Remarks on the Tripos To Topos Construction: extensionality, comprehensions, quotients and cauchy-complete objects

0.19 MB·

by  Fabio Pasquali

Remarks on the Tripos To Topos Construction:
extensionality, comprehensions, quotients and
cauchy-complete objects
4
Fabio Pasquali∗
1
0
2

b
e Abstract
F
We give a description of the tripos-to-topos constructions in terms of

4 four free constructions. We prove that these compose up to give a free
2 construction from the category of triposes and logical morphisms to the
categoryoftoposesandlogicalfunctors. Thenweshowthatothersimilar
]
T constructions, i.e. theone given by Frey in [2] and that of Carboni in [1]
are instances of thisone.
C
.
h
Introduction
t
a
m
Oneofthemostrelevantfeatureonemeetsindealingwiththetheoryoftriposes
[
is the Tripos To Topos construction. The Martin Hyland effective topos and

2 thetoposofsheavesoveralocale,althoughtheyhavequitedifferentproperties,
v e.g the former is not Grothendieck, they both are relevant instances of that
7 construction [8]. The construction was presented in [3], where triposes were
6
suitable Sets-indexed collections of Heyting algebras, and Pitts, in his PhD
8
thesis, generalized the theory to triposes over an arbitrary base [7].
7
. Loosely speaking, the Tripos To Topos construction can be seen as a way to
1
add to a tripos exactely those properties that one needs to obtain a subobjects
0
4 triposofanelementarytopos. Thosepropertiesare: havingcomprehensions(in
1 the sense of Lawvere), quotients, an extensional equality and cauchy-complete
: base (definitions are given in the notes). It turns out that each one of those
v
i properties can be freely add to a tripos, providing (by composition) an adjoint
X
situation between LTripos, the category of triposes and logical functors, and
r itssubcategoryontriposeswithcomprehensions,quotients,extensionalequality
a
and a cauchy-complete base. Moreover the latter is equivalent to LTopos, the
categoryofelementarytoposesandlogicalfunctors. Inotherwordsthere exists
an adjoint situation
--
LTripos mm ⊥ LTopos
in which the right adjoint is the obvious forgetful functor, while the reflector is
the functor that freely adds the four properties listed above.
∗Part of the research in this paper was carried out while the author worked at Utrecht
UniversityintheNWO-Project‘TheModelTheoryofConstructiveProofs’nr. 613.001.007.
1
1 Regular doctrines and Triposes
This section contains the basic definitions regarding doctrines and triposes and
categories of these, see [8] for details.
Let ISL be the category of inf-semilattices and homomorphisms.
Definition 1.1. A doctrine is a pair (C,P) where C is a non-empty category
with binary products and P a functor
P:Cop −→ISL
We shallrefer to C as the basecategoryofthe doctrine. We will often write
f∗ instead of P(f) to denote the action of the functor P on the morphism f of
C and we shall call it reindexing along f. Binary meets in inf-semilattices are
denoted by ∧. Elements in P(A) will often be called formulas over A and the
top element is denoted by ⊤ .
A
Definition 1.2. A doctrine is regular if for eacharrowf:X −→Y in C there
exists a functor ∃ ⊣f∗ satisfying
f
- Beck-Chevalley condition: i.e. for every pullback of the form
f
X //Y
g h
(cid:15)(cid:15) (cid:15)(cid:15)
Z //W
k
it holds that ∃ ◦g∗ =h∗◦∃
f k
- Frobenius Reciprocity: i.e. ∃ (α∧f∗β)=∃ α∧β.
f f
We denote by RD the category of regular doctrines and regular functors:
the objects are regular doctrines and the arrows are pairs (F,f)
Cop
◗◗◗◗P◗◗◗
((
Fop D(cid:27)(cid:27) opf♠♠♠ ♠R♠♠♠66ISL
where the functor F:C −→ D preserves binary products and f is a natural
transformation from the functor P to the functor R◦F which commutes with
left adjoints, i.e. it preserves the regular structures (∃, ∧, ⊤ and δ).
Definition 1.3. A regular doctrine (C,P) is a tripos if
- P factors through the category of Heyting algebras and homomorphisms
- foreverymorphismf inC thefunctor f∗ hasarightadjoint∀ satisfying
f
Beck-Chevalley condition
- for every X in C, there exists PX in C and a formula ∈ in P(X ×PX)
X
suchthatfor everyobjectY inC andformulaγ in P(X×Y) there exists
{γ}:Y −→PX such that (id ×{γ})∗ ∈ =γ.
