arXiv:2106.01878v1 [math.CT] 3 Jun 2021

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Chu representations of categories related to constructive

mathematics

Iosif Petrakis

Mathematics Institute, Ludwig-Maximilians-Universität München

[email protected]

Abstract

If C is a closed symmetric monoidal category, the Chu category ChupC, γq over C and an object γ ofit was defined by Chu in [5], as a ˚-autonomous category generated from C. In [6] Bishop introduced

the category PKJpXq of complemented subsets of a set X , in order to overcome the problems gener-

ated by the use of negation in constructive measure theory. In [34] Shulman mentions that Bishop’scomplemented subsets correspond roughly to the Chu construction. In this paper we explain thiscorrespondence by showing that there is a Chu representation (a full embedding) of PKJpXq intoChupSet, X ˆ Xq. A Chu representation of the category of Bishop spaces into ChupSet,Rq isshown, as the constructive analogue to the standard Chu representation of the category of topolog-ical spaces into ChupSet, 2q. In order to represent the category of predicates (with objects pairspX,Aq, where A is a subset of X , and the category of complemented predicates (with objects pairspX,Aq, where A is a complemented subset of X , we generalise the Chu construction by defining theChu category over a cartesian closed category C and an endofunctor on C. Finally, we introducethe antiparallel Grothendieck construction over a product category and a contravariant Set-valuedfunctor on it of which the Chu construction is a special case, in case C is a locally small, cartesianclosed category.

Keywords : category theory, Chu construction, Grothendieck construction, constructive mathemat-ics, complemented subsets, Bishop spaces.

1 Introduction

In category theory the Chu construction is a method of generating a ˚-autonomous category from aclosed symmetric monoidal category (CSMC) (see [31] and [18]). The ˚-autonomous categories providemodels for classical (multiplicative) linear logic (in [33]). The Chu construction was introduced by Chuin his Master’s Thesis, and appeared first in [5]. The category ChupSet,Xq was introduced by Lafontand Streicher in [16] under the name of games (semantics for linear logic). In a series of papers, seee.g., [30], Pratt and his collaborators applied the Chu construction on topics of theoretical computerscience (e.g., concurrency). The Chu construction has been applied to hardware verification, gametheory, fuzzy systems, and the foundations of quantum mechanics (see [1] and [2]). There is a richrepresentation theory connected to the Chu construction, as many important, and quite differentcategories are represented (fully embedded) into some Chu category. The representation of categoriesrelated to constructive mathematics into some Chu category is a major theme of this paper.

In [34], p. 8, Shulman mentions that “a notion corresponding roughly to the Chu construction wasalready introduced by (Bishop and Bridges in) [8] under the name complemented subset”. Actually,the notion of a complemented subset is already introduced by Bishop in [6], pp. 66-69, under the namecomplemented set. Here we explain in what sense Bishop’s notion corresponds to the Chu construction.Namely, there is a Chu representation of the thin category PKJpXq of complemented subsets of a setX into ChupSet,X ˆ Xq. Notice that Bishop’s motivation for introducing complemented (sub)sets isrooted to his need to overcome problems generated by the use of negation in basic set and measuretheory in a constructive setting (see [24], chapter 7, and [21]). Hence, the connection described herebetween the Chu construction and Bishop’s notion of complemented subsets seems to be accidental.

1

All notions and results presented here concern cartesian closed categories pcccq, although they canbe generalised to symmetric monoidal closed categories1. We structure this paper as follows:

• 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 theinduced boolean Chu representation of the category of information systems Inf .

• In section 5 we present the constructive normal Chu representation of the category of Bishopspaces Bis. This representation of Bis is the constructive analogue of the aforementioned Churepresentation of Top. The notion of a Bishop space is Bishop’s constructive, function-theoreticalternative 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 andof the category P

KJpXq 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 ourreconstruction 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 Chuconstruction.

• With the help of the generalised Chu construction we provide a generalised Chu representation ofthe 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 categoryand 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 proofwe 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 andD, respectively. To show that the Chu construction in Definition 2.1 is category, one uses the fact theproduct ˆ : C ˆ C Ñ C is a bifunctor (i.e., a functor). Moreover, if f : a Ñ a1 and g : b Ñ b1 in C1, thenf ˆ g : a ˆ b Ñ a1 ˆ b1, such that 1a ˆ 1b “ 1aˆb, and if f 1 : a1 Ñ a2 and g1 : b1 Ñ b2 in C1, then

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

If a1 “ a2 “ a and f 1 “ f “ 1a, by equation (1) we get

p1a ˆ g1q ˝ p1a ˆ gq “ p1a ˝ 1aq ˆ pg1 ˝ gq “ 1a ˆ pg1 ˝ gq. (2)

Similarly, if b1 “ b2 “ b and g1 “ g “ 1b, by equation (1) we get

pf 1 ˆ 1bq ˝ pf ˆ 1bq “ pf 1 ˝ fq ˆ p1b ˝ 1bq “ pf 1 ˝ fq ˆ 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

aˆj aˆd

cˆdcˆj

1aˆθ

φˆ1dφˆ1j

1cˆθ

p1c ˆ θq ˝ pφ ˆ 1jqp1q“ 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, γqover 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. Amorphism 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

aˆy aˆx

bˆy γ.

φ`ˆ1y f

1aˆφ´

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

ChuC0pγq “ ChupC, γq,

ChuC1pu : γ Ñ δq “ u˚ : ChupC, γq Ñ ChupC, δq,

pu˚q0pa, f, bq “ pa, u ˝ f, bq,

a ˆ b γ δf u

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 spacesin ChupSet,Xq, for some given set X, and if pφ`, φ´q : pA, f,Bq Ñ pC, g,Dq, then the commutativityof the rectangle

A ˆ D A ˆ B

C ˆ D X

φ`ˆidD f

idAˆφ´

g

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 pf : A Ñ pB Ñ Xq, where“ pfpaq

‰pbq “ fpa, bq,

for every a P A and b P B, is an injection. A Chu space pA, f,Bq in ChupSet,Xq is called extensional,if qf : B Ñ pA Ñ Xq, where “ qfpbq

‰paq “ fpa, bq,

for every b P B and a P A, is an injection. If pA, f,Bq is both separable and extensional, it is calledbiextensional. If B Ă XA and f : A ˆ B Ñ X is defined by fpa, bq “ bpaq, then pA, f,Bq is called anormal 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 objectspairs pa, F q, where a P C0 and F Ď C1pa, γq “ Hompa, γq, and a morphism h : pa, F q Ñ pb,Gq inAffpC, 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 injectiveon objects and faithful, and its is a representation, if it is a full embedding. If D is a Chu category andF is a representation, we call F a Chu representation. We call a Chu representation F strict, if F isinjective on arrows. We call a Chu representation boolean pnormalq, if F0paq is a Boolean pnormalqChu space, for every a P C0.

