedding of ChupC, γq into ChupC, Γγ q.

γ ` ` Proof. To show that E is a functor, it suﬃces to show that 1γ , φ , φ : pγ; a, f, bq Ñ pγ; c, g, dq. This follows from the fact that the commutativity of the following` upper˘ inner diagram implies the commutativity of the following outer diagram

´ 1aˆφ aˆd aˆb

` φ ˆ1d Γ0pxq

g 1γ

cˆd Γ0pyq. g

Clearly, Eγ is injective on objects and arrows, hence it is an embedding.

Proposition 8.3 (The generalised local Chu functor). The rule ChuC : FunpC, CqÑ Cat deﬁned by

C Chu0 pΓq“ ChupC, Γq,

C Chu1 pη : Γ ñ ∆q: ChupC, ΓqÑ ChupC, ∆q, C Chu1 pηq 0px; a, f, bq “ px; a, ηx ˝ f, bq, “ ‰ f aˆb Γ0pxq

ηx

∆0pxq

C 0 ` ´ 0 ` ´ Chu q 1 φ , φ , φ “ φ , φ , φ , “ ‰ ` ˘ ` ˘ C is a functor. Moroever, if ηx : Γ0pxq ãÑ ∆0pxq is a mono, for every x P C0, then Chu1 pηq is a full embedding of ChupC, Γq into ChupC, ∆q.

18 C 0 ` ´ Proof. To show that Chu1 is a functor, it suﬃces to show that if φ , φ , φ : px; a, f, bq Ñ py; c, g, dq 0 ` ´ in ChupC, Γq, then φ , φ , φ : px; a, η ˝ f, bq Ñ py; c, ηy ˝ g, d`q in ChupC˘, ∆q. This follows from the fact that commutativity` of˘ the following upper, inner diagram implies the commutativity of the following outer diagram

´ 1aˆφ aˆd aˆb

Γ0pxq

` η φ ˆ1d x

0 Γ1pφ q ∆0pxq

0 ∆1pφ q

cˆd Γ0pyq ∆0pyq g ηy

0 ´ 0 ´ ∆1pφ q ˝ ηx ˝ f ˝ p1a ˆ φ q“ ∆1pφ q ˝ ηx ˝ f ˝ p1a ˆ φ q “ 0 ‰ ´ “ ηy ˝ Γ1pφ q ˝ f ˝ p1a ˆ φ q ` “ ηy ˝ g“ ˝ φ ˆ 1d . ‰

`C ˘ If ηx : Γ0pxq ãÑ ∆0pxq is a mono, for every x P C0, then Chu1 pηq is injective on objects, and since it is C trivially injective on arrows, it is an embedding. In this case, Chu1 pηq is also full, as the commutativity of the above outer diagram implies the commutativity of the above, upper, inner diagram. As ηy is a mono, the resulted equality

0 ´ ` ηy ˝ Γ1pφ q ˝ f ˝ p1a ˆ φ q “ ηy ˝ g ˝ φ ˆ 1d “ ‰ ` ˘ 0 ´ ` implies the equality Γ1pφ q ˝ f ˝ p1a ˆ φ q“ g ˝ φ ˆ 1d. ` Deﬁnition 8.4. Let C, D be categories and F : C Ñ D a functor. If D is a generalised Chu category and F is a representation, we call F a generalised Chu representation. We call a generalised Chu representation F strict, if F is injective on arrows.

9 The generalised global Chu functor

The following fact is the generalised analogue to Lemma 3.1. Lemma 9.1. Let C, D be cartesian closed categories, Γ: C Ñ C, ∆: D Ñ D, F : C Ñ D such that F preserves products with pFabqa,bPC0 the canonical isomorphisms of F , and let η : F ˝ Γ ñ ∆ ˝ F

F CD

η Γ ùñ ∆ CD. F

The rule F˚ : ChupC, ΓqÑ ChupD, ∆q, deﬁned by

pF˚q0px; a, f, bq“ F0pxq; F0paq, ηx ˝ F1pfq ˝ Fab, F0pbq ` ˘

Fab F1pfq ηx F0paqˆ F0pbq F0pa ˆ bq F0pΓ0pxqq ∆0pF0pxqq

19 0 ` ´ pF˚q1 φ , φ , φ : F0paq, ηx ˝ F1pfq ˝ Fab, F0pbq Ñ F0pyq; F0pcq, ηy ˝ F1pgq ˝ Fcd, F0pdq , ` ˘ ` 0 ` ´ 0˘ ` ` ´ ˘ pF˚q1 φ , φ , φ “ F1pφ q, F1pφ q, F1pφ q , where φ0, φ`, φ´ : px; a, f, bq Ñ` py; c, g, dq ,˘ is a` functor. ˘ ` ˘ ˘ 0 ` ´ Proof. We show that pF˚q1 φ , φ , φ is well-deﬁned i.e., the following diagram commutes: ` ˘ ´ 1F0paqˆF1pφ q F0paqˆF0pdq F0paqˆF0pbq

