The Over-Topos at a Model in the General Case, Where the Base Topos in Which the Given Model Lives Is an Arbitrary Grothendieck Topos

Home , Category of elements, Stack (mathematics)

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In section 2, in preparation for the generalization of our construction to an arbitrary topos, we introduce a number of stacks that will be used in the sequel. Of particular relevance for our ∗ purposes is the notion of lifted topology on a category of the form (1F ↓ f ), where f : F→E is a geometric morphism, as the smallest topology which makes both projection functors to E and F comorphisms to the canonical sites on E and F; we provide a fully explicit description of this topology by providing a basis for it.

In section 3, we generalize the construction of the over-topos to a model in an arbitrary Grothendieck topos. For this, we construct a canonical stack associated with this model and apply Giraud’s general construction of the classifying topos of a stack to prove the desired universal property.

In section 4 we investigate the over-topos construction from a bicategorical point of view, relating it with classical constructions in the bicategory of Grothendieck toposes, as well as with the notion of totally connected topos. In the particular case of a set-based model, that is, of a point fM : Set → Set[T] of the classifying topos for T, the T-over-topos at M has as category of points the comma category (T[Set] ↓ M) and is shown to coincide with the colocalization of Set[T] at fM . This construction is dual, in a sense, to the notion of Grothendieck-Verdier localization at a point, which classiﬁes morphisms whose domain is a ﬁxed model.

1 Over-topos of a set-based T-model

We suppose in this section that E is equal to the topos Set of sets (within a fixed model of set theory). We first list some specific properties of the terminal object 1 of Set, we are going to make use of in this section, and which may fail in arbitrary toposes, as we are going to see in section 3 which will addres s the general case. First, Set is generated under coproducts by 1, that is, for any set X one has X ≃ a 1 HomSet(1,X) Moreover, 1 is projective, that is, any epimorphism X ։ 1 admits a section 1 → X, and indecomposable, which means that for any arrow from 1 to a coproduct ì∈I Xi, there is a section for at least one i ∈ I: Xi ` Xi i∈I

1 1 We are going to use these properties in order to deﬁne a cartesian, subcanonical site for the T-over-topos associated with M. This will involve the category of elements of M, and a certain topology related to the syntactic topology JT of T. Let CT denote the geometric syntactic of A~ T, and, for any object {~x .φ} of CT, that is, a geometric formula φ in the signature L in the A~ A~ context ~x of sort A~, J~x .φKM the interpretation of {~x .φ} in M. In particular the category R M of elements of M - seen as a geometric functor FM : CT → Set - has as objects the pairs A~ A~ ({~x .φ},~a), where ~a ∈ J~x .φKM is a global element of MA~ satisfying the formula φ, and as ~a1 ~a2 ~a1 ~a2 arrows ({~x1 .φ1}, ~a1) → ({~x2 .φ2}, ~a2) the arrows [θ]: {~x1 .φ1}→{~x2 .φ2}1 in CT such that JθKM ( ~a1)= ~a2, that is diagramatically in Set:

1 ~a1 ~a2

JθK A1 M A2 J~x1 .φ1KM J~x2 .φ2KM

We shall ﬁnd it convenient to present our Grothendieck topologies in terms of bases generating them. Recall that a basis B for a Grothendieck topology on a category C is a collection of presieves on objects of C (by a presieve we simply mean a small family of arrows with common codomain) satisfying the following properties (where we denote by B(c), for an object c of C, the collection of presieves in B on the object c):

2 (a) If f is the identity then {f} lies in B(cod(f)).

(b) If R ∈B(c) then for any arrow g : d → c in C there exists a presieve T in B(d) such that for each t ∈ T , g ◦ t factors through some arrow in R.

(c) B is closed under “multicomposition” of families; that is, given a presieve {fi : ci → c | i ∈ I} in B(c) and for each i ∈ I a presieve {gij : dij → ci | j ∈ Ii} in B(ci), the “multicomposite” presieve { fi ◦ gij : dij → c | i ∈ I, j ∈ Ii} belongs to B(c).

The Grothendieck topology generated by a basis has as covering sieves precisely those which contain a presieve in the basis. As it is well known, the Grothendieck topology JT on CT has as a basis the collection BT of small families {[θi]: {~xi.φi}→{~x.φ} | i ∈ I} such that the sequent

(φ ⊢~x _(∃~xi)θi(~xi, ~x)) i∈I is provable in T. The fact that the following definition is well-posed in ensured by the subsequent Lemma. Definition 1.1. Let M be a set-based model of a geometric theory T. We define the antecedents ant ant topology at M as the Grothendieck topology JM on R M generated by the basis BT consisting of the families θ A~i [ i] A~ (~b, {~x .φi}) −→ (~a, {~x .φ}) i i∈I,~b | JθiKM (~b)=~a

(indexed by the objects (~a, {~x.φ}) of R M and the families ([θi])i∈I in BT on {~x.φ}) consisting of A~ all the “antecedents” ~b of a given ~a ∈ J~x .φKM with respect to some [θi].

ant Lemma 1.2. The collection BT of sieves in R M is a basis for a Grothendieck topology. Proof. − Condition (a) is trivially satisﬁed.

B~ A~ − Condition (b): given an arrow [θ] : (~c, {y .ψ}) → (~a, {~x .φ}) in R M and a family [θi]T : A~i A~ A~ {~xi .φi}→{~x .φ})i∈I in BT on {~x .φ}, we have the following pullback squares in R M for each antecedent ~b of ~a:

~ ~ ~ B~ Ai ~ Ai ((~c, b), {~y , ~xi .θ(~y)= θi(~xi)}) (b, {~xi .φi})) y ∗ [θ θi] [θi]

[θ] (~c, {~yB~ .ψ}) (~a, {~xA~ .φ})

∗ B~ Note that the family {[θ θi] | i ∈ I} lies in BT({~y .ψ}) (as BT is stable under pullback). So the family ~ ~ ~ ∗ ~ B Ai B [θ θi] : ((~c, b), {~y , ~xi .θ(~y)= θi(~xi)}) → (~c, {~y .ψ})i∈I ant is the family of antecedents of ~c indexed by it, whence it lies in BT , as desired; indeed, ∗ ∗ Jθ θiKM = JθKM JθiKM since FM is cartesian.

− Condition (c) follows immediately from the fact that BT is a basis for JT.

A~ Remark. For a family ([θi])i∈I in BT({~x.φ}) and a global element ~a of the interpretation J~x .φKM of its codomain, the ﬁber at ~a of some JθiKM can be categorically characterized as the following pullback: −1 JθiKM (~a) 1 y ~a

Jθ K A~i i M A~ J~xi .φiKM J~x .φKM

3 As FM is a JT-continuous cartesian functor, it sends JT-covering families to jointly surjective families in Set, so by the stability of epimorphisms under pullback the global ﬁber of the cover at ~a is also an epimorphism: −1 hJθiKM ii∈I (~a) 1 y ~a

hJθ K i A~i i M i∈I A~ ` J~xi .φiKM J~x .φKM i∈I

Note that, by the stability of colimits under pullback, the global ﬁber of ~a decomposes as the coproduct −1 −1 hJθiKM ii∈I (~a) ≃ aJθiKM (~a). i∈I One can then identify the set of antecedents of ~a with the set of global elements ~b : 1 → −1 hJθiKM ii∈I (~a), which decomposes as the disjoint union of the sets of global elements of the ﬁbers −1 JθiKM (~a) (since 1 is indecomposable and coproducts are disjoint and stable under pullback). ant Note however that the projection R M → CT does not send JM -covers to JT-covers as ~a ∈ A~ A~i J~x .φKM may have antecedents in only some of the J~xi .φiKM .