All Chu representations included in this paper are going to be strict. If C is a ccc, let evγ,a : aˆγa Ñ

γ in C1 such that for every f : a ˆ b Ñ γ there is a unique pf : b Ñ γa with f “ evγ,a ˝`1a ˆ pf

˘. The

next result is also standard, and its proof is constructive. The normal Chu representation of Set

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

EC,γ0 paq “

`a, evγ,a, γ

a˘,

EC,γ1 pf : a Ñ bq “ pf, f´q :

`a, evγ,a, γ

a˘

Ñ`b, evγ,b, γ

b˘,

f´ “ ph : γb Ñ γa, h “ evγ,b ˝`f ˆ 1γb

˘,

aˆγa γ

aˆγb

bˆγb

evγ,a

1aˆph

fˆ1γb

evγ,b

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 : F0paq ˆ F0pbq Ñ F0pa ˆ bq, which is an isomorphism

4

a a ˆ b bpra prb

F0paq F0pa ˆ bq F0pbq

F0paq ˆ F0pbq.

F1ppraq F1pprbq

prF0paq prF0pbqFab

For every a, a1, b, b1 P C0 and every f : a Ñ a1, g : b Ñ b1 in C1 the following rectangle commutes

F0paˆbq F0pa1ˆb1q

F0paqˆF0pbq F0pa1qˆF0pb1q.

Fab Fa1b1

F1pfˆgq

F1pfqˆF1pgq

If G : D Ñ E also preserves products and pGcdqc,dPD0are the canonical isomorphisms Gcd : G0pcq ˆ

G0pdq Ñ 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

G0pF0paqˆF0pbqq

G0pF0paˆbqq G0pF0pbqqG0pF0paqq

prG0pF0paqq

prG0pF0paqq

G1pF1pprbqqG1pF1ppraqq

pG˝F qab

GF0paqF0pbq

G1pFabq

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

Lemma 3.1. Let F : C Ñ D be a product-preserving functor with pFabqa,bPC0the canonical isomor-

phisms of F , and let φ : F0pγq Ñ δ in D1. The rule F˚ : ChupC, γq Ñ ChupD, δq, defined by

pF˚q0pa, f, bq “`F0paq, φ ˝ F1pfq ˝ Fab, F0pbq

˘

F0paq ˆ F0pbq F0pa ˆ bq F0pγq δFab F1pfq φ

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

F0paqˆF0pdq F0paqˆF0pbq

F0pcqˆF0pdq δ.

F1pφ`qˆ1F0pdq φ˝F1pfq˝Fab

1F0paqˆF1pφ´q

φ˝F1pgq˝Fcd

5

By the commutativity of the following diagrams we have that

aˆd aˆb

cˆd γ

φ`ˆ1d f

1aˆφ´

g

F0paˆdq F0pcˆdq

F0paqˆF0pdq F0pcqˆF0pdq

F0paˆdq F0paˆbq

F0paqˆF0pdq F0paqˆF0pbq

Fad Fcd

F1pφ`ˆ1dq

F1pφ`qˆF1p1dq

Fad Fab

F1p1aˆφ´q

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`q ˆ F1p1dqs

“ φ ˝ F1pgq ˝ Fcd ˝ rF1pφ`q ˆ 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 showthat F ÞÑ F˚ is a functor on the category FunˆpC,Dq of product-preserving functors from C toD. What we showed though, in the previous lemma is that the pair pF, φq generated the functorF˚ : ChupC, γq Ñ ChupD, δq. Next we describe an instance of the (generalised) covariant Grothendieckconstruction 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 closedcategories 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, whereC is a cartesian closed category and γ P ObIdccCat

0pCq “ C0. A morphism pF, φq : pC, γq Ñ pD, δq is a

product-preserving functor F : C Ñ D and a morphism φ :“IdccCat

1 pF q‰0pγq Ñ δ i.e., φ : F0pγq Ñ δ. If

pG, θq : pD, δq Ñ pE , εq, then pG, θq ˝ pF, φq “`G ˝ F, θ ˝ G1pφq

˘. Moreover, 1pC,γq “

`IdC , 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, γq Ñ 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 isa monomorphism, then F˚ is a full embedding of ChupC, γq into ChupD, δq.

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

Chu1p1pC,γqq “ Chu1

`IdC , 1γ

˘““IdC

‰˚

“ 1ChupC,γq.

3One could have considered the cartesian closed functors i.e., the functors preserving the whole structure of a cartesianclosed 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

full embedding and φ a monomorphism. The equality`F0paq, φ ˝ F1pfq ˝ Fab, F0pbq

˘“

`F0pa1q, φ ˝

F1pf 1q ˝ Fa1b1 , F0pb1q˘

implies a “ a1, b “ b1, and as φ is a monomorphism and Fab an isomorphism,hence an epimorphism, we get F1pfq “ F1pf 1q, hence f “ f 1. The fact that F˚ is faithful and fullfollows immediately.

The local Chu functor is a special case of the global one. Namely,

Chu1pIdC , u : γ Ñ δq “ u˚ “ ChuC1puq : ChupC, γq Ñ 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 F0pγq

˘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 isnot 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

ETop0 pX,T q “ pX, PX,T , T q,

PX,T : X ˆ T Ñ 2,

PX,T px,Gq “

"1 , x P G

0 , x R G,

ETop1

`f : pX,T q

cntÝÑ pY, Sq

˘“`f,“E

Top1 pfq

‰´˘: pX, PX,T , T q Ñ pY, PY,S, Sq,

f´1 ““E

Top1 pfq

‰´: S Ñ T, U ÞÑ f´1pUq,

is a strict Chu representation of Top into ChupSet,2q.

Notice that although the proof of the previous proof is constructive, the definition of PX,T isclassical. One can show classically that the Chu space pX, PX,T , T q is separable if and only if thetopology T is T0. Clearly, pX, PX,T , T q is always extensional. The special properties of a topology T ona 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

ESet0 pXq “ pX, fX ,PpXqq,

fX : X ˆ PpXq Ñ 2,

fXpx,Aq “

"1 , x P A

0 , x R A,

ESet1

`f : X Ñ Y

˘“

`f, f´1

˘.

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

Top ChupSet,2q.

Set

ETop

∆ ESet

For all notions mentioned next we refer to [32], chapter 6. Recall that the Scott topology isHausdorff, 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 ,$Xq together with mor-phisms r : pX,ConX ,$Xq Ñ pY,ConY ,$Y q the approximable mappings i.e., appropriate relations r ĎConX ˆ Y . If s : pY,ConY ,$Y q Ñ pZ,ConZ,$Zq, the composition s ˝ r is defined by

Aps ˝ rqz :ô DBPConB

`ArB & Bsz

˘.

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

To show that $X pX,ConX ,$Xq Ñ pX,ConX ,$Xq we use the definition of an information system.To show that 1pX,ConX ,$Xq “ $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 ,$Xq “`|X|, SX

˘,

S1

`r : pX,ConX ,$Xq Ñ pY,ConY ,$Y q

˘“ |r| : |X| Ñ |Y |,

|r|pJq “ y P Y | DJ 1ĎfinJ

`J 1ry

˘(,

is a full embedding of Inf into Top. Consequently, ETop ˝ S : Inf Ñ ChupSet,2q is a a strict Churepresentation of Inf into ChupSet,2q.

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

| $X |pJq “ x P X | DJ 1ĎfinJ

`J 1 $X x

˘(.