ηx˝F1pfq˝Fab

` F1pφ qˆ1F0pdq ∆0pF0pxqq

0 ∆1pF1pφ qq

F0pcqˆF0pdq ∆0pF0pyqq. ηy˝F1pgq˝Fcd

Let 0 ´ A “ ∆1pF1pφ qq ˝ ηx ˝ F1pfq ˝ Fab ˝ r1F0paq ˆ F1pφ qs, ` B “ ηy ˝ F1pgq ˝ Fcd ˝ rF1pφ qˆ 1F0pdqs. By the deﬁnition of a morphism φ0, φ`, φ´ : px; a, f, bq Ñ py; c, g, dq we get

` 0 ˘ ´ ` ˘ Γ1pφ q ˝ f ˝ p1a ˆ φ “ g ˝ pφ ˆ 1dqñ

0 ´ ` p˚q F1pΓ1pφ qq ˝ F1pfq ˝ F1p1a ˆ φ q“ F1pgq ˝ F1pφ ˆ 1dq. ` ` As F1pφ ˆ 1dq ˝ Fad “ Fcd ˝ rF1pφ qˆ F1p1dqs, and since the following rectangle commutes

0 F1pΓ1pφ qq F0pΓ0pxqq F0pΓ0pyqq

ηx ηy

∆0pF0pxqq ∆0pF0pyqq, 0 ∆1pF1pφ qq

0 ´ A “ ∆1pF1pφ qq ˝ ηx ˝ F1pfq ˝ Fab ˝ rF1p1aqˆ F1pφ qs 0 ´ “ ∆1pF1pφ qq ˝ ηx ˝ F1pfq ˝ F1p1a ˆ φ q ˝ Fad 0 ´ “ ηy ˝ F1pΓ1pφ qq ˝ F1pfq ˝ F1p1a ˆ φ q ˝ Fad

p˚q ` “ ηy ˝ F1pgq ˝ F1pφ ˆ 1dq ˝ Fad ` “ ηy ˝ F1pgq ˝ Fcd ˝ rF1pφ qˆ F1p1dqs “ B.

The preservation of units and compositions by F˚ is immediate to show.

Next we define the appropriate category on which the generalised global Chu functor will be defined. Notice that this category is not a special case of the Grothendieck construction, but a variation of it. Definition 9.2 (The category of pairs of ccc’s and endofunctors). Let the category

EndpCq CPccCatÿ with objects pairs pC, Γq, where C in ccCat and Γ: C Ñ C an endofunctor on C, and morphisms pF, ηq: pC, Γq Ñ pD, ∆q, where F : C Ñ D is a product preserving functor and η : F ˝ Γ ñ ∆ ˝ F . If pG, θq: pD, ∆q Ñ pE, Eq, let pG, θq ˝ pF, ηq: pC, Γq Ñ pE, Eq be deﬁned by

20 F G C D E

η θ Γ ùñ ∆ ùñ E

CD E F G

pG, θq ˝ pF, ηq “ pG ˝ F,θ ˚ ηq, θ ˚ η : pG ˝ F q ˝ Γ ñ E ˝ pG ˝ F q,

pθ ˚ ηqa : G0pF0pΓ0paqqq Ñ E0pG0pF0paqqq,

pθ ˚ ηqa “ θF0paq ˝ G1pηaq

G1pηaq G0pF0pΓ0paqqq G0p∆0pF0paqqq

pθ˚ηqa θF0paq

E0pG0pF0paqqq

C Moreover, 1pC,Γq “ Id , 1Γ . ` ˘ First we explain why θ ˚ η is a natural transformation pG ˝ F q ˝ Γ ñ E ˝ pG ˝ F q. If f : a Ñ b in C1, then, as η : F ˝ Γ ñ ∆ ˝ F , the following left rectangle commutes:

F1pΓ1pfqq G1p∆1pF1pfqqq F0pΓ0paqq F0pΓ0pbqq G0p∆0pF0paqqq G0p∆0pF0pbqqq

ηa #1 ηb θF0paq #2 θF0pbq

∆0pF0paqq ∆0pF0pbqq E0pG0pF0paqqq E0pG0pF0pbqqq ∆1pF1pfqq E1pG1pF1pfqqq

By commutativity p#1q we get

p˚q G1pηbq ˝ G1pF1pΓ1pfqqq “ G1p∆1pF1pfqqq ˝ G1pηaq.