Proposition 1.3. The category R M of elements of M is geometric.

Proof. Actually all the properties we have to check are inherited from the geometricity of CT and the fact that M is a model of T:

− R M is cartesian: for any ﬁnite diagram D → R M, the underlying diagram in CT has a limit

A~d A~d lim{~xd .φd} = {(~xd )d∈D, ^ θδ(~ad)= ~ad′ } d∈D δ:d→d′

which is sent to a limit in Set by the cartesian functor FM ; note that an element of A~d A~d lim J~xd .φdKM is a family (~ad)d∈D with ~ad ∈ J~xd .φdKM and JθδKM (~ad) = ~ad′ for each d∈D transition morphism δ : d → d′ in D. This exactly says that

A~d A~d ((~ad)d∈D, lim {~xd .φd}) = lim (ad, {~xd .φd}). d∈D d∈D

~ B~ A~ − The image factorization in R M of an arrow [θ] : (b, {~y .ψ}) → (~a, {~x .φ}) is given by

[θ] (~b, {~yB~ .ψ}) (~a, {~xA~ .φ})

(~a, {∃~yθ(~y, ~x)})

and is easily seen to be pullback stable.

A~ A~ Subobjects in R M are arrows of the form (~a, {~x .φ}) ֒→ (~a, {~x .ψ}) with (φ ⊢T ψ). It thus − follows at once that lattices of subobjects are frames, as their ﬁnite meets (resp. arbitrary joins) are given by A~ A~ (~a, {~x . ^ φi}) (resp. (~a, {~x . _ φi})) i∈I i∈I

A~ A~ -for any ﬁnite family (resp. arbitrary family) {(~a, {~x .φi}) ֒→ (~a, {~x .φ}) | i ∈ I} of subob jects of a given (~a, {~xA~ .φ}).

T ant Deﬁnition 1.4. We deﬁne the -over-topos at M as the sheaf topos Sh(R M,JM ), and denote it by Set[M].

We shall now deﬁne a geometric theory TM whose models in any Grothendieck topos G will be all homomorphisms of T-models g : N → γ∗M, where γ is the unique geometric morphism G → Set; this theory will be classiﬁed by the T-over-topos at M.

4 A~ Deﬁnition 1.5. Let SetM be the language with a sort S(~a,{~xA~ .φ}) for each object (~a, {~x .φ}) of the category of elements of M and a function symbol

~a1, ~a2 fθ S A~ −→ S A~ ( ~a1,{ ~x1 1 .φ1}) ( ~a2,{ ~x2 2 .φ2})

~a1 ~a2 ~a1 ~a2 for each [θ(~x1 , ~x2 ]T : {~x1 .φ1}−→{~x2 .φ2} such that JθKM ( ~a1)= ~a2. c Let LM be the extension of the language L with a tuple of constant symbols c(~a,{xA~ .φ}) for each A~ c c (~a, {x .φ}) ∈ R M. There is a canonical LM -structure M extending M, obtained by interpreting each constant by the corresponding element of M. c We can naturally interpret SetM in LM by replacing, in the obvious way, each variable of sort S(~a,{~xA~ .φ}) appearing freely in a formula with the corresponding tuple of constants c(~a,{xA.φ}) and each function symbol with the corresponding T-provably functional formula. This yields, for each S~ ♯ formula-in-context {~z .ψ} over SetM a closed formula ψ . Let TM be the theory over SetM having as axioms all the geometric sequents

~ (φ ⊢ S ~ ψ) ~x (~a,{~xA.φ}) such that the corresponding sequent (φ♯ ⊢ ψ♯) is valid in M c. Given a Grothendieck topos G with global section functor γ, the model M is sent in G to a T-model γ∗M and for each {~xA.φ}, one has

A~ ∗ A~ J~x .φKγ∗M = γ (J~x .φKM )= a 1 A~ ~a:1→J~x .φKM

A~ ∗ A~ so any global element ~a :1 → J~x .φKM is sent into a global element γ (~a):1 → J~x .φKγ∗M in G. A~ Note that each element of J~x .φKγ∗M is counted exactly once in this coproduct. (since coproducts are disjoint in a topos). Theorem 1.6. Let T be a geometric theory and M a model of T in Set.

(i) The theory TM axiomatizes the T-model homomorphisms to (internalizations of) M; that is, for any Grothendieck topos G with global section functor γ : G → Set, we have an equivalence of categories ∗ TM [G] ≃ T[G]/γ M. (ii) There is a geometric functor T FU : C M → R M T classifying a M -model U internal to the category R M:

A~ S(~a,{~xA~ .φ}) 7→ (~a, {~x .φ}) ~ ~ ~a1, ~a2 A1 A2 fθ 7→ [θ] : ( ~a1, { ~x1 .φ1}) → ( ~a2, { ~x2 .φ2})

ant T (iii) The sheaf topos Set[M] = Sh(R M,JM ) is the classifying topos of M ; that is, for any Grothendieck topos G with global section functor γ : G → Set, we have an equivalence of categories Geom[G, Set[M]] ≃ T[G]/γ∗M. T ant (iv) The M -model U in R M as in (ii) is sent by the canonical functor R M → Sh(R M,JM ) to ‘the’ universal model of TM inside its classifying topos. (v) There is a full and faithful canonical functor

V T R M → C M ~ S ~ (~a, {~xA.φ}) 7→ {x (~a,{~xA.φ}) .⊤} ~a1, ~a2 [θ] 7→ fθ

ant T T which is a dense (cartesian but not geometric) morphism of sites (R M,JM ) → (C M ,J M ) ant such that FU ◦ V = 1R M ; in particular, JM is the topology induced by JTM via V and, V being full and faithful, it is subcanonical.