If x P | $X |pJq, then J 1 $X x, for some J 1 Ďfin 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 iff : |X| Ñ |Y |, then f “ |rf |, where Arfy :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 ,$Xq “ pY,ConY ,$Y q. If x P X, then txu P |Y |, hence txu Ď Y , andconsequently x P Y . Similarly, we get Y Ď X. If A P ConX , then

AX

“ tx P X | A $X xu P |Y |.

As A Ďfin AX

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

AY

“ ty P Y | A $Y yu P |Y | “ |X|.

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

Yi.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 Churepresentation 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 wepresent the normal Chu representation of the category of Bishop spaces. The notion of Bishop spaceis a constructive, function-theoretic alternative to the set-based notion of topological space, whichwas 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 proofof 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 functionsfrom X to R, by F

˚pXq the bounded elements of FpXq, and by ConstpXq the subset of FpXq of allconstant functions on X. If a P R, we denote by aX the constant function on X with value a. Wedenote 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 iscalled a modulus of continuity of φ on r´n, 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 denoteby 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

all extensional4 subsets of FpXq, we do not mention the embedding iFpXqF : F ãÑ FpXq, 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, whichis called a Bishop topology, or a topology of functions on X, that satisfies the following conditions:

pBS1q If a P R, then aX 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

X R

R.

f

F Q φ ˝ f φ P BicpRq

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 onX. Clearly, if F is a topology on X, then ConstpXq Ď F Ď FpXq, and the set of its bounded elements

4If 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 isdefined by separation, XP “ tx P X | P pxqu, its equality is inherited by that of X and the embedding of XP into X isdefined 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 Bishopspace, which we call the Bishop space of reals. If X is a metric space, the set CppXq of all weaklycontinuous functions of type X Ñ R, as it is defined in [8], p.76, is the set of pointwise continuousones. It is easy to see that the pair WpXq “ pX,CppXqq is Bishop space. Bishop calls CppXq the weaktopology on X, but here we avoid this term, since in [20] we use this term for the Bishop topologythat 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 uniformspace. 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 functionon R, by BS3 we get that if f P F then |f | P F . By BS2 and BS3, and using the following equalities

f ¨g “pf ` gq2 ´ f2 ´ g2

2P F,

f _ g “ maxtf, gu “f ` g ` |f ´ g|

2P F,

f ^ g “ mintf, gu “f ` g ´ |f ´ g|

2P F,

we get similarly that if f, g P F , then f ¨g, f _ g, f ^ g P F . Turning the definitional clauses of a Bishoptopology into inductive rules, Bishop defined in [6], p. 72, the least topology including a given subbaseF0. This inductive definition, which is also found in [8], p. 78, is crucial to the definition of new Bishoptopologies from given ones.

Definition 5.4. The category of Bishop spaces Bis is the subcategory of AffpSet,Rq with objects pairspX,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 amorphism from F to G in Bis, which is called a Bishop morphism, if @gPGpg ˝ h P F q

X Y

R.

h

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 subsetof FpXq, MorpF ,Gq is an extensional subset of FpX,Y q. Similarly to Top, the category Bis is notcartesian closed. The following Chu-representation of Bishop spaces is completely constructive, andits 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

EBis0 pX,F q “ pX, evX,F , F q,

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

BicpXqpfq ô @BPPpXqpboundedpBq ñ 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 abound for B Y tx0u. To avoid such a quantification, if x0 inhabits X, then for every bounded subset pB, x0

1,Mq ofX 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, thendpx, x0q ď dpx, x0

1q ` dpx01, x0q ď M ` dpx0

1, x0q, therefore x P rdx0ď ns, for some n ą M ` dpx0

1, x0q. Hence,

BicpXqpfq ô @nPNpf|rdx0ďns is uniformly continuousq,

since rdx0ď ns “ tx P dpx0, xq ď nu is trivially a bounded subset of X.

10

evX,F : X ˆ F Ñ R,

evX,F px, fq “ fpxq,

EBis1

`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

X ˆ G X ˆ F

Y ˆ G R

hˆidG evX,F

idXˆh˚

evY,G

evX,F

`pidX ˆ h˚qpx, gq

˘“ evX,F

`x, g ˝ h

˘

“ gphpxqq

“ evY,G

`hpxq, g

˘

“ evY,G

`ph ˆ idGqpx, gq

˘.

It is immediate to show that EBis is a functor, which is injective on objects and arrows. Next we showthat EBis 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

X ˆ G X ˆ F

Y ˆ G R

φ`ˆidG evX,F

idXˆφ´

evY,G

evX,F

`pidX ˆ φ´qpx, gq

˘“ evX,F

`x, φ´pgq

˘

““φ´pgq

‰pxqq

“ g`φ`pxq

˘

“ evY,G

`φ`pxq, g

˘

“ evY,G

`pφ` ˆ idGqpx, gq

˘.

From the resulting equality F Q φ´pgq “ g ˝ φ`, and since g P G is arbitrary, we conclude thatφ` 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.

Hence, EBis1

`φ`q “

`φ`, φ´

˘.

In [20] the mapping h˚ ““EBis

1 phq‰´

: G Ñ F is the ring homomorphism induced by h P MorpF ,Gq.Let the Chu space pX, evX,F , F q, and by Definition 2.3 let zevX,F : X Ñ pF Ñ Rq with zevX,F pxq “ px.Consequently, the Chu space pX, evX,F , F q is separable if and only if F separates the points of X:

px “FpF,Rqpx1 :ô @fPF

`pxpfq “R

px1pfq˘

ô @fPF

`fpxq “R fpx1q

˘

ô x “X x1.

11

If ~evX,F : F Ñ pX Ñ Rq with ~evX,F pxq “ qf , then pX, evX,F , F q is always extensional. Clearly, all theseproofs concerning the Chu space pX, evX,F , F q are constructive.

As in the case of the classical Chu representation of Top, the Chu representation of Bis does notinvolve 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

CTop0 pX,T q “ pX, evX , CpXqq,

evX : X ˆ CpXq Ñ R,

evXpx, fq “ fpxq,

CTop1

`h : pX,T q

cntÝÑ pY, Sq

˘“

`h, h˚

˘: pX, evX , CpXqq Ñ pY, evY , CpY qq,

h˚ : CpY q Ñ CpXq, h˚pgq “ g ˝ h,

is only an embedding of Top into ChupSet,Rq. To show that CTop is full, one needs to show thatif`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, thatφ` P CpX,Y q. One can show that φ` P CpX,Y q, if Y is completely regular i.e., a Hausdorff spaceY 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 aresult holds (classically), as one can show classically that the canonical topology of open sets inducedby some Bishop topology is completely regular. From the point of view of the theory of rings ofcontinuous functions, the restriction to crTop is not a loss of generality, as for every topological spaceX 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 qthere is a completely regular space pρX, ρT q and a continuous surjection τX : X Ñ ρX such that forevery completely regular space pY, Sq and continuous function f : X Ñ Y there is a unique continuousfunction ρf : ρX Ñ Y such that the following triangle commutes

X ρX

Y .

