aa r X i v : . [ m a t h . L O ] S e p A sheafification theorem for doctrines
Fabio Pasquali ∗ November 1, 2018
Abstract
We define the notion of sheaf in the context of doctrines. We provethe associate sheaf functor theorem. We show that grothendieck toposesand toposes obtained by the tripos to topos construction are instances ofcategories of sheaves for a suitable doctrine.
Introduction
In this paper we study the notions of sheaf in the context of doctrines. A doc-trine, whose definition was introduced by Lawvere [3, 4, 5], can be informallythough of as a category C with a chosen internal logic P . In this context wedefine an object of C to be a sheaf if it orthogonal to a class of morphisms of C , which satisfy appropriate conditions formulated in the logic of P . We shallrefer at those objects as P -sheaves and by we denote by Shv ( C , P ) the fullsubcategory of C on P -sheaves.For a suitable class of doctrines we give a proof of the associate sheaf func-tor theorem, i.e. of the fact that Shv ( C , P ) is a reflective subcategory of C .The proof is entirely internal in the logic of the doctrine and it is a generaliza-tion of the one given in [10] and sketched in [1], which is a partially internalproof, carried out in the internal logic of an arbitrary elementary topos.As an application, we show that grothendieck toposes and toposes obtainedby the tripos to topos construction (such as the effective topos [8]) are cate-gories of the form Shv ( C , P ) for a suitable doctrine.Sections 1 and 3 introduces doctrines and sheaves respectively. In section 4we prove the associate sheaf functor theorem for an appropriate class of doc-trine doctrines. In section 2 we give the definition of complete objects, whichturns out to be useful in discussing some application in section 5. Acknowledgments . The author is grateful to Giuseppe Rosolini, Jaap vanOosten and Ruggero Pagnan for their indispensable remarks. ∗ Part of the research in this paper was carried out while the author worked at UtrechtUniversity in the NWO-Project ‘The Model Theory of Constructive Proofs’ nr. 613.001.007. Preliminary definitions on doctrines
In this section we recall some definitions concerning doctrines. Our notationclosely follows the one in [6].Let
ISL be the category of inf-semilattices and homomorphisms between them.
Definition 1.1.
A doctrine is a pair ( C , P ) where C is a category with finiteproducts and P a functor P : C op −→ ISL
We shall refer to C as the base category of the doctrine. We will write f ∗ instead of P ( f ) to denote the action of the functor P on the morphism f of C , we will often call it reindexing along f . Binary meets in inf-semilattices aredenoted by ∧ . Elements in P ( A ) will often be called formulas over A and thetop element is denoted by ⊤ A . Example 1.2. If C is a category with finite limits, we shall denote by ( C , sub)the doctrine of subobjects of C . The functor sub maps every object A of C tosub( A ), the collection of subobjects over A . Top elements are represented byidentities, while binary meets and reindexing are provided by pullbacks. Definition 1.3.
A doctrine ( C , P ) is elementary existential if for each arrow f : X −→ Y in C there exists a functor ∃ f ⊣ f ∗ satisfying- Beck-Chevalley condition: i.e. for every pullback of the form X g (cid:15) (cid:15) f / / Y h (cid:15) (cid:15) Z k / / W it holds that ∃ f ◦ g ∗ = h ∗ ◦ ∃ k - Frobenius Reciprocity: i.e. ∃ f ( α ∧ f ∗ β ) = ∃ f α ∧ β . Remark 1.4.