X X
2
We shall abbreviate with δ the formula ∃ ⊤ of P(A×A) and call
A hidA,idAi A
it internal equality predicate over A. We say that PX is a weak power object
of X, moreover we say that a tripos has strong power objects if, for every A
in C, there exists an object PA in C such that PA is a weak power object of A
and
δPA =∀hπ2,π3i(hπ1,π2i∗ ∈A↔hπ1,π3i∗ ∈A)
We call LT the subcategoryof RDof triposesand logicalfunctors: the objects
are triposes and the arrows are those arrows (F,f) of RD such that f is an
homomorphismofHeytingalgebraswhichcommuteswithleftandrightadjoints,
i.e. it preserves all the first order structure (∀, ∃, →, ∧, ∨, ⊤, ⊥ and δ) and
F preserves weak power objects, i.e. F maps a weak power object of A into a
weak power object of FA.
2 Comprehensions
We recallinthissectionsomeknownconstructionsinvolvingthenotionofcom-
prehensions and related properties.
Definition2.1. Adoctrine(C,P)issaidtohavecomprehensionsifforevery
objectAofCandformulaαoverA,thereexistsamoprhism⌊α⌋:X −→Asuch
that⌊α⌋∗α=⊤ andforeveryf:Y −→Awithf∗α=⊤ thereexistsaunique
X X
morphism h:Y −→ X with ⌊α⌋◦h = f. Moreover ⌊α⌋ is said to be full if for
every formula β over A, ⌊α⌋∗α≤⌊α⌋∗β if and only if α≤β.
We denote by RD the subcategoryof RD whoseobjects areregulardoc-
(c)
trineswithfull comprehensionsandmorphismsarepairs(F,f)of RDinwhich
F preserves comprehensions. The inclusion of RD in RD has a left adjoint
(c)
c:RD−→RD .
(c)
We briefly recall it from [6]. Given a regular doctrine (C,P), we denote the
free regular doctrine with full comprehensions by (C ,P ). C is the category
c c c
whose objects are pairs (A,α) in which α is an element of P(A), while a mor-
phism f:(A,α) −→ (B,β) is a morphism f:A −→ B in C such that α ≤ f∗β.
Identity and composition are those of C.
The functor P is defined by the following assignment on objects of C
c c
P (A,α)={φ ǫ P(A) | φ≤α}
c
and by the following assignment on morphisms f:(A,α)−→(B,β) of C
c
P (f)(ψ)=f∗ψ∧α
c
It is straightforward to see that P (A,α) is an infsemilattice: if φ ≤ α and
c
ψ ≤α, then φ∧ψ ≤α, while the top element is α itself.
The unite of the adjunction is the family of morphisms (H,η) :(C,P) −→
P
(C ,P ) where H:C −→ C sends every object A to (A,⊤ ) and acts as the
c c c A
identity on morphisms, while η is the family of identity homomorphisms be-
tween inf-semilattices, since P (A,⊤ )=P(A).
c A
3
The left adjointrestrictsto the inclusionof LT into LT where the former
(c)
is categoryof triposes with full comprehensionsand logicalmorphisms preserv-
ing them. It is straightforward to see that for φ a formula over (A,α), the
functor ψ 7→(φ→ψ)∧α from P (A,α) to itself is right adjoint to ψ 7→ψ∧φ.
c
Moreovergivenf:(A,α)−→(B,β),therightadjointtoP (f)is∀ (α→−)∧β.
c f
Finally the following assignments
P(A,α)=(P(A),∀ (∈ →π∗α))
π2 A 1
∈ =∈ ∧ π∗∀ (∈ →π∗α))
(A,α) A 2 π2 A 1
determine a weak power object of (A,α), making (C ,P ) a tripos.
c c
It is immediate to see that (H,η) :P −→ P is a logical morphism by replac-
P c
ing,inthe assignmentsabove,αandβ with⊤ and⊤ respectively. Moreover
A B
if (F,f):(C,P) −→ (D,R) is logical, its unique extension (F,f):(C ,P ) −→
c c
(D,R) is also logical. Then the restriction of the functor c to LT, which we
denote also by c, fits in the following commutative diagram
c
,,
RDOO kk ⊥ RD(c)
OO
(cid:31)? c (cid:31)?
,,
LT jj ⊥ LT(c)
3 Quotients
Beforerecallingfrom[5]thedefinitionofquotients,weneedthenotionofequiv-
alence relation in a regular doctrine.