τX

ρff

Proposition 5.6 (Chu representation of crTop). The functor EcrTop : crTop Ñ ChupSet,Rq, where

EcrTop0 pX,T q “ pX, evX , CpXqq,

evX : X ˆ CpXq Ñ R,

evXpx, fq “ fpxq,

EcrTop1

`h : pX,T q

cntÝÑ pY, Sq

˘“

`h, h˚

˘: pX, evX , CpXqq Ñ pY, evY , CpY qq,

h˚ : CpY q Ñ CpXq, 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 ifthe 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 afamily of zero sets of X (see [12], p. 38). As

`φ`

˘´1`ζpgq

˘“ tx P X | φ`pxq P ζpfqu

“ tx P X | gpφ`pxqq “ 0u

“ ζpg ˝ φ`q,

and g ˝ φ` P CpXq, we conclude that`φ`

˘´1`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 topologyof C i.e., the smallest topology τpCq that turns all elements of C into continuous functions, is equalto 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 qwith τpCq “ S, then a function φ` : pX,T q Ñ pY, Sq is continuous if and only if g ˝ φ` P CpXq, forevery g P C. A generalisation of the proof of Proposition 5.5 follows next. Its proof is identical to theproof 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 qdefines 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.

Definition 6.1. Let pX,“Xq be a set. A subset of X is a pair pA, iXA q, where pA,“Aq is a set andiXA : A ãÑ X is an embedding (i.e., an injection) of A into X. If pA, iXA q and pB, iXB q are subsets of X,then A is a subset of B, in symbols pA, iXA q Ď pB, iXB q, or simpler A Ď B, if there is f : A Ñ B suchthat the following diagram commutes

A B

X.

f

iXA iXB

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

pA, iXA q “PpXq pB, iXB 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 hasobjects 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 showthat f : A Ď B is an embedding, and that the category PpXq is thin.

Proposition 6.2 (Chu-representation of PpXq). If pX,“Xq is a set, the functor EX : PpXq ÑChupSet,Xq, defined by

EX0

`A, iXA

˘“`A, IXA ,1

˘,

IXA : A ˆ 1 Ñ X, IXA pa, 0q “ iXA paq; a P A,

EX1

`f :

`A, iXA

˘Ñ

`B, iXB

˘˘“ pf, id1q :

`A, IXA ,1

˘Ñ

`B, IXB ,1

˘,

is a strict Chu representation of PpXq into ChupSet,Xq.

Proof. If f :`A, iXA

˘Ñ

`B, iXB

˘, then by the commutativity of the following triangle we get the com-

mutativity of the following rectangle

A ˆ 1 A ˆ 1

B ˆ 1 X

A B

X

fˆid1 IXA

idAˆid1

IXB

f

iXBiXA

13

thus EX1 pfq :

`A, IXA ,1

˘Ñ

`B, IXB ,1

˘. Clearly, EX is a functor injective on objects and arrows, hence

an embedding. Moreover, by the commutativity of the above rectangle we get the commutativity ofthe above triangle. Hence, if pf, id1q :

`A, IXA ,1

˘Ñ

`B, IXB ,1

˘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 withcodomain 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 suchthat the following triangle commutes

a b

γ.

f

i j

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, γq Ñ ChupC, γq,defined by

ESubpC,γq0

`i : a ãÑ γ

˘“

`a, i ˝ pra, 1

˘,

a ˆ 1 a xpra i

ESubpC,γq1

`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.

Proof. The morphism pra is an iso, hence a mono. To show that ESubpC,γq1 pfq :

`a, i ˝ pra, 1

˘Ñ`

b, j ˝ prb, 1˘, we show that the following diagram commutes

aˆ1 aˆ1

bˆ1 γ

fˆ11 i˝pra

1aˆ1

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

b b ˆ 1 1.

a ˆ 1

a 1

pra

prb

f

pr1

fˆ11

pr1

11

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 ESubpC,γq 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,“Xq be a set. An inequality on X, or an apartness relation on X, is a relationx ‰X y such that the following conditions are satisfied:

pAp1q @x,yPX

`x “X y & x ‰X y ñ K

˘.

pAp2q @x,yPX

`x ‰X y ñ y ‰X x

˘.

pAp3q @x,yPX

`x ‰X y ñ @zPXpz ‰X x _ z ‰X yq

˘.

We write pX,“X ,‰Xq to denote the equality-inequality structure of a set X. If`A, iXA

˘is a subset of

X, the canonical inequality on A induced by ‰X is defined by

a ‰A a1 :ô iXA paq ‰X iXA pa1q,

for every a, a1 P A. If pY,“Y ,‰Y q is a set with inequality, a function f : X Ñ Y is called stronglyextensional, if fpxq ‰Y fpx1q ñ x ‰X x1, for every x, x1 P X.

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

Proof. If x, y P X such that x ‰ y, and if x1, y1 P X such that x1 “X x and y1 “X y, we show thatx1 ‰ y1. By pAp3q we get x1 ‰ x, which is excluded from pAp1q, or x1 ‰ 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.

Definition 7.3. Let pX,“X ,‰Xq be a set, and pA, iXA q, pB, iXB q Ď X. We say that A and B are disjointwith respect to ‰X , in symbols AKJ‰X

B, if

A KJ‰X

B :ô @aPA@bPB

`iXA paq ‰X iXB pbq

˘.

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

Clearly, if AKJB, then AXB is not inhabited. The positive disjointness of subsets of X induces thenotion 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.

Definition 7.4. A complemented subset of a set pX,“X ,‰Xq is a pair A :“ pA1, A0q, where pA1, iXA1q

and pA0, iXA0q are subsets of X such that A1KJA0. If DompAq :“ A1 Y A0 is the domain of A, the

indicator function, or characteristic function, of A is the operation χA : DompAq ù 2 defined by

χApxq :“

"1 , x P A1

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,

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

Clearly, A “PKJpXq B ô A1 “PpXq B

1 & A0 “PpXq B0. Notice that if f1 : A

1 Ď B1 and f0 : B0 Ď

A0, then f1, f0 are strongly extensional functions. E.g., if f1pa1q ‰B1 f1pa11q, for some a1, a1

1 P A1,then from the definition of the canonical inequality ‰B1 this means that iX

B1

`f1pa1q

˘‰X iX

B1

`f1pa1

1q˘.

By the extensionality of ‰X we get iXA1pa1q ‰ iX

B1pa11q :ô a1 ‰A1 a1

1.

Definition 7.5. If pX,“X ,‰Xq is a set, the category PKJpXq has objects the complemented subsets of

X and a morphism f : A Ñ B is a pair f “ pf1, f0q : A Ď B i.e., f1 : A1 Ď B1 and f0 : B0 Ď A0. The

unit morphism 1A of A is the pair pidA1 , idA0q, and if g “ pg1, g0q : B Ď C, then g˝f :“ pg1˝f1, f0˝g0q

15

A1 B1

X

C1 C0 B0

X

A0

f1

iXB1iX

A1 iXC1

g1 g0

iXB0 iX

A0iXC0

f0

Clearly, the category PKJpXq is thin.

Proposition 7.6 (Chu representation of PKJpXq). If pX,“X ,‰Xq is a set with an inequality, then thefunctor EX : PKJpXq Ñ ChupSet,X ˆ Xq, defined by

EX

0

`A1, iXA1 , A

0, iXA0

˘“`A1, iXA1 ˆ iXA0 , A

0˘,

EX

1

`pf1, f0q : A Ñ B

˘“ pf1, f0q :

`A1, iXA1 ˆ iXA0 , A

0˘

Ñ`B1, iXB1 ˆ iXB0 , B

0˘,

is a strict Chu representation of PKJpXq into ChupSet,X ˆ Xq.