As θ : G ˝ ∆ ñ E ˝ G, and F1pfq: F0paq Ñ F0pbq in D1, the above right rectangle commutes. The commutativity of the following rectangle diagram follows:

G1pF1pΓ1pfqqq G0pF0pΓ0paqqq G0pF0pΓ0pbqqq

pθ˚ηqa pθ˚ηqb

E0pG0pF0paqqq E0pG0pF0pbqqq, E1pG1pF1pfqqq

pθ ˚ ηqb ˝ G1pF1pΓ1pfqqq “ θF0pbq ˝ G1pηbq ˝ G1pF1pΓ1pfqqq p˚q “ θF0pbq ˝ G1p∆1pF1pfqqq ˝ G1pηaq p#2q “ E1pG1pF1pfqqq ˝ θF0paq ˝ G1pηaq

“ E1pG1pF1pfqqq ˝ pθ ˚ ηqa. C C If pF, ηq: pC, Γq Ñ pD, ∆q, then pF, ηq ˝ 1pC,Γq “ pF, ηq ˝ Id , 1Γ “ pF ˝ Id , η ˚ 1Gq “ pF, ηq, as D D η ˚ 1G “ η. Similarly, if 1pD,∆q ˝ pF, ηq “ Id , 1∆ “ pId` ˝ F, 1˘∆ ˚ ηq “ pF, ηq, as 1∆ ˚ η “ η. If pG, θq: pD, ∆q Ñ pE, Eq and pH, ρq: pE, Eq Ñ` pZ,Zq˘, then pH, ρq ˝ rpG, θq ˝ pF, ηqs “ H ˝ pG ˝ F q, ρ ˚ pθ ˚ ηq , ` ˘ rpH, ρq ˝ pG, θqs ˝ pF, ηq“ pH ˝ Gq ˝ F, pρ ˚ θq ˚ η , and as ρ ˚ pθ ˚ ηq “ pρ ˚ θq ˚ η, we get pH, ρq ˝ rpG, θ`q ˝ pF, ηqs “ rpH, ρq ˝ pG,˘ θqs ˝ pF, ηq.

21 Theorem 9.3 (The generalised global Chu functor). The rule

CHU: EndpCqÑ Cat, CPccCatÿ

CHU0pC, Γq“ ChupC, Γq,

CHU1 F, ηq: pC, Γq Ñ pD, ∆q : ChupC, ΓqÑ ChupD, ∆q, ` ˘ CHU1 F, ηq“ F˚, ` where F˚ is deﬁned in Lemma 9.1, is a functor. Moreover, if F : C Ñ D is a full embedding and ηa is a monomorphism, for every a P C0, then F˚ is a full embedding of ChupC, Γq into ChupD, ∆q.

Proof. By Lemma 9.1 CHU1pF, φq is well-deﬁned. Clearly,

CHU 1 CHU IdC, 1 IdC 1 . 1p pC,Γqq“ 1 Γ “ ˚ “ CHUpC,Γq ` ˘ “ ‰ If pG, θq: pD, ∆q Ñ pE, Eq, we show that CHU1pG ˝ F,θ ˚ ηq “ pG ˝ F q˚ “ G˚ ˝ F˚ “ CHU1pG, θq ˝ CHU1pF, ηq. If A “ pG ˝ F q˚ 0px; a, f, bq and B “ pG˚q0 pF˚q0px; a, f, bq , then “ ‰ ` ˘ A “ G0pF0pxqq; G0pF0paqq, pθ ˝ ηqx ˝ G1pF1pfqq ˝ pG ˝ F qab, G0pF0pbqq ` ˘ “ G0pF0pxqq; G0pF0paqq,θF0pxq ˝ G1pηxq ˝ G1pF1pfqq ˝ G1pFabq ˝ GF0paqF0pbq, G0pF0pbqq ` ˘ “ G0pF0pxqq; G0pF0paqq,θF0pxq ˝ G1 ηx ˝ F1pfq ˝ Fab ˝ GF0paqF0pbq, G0pF0pbqq ` “ ‰ ˘ “ pG˚q0 F0pxq; F0paq, ηx ˝ F1pfq ˝ Fab, F0pbq “ B. ` ˘