5 Proof. The proof will proceed as follows. We shall ﬁrst establish, for any Grothendieck topos G, an equivalence, natural in G, between the category T[G]/γ∗M of T-model homomorphisms to ∗ ant γ M and the category Flat ant ( M, G) of J -continuous ﬂat functors M → G. Next, we shall JM R M R ∗ establish an equivalence between T[G]/γ M and TM [G] (natural in G), obtaining in the process an explicit axiomatization for the theory TM ; moreover, we will show that the resulting equivalence T Flat ant ( M, G) ≃ M [G] ≃ CartJT (CT , G) JM R M M is induced, on the one hand, by composition with the cartesian cover-preserving functor V , and on the other hand, by composition with the geometric functor FU . From this we shall deduce that we have an equivalence of toposes

ant Sh Sh T T (R M,JM ) ≃ (C M ,J M )

ant T T whose two functors are induced by the morphisms of sites V : (R M,JM ) → (C M ,J M ) and T T ant FU : (C M ,J M ) → (R M,JM ). This in turn implies (by Proposition 5.3 [3]) that the morphism ant of sites V is dense and that JM is the Grothendieck topology on R M induced by the syntactic ∼ topology JTM . Since, as it is easily seen, FU ◦ V = 1R M , the functor V is full and faithful and ant therefore the subcanonicity of JTM entails that of JM . The above equivalence of toposes also implies, by the syntactic construction of ‘the’ universal model of a geometric theory inside its T ant classifying topos, that the M -model U is sent by the canonical functor R M → Sh(R M,JM ) to ‘the’ universal model for TM inside this topos, thus completing the proof of the theorem. Let N be in T[G] and g N −→ γ∗M be a L-structure homomorphism in G. By the categorical equivalence between models of a geometric theory and cartesian cover-preserving functors on its syntactic site, g is the same thing as a natural transformation g ~ A~ {~xA.φ} A~ J~x .φKN J~x .φKγ∗(M)

A~ (for {~x .φ}∈CT). A~ Note that J~x .φKN is the disjoint union

A~ ~a J~x .φK = N ~ N a {~xA.φ} A~ ~a∈J~x .φKM of the ﬁbers: N~a 1 {~xA~ .φ} y γ∗(~a)

g ~ A~ {~xA.φ} A~ J~x .φKN J~x .φKγ∗(M)

~a Thus g ~ yields a family of objects (N ) ~ indexed by the category of {~xA.φ} {~xA~ .φ} ({~xA.φ},~a)∈R M elements of M. The naturality of g implies that for any morphism in R M corresponding to a [θ] in CT, one has a unique arrow ~ ~a1, ~a2 N[θ] ~a1 ~a2 N ~a1 → N ~a2 {~x1 .φ1} {~x2 .φ2} making the following diagram commute:

~a1 N ~a 1 1 ~a , ~a {~x1 .φ1} ~ 1 2 N[θ] ∗ y γ ( ~a1)

~a2 N ~a 1 {~x 2 .φ } 2 2 y

~a1 g{~x .φ1} ∗ ~a1 1 ~a1 γ ( ~a2) J~x1 .φ1KN J~x1 .φ1Kγ∗(M) γ∗M([θ]) N θ g ~a ([ ]) {~x 2 .φ } A2 2 2 ~a2 Jx2 .φ2KN J~x2 .φ2Kγ∗(M)

6 ant Let us show that g deﬁnes a JM -continuous ﬂat functor: g R M → G (~a, {~xA~ .φ}) 7→ N~a {~xA~ .φ} ~a1, ~a2 [θ] 7→ N[θ]

ant We have to check that g preserves the terminal object and pullbacks, and that it sends JM - covering families to jointly epimorphic families. These are actually consequences of N being a T-model:

− For the terminal object, observe that both J[].⊤KN and J[].⊤Kγ∗M are equal to the terminal ∗ object 1, whence N{[].⊤} = 1 too. − For pullbacks, one knows that

~a1 ~a2 ~a1 ~a2 J~x , ~x .θ (~x )= θ (~x )K ≃ J~x .φ K × A~ J~x .φ K 1 2 1 1 2 2 N 1 1 N J~x .φKN 2 2 N

~a1 ~a2 as N is a model; then for ~a1 ∈ J~x1 .φ1KM , ~a2 ∈ J~x2 .φ2KM such that Jθ1KM ( ~a1) = ~c = Jθ2KM ( ~a2), one has

∗ ∗ ~a1, ~a2 γ ( ~a1),γ ( ~a2) ~a1 ~a2 N ~a1 ~a2 =1 × ~a1 ~a2 J~x1 , ~x2 .θ1(~x1)= θ2(~x2)KN {~x1 ,~x2 .θ1(~x1)=θ2(~x2)} J~x1 ,~x2 .θ1(~x1)=θ2(~x2)Kγ∗M ∗ ∗ γ ( ~a1),γ ( ~a2) ~a1 ~a2 ≃ 1 × ~a ~a (J~x .φ KN × A~ J~x .φ KN ) 1 2 1 1 J~x .φKN 2 2 J~x1 ,~x2 .θ1(~x1)=θ2(~x2)Kγ∗M ∗ ∗ γ ( ~a1) ~a γ ( ~a2) ~a 1 ∗ 2 ≃ (1 × ~a J~x .φ1KN ) × γ (~c) A~ (1 × ~a J~x .φ2KN ) 1 1 1× J~x .φKN 2 2 J~x1 .φ1Kγ∗M A~ J~x2 .φ2Kγ∗M J~x .φKγ∗M

~a1 ~a2 ≃ N ~a × ~c N ~a . 1 N ~ 2 {~x1 .φ1} {~xA.φ} {~x2 .φ2}

ant − For JM -continuity: let ~ ~ Ai A ([θi] : (b, {~xi .φi})→(~a, {x .φ}))~ −1 b∈hJθiKM ii∈I (~a) ant ∗ be a family in BT . As γ M and N are T-models, they both send ([θi])i∈I to epimorphic families in G. On the other hand, one can express the ﬁber of a along the coproduct map −1 hJθiKM ii∈I as −1 A~i a 1 ≃ aJθiKM (~a) → aJ~xi .φiKM . −1 i∈I i∈I hJθiKM ii∈I (~a) Moreover, this pullback is preserved by γ∗, which sends it to

∗ −1 ∗ −1 ∗ γ (hJθiKM ii∈I (~a)) ≃ a γ (JθiKM (~a)) ≃ a γ (1) ≃ a 1 i∈I −1 −1 hJθiKM ii∈I (~a) hJθiKM ii∈I (~a)

−1 which is the hJθiKM ii∈I (~a)-indexed coproduct of the terminal object of G. Then by the stability of coproducts under pullbacks one has

∗ −1 A~ A~ γ (hJθiKM i (~a)) × A~ J~x .φKN ≃ ( 1) × A~ J~x .φKN i∈I J~x .φKγ∗M a J~x .φKγ∗M ~ −1 b∈hJθiKM ii∈I (~a) ~b ~ = a N Ai . {~x .φi} ~ −1 i b∈hJθiKM ii∈I (~a)

So in the diagram

~b ∗ −1 ~ −1 N A~ γ (hJθiKM ii I (~a)) `b∈hJθiKM ii∈I (~a) i ∈ {~xi .φi}

N~a 1 {~xA~.φ}

ì∈I g A~ {~x i .φ } ∗ A~i i i A~i γ (~a) ì∈I J~xi .φiKN ì∈I J~xi .φiKγ∗(M)

g ~ A~ {~xA.φ} A~ J~x .φKN J~x .φKγ∗(M)

7 the front, right and back squares are pullbacks, whence the left square is a pullback too: but this forces the upper left arrow

~b ~a ~ ~ a N Ai → N A {~x .φi} {~x .φ} ~ −1 i b∈hJθiKM ii∈I (~a) to be an epimorphism by the stability of epimorphisms under pullback in G.

a The data of the N{~xA.φ}’s with their transitions morphisms deﬁne a SetM -structure Sg:

~a S ~ 7→ N (~a,{~xA.φ}) {~xA~ .φ} ~a1, ~a2 ~a1, ~a2 ~a1 ~a2 fθ 7→ N[θ] : N ~a1 → N ~a2 {~x1 .φ1} {~x2 .φ2}

This structure is actually a TM -model. Indeed, this follows at once from the fact that the interpretation in Sg of any geometric formula ψ over the signature SM can be expressed as a ♯ c c pullback of the interpretation of the corresponding formula ψ in the LM -structure M , as in the following diagram:

S A~ S A~ ( ~a1,{ ~x1 1 .φ1}) (~an,{ ~xn n .φn}) ♯ c [[z1 ,...,zn .ψ]]Sg [[[].ψ ]]M

S A~ S A~ ( ~a1,{ ~x1 1 .φ1}) (~an,{ ~xn n .φn}) [[z1 ,...,zn .⊤]]Sg 1 y

< ~a1,..., ~an>

~ ~ gA1×...×gAn A~1 A~n A~1 A~n [[ ~x1 ,..., ~xn .⊤]]N [[ ~x1 ,..., ~xn .⊤]]γ∗(M)

This analysis shows that a simple axiomatization for the theory TM may be obtained by phras- T ing in logical terms the property that the functor V : R M →C M in the statement of the theorem be cartesian and cover-preserving. Recalling the well-known characterizations of pullbacks and terminal objects in the internal language of a topos, this leads to the following axioms for TM :

S(~a,{[].⊤}) ⊤⊢[] (∃x )⊤; ′ S S ⊤⊢x (~a,{[].⊤}) ,x′ (~a,{[].⊤}) (x = x );

~a1,~c ( ~a1, ~a2), ~a1 ~a2,~c ( ~a1, ~a2), ~a2 ~ ~ ~ ~ ⊤⊢ S ,{x′ ~a1 ,x′ ~a2 .θ (x′ )=θ (x′ )}) f (f (z)) = f (f (z))) , ( ~a , ~a 1 2 1 1 2 2 θ1 ~′ θ1 ~′ z 1 2 ~x1=x1 ~x2=x2

( ~a1, ~a2), ~a1 ( ~a1, ~a2), ~a1 ′ ( ~a1, ~a2), ~a2 ( ~a1, ~a2), ~a2 ′ ′ f (z)= f (z )) ∧ (f (z)= f (z )) ⊢ ′ z = z , ~′ ~′ ~′ ~′ z,z ~x1=x1 ~x1=x1 ~x2=x2 ~x2=x2

~a1,~c ~a2,~c ( ~a1, ~a2), ~a1 ( ~a1, ~a2), ~a2 f (y1)= f (y2) ⊢ S ~ S ~ (∃z)((f (z)= y1)∧(f (z)= y2)) θ1 θ2 ( ~a ,{ ~x A1 .φ }) ( ~a ,{ ~x A2 .φ }) ~′ ~′ 1 1 1 2 2 2 ~x1=x1 ~x2=x2 y1 ,y2

~a1 ~a2 for any ~a1 ∈ J~x1 .φ1KM , ~a2 ∈ J~x2 .φ2KM such that Jθ1KM ( ~a1)= ~c = Jθ2KM ( ~a2);

S A~ ~ (b~ ,{~x i .φ }) bi,~a ~ i i i ⊤⊢ S ~a,{~xA.φ})) _ (∃z )(fθ (z)= y) y ( i ~ −1 i∈I,bi∈JθiKM (~a)

A~i A~ A~ A~ for any family {[θi]: {~xi .φi}→{~x .φ} | i ∈ I}∈BT({~x .φ}) and any ~a ∈ J~x .φKM . ∗ The ﬁrst two axioms express the fact that N{[].⊤} = 1, the following three express the fact that ~a1 ~a2 for any ~a1 ∈ J~x1 .φ1KM , ~a2 ∈ J~x2 .φ2KM such that Jθ1KM ( ~a1)= ~c = Jθ2KM ( ~a2), we have a pullback square ( ~a1, ~a2), ~a1 N ~ ~x =x′ ~a1, ~a2 1 1 ~a1 N N ~a ~′ ~a1 ~′ ~a2 ~′ ~′ 1 {x1 ,x2 .θ1(x1)=θ2(x2)} {~x1 .φ1}

( ~a , ~a ), ~a 1 2 2 ~a1,~c N ~ N ~x =x′ θ 2 2 1

~a2,~c Nθ N ~a2 2 N~c ~a2 {~xA~ .φ} {~x2 .φ2}

8 and the last one corresponds to the property of V being cover-preserving.

It is immediate to see that the JTM -continuous cartesian functor FSg : CTM → G corresponding to the TM -model Sg is given by g ◦ FU and that, conversely, the ﬂat functor g can be recovered T from the M -model Sg as the composite FSg ◦ V . Let us now show that, in the other direction, given families

~a N ~ ~xA.φ ~ { 1 } (~a,{~xA.φ})∈R M and ~a1, ~a2 ~a1 ~a2 f : N ~a → N ~a θ 1 2 [θ]:{ ~x1.φ1}→{ ~x1.φ1}∈CT with [[θ]]M ( ~a1)= ~a2 {~x1 .φ1} {~x2 .φ2} ant respectively of objects and arrows in G deﬁning a JM -continuous ﬂat (equivalently, cartesian) T T functor G : R M → G, or, equivalently a M -model in G, we can associate with it a -model N in G and a homomorphism of T-models g : N → γ∗(M). A A ~a First, for each each {~x .φ} and ~a ∈ J~x .φKM , we set g{~xA.φ} equal to the composite arrow

N~a ! 1 {~xA~ .φ} γ∗(~a) g~a {~xA.φ} A~ J~x .φKγ∗(M)

A T ~ Then we deﬁne, for each object {~x .φ} of C , g{~xA.φ} as the arrow determined by the universal property of the coproduct as in the following diagrams (where the vertical arrows are the canonical coproduct inclusions): ~a N A~ {~x .φ} ~a g ~ {~xA.φ}

~a A~ N ~ J~x .φK ∗ A g ~ γ (M) `A {~x .φ} {~xA.φ} ~a∈J~x .φKM

We need to ensure that:

− we have a JT-continuous cartesian functor:

FN CT −→ G A~ ~a {~x .φ} 7→ N A~ = N {~x .φ} ` {~xA~ .φ} A~ a∈Jx .φKM ~ ~ A1 A2 ~a1,[[θ]]M ( ~a1) ~a1 ~a2 θ : { ~x1 .φ1}→{ ~x2 .φ2} 7→ f : N ~ → N ~ A1 A2 `~ `~ { ~x1 .φ1} `~ { ~x2 .φ2} A1 A1 A1 ~a1∈Jx1 .φKM ~a1∈Jx1 .φKM ~a2∈Jx2 .φKM

and hence a T-model N in G; − we have a natural transformation

∗ ~ ~ g = (g{~xA.φ}){~xA.φ}∈CT : N → γ M.

This amounts to the following conditions:

− If {[].⊤} is the terminal object of the syntactic site, then its interpretation in M has exactly one global element 1 → J[], ⊤KM = 1 so requiring FN to preserve the terminal object is ∗ equivalent to demanding N{[].⊤} = 1, which is ensured by the fact that the functor G preserves the terminal object by our hypotheses.