Proof. Let iXA1 ˆ iX

A0 : A1 ˆ A0 Ñ X ˆ X where

“iXA1 ˆ iX

A0

‰pa1, a0q “

`iXA1pa1q, iX

A0pa0q˘, for every

pa1, a0q P A1 ˆ A0. If pf1, f0q : A Ñ B, then pf1, f0q :`A1, iX

A1 ˆ iXA0 , A

0˘

Ñ`B1, iX

B1 ˆ iXB0 , B

0˘

is amorphism in ChupSet,X ˆ Xq, as the commutativity of the following rectangle

A1ˆB0 A1ˆA0

B1ˆB0 XˆX

f1ˆidB0 iXA1

ˆiXA0

idA1ˆf0

iXB1

ˆiXB0

follows from the commutativity of the following two triangles

A1 B1

X

B0 A0

X

f1

iXB1iX

A1

f0

iXB0 iX

A0

“`iXA1 ˆ iXA0

˘˝`1A1 ˆ f0

˘‰pa1, b0q “

“iXA1 ˆ iXA0

‰pa1, f0pb0qq

“`iXA1pa1q, iXA0pf0pb0qq

˘

“`iXB1pf1pa1qq, iXB0pb0q

˘

““iXB1 ˆ iXB0

‰`f1pa1q, b0

˘

““`iXB1 ˆ iXB0

˘˝`f1 ˆ 1B0

˘‰pa1, b0q.

Clearly, EX is a functor injective on objects and arrows, hence an embedding. It is also full, as theabove equalities also show that the commutativity of the above rectangle implies the commutativity ofthe above triangles. hence, if pf1, f0q :

`A1, iX

A1 ˆ iXA0 , A

0˘

Ñ`B1, iX

B1 ˆ iXB0 , B

0˘

in ChupSet,X ˆXq,then pf1, f0q : A Ñ B.

Consequently, one can identify PKJpXq with the full subcategory of ChupSet,X ˆXq with objects

triplets`A1, iX

A1 ˆ iXA0 , A

0˘, where iX

A1 : A1 ãÑ X and iX

A0 : A0 ãÑ X such that @a1PA1@a0PA0

`iXA1pa1q ‰X

iXA0pa0q

˘. Notice that the Chu category ChupSet,X ˆXq “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 ˆXq,

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 comple-mented predicates Pred‰, defined in the following two sections, respectively, we generalise the Chuconstruction. 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 endofunctoron 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 : aˆ b Ñ Γ0pxq P C1. A morphism φ : px; a, f, bq Ñ py; c, g, dq in ChupC,Γq, or a Chu transform,is a triplet φ “

`φ0, φ`, φ´

˘, where φ0 : x Ñ y, φ` : a Ñ c and φ´ : d Ñ b are in C1 such that the

following diagram commutes

aˆd aˆb

Γ0pxq

Γ0pyq.cˆd

φ`ˆ1d

f

1aˆφ´

Γ1pφ0q

g

If θ “`θ0, θ`, θ´

˘: py; c, g, dq Ñ pz; i, h, jq, let θ ˝ φ “

`θ0 ˝ φ0, θ` ˝ φ`, φ´ ˝ θ´

˘

aˆd aˆb

Γ0pxq

Γ0pyqcˆd

Γ0pzq

iˆj.cˆj

aˆj

φ`ˆ1d

f

1aˆφ´

Γ1pφ0q

g

Γ1pθ0q

1cˆθ´

θ`ˆ1j

h

1aˆpφ´˝θ´q

pθ`˝φ`qˆ1j

Γ1pθ0˝φ0q

Moreover, 1px;a,f,bq “ p1x, 1a, 1bq

aˆb aˆb

Γ0pxq

Γ0pxq.aˆb

1aˆ1b

f

1aˆ1b

Γ1p1xq“1Γ0pxq

f

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

17

triangle as follows:

Γ1pθ0 ˝ φ0q ˝ f ˝ r1a ˆ pφ´ ˝ θ´qs “ Γ1pθ0q ˝ Γ1pφ0q ˝ f ˝ r1a ˆ pφ´ ˝ θ´qs

p2q“ Γ1pθ0q ˝

“Γ1pφ0q ˝ f ˝ p1a ˆ φ´q

‰˝ p1a ˆ θ´q

“ Γ1pθ0q ˝ g ˝ pφ` ˆ 1dq ˝ p1a ˆ θ´q

p4q“ Γ1pθ0q ˝ g ˝ p1c ˆ θ´q ˝ pφ` ˆ 1jq

““Γ1pθ0q ˝ g ˝ p1c ˆ θ´q

‰˝ pφ` ˆ 1jq

““h ˝ pθ` ˆ 1jq

‰˝ pφ` ˆ 1jq

p3q“ h ˝

“pθ` ˝ φ`q ˆ 1j

‰.

Proposition 8.2. Let Γγ : C Ñ C the constant endofunctor with value γ i.e., Γγ0paq “ γ, for every

a P C0, and Γγ1pfq “ 1γ , for every f P C1. The functor Eγ : ChupC, γq Ñ ChupC,Γγq, defined by

Eγ0 pa, f, bq “ pγ; a, f, bq,

Eγ1

``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,Γγq.

Proof. To show that Eγ 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 thecommutativity of the following outer diagram

aˆd aˆb

Γ0pxq

Γ0pyq.cˆd

φ`ˆ1d

f

1aˆφ´

1γ

g

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, Cq Ñ Cat defined by

ChuC0pΓq “ ChupC,Γq,

ChuC1pη : Γ ñ ∆q : ChupC,Γq Ñ ChupC,∆q,

“ChuC

1 pηq‰0px; a, f, bq “ px; a, ηx ˝ f, bq,

aˆb Γ0pxq

∆0pxq

f

ηx

“ChuCq

‰1

`φ0, φ`, φ´

˘“`φ0, φ`, φ´

˘,

is a functor. Moroever, if ηx : Γ0pxq ãÑ ∆0pxq is a mono, for every x P C0, then ChuC1pηq is a full

embedding of ChupC,Γq into ChupC,∆q.

18

Proof. To show that ChuC1 is a functor, it suffices to show that if

`φ0, φ`, φ´

˘: px; a, f, bq Ñ py; c, g, dq

in ChupC,Γq, then`φ0, φ`, φ´

˘: px; a, η ˝ f, bq Ñ py; c, ηy ˝ g, dq in ChupC,∆q. This follows from

the fact that commutativity of the following upper, inner diagram implies the commutativity of thefollowing outer diagram

aˆd aˆb

Γ0pxq

∆0pxq

∆0pyqΓ0pyqcˆd

φ`ˆ1d

f

ηx

Γ1pφ0q

1aˆφ´

∆1pφ0q

g ηy

∆1pφ0q ˝ ηx ˝ f ˝ p1a ˆ φ´q ““∆1pφ0q ˝ ηx

‰˝ f ˝ p1a ˆ φ´q

“ ηy ˝“Γ1pφ0q ˝ f ˝ p1a ˆ φ´q

‰

“ ηy ˝ g ˝`φ` ˆ 1d

˘.