In a given regular doctrine (C,P), an equivalence relation ρ over an object
A of C is a formula over A×A such that the following conditions hold
δ ≤ρ
A
ρ=hπ ,π i∗ρ
2 1
hπ ,π i∗ρ∧hπ ,π i∗ρ≤hπ ,π i∗ρ
1 2 2 3 1 3
Definition 3.1. A regular doctrine (C,P) is said to have quotients if for
every A in C and every equivalence relation ρ over A there exists a morphism
q:A −→ A/ρ such that ρ ≤ (q×q)∗δ and for every morphism f:A −→ Y
A/ρ
such that ρ ≤ (f ×f)∗δ there exists a unique h:A/ρ −→ Y with h◦q = f.
Y
A/ρ is said to be an effective quotients if ρ=(q×q)∗δ .
A/ρ
We denote by RD the subcategory of RD whose object are regular doc-
(c,q) (c)
trineswithfullcomprehensionsandeffectivequotientsandmorphismsarethose
morphisms of RD which preserve quotients.
(c)
Maietti and Rosolini in [5] proved that the inclusion of RD in RD has
(c,q) (c)
a left adjoint q:RD −→ RD . Given a regular doctrine (C,P) we shall
(c) (c,q)
4
denote the free regular doctrine with effective quotients by (C ,P ). C is the
q q q
categorywhoseobjectsarepairs(A,ρ)inwhichAisinCandρisanequivalence
relation on A. An arrow f:(A,ρ) −→(B,σ) is an arrow f:A−→ B in C such
that ρ≤(f ×f)∗σ. The functor P is determined by the assignment
q
P (A,ρ)={φ ǫ P(A) | π∗φ∧ρ≤π∗φ}
q 1 2
andbyP (f)=f∗. Theuniteoftheadjunctionisthefamily(∇,ζ) :(C,P)−→
q P
(C ,P ) where ∇ is the identity on morphisms and sends an object A of C to
q q
(A,δ )inC ,whileζisthefamilyofidentityhomomorphisms,sinceP (A,δ )=
A q q A
P(A).
As before, the left adjoint restricts to the inclusion of LT into LT ,
(c,q) (c)
where the former is the category of triposes with full comprehensions and ef-
fective quotients and morphisms are those morphisms of LT which preserve
(c)
quotients. The proof that the above quotients completion preserves rights ad-
joints (→ and ∀) is in [5]. If P(X) is a weak power object of X in C, then a
power objects of (X,ρ) in C is
q
(P(X),∀ (hπ ,π i∗ ∈ ↔hπ ,π i∗ ∈ ))
hπ2,π3i 1 2 X 1 3 X
while
∈ = ∈ ∧ ∀ (hπ ,π i∗ρ→hπ ,π i∗ ∈ )
(X,ρ) X hπ2,π3i 1 2 1 3 X
Moreover ∇P(X) is a strong power object of ∇X. Therefore (∇,ζ) is a mor-
phism in LT . It is straightforward to prove that if (F,f):(C,P) −→ (D,R)
c
is a logical morphism, then its unique extension (F,f):(C ,P ) −→ (D,R) is
q q
logical. Hence we have that the restriction of q to LT fits into the following
(c)
diagram
q
--
RD(c) ll ⊥ RD(c,q)
OO OO
(cid:31)? q (cid:31)?
,,
LT(c) ll ⊥ LT(c,q)
4 Extensionality
The notionof extensionalityprovidesa link between the internalequality pred-
icate of a doctrine and the actual equality of terms seen as morphisms of the
base category.
Definition 4.1. A regular doctrine (C,P) is extensional if for every A in C
and every pairs of parallel morphisms f,g:X −→A, it holds that
f =g if and only if ⊤ ≤hf,gi∗δ
X A
Given a regular doctrine (C,P), we denote by (C ,P ) the regular doctrine
e e
whereC hasthe sameobjectsofC andmorphism[f]:X −→Aareequivalence
e
classes of morphisms of C with respect to the equivalence relation
f ∼g if and only if ⊤ ≤hf,gi∗δ
X A
5
thefunctorP isdefinedbyP (A)=P(A)while[f]∗isf∗. Authorsin[6]proved
e e
that reindexing does not depend on the choice of representatives. (C ,P ) is
e e
called the extensional collapse of (C,P).