− Pullbacks in CT ~ ~ ~ A1 A2 A1 { ~x1 , ~x2 .θ1( ~x1)= θ2( ~x2)}{ ~x1 .φ1} y [θ1]

[θ2] A~2 A~ { ~x2 .φ2}{~x .φ}

9 are sent by FM to pullbacks in Set

~ ~ ~ A1 A2 A1 J ~x1 , ~x2 .θ1( ~x1)= θ2( ~x2)KM J ~x1 .φ1KM y Jθ1KM

~ Jθ2KM ~ A2 A J ~x2 .φ2KM J~x .φKM ,

A~1 A~2 A~1 where elements of J ~x1 , ~x2 .θ1( ~x1) = θ2( ~x2)KM are pairs ( ~a1, ~a2):1 → J ~x1 .φ1KM × ~ A2 J ~x2 .φ2KM such that Jθ1KM ( ~a1)= ~a = Jθ2KM ( ~a2). Now, if all the squares of the form

~a1, ~a2 ~a1 N A~ A~ N A~ { ~x1 1 , ~x2 2 .θ1( ~x1)=θ2( ~x2)} { ~x1 1 .φ1}

N ~a1,~a [θ1] N ~a2,~a [θ ] ~a2 2 ~a N A~ N A~ { ~x2 2 .φ2} {~x .φ}

are pullbacks, which is the case since the functor G preserves pullbacks by our hypotheses, then, by the stability of coproducts along pullbacks, we have

~a1, ~a2 ~ ~ N A1 A2 = a N A~ A~ { ~x1 , ~x2 .θ1( ~x1)=θ2( ~x2)} { ~x1 1 , ~x2 2 .θ1( ~x1)=θ2( ~x2)} A~ A~ J ~x1 1 , ~x2 2 .θ1( ~x1)=θ2( ~x2)KM

~a1 ~a2 = N ~ × ~a N ~ a A1 N ~ A2 { ~x1 .φ1} {~xA.φ} { ~x2 .φ2} A~ A~ J ~x1 1 , ~x2 2 .θ1( ~x1)=θ2( ~x2)KM

~a1 ~a2 ~a = a N A~ × N a N A~ { ~x1 1 .φ1} ` A~ { ~x2 2 .φ2} ~ {~x .φ} A~ ~a∈J~xA.φK A~ J ~x1 1 .φ1KM M J ~x2 2 .φ2KM

~ ~ = N A1 ×N ~ N A2 . { ~x1 .φ1} {~xA.φ} { ~x2 .φ2}

A~i A~ − JT-continuity: given a small JT-cover ([θi]: {~xi .φi}→{~x .φ})i∈I , since M is a T-model, the corresponding functor FM : CT → G sends it to a jointly epimorphic family (JθiKM : A~i A~ A A~i J~xi .φiKM → J~x .φKM )i∈I ; so each ~a ∈ Jx .φKM has an antecedent ~b ∈ J~xi .φiKM for some i ∈ I. Therefore for each ~a ∈ J~x.φK the cocone of ﬁbers

~ N b,~a ~b [θi] ~a N A~ −→ N A~ ~ { ~xi i .φi} {~x .φ}i∈I, JθiKM (b)=~a in each of its antecedents is jointly epimorphic in G if and only if the coproduct of ﬁbers

N[θi] N A~ −→ N A~ { ~xi i .φi} {~x .φ}i∈I

ant is. But this exactly amounts to requiring the following functor to be JM -continuous:

G R M → G (~a, {~xA~ .φ}) 7→ N~a {~xA~ .φ} ~a1, ~a2 [θ] 7→ N[θ]

The naturality of g follows immediately from the definition of the functor FN on the arrows of CT. ant To conclude our proof of the categorical equivalence between flat JM -continuous functors on T ∗ R M and -model homomorphisms to γ (M), we have to check that the two functors defined above are mutually quasi-inverse. For the construction starting from a homomorphism of L-structures between T-models g : N → γ∗M, observe that as the codomain of g decomposes as the coproduct A~ of all elements J~x .φKγ∗M = A~ 1, by the stability of coproducts under pullbacks one `~a:1→J~x .φKM has ~a N ~ ≃ N ~ . {~xA.φ} a {~xA.φ} A ~a∈J~x .φKM ant For the converse process, if one starts with a JM -continuous flat functor N(−) : R M → G and A~ ∗ defines, for each {~x .φ}, N{~xA~ .φ} as the above coproduct, then pulling it back along γ (~a):1 →

10 A~ J~x .φKγ∗M , one has (again, by the stability of coproducts under pullback)

∗ ∗ ~b ∗ ∗ ~b γ (~a) ( N ~ ) ≃ γ (~a) (N ~ ). a {~xA.φ} a {~xA.φ} A A ~b∈J~x .φKM ~b∈J~x .φKM

A~ For any element ~b of J~x .φKM , we have the following pullback squares:

∗ ∗ ~ ∗ ∗ ~b γ (~a),γ (b) γ (~a) (N A~ ) 1 × A~ 1 1 {~x .φ} J~x .φKγ∗M y y γ∗(~a)

∗ ~ ~b γ (b) A~ N ~ 1 J~x .φKγ∗M {~xA.φ}

Now, there are two possible values for the pullback on the right-hand side:

01 11 y y ∗ ∗ whenever ~a16= ~a2 γ ( ~a2) whenever ~a1= ~a2 γ ( ~a2) ∗ ∗ γ ( ~a1) A~ γ ( ~a1) A~ 1 J~x .φKγ∗M 1 J~x .φKγ∗M

~ ∗ ∗ ~b ∼ So the middle pullback is the initial object whenever ~a 6= b, whence γ (~a) (N ~ ) = 0; on the {~xA.φ} ∗ ∗ ~b ∼ ~a ~ other hand, γ (~a) (N ~ ) = N ~ whenever ~a = b. This clearly implies our thesis. {~xA.φ} {~xA.φ} Remark. We have made use of specific properties of Set at several steps of the above constructions and proofs: − In the definition of the antecedent topology, we only had to consider global elements of the A~ sets J~x .φKM . This is because 1 is a generator of Set, so that generalized elements would just be coproducts of global elements. − As a consequence, the very notion of antecedent element is simplified. As we shall see in section 4, considering antecedents of generalized elements gives rise to complications when considering jointly epimorphic families, as antecedents may be indexed by other objects of the topos than the domain of the generalized element we search antecedents of. In this case, we just had to consider the global elements of the fiber of a global element; in other words, the antecedent topology exists already in the comma category (1 ↓ FM ); in the general case, it will be scattered on the fibers of a comma (y ↓ FM ), for y the Yoneda embedding of a small, cartesian subcanonical site for the given topos. − We also used that 1 was indecomposable to retrieve global elements of the fibers of a jointly surjective family. This is not anymore a valid argument in an arbitrary Grothendieck topos.