For every object A in C the formula ∃ ∆ A ⊤ A is in P ( A × A ).We will abbreviate it by δ A and we shall refer at it as the equality predicate over A . The formula δ A is substitutive , i.e. for every X in C and every φ in P ( X × A ) it holds that h π , π i ∗ φ ∧ h π , π i ∗ δ A ≤ h π , π i ∗ φ where π , π and π are projections from X × A × A . Moreover for a morphism f : A −→ B in C and α a formula in P ( A ) we have ∃ f α = ∃ π (( id B × f ) ∗ δ B ∧ π ∗ φ )where π and π are projections from B × A [5].Given a doctrine ( C , P ), an object A of C and a formula α in P ( A ), wesay that the doctrine has a comprehension of α if there exists a morphism ⌊ α ⌋ : X −→ A such that ⊤ X ≤ ⌊ α ⌋ ∗ α and for every f : Y −→ A such that ⊤ Y ≤ f ∗ α there exists a unique h : Y −→ X with ⌊ α ⌋ ◦ h = f .2 efinition 1.5. In a elementary existential doctrine ( C , P ) a morphism f issaid to have an image if there exists a comprehension of ∃ f ⊤ X . Definition 1.6.
A doctrine ( C , P ) has power objects if for every X in C ,there exists P ( X ) in C and a formula ∈ X in P ( X × P ( X )) such that for everyobject Y in C and formula γ in P ( X × Y ) there exists a unique morphism { γ } : Y −→ P ( X ) such that γ = ( id X × { γ } ) ∗ ∈ X . Remark 1.7.
If a doctrine ( C , P ) has power objects, then for every morphism f : A −→ P ( B ) we have that f = { ( id B × f ) ∗ ∈ B } . Example 1.8.
The subjects doctrine of an elementary topos E is an elementaryexistential doctrine with power objects [1]. Also localic triposes, i.e. triposes ofthe form (Sets , H ( − ) ), for H a locale, are elementary existential doctrines withpower objects [ ? ]. But the realizability triposes are not, since arrows of the form { γ } need not to be unique [1].Some of the definitions and some the proofs that follows in the next sectionsare more readable if written in the internal language and we introduce itrecalling the definition of Pitts in [ ? ]: let ( C , P ) be an elementary existentialdoctrine; let Σ P be the signature which has a sort for each object of C , an n -aryfunction symbols for each morphism in C of the form X × X × ...X n −→ X andan n -ary relation symbols for each element of P ( X × X × ...X n ); the internallanguage of ( C , P ) is made by those terms and formulas over Σ P . Thus, for anelementary existential doctrine, formulas in the internal language are written inthe regular fragment of logic (i.e ∧ , ⊤ , = and ∃ ). In an elementary existential doctrine ( C , P ) a formula F in P ( Y × A ) is afunctional relation from Y to A if F ( y, a ) ∧ F ( y, a ′ ) ≤ δ A ( a, a ′ ) ⊤ Y ( y ) = ∃ a : Y. F ( y, a )We shall denote by Map ( C , P ) the category whose objects are those of C andwhose morphisms are functional relations of ( C , P ): identities are internal equal-ities and the composition is the usual composition of relations, i.e. if F is in P ( A × B ) and G in P ( B × C ), the composition G ◦ F is the formula of P ( A × C )determined by ∃ b : B. ( F ( a, b ) ∧ G ( b, c ))There exists a functor Γ: C −→ Map ( C , P ) which is the identity on objects anmaps a morphism f : Y −→ A to the formula Γ f = ( f × id A ) ∗ δ A in P ( Y × A )which is a functional relation from Y to A . Γ f is said the internal graph of f . Definition 2.1.