The morphism (L,λ) :(C,P) −→ (C ,P ), where L(f) = [f] and λ is the
P e e
familyofidentityhomomorphismsbetweeninfsemilattices,constitutestheunite
of the adjunction
e
,,
RDkk ⊥ RD(e)
It is immediate to see that the extensional collapse can be restricted to the
categoryofregulardoctrines withcomprehensionsandquotients andalsoto its
subcategory of triposes; in other words we have the following diagram
e
--
RD(c,q) mm ⊥ RD(c,q,e)
OO OO
(cid:31)? e (cid:31)?
--
LT(c,q) ll ⊥ LT(c,q,e)
We conclude the section by proving two lemmas, which require the notions
of comprehensions, quotients and extensionality seen so far and which turn out
to be useful in the proof of proposition 6.1.
Lemma4.2. Inanextensionalregulardoctrine(C,P)withfullcomprehensions
and quotients, the following hold:
i) C has equalizers
ii) if f:X −→Y in C is mono, then (f ×f)∗δ =δ
Y X
iii) ifρisanequivalencerelationonA,thenthecanonicalquotientsmorphism
q:A−→A/ρ is internally surjective, i.e. ⊤ =∃ ⊤
A/ρ q A
Proof. i) for every pair of parallel arrows f,g:A⇒B in C the following
⌊hf,gi∗δ ⌋:E −→A⇒B
B
is an equalizer diagram. The composition is equal because of extensionality.
The universalproperty of equalizerscomes from the universalproperty of com-
prehensions. ii) Suppose f:X −→ Y is mono. By i) C has pullbacks, thus
considerthat the pullback off alongitself is isomorphic to X. iii) Suppose ρ is
anequivalencerelationonA; considerthe canonicalquotientmapq:A−→A/ρ
and the diagram
q
A //A/ρ
❄
❄h❄❄❄❄❄❄(cid:31)(cid:31) ④④④④④⌊④∃④q④⊤== A⌋
X
where h is the unique arrow which comes from the universal property of com-
prehensions since ⊤ ≤q∗∃ ⊤ . We have then
A q A
ρ≤(q×q)∗δ =(h×h)∗(⌊∃ ⊤ ⌋×⌊∃ ⊤ ⌋)∗δ =(h×h)∗δ
A/ρ q A q A A/ρ X
6
where the last equality holds by ii) since every comprehension arrow is mono.
Thus, by the universal property of quotients, we have a unique morphism
k:A/ρ−→X with k◦q =h. Then the two following diagram commutes
A
❈
④④q④④④④④ h❈❈❈❈q❈❈❈
}}④ (cid:15)(cid:15) !!
A/ρ //X // A/ρ
k ⌊∃q⊤A⌋
and, by universal properties of quotients, we have that ⌊∃ ⊤ ⌋◦k = id .
q A A/ρ
Thus ⌊∃ ⊤ ⌋ ◦ k ◦ ⌊∃ ⊤ ⌋ = ⌊∃ ⊤ ⌋, from which k ◦ ⌊∃ ⊤ ⌋ = id sinse
q A q A q A q A X
comprehensions are mono. Fullness leads to ⊤ =∃ ⊤ .
A/ρ q A
Lemma 4.3. If (C,P) is an extensional tripos with comprehensions and effec-
tive quotients, then it has strong power objects.
Proof. Suppose P(A) is a weakpowerobject ofA inC andlet P(A) denote the
quotient r:P(A)−→P(A)=P(A)/⇔ where
A
⇔ =∀ (hπ ,π i∗ ∈ ↔hπ ,π i∗ ∈ )
A hπ2,π3i 1 2 A 1 3 A
For every B and φ over A×B we denote by χ the morphism r◦{φ} and by
φ
in the following formula over A×P(A)
A
in =∃ (hπ ,π i∗ ∈ ∧ hrπ ,π i∗δ )
A hπ1,π3i 1 2 A 2 3 P(A)
Byeffectivenessofquotientsandinternalsurjectivityofr,wehavethefollowing
equalities
(id ×r◦{φ})∗in =∃ (hπ ,π i∗ ∈ ∧ hπ ,{φ}◦π i∗(r×r)∗δ )
A A hπ1,π3i 1 2 A 2 3 P(A)
=∃ (hπ ,π i∗ ∈ ∧ hπ ,{φ}◦π i∗ ⇔ )
hπ1,π3i 1 2 A 2 3 A
=φ∧∃ hπ ,{φ}◦π i∗ ⇔ = φ
hπ1,π3i 2 3 A
5 Cauchy-completeness
Inthissectionweintroducethedefinitionofcauchy-completeobjectsinaregular
doctrine (C,P). Recall that, given a regular doctrine (C,P), a formula F in
P(Y ×A) is functional from Y to A if it holds that
⊤ ≤∃ F and hπ ,π i∗F ∧hπ ,π i∗F ≤hπ ,π i∗δ
Y π2 1 2 1 3 2 3 A
Definition 5.1. Given a regular doctrine (C,P), an object A is said cauchy-
complete if for every Y and formula F in P(Y ×A) which is functional from Y
to A there exists a morphism f:Y −→A such that (f ×id )∗δ =F.