0 ` ´ 0 ` ´ The equality rpG ˝ F q˚s1pφ , φ , φ q “ pG˚q1 pF˚q1pφ , φ , φ q follows immediately. Let F : C Ñ D be a full embedding and ηa a monomorphism,` for every a P C0. The˘ equality F0pxq; F0paq, ηx ˝ F1pfq˝ 1 1 1 1 1 1 1 Fab, F0pbq “ F0px q; F0pa q, ηx1 ˝ F1pf q ˝ Fa1b1 , F0pb q implies x “ x , a “ `a , b “ b , and as ηx is a 1 1 monomorphism˘ ` and Fab an isomorphism, hence an epimorphism,˘ we get F1pfq“ F1pf q, hence f “ f . The fact that F˚ is faithful and full follows immediately.

The local generalised Chu functor is a special case of the global one. Namely, if D “ C, F “ IdC, and Γ, ∆: C Ñ C, and if η : IdC ˝ Γ ñ ∆ ˝ IdC i.e., η : Γ ñ ∆, then

ChuC η IdC . 1 p q“ ˚ “ ‰ 10 A generalised Chu representation of the category of predicates

Predicates on sets were organised in a category that was called Pred in [15], in order to describe the logic and type theory of standard sets in ﬁbred form. Here we present this category within BST.

X Deﬁnition 10.1. The objects of the category of predicates Pred are triplets pX, iA , Aq, where X is X X Y a set and pA, iA q is a subset of X. If pX, iA , Aq and pY, iB ,Bq are objects of Pred, a morphism X Y 0 ` 0 u: pX, iA , Aq Ñ pY, iB,Bq in Pred is a pair of functions u “ u , u , where u : X Ñ Y and u` : A Ñ B such that the following diagram commutes ` ˘

u` A B

X Y iA iB X Y . u0

0 ` Y Z X Z 0 If v “ v , v : pY, iB ,Bq Ñ pZ, iC ,Cq, let v ˝ u: pX, iA , Aq Ñ pZ, iC ,Cq, defined by v ˝ u “ v ˝ 0 ` ` u , v ˝ù . Moreover,˘ 1 X “ idX , idA . ` pX,iA ,Aq ˘ ` ˘ 22 X In [15], p. 11, the embedding iA : A Ñ X is omitted for simplicity, and a morphism u is just a function u0 : X Ñ Y such that 0 X Y @aPADbPB u piA paqq “Y iBpbq . It is immediate to see that to each a P A there` is a unique (up to˘ the equality of B) b P B such that 0 X Y u piA paqq “Y iBpbq. By Myhill’s principle of non-choice (or unique choice), introduced in [19], there is a (necessarily) unique map u` that makes the above diagram commutative. As this principle is avoided in BST, we prefer to present a morphism u in Pred as a pair pu0, u`q. It is immediate to see that if u0 is an embedding, then u` is an embedding, and if u0 is strongly extensional, then u` is also strongly extensional. For a specific set X the “fibre” category PredX is the subcategory of Pred with X X X objects triplets of the form pX, A, iA q with X fixed, while a morphism u: pX, A, iA q Ñ pX,B,iB q is a pair pidX , uAB q, and the required commutativity of the following diagram

uAB A B

X X iA iB X X idX

expresses that uAB : A Ď B. Hence PredX is identified with the category PpXq. Proposition 10.2 (Generalised Chu representations of Set and Pred). (i) The functor ESet : Set Ñ ChupSet, Idq, defined by Set X 1 E0 pXq“ X; X,IX , , X 1 ` X ˘ IX : X ˆ Ñ Id0pXq“ X, IX px, 0q“ x; x P X, Set X Y 0 ` X 1 Y 1 E1 f : X Ñ Y “ pf,f, id1q: X,IA , A Ñ Y, iB ,B “ u , u , id1 : X; X,IX , Ñ Y ; Y,IY , , is a strict` generalised˘ Chu representation` ˘ of Set` into Chu˘˘ pSet` , Idq. ˘ ` ˘ ` ˘ (ii) The functor EPred : Pred Ñ ChupSet, Idq, defined by Pred X X 1 E0 X, iA , A “ X; A, IA , , ` ˘ ` ˘ X 1 X X IA : A ˆ Ñ Id0pXq“ X, IA pa, 0q“ iA paq; a P A, Pred 0 ` X Y 0 ` X 1 Y 1 E1 u “ u , u : X,IA , A Ñ Y,IB ,B “ u , u , id1 : X; A, IA , Ñ Y ; B,IB , , is a strict` generalised` Chu˘ ` representation˘ ` of Pred˘˘into` ChupSet,˘Id`q. ˘ ` ˘