If ηx : Γ0pxq ãÑ ∆0pxq is a mono, for every x P C0, then ChuC1 pηq is injective on objects, and since it is

trivially injective on arrows, it is an embedding. In this case, ChuC1pηq is also full, as the commutativity

of the above outer diagram implies the commutativity of the above, upper, inner diagram. As ηy is amono, the resulted equality

ηy ˝“Γ1pφ0q ˝ f ˝ p1a ˆ φ´q

‰“ ηy ˝ g ˝

`φ` ˆ 1d

˘

implies the equality Γ1pφ0q ˝ 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 categoryand F is a representation, we call F a generalised Chu representation. We call a generalised Churepresentation 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,bPC0the canonical isomorphisms of F , and let η : F ˝ Γ ñ ∆ ˝ F

C D

C D.

F

F

Γ ∆ηùñ

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

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

˘

F0paq ˆ F0pbq F0pa ˆ bq F0pΓ0pxqq ∆0pF0pxqqFab F1pfq ηx

19

pF˚q1`φ0, φ`, φ´

˘:`F0paq, ηx ˝ F1pfq ˝ Fab, F0pbq

˘Ñ

`F0pyq;F0pcq, ηy ˝ F1pgq ˝ Fcd, F0pdq

˘,

pF˚q1`φ0, φ`, φ´

˘“`F1pφ0q, F1pφ`q, F1pφ´q

˘,

where`φ0, φ`, φ´

˘: px; a, f, bq Ñ py; c, g, dq

˘, is a functor.

Proof. We show that pF˚q1`φ0, φ`, φ´

˘is well-defined i.e., the following diagram commutes:

F0paqˆF0pdq F0paqˆF0pbq

∆0pF0pxqq

∆0pF0pyqq.F0pcqˆF0pdq

F1pφ`qˆ1F0pdq

ηx˝F1pfq˝Fab

1F0paqˆF1pφ´q

∆1pF1pφ0qq

ηy˝F1pgq˝Fcd

LetA “ ∆1pF1pφ0qq ˝ ηx ˝ F1pfq ˝ Fab ˝ r1F0paq ˆ F1pφ´qs,

B “ ηy ˝ F1pgq ˝ Fcd ˝ rF1pφ`q ˆ 1F0pdqs.

By the definition of a morphism`φ0, φ`, φ´

˘: px; a, f, bq Ñ py; c, g, dq

˘we get

Γ1pφ0q ˝ f ˝ p1a ˆ φ´ “ g ˝ pφ` ˆ 1dq ñ

p˚q F1pΓ1pφ0qq ˝ F1pfq ˝ F1p1a ˆ φ´q “ F1pgq ˝ F1pφ` ˆ 1dq.

As F1pφ` ˆ 1dq ˝ Fad “ Fcd ˝ rF1pφ`q ˆ F1p1dqs, and since the following rectangle commutes

F0pΓ0pxqq F0pΓ0pyqq

∆0pF0pyqq,∆0pF0pxqq

F1pΓ1pφ0qq

ηyηx

∆1pF1pφ0qq

A “ ∆1pF1pφ0qq ˝ ηx ˝ F1pfq ˝ Fab ˝ rF1p1aq ˆ F1pφ´qs

“ ∆1pF1pφ0qq ˝ ηx ˝ F1pfq ˝ F1p1a ˆ φ´q ˝ Fad

“ ηy ˝ F1pΓ1pφ0qq ˝ F1pfq ˝ F1p1a ˆ φ´q ˝ Fad

p˚q“ ηy ˝ F1pgq ˝ F1pφ` ˆ 1dq ˝ Fad

“ ηy ˝ F1pgq ˝ Fcd ˝ rF1pφ`q ˆ 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 categoryÿ

CPccCat

EndpCq

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

20

C D E

EC D

F G

F G

Γ ∆ Eηùñ

θùñ

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

G0pF0pΓ0paqqq G0p∆0pF0paqqq

E0pG0pF0paqqq

G1pηaq

θF0paqpθ˚ηqa

Moreover, 1pC,Γq “`IdC , 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 inC1, then, as η : F ˝ Γ ñ ∆ ˝ F , the following left rectangle commutes:

F0pΓ0paqq F0pΓ0pbqq

∆0pF0pbqq∆0pF0paqq

G0p∆0pF0paqqq G0p∆0pF0pbqqq

E0pG0pF0pbqqqE0pG0pF0paqqq

F1pΓ1pfqq

ηb#1ηa

∆1pF1pfqq

G1p∆1pF1pfqqq

θF0pbq#2θF0paq

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. Thecommutativity of the following rectangle diagram follows:

G0pF0pΓ0paqqq G0pF0pΓ0pbqqq

E0pG0pF0pbqqq,E0pG0pF0paqqq

G1pF1pΓ1pfqqq

pθ˚ηqbpθ˚ηqa

E1pG1pF1pfqqq

pθ ˚ ηqb ˝ G1pF1pΓ1pfqqq “ θF0pbq ˝ G1pηbq ˝ G1pF1pΓ1pfqqq

p˚q“ θF0pbq ˝ G1p∆1pF1pfqqq ˝ G1pηaq

p#2q“ E1pG1pF1pfqqq ˝ θF0paq ˝ G1pηaq

“ E1pG1pF1pfqqq ˝ pθ ˚ ηqa.

If pF, ηq : pC,Γq Ñ pD,∆q, then pF, ηq ˝ 1pC,Γq “ pF, ηq ˝`IdC , 1Γ

˘“ pF ˝ IdC , η ˚ 1Gq “ pF, ηq, as

η ˚ 1G “ η. Similarly, if 1pD,∆q ˝ pF, ηq “`IdD, 1∆

˘“ pIdD ˝ 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, θq ˝ pF, ηqs “ rpH, ρq ˝ pG, θqs ˝ pF, ηq.

21

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

CHU :ÿ

CPccCat

EndpCq Ñ Cat,

CHU0pC,Γq “ ChupC,Γq,

CHU1

`F, ηq : pC,Γq Ñ pD,∆q

˘: ChupC,Γq Ñ 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 isa 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,

CHU1p1pC,Γqq “ CHU1

`IdC , 1Γ

˘““IdC

‰˚

“ 1CHUpC,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.

The equality rpG ˝ F q˚s1pφ0, φ`, φ´q “ pG˚q1`pF˚q1pφ0, 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 ˝

Fab, F0pbq˘

“`F0px1q;F0pa1q, ηx1 ˝ F1pf 1q ˝ Fa1b1 , F0pb1q

˘implies x “ x1, a “ a1, b “ b1, and as ηx is a

monomorphism and Fab an isomorphism, hence an epimorphism, we get F1pfq “ F1pf 1q, hence f “ f 1.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

ChuC1pηq “

“IdC

‰˚.

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 thelogic and type theory of standard sets in fibred form. Here we present this category within BST.

Definition 10.1. The objects of the category of predicates Pred are triplets pX, iXA , Aq, where X isa set and pA, iXA q is a subset of X. If pX, iXA , Aq and pY, iYB , Bq are objects of Pred, a morphismu : pX, iXA , Aq Ñ pY, iYB , Bq in Pred is a pair of functions u “

`u0, u`

˘, where u0 : X Ñ Y and

u` : A Ñ B such that the following diagram commutes

X Y .