A A
We shall say that a regular doctrine is cauchy-complete if every objects of
the base is cauchy-complete.
7
Remark 5.2. The term cauchy-complete is usually introduced in terms of left
adjoints. It can be proved that a formula is functional if and only if it is a left
adjoint (in the logic of the doctrine). The non trivial part of this is in [9].
Lemma 5.3. In an extensional cauchy-complete regular doctrine (C,P) with
full comprehensions, a morphism f:A −→ B in C is mono if and only if it is
the comprehensions of some formula β over B.
Proof. Every comprehension morphism is mono. For the converse, suppose
f:A −→ B is mono and consider the formula ∃ ⊤ over B. Since ⊤ ≤
f A A
f∗∃ ⊤ , by the universal property of comprehensions there exists a morphism
f A
k with ⌊∃ ⊤ ⌋ ◦ k = f. Moreover (⌊∃ ⊤ ⌋ × f)∗δ is functional from the
f A f A B
domain of ⌊∃ ⊤ ⌋ to A, then, by cauchy-completeness, it is the internal graph
f A
of a morphism k′. By extensionality k′ is the unique such a morphism and it is
straightforwardto see that k′ is the inverse of k.
After lemma 5.3 we can formulate the following proposition, whose proof
can be found in [4].
Proposition5.4. Anextensionalcauchy-completeregulardoctrine(C,P)with
full comprehensions is isomorphic in RD to the subobjects doctrines (C,sub).
Given a regular doctrine (C,P) we shall denote the free cauchy-complete
regular doctrine on (C,P) by (C ,P ). C has the same objects as C, while
l l l
morphisms from A to B are functional formulas from A to B: identities are
internalequalitiesandthecompositionistheusualcompositionofformulas,i.e.
if φ is in P(A×B) and ψ in P(B×C), the composition ψ◦φ is the formula of
P(A×C) defined by
∃ (hπ ,π i∗φ∧hπ ,π i∗ψ)
hπ1,π3i 1 2 2 3
The action of the functor P is defined by the following assignments: P(A) =
l l
P(A) and P(φ):P (B)−→P (A) is the functor which maps every formula β in
l l l
P (B) in the following formula of P (A)
l l
∃ (φ∧π∗β)
π1 2
Frobenius reciprocity and the fact that φ is functional ensure that P(φ) is an
l
homomorphism of inf-semilattices. Moreover the assignment
∃ α=∃ (φ∧π∗α)
φ π2 1
produces a left adjoint to P (φ) which satisfies Beck-Chevally condition and
l
Frobenius-Reciprocity.
The morphism (Γ,γ) :(C,P) −→ (C ,P) where Γ is the functor from C
P l l
to C which maps every morphism f:A −→ B of C to the formula Γf =
l
(f×id )∗δ andγ is the family ofidentity homomorphismsofinf-semilattices,
B B
constitutes the unite of the following adjunction
l
--
RD(e) ll ⊥ RD(e,l)
8
where RD is the full subcategoryof RD on extensionalcauchy-complete
(e,l) (e)
regular doctrines.
In fact, for every morphism (F,f):(C,P) −→ (D,R) in RD , where (D,R)
(e)
is cauchy-complete, define a morphism (F,f):(C ,P ) −→ (D,R) where f = f
l l
and the functor F:C−→C is given by the assignment
l
A FA
φ 7→ φF
(cid:15)(cid:15) (cid:15)(cid:15)
B FB
in which φ is the morphism of D such that its internal graph is equal to
F
(hFπ ,Fπ i−1)∗f φ. SinceφisfunctionalfromAtoB andf preserves
A B A×B A×B
the regular structures, (hFπ ,Fπ i−1)∗f φ is functional from FA to FB.
A B A×B
Cauchy-completenessandextensionalityof(D,R)ensurethattheformulaisthe
graph of a unique morphism φ .