(iii) If F : Set Ñ Pred is the full embedding of Set into Pred, deﬁned by F0pXq “ pX, idX , Xq and Fipf : X Ñ Y q “ pf,fq, the following diagram commutes

ESet Set ChupSet,Idq

F Id

Pred ChupSet,Idq. EPred

0 ` X Y 0 ` X 1 Proof. We show only (ii). If u “ u , u : X, iA , A Ñ Y, iB,B , then u , u , id1 : X; A, IA , Ñ Y 1 Y ; B,IB , , as the commutativity` of the˘ ` rectangle˘ ` ˘ ` ˘ ` ˘ ` ˘ u` A B

X Y iA iB X Y u0

23 implies the commutativity of the following diagram

idAˆid1 Aˆ1 Aˆ1

X IA

u`ˆid1 X

0 0 Id1pu q“u

Bˆ1 Y Y IB

0 X 0 X Y ` Y ` u IA pa, 0q “ u iA paq “ iB u paq “ IB u paq, 0 . ` ˘ ` ˘ ` ˘ ` ˘ Clearly, EPred is injective on objects and arrows, hence EPred is an embedding. It is also full, as if 0 ` X 1 Y 1 0 ` X Y u , u , id1 : X; A, IA , Ñ Y ; B,IB , , then u “ u , u : X, iA , A Ñ Y, iB,B , because the `commutativity˘ ` of the last˘ diagram` implies˘ the commutativit` y˘ of` the ﬁrst rectangle.˘ ` ˘

Deﬁnition 10.3. If C is a category, the category PredpCq of C has objects pairs px, i: a ãÑ xq, where 0 ` x P C0 and i P C1pa, xq is a monomorphism, and morphisms pf ,f q: px, i: a ãÑ xq Ñ py, j : b ãÑ yq ` with j ˝ f “ f0 ˝ i

f ` a b i j x y. f 0

If pg0, g`q: py, j : b ãÑ yq Ñ pz, k : e ãÑ zq, then pg0, g`q ˝ pf 0,f `q “ pg0 ˝ f 0, g` ˝ f `q. Moreover, 1px,i: aãÑxq “ p1x, 1aq. Proposition 10.4 (Generalised Chu representation of PredpCq). If C is a ccc, the functor

C C EPredp q : PredpCqÑ ChupC, Id q,

PredpCq E0 x, i: a ãÑ x “ x; a, i ˝ pra, 1 , ` ˘ ` ˘ pra i a ˆ 1 a x

PredpCq 0 ` 0 ` E1 f ,f : x, i: a ãÑ x Ñ y, j : b ãÑ y “ f ,f , 11 : x; a, i ˝ pra, 1 Ñ y; b, j ˝ prb, 1 , `` ˘ ` ˘ ` ˘˘ ` ˘ ` ˘ ` ˘ is a strict generalised Chu representation of PredpCq into ChupC, IdCq.

PredpCq 0 ` Proof. The morphism pra is an iso, hence a mono. To show that E1 f ,f : x; a, i ˝ pra, 1 Ñ y; b, j ˝ prb, 1 , we show that the following diagram commutes ` ˘ ` ˘ ` ˘ 1aˆ1 aˆ1 aˆ1

i˝pra

f `ˆ11 x

f 0

bˆ1 y j˝prb

24 0 0 f ˝ pi ˝ praq “ pf ˝ iq ˝ pra ` “ pj ˝ f q ˝ pra ` “ j ˝ pf ˝ praq ` “ j ˝ rprb ˝ pf ˆ 11qs ` “ pj ˝ prbq ˝ pf ˆ 11q

` ` as the equality f ˝pra “ prb ˝pf ˆ11q follows as in the proof of Proposition 6.4. If x; a, i˝pra, 1 “ y; b, j ˝ prb, 1 , then x “ y, a “ b, and i ˝ pra “ j ˝ pa. As pra is a mono, we` get i “ j, and˘ hence` EPredpC˘q is injective on objects. It is trivially injective on arrows. To show that it is full, let 0 ` ´ ´ pφ , φ , φ q: x; a, i˝pra, 1 Ñ y; b, j ˝prb, 1 . Clearly, φ “ 11. Moreover, by the previous equalities 0 ` 0 ` we get pφ ˝ i`q ˝ pra “ pj ˝˘φ q` ˝ pra, and since˘ pra is a mono, we conclude that φ ˝ i “ j ˝ φ i.e., pφ0, φ`q: x, i: a ãÑ x Ñ y, j : b ãÑ y . ` ˘ ` ˘ 11 A generalised Chu representation of the category of complemented predicates