A B

iXA iYB

u`

u0

If v “`v0, v`

˘: pY, iYB , Bq Ñ pZ, iZC , Cq, let v ˝ u : pX, iXA , Aq Ñ pZ, iZC , Cq, defined by v ˝ u “

`v0 ˝

u0, v` ˝ u`˘. Moreover, 1pX,iXA ,Aq “

`idX , idA

˘.

22

In [15], p. 11, the embedding iXA : A Ñ X is omitted for simplicity, and a morphism u is just afunction u0 : X Ñ Y such that

@aPADbPB

`u0piXA paqq “Y iYBpbq

˘.

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 thatu0piXA paqq “Y iYBpbq. By Myhill’s principle of non-choice (or unique choice), introduced in [19], thereis a (necessarily) unique map u` that makes the above diagram commutative. As this principle isavoided in BST, we prefer to present a morphism u in Pred as a pair pu0, u`q. It is immediate to seethat if u0 is an embedding, then u` is an embedding, and if u0 is strongly extensional, then u` is alsostrongly extensional. For a specific set X the “fibre” category PredX is the subcategory of Pred withobjects triplets of the form pX,A, iXA q with X fixed, while a morphism u : pX,A, iXA q Ñ pX,B, iXB q is apair pidX , uABq, and the required commutativity of the following diagram

X X

A B

iXA iXB

uAB

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 ÑChupSet, Idq, defined by

ESet0 pXq “

`X;X, IXX ,1

˘,

IXX : X ˆ 1 Ñ Id0pXq “ X, IXX px, 0q “ x; x P X,

ESet1

`f : X Ñ Y

˘“ pf, f, id1q :

`X, IXA , A

˘Ñ

`Y, iYB , B

˘˘“`u0, u`, id1

˘:`X;X, IXX ,1

˘Ñ

`Y ;Y, IYY ,1

˘,

is a strict generalised Chu representation of Set into ChupSet, Idq.

(ii) The functor EPred : Pred Ñ ChupSet, Idq, defined by

EPred0

`X, iXA , A

˘“`X;A, IXA ,1

˘,

IXA : A ˆ 1 Ñ Id0pXq “ X, IXA pa, 0q “ iXA paq; a P A,

EPred1

`u “

`u0, u`

˘:`X, IXA , A

˘Ñ

`Y, IYB , B

˘˘“`u0, u`, id1

˘:`X;A, IXA ,1

˘Ñ

`Y ;B, IYB ,1

˘,

is a strict generalised Chu representation of Pred into ChupSet, Idq.

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

Pred ChupSet,Idq.

Set ChupSet,Idq

F Id

ESet

EPred

Proof. We show only (ii). If u “`u0, u`

˘:`X, iXA , A

˘Ñ

`Y, iYB , B

˘, then

`u0, u`, id1

˘:`X;A, IXA ,1

˘Ñ`

Y ;B, IYB ,1˘, as the commutativity of the rectangle

X Y

A B

iXA iYB

u`

u0

23

implies the commutativity of the following diagram

Aˆ1 Aˆ1

X

YBˆ1

u`ˆid1

IXA

idAˆid1

Id1pu0q“u0

IYB

u0`IXA pa, 0q

˘“ u0

`iXA paq

˘“ iYB

`u`paq

˘“ IYB

`u`paq, 0

˘.

Clearly, EPred is injective on objects and arrows, hence EPred is an embedding. It is also full, as if`u0, u`, id1

˘:`X;A, IXA ,1

˘Ñ

`Y ;B, IYB ,1

˘, then u “

`u0, u`

˘:`X, iXA , A

˘Ñ

`Y, iYB , B

˘, because the

commutativity of the last diagram implies the commutativity 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, wherex P C0 and i P C1pa, xq is a monomorphism, and morphisms pf0, f`q : px, i : a ãÑ xq Ñ py, j : b ãÑ yqwith j ˝ f` “ f0 ˝ i

x y.

a b

i j

f`

f0

If pg0, g`q : py, j : b ãÑ yq Ñ pz, k : e ãÑ zq, then pg0, g`q ˝ pf0, f`q “ pg0 ˝ f0, g` ˝ f`q. Moreover,1px,i : aãÑxq “ p1x, 1aq.

Proposition 10.4 (Generalised Chu representation of PredpCq). If C is a ccc, the functor

EPredpCq : PredpCq Ñ ChupC, IdCq,

EPredpCq0

`x, i : a ãÑ x

˘“`x; a, i ˝ pra, 1

˘,

a ˆ 1 a xpra i

EPredpCq1

``f0, f`

˘:`x, i : a ãÑ x

˘Ñ

`y, j : b ãÑ y

˘˘“`f0, f`, 11

˘:`x; a, i ˝ pra, 1

˘Ñ

`y; b, j ˝ prb, 1

˘,

is a strict generalised Chu representation of PredpCq into ChupC, IdCq.

Proof. The morphism pra is an iso, hence a mono. To show that EPredpCq1

`f0, f`

˘:`x; a, i ˝ pra, 1

˘Ñ`

y; b, j ˝ prb, 1˘, we show that the following diagram commutes

aˆ1 aˆ1

x

ybˆ1

f`ˆ11

i˝pra

1aˆ1

f0

j˝prb

24

f0 ˝ pi ˝ praq “ pf0 ˝ 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 EPredpCq is injective on objects. It is trivially injective on arrows. To show that it is full, letpφ0, φ`, φ´q :

`x; a, i˝pra, 1

˘Ñ

`y; b, j˝prb, 1

˘. Clearly, φ´ “ 11. Moreover, by the previous equalities

we get pφ0 ˝ iq ˝ pra “ pj ˝ φ`q ˝ pra, and since pra is a mono, we conclude that φ0 ˝ 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 acategory Pred‰. Its subcategory Pred‰

se is formed by considering in the definition of the morphismsin Pred‰ strongly extensional functions. The motivation behind the next definition is to get a strictgeneralised Chu representation of Pred‰pSetq into the Chu category over Set and the endofunctorId2 : Set Ñ Set, defined by

Id20pXq “ X ˆ X,

Id21pf : X Ñ Y q : X ˆ X Ñ Y ˆ Y,

rId21pfqspx, x1q “`fpxq, fpx1q

˘.

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 extensionalfunctions between them, and A :“ pA1, A0q is a complemented subset of X. If pX,Aq and pY,Bq areobjects 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

X Y

A1 B1

Y X.

B0 A0

iXA1 iY

B1

u`

u0

iYB0 iX

A0

u´

u0

If u “`v0, v`, v´

˘: pY,Bq Ñ pZ,Cq, we define the composite morphism v ˝ u : pX,Aq Ñ pZ,Cq by

v ˝ u “`v0 ˝ u0, 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,

EPred‰pSetq0

`X,A

˘“

`X;A1, iXA1 ˆ iXA0 , A

0˘,

iXA1 ˆ iXA0 : A1 ˆ A0 Ñ Id20pXq “ X ˆ X,

EPred‰pSetq1

`u0, u`, u´

˘“`u0, u`, u´

˘:`X;A1, iXA1 ˆ iXA0 , A

0˘

Ñ`Y ;B1, iYB1 ˆ iYB0 , B

0˘,

where`u0, u`, u´

˘:`X,A

˘Ñ

`Y,B

˘, is a strict generalised Chu representation of Pred‰pSetq into

ChupSet, Id2q.