2 An interlude on stacks

We will turn in the next section to the construction of the over-topos at a model in the general case, where the base topos in which the given model lives is an arbitrary Grothendieck topos. We chose to treat this general case separately as it requires Giraud’s theory of the classifying topos of a stack, while the set-valuated case required more conventional tools. In this section we present a number of results on stacks that we shall need in our analysis. Given a pseudofunctor I : Cop → Cat, we will denote by πI the canonical projection functor G(I) →C, where G(I) is the category obtained from I by applying the Grothendieck construction. Gir Given a Grothendieck topology J on C, there is a smallest topology JI on G(I) which makes πI a comorphism of sites to (C,J); this topology, which we call the Giraud topology, has as covering sieves those which contain cartesian lifts of J-covering families in C. Let f : F→E be a geometric morphism. There are three pseudofunctors naturally associated with it:

− We deﬁne op tf : F → Cat as the functor sending any object F of F to the category (F ↓ f ∗) and any arrow v : F → F ′ to the functor (F ′ ↓ f ∗) → (F ↓ f ∗) induced by composition with v.

11 − We deﬁne op rf : E → Cat as the pseudofunctor sending an object E of E to F/f ∗(E) and an arrow u : E → E′ to the pullback functor (f ∗(u))∗ : F/f ∗(E′) →F/f ∗(E). − We deﬁne sf : E → Cat as the functor sending an object E of E to F/f ∗(E) and an arrow u : E → E′ to the functor ∗ ∗ ′ ∗ Σf ∗(u) : F/f (E) → F/f (E ) induced by composition with f (u) (which is left adjoint to the pullback functor (f ∗(u))∗ : F/f ∗(E′) →F/f ∗(E)).

∗ ∗ By the adjunction between Σf ∗(u) and (f (u)) (for any arrow u in E), the fibration to E associated with rf coincides with the opfibration associated with sf ; in particular, this functor is both a fibration and an opfibration. This fibration is a stack for the canonical topology on E by the results in [5]. Note that the domain of this fibration also admits a canonical functor to F, which is precisely the fibration associated with tf .

Proposition 2.1. tf is a (split) stack for the canonical topology on F.

Proof. Let {fi : Fi → F | i ∈ I} be an epimorphic family in F and

∗ ∗ ∼ ′ ∗ {Ai = (Fi, Ei, αi : Fi → f (Ei)) | i ∈ I}, {fij : πi (Ai) −→ πj (Aj ) | (i, j) ∈ I × I},

′ where πi and πj are deﬁned, for each (i, j) ∈ I × I, by the pullback square

πi Fi ×F Fj Fi

′ πj fi

Fj F, fj be a collection of descent data indexed by it. For any (i, j) ∈ I,

∗ ∗ πi (Ai) = (Fi ×F Fj , Ei, αi ◦ πi : Fi ×F Fj → f (Ei)),

′ ∗ ′ ∗ πj (Aj ) = (Fi ×F Fj , Ej , αj ◦ πj : Fi ×F Fj → f (Ej )).

So the isomorphism fij actually identiﬁes with an isomorphism

fij : Ei −→∼ Ej in E such that the following triangle commutes:

αi◦πi ∗ Fi ×F Fj f (Ei)

∗ f (fij ) ′ αj ◦πj ∗ f (Ej )

Note that the fact that tf is split easily implies that

− for any i ∈ I, fii =1Ai ,

− for any i, j, k ∈ I, fjk ◦ fij = fik.

Let E be the colimit of the diagram in E having the Ei’s as vertices and the fij ’s as edges between them (with the above relations); the above identities actually imply that, for each i ∈ I, E =∼ Ei. The commutativity of the above triangles ensures that we have a cocone from the Fi’s ∗ ∗ to f (E) whose legs are given by the composites of the arrows αi with the image under f of the ∗ corresponding canonical colimit arrow Ei → E. Since the representable HomF (−,f (E)) is a sheaf for the canonical topology on F, it follows that there is a unique arrow F → f ∗(E) which restricts on the Fi’s to the legs of this cocone, and which therefore provides the required ‘amalgamation’ for our descent data. The uniqueness of the amalgamation (up to isomorphism) also follows at once from the sheaf property.

12 Corollary 2.2. Let M be a model of a geometric theory T in a Grothendieck topos E. Then the functor op tfM : E → Cat associated with the geometric morphism fM : E → Sh(CT,JT) corresponding to M via the universal property of the classifying topos for T is a stack for the canonical topology on E. In particular, if (C,J) is a site of deﬁnition for E then the functor

M Cop −→ Cat c 7−→ (c ↓ FM ), u u∗ c1 → c2 7−→ (c2 ↓ FM ) → (c1 ↓ FM )

∗ where u : (c2 ↓ FM )→(c1 ↓ FM ) is the pre-composition functor sending any generalized element A~ A~ a : c2 → J~x .φKM to a ◦ u : c1 → J~x .φKM is a stack for the topology J, where FM is the functor CT →E taking the interpretations of formulae in the model M.

∗ ∗ The category (1F ↓ f ) = G(rf ) = G(tf ) has as objects the triplets (F,E,α : F → f (E)) (where E ∈E, F ∈F and α is an arrow in F) and as arrows (F,E,α : F → f ∗(E)) → (F ′, E′, α′ : F ′ → f ∗(E′)) the pairs of arrows (v : F → F ′,u : E → E′) in F and E such that f ∗(u) ◦ α = α′ ◦ v; the functors πrf and πtf are respectively the canonical projection functors from this category to E and F. Since via these functors the category G(rf )= G(tf ) is fibered both over E and over F, it is natural to consider the smallest Grothendieck topology on it which makes πrf and πtf comorphisms of sites when E and F are endowed with their canonical topologies, in other words the join of the Giraud topologies on it induced by the canonical topologies on E and F. As we shall see, this topology will play a crucial role in connection with our construction of the over-topos. To this end, we more generally describe, for any category C and basis B for a Grothendieck topology on C such ′ ′ that (C,JB) is a site of definition for E, a basis for the Grothendieck topology on G(rf ), where rf is the ‘restriction’ of rf to C (that is, the composite of rf with the opposite of the canonical functor C→E), which is generated by the Giraud topology induced by the canonical topology on F and by the Giraud topology induced by JB. This result will be applied subsequently to the syntactic site (CT,JT) of definition of a geometric theory T inside its classifying topos. Definition 2.3. Given a Grothendieck topology J on C (resp. a basis B for a Grothendieck topology on C) such that (C,J) (resp. (C,JB)) is a site of definition for the topos E, we shall call ′ the Grothendieck topology on G(rf ) generated by the Giraud topology induced by the canonical topology on F and by the Giraud topology induced by J (resp. by JB) the (f,J)-lifted topology (resp. the (f, B)-lifted topology) and shall denote it by L(f,J) (resp. L(f,B)). Theorem 2.4. Let f : F →E be a geometric morphism and B be a basis for a Grothendieck topology on C such that (C,JB) is a site of definition for E. Then, with the above notation, the ′ (f, B)-lifted Grothendieck topology on G(rf ) has as a basis the collection of multicomposites of a ′ Gir family of cartesian lifts (with respect to rf ) of arrows in a family of B with JF -covering families, that is, the collection of families

(u , (ξ ,b )) ij i fij ((dij , (ci,bij )) (F, (c,a)))i∈I,j∈Ji where (ξi : ci → c)i∈I is a family in B(c) and the families