Here we organise the complemented predicates on sets that are equipped with a fixed inequality in a ‰ ‰ category Pred . Its subcategory Predse is formed by considering in the definition of the morphisms in Pred‰ strongly extensional functions. The motivation behind the next definition is to get a strict generalised Chu representation of Pred‰pSetq into the Chu category over Set and the endofunctor Id2 : Set Ñ Set, defined by 2 Id0pXq“ X ˆ X, 2 Id1pf : X Ñ Y q : X ˆ X Ñ Y ˆ Y, 2 1 1 rId1pfqspx,x q“ fpxq,fpx q . This result is in complete analogy to the full embedding` of Pred˘ into ChupSet, Idq. Definition 11.1. The category Pred#pSetq of complemented predicates has objects pairs pX, Aq, where X is in Set#, the category of sets equipped with a fixed inequality and strongly extensional functions between them, and A :“ pA1, A0q is a complemented subset of X. If pX, Aq and pY, Bq are objects of Pred#, a morphism u: pX, Aq Ñ pY, Bq is a triplet u “ u0, u`, u´ , where u0 : X Ñ Y , u` : A1 Ñ B1, and u´ : B0 Ñ A0 such that the following rectangles commute` ˘

u` u´ A1 B1 B0 A0

X Y Y X iA1 iB1 iB0 iA0 X Y Y X. u0 u0

If u “ v0, v`, v´ : pY, Bq Ñ pZ, Cq, we deﬁne the composite morphism v ˝ u: pX, Aq Ñ pZ, Cq by 0 0 ` ` ´ ´ v ˝ u “` v ˝ u , v ˘ ˝ u , u ˝ v . Moreover, 1pX,Aq “ idX , idA1 , idA0 . ` ˘ ` ˘ Proposition 11.2 (Generalised Chu representation of Pred‰pSetq). The functor

‰ EPred pSetq : Pred‰ Ñ ChupSet, Id2q,

Pred‰pSetq A 1 X X 0 E0 X, “ X; A , iA1 ˆ iA0 , A , X X ` 1 ˘ 0 ` 2 ˘ iA1 ˆ iA0 : A ˆ A Ñ Id0pXq“ X ˆ X, Pred‰pSetq 0 ` ´ 0 ` ´ 1 X X 0 1 Y Y 0 E1 u , u , u “ u , u , u : X; A , iA1 ˆ iA0 , A Ñ Y ; B , iB1 ˆ iB0 ,B , where u0, u`, u´ `: X, A Ñ˘ Y,`B , is a strict˘ ` generalised Chu representation˘ ` of Pred‰pSet˘ q into ChupSet` , Id2q. ˘ ` ˘ ` ˘

25 0 ` ´ A B 0 ` ´ 1 X X 0 1 Y Proof. If u , u , u : X, Ñ Y, , then u , u , u : X; A , iA1 ˆ iA0 , A Ñ Y ; B , iB1 ˆ Y 0 iB0 ,B , as` the commutativity˘ ` ˘ of the` following˘ two` rectangles˘ ` ˘ ` ˘ u` u´ A1 B1 B0 A0

X Y Y X iA1 iB1 iB0 iA0 X Y Y X. u0 u0 implies the commutativity of the following diagram

´ idA1 ˆu A1ˆB0 A1ˆA0

iX ˆiX A1 A0

` u ˆidB0 XˆX

2 0 Id1pu q

B1ˆB0 Y ˆY iY ˆiY B1 B0

2 0 X X ´ 1 0 2 0 X 1 X ´ 0 Id1pu q iA1 ˆ iA0 idA1 ˆ u a , b “ Id1pu q iA1 pa q, iA0 pu pb qq “` ˘` ˘` ˘‰ 0 X “` 1 0 X ´ 0 ˘‰ “ u piA1 pa qq, u piA0 pu pb qqq ` Y ` 1 Y 0 ˘ “ iB1 pu pa qq, iB0 pb q ` Y Y ` 1 ˘0 “ iB1 ˆ iB0 u pa q, b q “ Y Y ‰` ` ˘ 1 0 “ iB1 ˆ iB0 u ˆ idB0 pa , b q. “ ‰` ˘ ‰ ‰ Clearly, EPred is injective on objects and arrows, hence EPred is an embedding. It is also full, as 0 ` ´ 1 X X 0 1 Y Y 0 0 ` ´ A B if u , u , u : X; A , iA1 ˆ iA0 , A Ñ Y ; B , iB1 ˆ iB0 ,B , then u , u , u : X, Ñ Y, , because` the commutativity˘ ` of the last˘ diagram` implies the commutativity˘ ` of the above˘ ` two˘ rectangles.` ˘