25

Proof. If`u0, u`, u´

˘:`X,A

˘Ñ

`Y,B

˘, then

`u0, u`, u´

˘:`X;A1, iX

A1 ˆ iXA0 , A

0˘

Ñ`Y ;B1, iY

B1 ˆiYB0 , B

0˘, as the commutativity of the following two rectangles

X Y

A1 B1

Y X.

B0 A0

iXA1 iY

B1

u`

u0

iYB0 iX

A0

u´

u0

implies the commutativity of the following diagram

A1ˆB0 A1ˆA0

XˆX

Y ˆYB1ˆB0

u`ˆidB0

iXA1

ˆiXA0

idA1ˆu´

Id21

pu0q

iYB1

ˆiYB0

Id21pu0q“`iXA1 ˆ iXA0

˘`idA1 ˆ u´

˘`a1, b0

˘‰“ Id21pu0q

“`iXA1pa1q, iXA0pu´pb0qq

˘‰

“`u0piXA1pa1qq, u0piXA0pu´pb0qqq

˘

“`iYB1pu`pa1qq, iYB0pb0q

˘

““iYB1 ˆ iYB0

‰`u`pa1q, b0q

˘

““iYB1 ˆ iYB0

‰`u` ˆ idB0

˘pa1, b0q.

Clearly, EPred‰is injective on objects and arrows, hence EPred‰

is an embedding. It is also full, asif`u0, u`, u´

˘:`X;A1, iX

A1 ˆ iXA0 , A

0˘

Ñ`Y ;B1, iY

B1 ˆ iYB0 , B

0˘, then

`u0, u`, u´

˘:`X,A

˘Ñ

`Y,B

˘,

because the commutativity of the last diagram implies the commutativity of the above two rectangles.

12 The Chu construction and the antiparallel Grothendieck construc-

tion

So far, we related the two constructions through the domain of the global Chu functor. The domainof the generalised global Chu functor has also some affinity to the Grothendieck construction. Nextwe discuss the relation between the two constructions themselves. A first result in this direction isthe following result of Abramsky in [2], p. 14. Notice that instrumental to the proof of his result is acontravariant, or reverse, definition of the arrows in the Grothendieck category. Namely, if P : Cop ÑCAT, where CAT is the category of (large) categories, an arrow pf, φq : pa, xq Ñ pb, yq in the categoryGrothpC, P q, where x, y are objects in P0paq and P0pbq, respectively, is an arrow f : b Ñ a in C andan arrow φ : rP1pfqs0pxq Ñ y in P0pbq. In the literature the standard approach to the definition of thecategory 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 definitionof the arrows in GrothpC, P q is necessary to Abramsky’s result. Next follows the generalisation ofAbramsky’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, xq Ñ

26

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

h˚ : ChuxpC, γq Ñ Chux1pC, γq,

where h˚0pa, f, xq “

`a, f ˝ p1a ˆ hq, x1

˘and h˚

1

`φ`, 1x

˘“

`φ`, 1x

˘. If Chuγ : Cop Ñ CAT is the

contravariant functor defined byC0 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, orthe antiparallel category of elements, on the product category, in case the ccc C is locally small. In thenext definition we could consider a product C ˆ D instead of a product C ˆ C, and more options occurif larger products of categories are considered. If C is a category, a, b P C0, and S : pC ˆ Cqop Ñ Set acontravariant functor on C ˆ C, let the induced contravariant functors

Sa : Cop Ñ Set, Sapcq “ S0pa, cq Sapg : c Ñ c1q “ S1p1a, gq : S0pa, c1q Ñ S0pa, cq,

bS : Cop Ñ Set, bSpcq “ S0pc, bq bSpg : c Ñ c1q “ S1pg, 1bq : S0pc1, bq Ñ S0pc, bq.

Definition 12.2. Let C be a category and S : pC ˆ Cqop Ñ Set. The pcontravariantq antiparallelGrothendieck category GrothÔ

`CˆC, 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´qspuq “ rySpφ`qspvq

S0pb, yq S0pa, yq.

S0pa, xq

ySpφ`q

Sapφ´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:

S0pc, zq S0pb, zq S0pa, zq.

S0pb, yq S0pa, yq

S0pa, xq

S1pθ`, 1zq S1pφ`, 1zq

S1p1b, θ´q S1p1a, θ

´q

S1pφ`, 1yqS1p1a, φ

´q

27

rS1p1a, φ´ ˝ θ´qspuq “

“S1

`p1a, φ

´q ˝ p1a, θ´q˘‰

puq

““S1p1a, θ

´q ˝ S1p1a, φ´q‰puq

“ rS1p1a, θ´qs

`“S1p1a, φ

´q‰puq

˘

p5q“ rS1p1a, θ

´qs`“S1pφ`, 1yq

‰pvq

˘

““S1

`pφ`, 1yq ˝ p1a, θ

´q˘‰

pvq

““S1pφ` ˝ 1a, 1y ˝ θ´q

‰pvq

““S1pφ`, θ´q

‰pvq

““S1p1b ˝ φ`, θ´ ˝ 1zq

‰pvq

““S1

`p1b, θ

´q ˝ pφ`, 1zq˘‰

pvq

“ rS1pφ`, 1zqs`“S1p1b, θ

´q‰pvq

˘

p6q“ rS1pφ`, 1zqs

`“S1pθ`, 1zq

‰pwq

˘

““S1

`pθ`, 1zq ˝ pφ`, 1zq

˘‰pwq

“ rS1pθ` ˝ 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. IfC is a locally small ccc, we have the Set-valued contravariant functor

Homp´ˆ´, γq˘: pC ˆ Cqop Ñ Set,

pa, bq ÞÑ Hompa ˆ b, γq,

Homp´ˆ´, γq˘1pφ` : a Ñ a1, φ´ : b Ñ b1q : Hompa1 ˆ b1, γq Ñ Hompa ˆ b, γq,

“Homp´ˆ´, γq

˘1

`φ`, φ´

˘‰phq “ h ˝

`φ` ˆ φ´

˘

aˆb a1ˆb1 γ.φ`ˆφ´ h

“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 antiparallelGrothendieck category GrothÔ

`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 moreinteresting, as the functor S is only Set-valued, and not CAT-valued. Next we describe the globalversion of the functor Homp´ˆ´, γq.

Proposition 12.4. If C is a locally small ccc, the functor

Homp´ˆ´, ´q : C Ñ Fun`pC ˆ Cqop,Set

˘,

“Homp´ˆ´, ´q

‰0pγq “ Homp´ˆ´, γq,

“Homp´ˆ´, ´q

‰1pf : γ Ñ γ1q “ ηf : Homp´ˆ´, γq ñ Homp´ˆ´, γ

1q,

ηf

pa,bq: Hompa ˆ b, γq Ñ Hompa ˆ b, γ1q,

ηf

pa,bqphq “ f ˝ h

28

aˆb γ γ1.h f

etaf

pa,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 Governmentand the Länder.

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