An object A of C is said to be complete if for every Y in C and for every functional relation F from Y to A there exists a unique morphism f : Y −→ A such that Γ f = F
3e shall denote by
Map c ( C , P ) the full subcategory of Map ( C , P ) on com-plete objects. There exists a functor ∇ : Map c ( C , P ) −→ C which is the identity on objects and maps every functional relation F to theunique morphism of C whose internal graph is F , which fits in the followingcommutative diagram Map c ( C , P ) (cid:31) (cid:127) / / ∇ ( ( PPPPPPPPPPPPP
Map ( C , P ) C Γ O O Remark 2.2. (Rosolini [9]) Given an elementary existential doctrine ( C , P ),we say that a formula L in P ( Y × A ) is a left adjoint if there exists a formula R in P ( A × Y ) such that δ Y ( y, y ′ ) ≤ ∃ a : A. ( L ( y, a ) ∧ R ( a, y ′ )) ∃ y : Y. ( R ( a, y ) ∧ L ( y, a ′ )) ≤ δ A ( a, a ′ )Thus an object A is complete if and only if for every Y and every left adjointformula L in P ( Y × A ) there exists a unique f : Y −→ A such that Γ f = L .In fact, given a compete object A and formulas L and R as above L ( y, a ) ∧ R ( a, y )is functional from Y to A , then, since A is complete, there exists a uniquemorphism f : Y −→ A such that δ A ( f ( y ) , a ) = L ( y, a ) ∧ R ( a, y )from which δ A ( f ( y ) , a ) ≤ L ( y, a ). Moreover ⊤ Y × Y = R ( f ( y ) , y ), hence L ( y, a ) ≤ R ( f ( y ) , y ) ∧ L ( y, a ) ≤ ∃ z : Y.R ( f ( y ) , z ) ∧ L ( z, a ) ≤ δ A ( f ( y ) , a )and therefore L ( y, a ) = δ B ( f ( y ) , a ). Conversely, every functional relation F isleft adjoint to F op = h π , π i ∗ F . Definition 3.1.
In an elementary existential doctrine ( C , P ) a morphism f : A −→ B of C is said to be internally bijective whenever δ A = ( f × f ) ∗ δ B ⊤ B = ∃ f ⊤ A In the internal language the first condition is δ A ( a, a ′ ) = δ B ( f ( a ) , f ( a ′ ))and when it holds we shall say that f is internally injective . On the otherhand the second condition is ⊤ B ( b ) = ∃ a : A. δ B ( f ( a ) , b )and when it holds we shall say that f is internally surjective .4 xample 3.2. Given an elementary topos E and its doctrine of subobjects( E , sub), internally injective morphisms are the monomorphisms and internallysurjective morphisms are the epimorphisms. Therefore the class of internallybijective arrows is the class of isomorphisms [2]. Definition 3.3.
In an elementary existential doctrine ( C , P ) an object A of C is a P - sheaf if for every span in C of the form Y X d o o q / / A with d internally bijective, there exists a unique h : Y −→ A with h ◦ d = q .In other words, A is a P -sheaf if it is orthogonal to the class of internally bi-jective morphisms of ( C , P ). We shall denote by Shv ( C , P ) the full subcategoryof C on P -sheaves. Proposition 3.4.
In every elementary existential doctrine ( C , P ) if an objectis complete, then it is a P -sheaf. Proof.
Suppose A is complete and take the following span in C Y X d o o q / / A with d internally bijective and define F in P ( Y × A ) as F ( y, a ) = ∃ x : X. δ A ( q ( x ) , a ) ∧ δ Y ( d ( x ) , y )by Frobenius Reciprocity we have that F ( y, a ) ∧ F ( y, a ′ ) ≤ ∃ x x ′ : X. δ A ( q ( x ) , a ) ∧ δ Y ( d ( x ) , d ( x ′ )) ∧ δ A ( q ( x ′ ) , a ′ )which, by internal injectivity of d and repetitive use of substitutivity of δ , bringsto F ( y, a ) ∧ F ( y, a ′ ) ≤ δ A ( a, a ′ ). Moreover ∃ a : A. F ( y, a ) = ∃ x : X. ∃ a : A. δ A ( q ( x ) , a ) ∧ δ Y ( d ( x ) , y )= ∃ x : X. δ Y ( d ( x ) , y ) = ⊤ Y again using Frobenius reciprocity and internal surjectivity of d .Then, since A is complete, there exists a unique morphism h : Y −→ A such that F ( y, a ) = δ A ( h ( y ) , a ).Because of internal injectivity of d we have δ A ( h ( d ( x )) , a ) = ∃ x ′ : X. δ A ( q ( x ′ ) , a ) ∧ δ X ( x ′ , x ) = δ A ( q ( x ) , a )thus q and h ◦ d have the same internal graph, therefore they are equal bycompleteness of A .As a corollary of 3.4, we have that there exists a functor U : Map c ( C , P ) −→ Shv ( C , P ) which is the identity on objects and maps every functional relation F to the unique morphism f such that Γ f = F . Thus the functor ∇ : Map c ( C , P ) −→ factors through the inclusion of Shv ( C , P ) in C as in the following commu-tative diagram Map c ( C , P ) (cid:31) (cid:127) / / ∇ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ U (cid:15) (cid:15) Map ( C , P ) Shv ( C , P ) (cid:31) (cid:127) / / C Γ O O An elementary existential doctrine ( C , P ) is said to admit sheafification if P - Shv is reflective. In the next section we will introduce a class of doctrines whichadmit sheafification. For those doctrines it holds also that both the functor U and the inclusion of Map c ( C , P ) into Map ( C , P ) are equivalences. This factturns out to be useful to prove in section 5 that every topos that comes from atripos via the tripos to topos construction is a category of the form Shv ( C , P )for some suitable doctrine ( C , P ). Definition 4.1.
An elementary existential doctrine ( C , P ) is said to have sin-gletons ifi) ( C , P ) has power objectsii) arrows of the form { δ A } have images and are internally injectiveiii) for every morphism g : Y −→ P ( A ) it holds that ( id A × g ) ∗ ∈ A is a func-tional relation from Y to A if and only if g ∗ ( ∃ { δ A } ⊤ A ) = ⊤ Y . Example 4.2.
The doctrine of subobjects of an elementary topos ( E , sub) hassingletons. In fact for every A the arrow δ A is represented by the diagonal∆ A : A −→ A × A and { δ A } is the unique arrow that makes the following apullback A ∆ A (cid:15) (cid:15) / / . ∈ A (cid:15) (cid:15) A × A id A ×{ δ A } / / A × P ( A ) { δ A } is mono and therefore internally injective in ( E , sub). The doctrine has allimages, given by the epi-mono factorization of E . Given a morphism g : Y −→ P ( A ), the condition g ∗ ( ∃ { δ A } ⊤ A ) = ∃ π ( { δ A } × g ) ∗ δ P ( A ) = ⊤ Y can be writteninternally as ∃ a : A. ∀ x : A. ( δ A ( a, x ) ↔ x ∈ A g ( y ))this proposition can be easily seen to be equivalent to the following ∃ a : A. ( a ∈ A g ( y ) ∧ ∀ x : A. x ∈ A g ( y ) → δ A ( a, x ))which expresses the fact that ( id A × g ) ∗ ∈ A is a functional relation [2].The following three lemmas are instrumental to prove in 4.6 that the con-dition of having singletons is sufficient for an elementary existential doctrine toadmit sheafification. Thus in the rest of the section we will assume to work with6n elementary existential doctrine ( C , P ) with singletons, we shall abbreviatethe formula ∃ { δ A } ⊤ A with σ A and denote by S A the image of { δ A } , i.e. thedomain of the arrow ⌊ σ A ⌋ : S A −→ P ( A ) Lemma 4.3.
For every object A in C , there exists a morphism η A : A −→ S A such that η A is internally bijective. Proof.
We trivially have that ⊤ A = { δ A } ∗ σ A , therefore, by the universal prop-erty of comprehensions, there exists η A : A −→ S A with ⌊ σ A ⌋ ◦ η A = { δ A } .Internally injectivity of η A follows from internal injectivity of both { δ A } and ⌊ σ A ⌋ . Moreover, by definition of comprehension, we have that ⊤ S A ( s ) = ∃ a : A. δ P ( A ) ( { δ A } ( a ) , ⌊ σ A ⌋ ( s ))= ∃ a : A. δ P ( A ) ( ⌊ σ A ⌋ ( η A ( a )) , ⌊ σ A ⌋ ( s ))= ∃ a : A. δ S A ( η A ( a ) , s )which proves the internal surjectivity of η A . Lemma 4.4.