12 The Chu construction and the antiparallel Grothendieck construction

So far, we related the two constructions through the domain of the global Chu functor. The domain of the generalised global Chu functor has also some affinity to the Grothendieck construction. Next we discuss the relation between the two constructions themselves. A first result in this direction is the following result of Abramsky in [2], p. 14. Notice that instrumental to the proof of his result is a contravariant, or reverse, definition of the arrows in the Grothendieck category. Namely, if P : Cop Ñ CAT, where CAT is the category of (large) categories, an arrow pf, φq: pa, xq Ñ pb, yq in the category GrothpC, P q, where x,y are objects in P0paq and P0pbq, respectively, is an arrow f : b Ñ a in C and an arrow φ: rP1pfqs0pxqÑ y in P0pbq. In the literature the standard approach to the definition of the category of elements or of the Grothendieck category is is the covariant definition of the arrow pf, φq, where f : a Ñ b and φ: x Ñ rP1pfqs0pyq. As we explain also later in this section, this reverse definition of the arrows in GrothpC, P q is necessary to Abramsky’s result. Next follows the generalisation of Abramsky’s result on an arbitrary ccc.

Proposition 12.1 (Abramsky 2018). Let C be a ccc and γ P C0. If x P C0, let ChuxpC, γq be the subcat- ` egory of ChupC, γq with objects triplets of the form pa, f, xq and morphisms the pairs pφ , 1xq: pa, f, xqÑ

26 pb, g, xq. If h: x1 Ñ x, let the functor

˚ h : ChuxpC, γqÑ Chux1 pC, γq,

˚ 1 ˚ ` ` γ op where h0 pa, f, xq “ a, f ˝ p1a ˆ hq,x and h1 φ , 1x “ φ , 1x . If Chu : C Ñ CAT is the contravariant functor` deﬁned by ˘ ` ˘ ` ˘ C0 Q x ÞÑ ChuxpC, γq, Chuγph: x1 Ñ xq“ h˚, then the category GrothpC, Chuγq is the Chu category ChupC, γq.

Proof. See [10].

The Chu construction can be seen as a special case of the antiparallel Grothendieck construction, or the antiparallel category of elements, on the product category, in case the ccc C is locally small. In the next deﬁnition we could consider a product C ˆ D instead of a product C ˆ C, and more options occur op if larger products of categories are considered. If C is a category, a, b P C0, and S : pC ˆ Cq Ñ Set a contravariant functor on C ˆ C, let the induced contravariant functors

op 1 1 Sa : C Ñ Set, Sapcq“ S0pa, cq Sapg : c Ñ c q“ S1p1a, gq: S0pa, c qÑ S0pa, cq,

op 1 1 bS : C Ñ Set, bSpcq“ S0pc, bq bSpg : c Ñ c q“ S1pg, 1bq: S0pc , bqÑ S0pc, bq. Deﬁnition 12.2. Let C be a category and S : pC ˆ Cqop Ñ Set. The pcontravariantq antiparallel Ô Grothendieck category Groth C ˆC,S has objects triplets pa, x, uq, where a, x P C0 and u P S0pa, xq, and morphisms pairs φ`, φ´ :`pa, x, uq˘ Ñ pb, y, vq, where φ` : a Ñ b and φ´ : y Ñ x are morphisms ´ ` in C such that rSapφ qsp` uq “˘ rySpφ qspvq

S0pa, xq

´ Sapφ q

S0pb, yq S0pa, yq. ` ySpφ q

` ´ ` ´ ` ´ “ ` ´ ´ If θ ,θ : pb, y, vq Ñ pc, z, wq, let θ ,θ ˝ φ , φ “ θ ˝φ , φ θ . Moreover, 1pa,x,uq “ p1a, 1xq ` ˘ ` ˘ ` ˘ ` ˘ To justify the composition of morphisms in GrothÔ C ˆ C,S , let the equalities: ` ˘ ´ ` S1p1a, φ qspuq “ rS1pφ , 1yqspvq (5)