F
Moreover (F,f) uniquely makes the following diagram commute
(F,f)
Cop ++Dop
❋❋❋❋P❋❋"" ||①①①①R①① NN
ISL
OO
(Γ,γ) (F,f)
Pl
%%
Cop
l
This is due to the fact that F ◦ Γ is trivially equal to F on objects and if
h:A −→ B is a morphism in C, then F(Γ(h)) is the morphism of D whose
graph is (hFπ ,Fπ i−1)∗f (h×id )∗δ but
A B A×B B B
f (h×id )∗δ =F(h×id )∗f δ =hFπ ,Fπ i∗(Fh×id )∗δ
A×B B B FB B×B B A B FB FB
therefore (hFπ ,Fπ i−1)∗f Γ(h) is the graph of Fh and by extensionality
A B A×B
we have F(Γ(h))=Fh.
The following lemma is instrumental to prove that the previous adjunction
restricts to categories of richer doctrines.
Lemma 5.5. Given a regular doctrine (C,P)
i) if (C,P) has full comprehensions, so has (C ,P )
l l
ii) if (C,P) has effective quotients, so has (C ,P )
l l
iii) if (C,P) is a tripos, so is (C ,P)
l l
Proof. i) suppose α is a formula in P(A) = P(A), since (C,P) has full com-
l
prehensions, there exists the morphism ⌊α⌋:X −→ A in C with ⌊α⌋∗α = ⊤ .
X
Then Γ⌊α⌋ is a morphism from X to A in C such that P (Γ⌊α⌋)α = ⊤ .
l l X
Note that, since comprehension are full in (C,P), we have that ∃ ⊤ =
⌊α⌋ X
9
α, thus for every formula of (C ,P), which is functional from Y to A with
l l
P (F)α=⊤ , (id ×⌊α⌋)∗F is functional from Y to X, moreover the compo-
l Y Y
sition Γ⌊α⌋◦(id ×⌊α⌋)∗F is equal to F. ii) If q:A −→ A/ρ is the quotient
Y
in (C,P) of an equivalence relation ρ over A, then Γq:A −→ A/ρ is the quo-
tient of ρ in (C ,P), in fact, for every functional formula F from A to B with
l l
ρ≤(F×F)∗δ ,wehavethatξ =∃ (hπ ,π i∗Γq∧hπ ,π i∗F)isfunctional
B hπ2,π3i 1 2 1 3
from A/ρ to B with ξ◦Γq =F. iii) Straightforward.
Note that that ΓP(A) is exactly P(A) and the same holds for δΓP(A), since
γ is a family of identitiy morphisms. Moreover a power object of A in (C,P)
is also a power objects of A in (C ,P ), in fact for an object φ over A×B,
l l
the morphism Γ{φ} is such that P (id ×Γ{φ})∈ =φ. A consequence of this
l A A
is that the unite (Γ,γ) preserves power objects. Moreover in (F,f) is a logic
morphism,itsuniqueextension(F,f)islogictoo. Thentheleftadjointfunctor
l:RD −→RD restricts to the following commutative diagram
(e) (e,l)
l
--
RD(c,q,e) mm ⊥ RD(c,q,e,l)
OO OO
(cid:31)? l (cid:31)?
--
LT(c,q,e) mm ⊥ LT(c,q,e,l)
6 The Tripos to Topos construction
Gluing together the diagrams obtained in the previous sections we have
c q e l
,, -- -- --
RDOO kk ⊥ RD(c) ll ⊥ RD(c,q) mm ⊥ RD(c,q,e) mm ⊥ RD(c,q,e,l)
OO OO OO OO
(cid:31)? c (cid:31)? q (cid:31)? e (cid:31)? l (cid:31)?
,, ,, -- --
LT jj ⊥ LT(c) ll ⊥ LT(c,q) ll ⊥ LT(c,q,e) mm ⊥ LT(c,q,e,l)
It has been remarked in [5] that the action on objects of the functor from LT
to RD is, up to equivalence, the tripos to topos construction of Hyland-
(c,q,e,l)
Johnstone and Pitts [8]. In this section we show that the category LT is
(c,q,e,l)
equivalent to LTopos, the category of elementary toposes and logic functors.
Proposition 6.1. A non-empty category C with binary products is an ele-
mentary topos if and only if there exists an cauchy-complete extensional tripos
(C,P) with full comprehensions and effective quotients.
Proof. Suppose C is an elementary topos, then consider its subobjects tripos
(C,sub). In a suobjects doctrine every formula is its own full comprehension.
Effective quotients comes from exactness of C. Every functional formula is the
graph of a unique morphism [4], then (C,sub) is also cauchy-complete and ex-
tentional.
Conversely, if (C,P) is an object of LT , then C has
(c,q,e,l)
10