For every A in C it holds that δ S A ( η A ( a ) , s ) = a ∈ A ⌊ σ A ⌋ ( s ). Proof.
Since { δ A } = ⌊ σ A ⌋ ◦ η A , we have δ A = ( id A × { δ A } ) ∗ ∈ A = ( id A × η A ) ∗ ( id A × ⌊ σ A ⌋ ) ∗ ∈ A therefore ∃ id A × η A δ A ≤ ( id A × ⌊ σ A ⌋ ) ∗ ∈ A This inequality can be written internally as ∃ x, y : A. δ A ( x, a ) ∧ δ S A ( η A ( x ) , s ) ∧ δ A ( x, y ) ≤ a ∈ A ⌊ σ A ⌋ ( s )by Frobenius reciprocity and substitutivity of δ A , the left hand side of the in-equality is equal to δ S A ( η A ( a ) , s ), which is a functional relation from S A to A ,as follows from the fact that η A is internally bijective.Thus to prove the lemma it suffices to prove that also the right hand side of theinequality is a functional relation from S A to A . This is true since, by definitionof comprehension, we have ⊤ S A ( s ) = σ A ( ⌊ σ A ⌋ ( s )) and recalling that σ A is ashorthand for ∃ { δ A } ⊤ A , we have that ⊤ S A ( s ) = ∃ { δ A } ⊤ A ( ⌊ σ A ⌋ ( s )) = ∃ a : A. δ P ( A ) ( { δ A } ( a ) , ⌊ σ A ⌋ ( s ))Now apply point iii) of definition 4.1 on a ∈ A ⌊ σ A ⌋ ( s ): S A −→ P ( A ). Lemma 4.5.
For every functional relation F from Y to A there exists a uniquemorphism h : Y −→ S A with F ( y, a ) = δ S A ( h ( y ) , η A ( a )) Proof.
Suppose F in P ( Y × A ) is functional from Y to A . Then F op = h π , π i ∗ F determines a morphism { F op } : Y −→ P ( A ) sucht that ∃ a : A. a ∈ A { F op } ( y ) = ⊤ Y ( y )7 ∈ A { F op } ( y ) ∧ a ′ ∈ A { F op } ( y ) ≤ δ A ( a, a ′ )then, by point iii) of definition 4.1, { F op } ∗ σ A = ⊤ Y and therefore, by theuniversal property of comprehension, there exists h : Y −→ S A with ⌊ σ A ⌋ ◦ h = { F op } . Thus, by lemma 4.4 F ( y, a ) = a ∈ A { F op } ( y ) = a ∈ A ⌊ σ A ⌋ ( h ( y )) = δ S A ( η A ( a ) , h ( y ))To prove uniqueness of h , note that for every g such that δ S A ( η A ( a ) , g ( y )) = F ( y, a ), it holds that a ∈ A ⌊ σ A ⌋ ( g ( y )) = δ S A ( η A ( a ) , g ( y )) = δ S A ( η A ( a ) , h ( y )) = a ∈ A ⌊ σ A ⌋ ( h ( y ))Recall that the formula a ∈ A ⌊ σ A ⌋ ( g ( y )) corresponds to ( id A × ⌊ σ A ⌋ ◦ g ) ∗ ∈ A ,then from the previous equality we have( id A × ⌊ σ A ⌋ ◦ g ) ∗ ∈ A = ( id A × ⌊ σ A ⌋ ◦ h ) ∗ ∈ A and therefore ⌊ σ A ⌋ ◦ g = { ( id A × ⌊ σ A ⌋ ◦ g ) ∗ ∈ A } = { ( id A × ⌊ σ A ⌋ ◦ h ) ∗ ∈ A } = ⌊ σ A ⌋ ◦ h since comprehension morphisms are mono, g = h .We now want to prove that for an elementary existential doctrine ( C , P )with singletons, the inclusion of the category Shv ( C , P ) in C has a left adjoint,whose unite is the family of morphisms of the form η A : A −→ S A . In order toprove this, knowing that, by lemma 4.3, every η A is internally bijective, it isenough to show that for every A in C and every span of the form Y X d o o q / / S A where d is internally bijective, there exists h : Y −→ S A with h ◦ d = q . Thisinfact shows that S A is a P -sheaf and moreover replacing d with η X we havethat every q : X −→ S A has a unique extention to S X , proving that Shv ( C , P )is reflective. Proposition 4.6.