´ ` S1p1b,θ qspvq “ rS1pθ , 1zqspwq. (6) ´ ´ ` ` We show the equality S1p1a, φ ˝ θ qspuq “ rS1pθ ˝ φ , 1zqspwq as follows:

S0pa, xq

´ S1p1a, φ q ` S1pφ , 1yq S0pb, yq S0pa, yq

´ ´ S1p1b,θ q S1p1a,θ q

S0pc, zq S0pb, zq S0pa, zq. ` ` S1pθ , 1zq S1pφ , 1zq

27 ´ ´ ´ ´ rS1p1a, φ ˝ θ qspuq“ S1 p1a, φ q ˝ p1a,θ q puq ´ ´ “ “S1p`1a,θ q ˝ S1p1a, φ ˘‰q puq ´ ´ “ r“S1p1a,θ qs S1p1a, φ q‰ puq

p5q ´ `“ ` ‰ ˘ “ rS1p1a,θ qs S1pφ , 1yq pvq ` ´ “ S1 pφ , 1yq`“ ˝ p1a,θ q p‰vq ˘ ` ´ “ “S1p`φ ˝ 1a, 1y ˝ θ q p˘‰vq ` ´ “ “S1pφ ,θ q pvq ‰ ` ´ “ “S1p1b ˝ φ ,θ‰ ˝ 1zq pvq ´ ` “ “S1 p1b,θ q ˝ pφ , 1z‰q pvq ` ´ “ r“S1p`φ , 1zqs S1p1b,θ ˘‰q pvq

p6q ` `“ ` ‰ ˘ “ rS1pφ , 1zqs S1pθ , 1zq pwq ` ` “ S1 pθ , 1zq ˝`“ pφ , 1zq p‰wq ˘ ` ` “ r“S1p`θ ˝ φ , 1zqspwq. ˘‰

The parallel Grothendieck construction on C ˆ C and S, with φ`, φ´ : pa, x, uq Ñ pb, y, vq is a pair of morphisms φ` : a Ñ b and φ´ : x Ñ y in C is the standard category` of˘ elements over C ˆ C and S. If C is a locally small ccc, we have the Set-valued contravariant functor

op Homp´ˆ´, γq : pC ˆ Cq Ñ Set, ˘ pa, bq ÞÑ Hompa ˆ b, γq, ` 1 ´ 1 1 1 Homp´ˆ´, γq 1pφ : a Ñ a , φ : b Ñ b q: Hompa ˆ b , γqÑ Hompa ˆ b, γq, ˘ ` ´ ` ´ Homp´ˆ´, γq 1 φ , φ phq“ h ˝ φ ˆ φ “ ˘ ` ˘‰ ` ˘ φ`ˆφ´ h aˆb a1ˆb1 γ.

` ´ Homp´ˆ´,γq 1 φ ,φ phq “ ˘ ` ˘‰ Proposition 12.3. If C is a locally small ccc and γ P C0, the Chu category ChupC, γq is the antiparallel Ô Grothendieck category Groth C ˆ C, Homp´ˆ´, γq . ` ˘ Proof. In this case the deﬁning equality (5) takes the form

´ ´ Homp´ˆ´, γq 1 1a, φ pfq“ Homp´ˆ´, γq 1 φ `, 1y pgq “ ˘ ` ˘‰ “ ˘ ` ˘‰ ´ ` i.e., f ˝ p1a ˆ φ q“ g ˝ pφ ˆ 1yq.

In relation to Abramsky’s result, and for a locally small ccc C the previous result is maybe more interesting, as the functor S is only Set-valued, and not CAT-valued. Next we describe the global version of the functor Homp´ˆ´, γq. Proposition 12.4. If C is a locally small ccc, the functor

op Homp´ˆ´, ´q: C Ñ Fun pC ˆ Cq , Set , ` ˘ Homp´ˆ´, ´q 0pγq“ Homp´ˆ´, γq, “ 1 ‰ f 1 Homp´ˆ´, ´q 1pf : γ Ñ γ q“ η : Homp´ˆ´, γqñ Homp´ˆ´, γ q, “ ‰ f : Hom Hom 1 ηpa,bq pa ˆ b, γqÑ pa ˆ b, γ q, f ηpa,bqphq“ f ˝ h

28 h f aˆb γ γ1.

f etapa,bqphq is an embedding. of C into Fun pC ˆ Cqop, Set . ` ˘

Acknowledgments Our research was supported by LMUexcellent, funded by the Federal Ministry of Education and Re- search (BMBF) and the Free State of Bavaria under the Excellence Strategy of the Federal Government and the Länder.

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