Every elementary existential doctrine with singletons admitssheafification.
Proof.
Consider the span above and the following diagram X d (cid:15) (cid:15) q / / S A ⌊ σ A ⌋ / / P ( A ) Y { ξ op } ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ where ξ ( y, a ) = ∃ x : X. δ Y ( y, d ( x )) ∧ δ S A ( q ( x ) , η A ( a )) and ξ op = h π , π i ∗ ξ .We need prove that there exists a unique h : Y −→ S A with h ◦ d = q .To prove the existence of h , note that, by Frobenius Reciprocity and the factthat η A and d are internally surjective, ⊤ Y ( y ) = ∃ a : A. ξ ( y, a ). Again by8robenius Reciprocity and by internal injectivity of η A and d we have that ξ ( y, a ) ∧ ξ ( y, a ) ≤ δ A ( a, a ′ ). Therefore ξ is functional from Y to A and by lemma4.5 there exists a unique h : Y −→ S A , such that ξ ( y, a ) = δ S A ( h ( y ) , η A ( a )).Once we have the morphism h : Y −→ S A , we need show that it makes thediagram commutes. Note, by using internal injectivity of d , that ξ ( d ( x ) , a ) = δ S A ( f ( x ) , η A ( a )) which means that δ S A ( h ( d ( x )) , η A ( a )) = δ S A ( f ( x ) , η A ( a )) andthus, by lemma 4.5, h ◦ d = f .Thus the diagram at the end of section 3 becomes Map c ( C , P ) (cid:31) (cid:127) / / L & & U (cid:15) (cid:15) Map ( C , P ) Shv ( C , P ) (cid:27) { ⊥ C Γ O O q q Moreover, the fact that for an elementary existential doctrine ( C , P ) with sin-gletons the category Shv ( C , P ) is reflective allows to characterize P -sheaves ascomplete objects, as in the following corollary. Corollary 4.7.
Let ( C , P ) be an elementary existential doctrine with single-tons, then the functor U is an equivalence. Proof. U is trivially full and faithful. To prove that it is essentially surjective,take A in Shv ( C , P ) and a functional relation F from Y to A . Then A is acomplete object if there exists a morphism in C from Y to A whose internalgraph is F . By lemma 4.5 there exists a morphism h : Y −→ S A in C such that F = ( h × η A ) ∗ δ S A . Since Shv ( C , P ) is reflective, η A is an isomorphism, then η − A ◦ h : Y −→ A is the desired morphism. Corollary 4.8.
Let ( C , P ) be an elementary existential doctrine with single-tons, then the inclusion of Map c ( C , P ) into Map ( C , P ) is an equivalence. Proof.
Since the inclusion in full, we only have to show that it is also essen-tially surjective. Suppose A is in Map ( C , P ), then by 4.7 we have that S A isin Map c ( C , P ). Since η A : A −→ S A is internally bijective it straightforwardto show that the functional relation Γ η A from A to S A is an isomorphism in Map ( C , P ).Thus the diagram above reduces to Map ( C , P ) ≃ Map c ( C , P ) ≃ Shv ( C , P ) (cid:21) (cid:21) ⊢ C U U We discuss in this section some relevant examples.Let E be an elementary topos and j a Lawvere-Thierney topology. Denote9y E j the sub-topos of j -sheaves, i.e. the full subcategory of E on those objectswhich are orthogonal to j -dense monomoprhisms [2]. Denote also by ( E , cl j ) thedoctrine of j -closed subobjects. The doctrine ( E , cl j ) is an elementary existen-tial doctrine with singletons [1], therefore by 4.6 it admits sheafification. Weshow that E j is equivalent to Shv ( E , cl j ).The class of internally bijective morphisms in ( E , cl j ) is the class of j -bidensearrows of E . Thus the category Shv ( E , cl j ) is the full sub-category of E on thoseobjects which are orthogonal to all j -bidense arrows. Since every j -dense monois in particular j -bidense, Shv ( E , cl j ) is a full subcategory of E j . To prove thatthe inclusion is also essentially surjective, take A in E j and denote by L the as-sociated sheaf functor. L is known to be left adjoint to the inclusion of E j in E and denote by l the unite of the adjunction. Take any morphism q : X −→ A andany j -bidense morphism d : X −→ Y then consider the following commutativediagram Y l Y (cid:15) (cid:15) X d o o l X (cid:15) (cid:15) q / / A l A (cid:15) (cid:15) LY LX Ld o o Lq / / LA Since A is a j -sheaf, l A is iso. Also Ld is iso since d is j -bidense [2], thus inparticular is a j -dense mono. So there exists h : LY −→ A with h ◦ Ld = l − A ◦ Lq .Then we have a morphism h ◦ l Y : Y −→ A making the diagram commute.To show its uniqueness, suppose two morphisms g and f are such that g ◦ d = q = f ◦ d , then, since d is bidense, it is j -true that their graphs are equal, bythe same argument as 3.4, and therefore it is j -true that ⊤ Y = δ A ( g ( y ) , f ( y )).Since A is a j -sheaf and therefore a separated object, its diagonal is closed andthen g = f [2].With the next class of example we want to show that given a tripos ( C , P ),the topos C [ P ], i.e. the topos obtained from ( C , P ) by the tripos to topos con-struction [8], is a category of the form Shv ( C , P ) for an appropriate doctrine( C , P ) built out of ( C , P ). For the definition of tripos, the reader is referred to[8].Suppose ( C , P ) is a tripos and consider the pair ( C , P ) obtained from ( C , P ) byfreely adding comprehensions, extensional equality and quotients: details canbe found in [6], nevertheless we give an explicit description of ( C , P ).Objects of C are pairs ( A, ρ ) where A is an object of C and ρ is a partial equiv-alence relation over A , i.e. an formula of P ( A × A ) such that ρ ( x, y ) = ρ ( y, x )and ρ ( x, y ) ∧ ρ ( y, z ) ≤ ρ ( x, z ).A morphism [ f ]: ( A, ρ ) −→ ( B, σ ) is an equivalence class of morphisms f : A −→ B of C such that ρ ≤ ( f × f ) ∗ σ , with respect to the following equivalence relation f ∼ g if and only if ⊤ A ≤ h f, g i ∗ σ For an object (
A, ρ ) in C we have that P ( A, ρ ) = { φ ǫ P ( A ) | φ ( x ) ≤ ρ ( x, x ) and φ ( x ) ∧ ρ ( x, y ) ≤ φ ( y ) } f ]: ( A, ρ ) −→ ( B, σ ) we have that P [ f ] is give by the assigment φ f ∗ φ ∧ ∆ ∗ A ρ for φ in P ( A, ρ ). The doctrine ( C , P ) is a tripos with power objects [7].It is straightforward to see that the topos C [ P ] obtained from ( C , P ) via the tri-pos to topos construction is Map ( C , P ), then by 4.8 we have C [ P ] ≃ Shv ( C , P ). References [1] B. Jacobs.
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