∗ (bij : dij → f (ci) ×f ∗(c) F )j∈Ji f are epimorphic in E for each i ∈ I:

uij dij b fij

∗ πi f (ci) ×f ∗(c) F F bij y ai a ∗ ∗ f (ξi) ∗ f (ci) f (c)

13 Proof. Note that the collection of arrows

(u , (ξ ,b )) ij i fij (dij , (ci,bij )) (F, (c,a)) i∈I,j∈Ji is the multicomposite of the family

∗ ∗ (πi, (ξi, 1)) : (f (ci) ×f (c) F, (ci,ai)) → (F, (c,a))i∈I , each of whose arrows is cartesian with respect to the ﬁbration rf to E, with the families

∗ ∗ (bij , (1ci , bij )) : (dij , (ci,bij )) → (f (ci) ×f (c) F, (ci,ai))j∈J f f i

(for i ∈ I), each of whose arrows is cartesian with respect to the fibration tf to F (and vertical with respect to the fibration rf ). The first condition in the definition of basis is clearly satisfied, since B is a basis. The second condition follows from the stability under pullback of epimorphic families in F as well as of families in B, in light of the compatibility of multicomposition with respect to pullback. It remains to show that the collection of families specified in the theorem is closed with respect to multicomposition. Let (u , (ξ ,b )) ij i fij (dij , (ci,bij )) (F, (c,a)) i∈I, j∈Ji and, for each i ∈ I, j ∈ Ji,

ij ij ij ij ij ij (ukl, (ξk ,bfkl)) (dkl, (ck ,bkl)) ((dij , (ci,bij )) ij ij k∈K ,l∈Lk

uij dij kl kl ij bfkl

∗ ij ∗ f (ck ) ×f (ci) dij dij ij bkl y bij f ∗(ξij ) ∗ ij k ∗ f (ck ) f (ci) be families satisfying the conditions in the theorem. We want to prove that their multicomposite also satisﬁes these conditions. As B is a basis for a Grothendieck topology, the family

ij ij ξi◦ξk c c ij k i∈I,j∈Ji,k∈K is in B. Consider the following pullback diagrams:

∗ ij ∗ f (ck ) ×f ∗(c) F f (ci) ×f ∗(c) F F y y a

f ∗(ξij ) ∗ ∗ ij k ∗ f (ξi) ∗ f (ck ) f (ci) f (c)

∗ ij f (ξi◦ξk )

14 For each i ∈ I and j ∈ Ji, in the diagram

ij hukli k∈Kij ij l∈L k

ij ∗ ij ` dkl f (ck ) ×f ∗(c) dij dij k∈Kij ij y l∈L k

∗ ij ∗ f (ck ) ×f ∗(c) F f (ci) ×f ∗(c) F y ij hbkli k∈Kij ij l∈L k ∗ ij ∗ f (ck ) ∗ ij f (ci) f (ξk ) both the front lower and back squares are pullbacks, hence so is the top square. Then in the diagram

ij hhukli k∈Kij i i∈I ij j∈J l∈L i k

ij ∗ ij ` ` dkl ` ` (f (ck ) ×f ∗(c) dij ) ` dij i∈I k∈Kij i∈I k∈Kij i∈I j∈J ij j∈J j∈J i l∈L i i hb i k y fij i∈I j∈Ji

∗ ij ∗ ` ` (f (ck ) ×f ∗(c) F ) ` (f (ci) ×f ∗(c) F ) i∈I k∈Kij i∈I j∈J i y ij hhbkli k∈Kij i i∈I ij j∈J l∈L i k ∗ ij ∗ f (ck ) ij f (ci) ` ` f ∗ ξ ` i∈I k∈Kij ` h ( k )i j∈Ji i∈I j∈Ji i∈I k∈Kij the upper square is a pullback, whence the left-hand arrow in it is an epimorphism (as it is the pullback of the epimorphism hbij i i∈I ). Therefore the arrow f j∈Ji

ij ij ∗ ij hhbkli k∈Kij i i∈I : a a dkl → a a (f (ck ) ×f ∗(c) F ) ij j∈J f l∈L i k i∈I k∈Kij i∈I k∈Kij j∈J ij j∈J i l∈L i k is also an epimorphism, as it is the composite of two epimorphisms. But this is precisely the arrow corresponding to our multicomposite family, whence we can conclude that the latter satisﬁes the conditions of the theorem, as required (as coproducts are disjoint in a topos, a coproduct of arrows is an epimorphism if and only if each of the arrows are). We now apply the theorem to the geometric morphism to the classifying topos of a geometric theory T, represented as the topos of sheaves on its geometric syntactic site, induced by a model of T:

Corollary 2.5. Let M be a model of a geometric theory T in a Grothendieck topos E and fM : E → Sh(CT,JT) the geometric morphism corresponding to it via the universal property of the classifying topos for T. Then we have:

(i) The (fM ,JT)-lifted topology L(fM ,JT) on (1E ↓ FM ) has as a basis the collection of families

~ (uij , ([θi]T,bij )) Ai f A~ (dij ({~x .φi} , JθiKM (a))) (e, ({~x .φ} ,a) i i∈I,j∈Ji

A~i A~ A~ where e ∈E, ([θi]T : {~xi .φi}→{~x .φ})i∈I is a family in BT({~x .φ}) and the families

−1 bij : dij → JθiKM (a)j∈J f i

15 are epimorphic in E for each i ∈ I:

u dij ij b fij

−1 πi JθiKM (a) e bij y JθiKM (a) a

Jθ K A~i i M A~ J~xi .φiKM J~x .φKM

(ii) For any separating set C for E, the canonical functor C→E induces a L(fM ,JT)-dense functor (iC ↓ FM ) → (1E ↓ FM ), where iC is the canonical embedding of C in E, and the Grothendieck

topology induced by L(fM ,JT) on the category (iC ↓ FM ) has as a basis the collection of families

~ (uij , ([θi]T,bij )) Ai f A~ (dij , ({~x .φi} , JθiKM (a))) (c, ({~x .φ} ,a) i i∈I,j∈Ji

A~i A~ A~ where c, dij ∈ C for each i and j, ([θi]T : {~xi .φi}→{~x .φ})i∈I is a family in BT({~x .φ}) and the families −1 bij : dij → JθiKM (a)j∈J f i are epimorphic in E for each i ∈ I:

u dij ij b fij

−1 πi JθiKM (a) c bij y JθiKM (a) a

Jθ K A~i i M A~ J~xi .φiKM J~x .φKM

(iii) If E is the topos Set of sets and 1 : {∗} → E is the functor from the one-object and one-arrow category {∗} to E picking the terminal object of Set, the embedding (R M)=(1{∗} ↓ FM ) → (1E ↓ FM ) induced by the functor 1 is L(fM ,JT)-dense and the topology induced by L(fM ,JT) on (R M) is the antecedent topology of Definition 1.1. Proof. (i) This is the particular case of Theorem 2.4 by taking E to be the classifying topos of T and (C,J) the geometric syntactic site of T. (ii) This easily follows from the definition of induced topology on a full dense subcategory. (iii) This is the particular case of (ii) when E is Set and C is the category {∗}, regarded as a separating set for E through the embedding 1 : {∗} → E. We now fix a Grothendieck topos E and a small subcanonical cartesian site (C,J) of definition for E. Definition 2.6. A cartesian stack is a stack M on (C,J) such that any M(c) is a cartesian category, and such that any transition functor M(u), for u a morphism in C, is cartesian. A morphism of cartesian stacks is a morphism of stacks α : M1 → M2 such that in any c, αc : M1(c) → M2(c) is cartesian. We shall denote by Stcart(C,J) the category of stacks on a site (C,J) and morphisms of cartesian stacks between them. The following lemma, whose proof is straightforward, expresses cartesian lifts in cartesian fibrations as pullback squares, and will serve for describing one half of the correspondence provided by Giraud’s construction of the classifying topos of a stack recalled below:

Lemma 2.7. Let M be a cartesian stack, with 1M(c) the terminal object of the ﬁber of M at c. M Then, in the Grothendieck ﬁbration πM : R →C, for any arrow u : c1 → c2 and any object a in M(c2), the following square is a pullback:

(u,1M(u)(a) ) (c1, M(u)(a)) (c2,a) y (1c1 ,!M(u)(a) ) (1c2 ,!a) (u,11M ) (c1 ) (c1, 1M(c1)) (c2, 1M(c2))

16 Proposition 2.8. For any cartesian stack on a site M on (C,J) whose underlying category C is M cartesian, R is a cartesian category. M Proof. We must prove that R is cartesian. As we shall see, the considered property holds M M globally on R out of holding in a specific fiber. Let (ci,ai)i∈I a finite diagram in R ; then, as C is cartesian, we can compute the limit pi : limi∈I ci → ci in C. Moreover, M(limi∈I ci) is cartesian, so the finite limit limi∈I M(pi)(ai) also exists in M(limi∈I ci), providing a cone

((pi, πi) : (lim ci, lim M(pi)(ai)) → (ci,ai))i∈I i∈I i∈I M M M in R , where πi : limi∈I (pi)(ai) → (pi)(ai) is the projection in the ﬁber. Now for any other cone ((vi, qi) : (c,a) → (ci,ai))i∈I , with qi : a → M(vi)(ai), there exists a unique arrow w : c → limi∈I ci by the universal property of the limit in C; but, as the transition functors M(pi) are cartesian, we have

M(w)(lim M(pi)(ai)) ≃ lim M(w)M(pi)(ai) ≃ lim M(vi)(ai) i∈I i∈I i∈I inducing a unique arrow f : a → M(w)(limi∈I M(pi)(ai)), so that there is a unique factorization in M R (w,f) (c,a) (lim ci, lim M(pi)(ai)) i∈I i∈I

(vi,qi) (pi,πi) (ci,ai) as desired. Let us recall from [5] the following fundamental classification result: Theorem 2.9. Let M be a cartesian stack on a cartesian site (C,J) of definition for a topos E, Gir M M Gir and JM be the Giraud topology induced by J. Then the sheaf topos E[ ]= Sh(R ,JM ), with its M M Gir canonical morphism pM : E[ ] →E induced by the comorphism of sites πM : (R ,JM ) → (C,J), is the classifier of M, in the sense that the following universal property holds: for any geometric morphism g : G→E, ∗ GeomE [g,pM] ≃ Stcart(C,J)[M, G/g ].

Given a triangle f G E[M]

g pM E of geometric morphisms, the cartesian morphism of stacks α : M → G/g∗ associated with g as in the theorem can be described as follows. ∗ ∼ ∗ ∗ ∗ We have g = f pM. The restriction to C of the functor pM is just the terminal section 1M(−), so we have a commutative (up to isomorphism) triangle

∗ M f R G

∗ 1M(−) g C

M M Now, for any (c,a) in R , there is a unique arrow !a : a → 1M(c) in the ﬁber (c), which is ∗ ∗ ∗ ∗ ∗ sent by f to an arrow f (1c, !a): f (c,a) → f (c, 1M(c)) ≃ g (c). This yields, for each c ∈ C, a functor ∗ αc : M(c) → G/g (c) ∗ ∗ ∗ a 7→ f (1c, !a): f (c,a) → g (c)

17 ∗ The functors (αc : M(c) → G/g (c))c∈C actually deﬁne a natural transformation, that is, all the squares of the following form commute up to isomorphism:

α c2 ∗ M(c2) G/g (c2)

M(u) (g∗(u))∗ α c1 ∗ M(c1) G/g (c1)

M Indeed, by Lemma 2.7 the following square is a pullback in R :

(u,1M(u)(a) ) (c1, M(u)(a)) (c2,a) y (1c1 ,!M(u)(a) ) (1c2 ,!a) (u,11M ) (c1 ) (c1, 1M(c1)) (c2, 1M(c2))

It thus follows that, the functor f ∗ being cartesian, the following square is also a pullback:

∗ f (u,1M ) ∗ (u)(a) ∗ f (c1, M(u)(a))) f (c2,a) y M αc1 ( (u)(a)) αc2 (a) ∗ ∗ g u ∗ g (c1) g (c2)

M ∗ ∗ So αc1 ( (u)(a)) = (g u) αc2 (a), which is precisely the content of the naturality condition.

The cartesianness of α is actually inherited from that of f ∗, as α acts as a restriction of f ∗ on each ﬁber in light of Proposition 2.8.

3 The T-over-topos of a model in an arbitrary topos

Now we turn to the construction of the over-topos uM : E[M] →E associated with a T-model M in an arbitrary topos E. Recall that the desired formula will be, for any E-topos g : G→E,

∗ GeomE [g,uM ] ≃ T[G]/g (M).

We suppose that we are given a cartesian small subcanonical site of deﬁnition (C,J) for E. Note that the canonical embedding iC of C into E identiﬁes C with a separating set of objects for E and J with the Grothendieck topology induced on it by the canonical topology on E.

In the case where E = Set, as Set is generated by 1 under coproducts, the global elements A~ A~ 1 → J~x .φKM are suﬃcient to generate all the generalized elements X ≃ `X 1 → J~x .φKM . In the general case, global elements must be replaced by generalized elements, possibly restricting to those whose domain is an object of C, which we call basic generalized elements. Indeed, we shall replace the category of global elements of M by an indexed category of basic generalized elements of interpretations in M of geometric formulas over the language of T, that is, with the comma category (iC ↓ FM ), where FM : (CT,JT) → E is the JT-continuous cartesian functor sending a A~ A~ geometric formula-in-context {~x .φ} to its interpretation J~x .φKM in M. This deﬁnes a prestack

M Cop −→ Cat c 7−→ (c ↓ FM ), u u∗ c1 → c2 7−→ (c2 ↓ FM ) → (c1 ↓ FM )

∗ A~ where u : (c2 ↓ FM )→(c1 ↓ FM ) is the pre-composition functor sending some a : c2 → J~x .φKM A~ to a ◦ u : c1 → J~x .φKM .

Proposition 3.1. M is a cartesian stack on (C,J). That is, for each c in C, (c ↓ FM ) is cartesian, ′ and for any arrow u : c → c in C, the transition functor (u ↓ FM ) is cartesian.

Proof. This is a consequence of FM being cartesian: indeed any ﬁnite diagram

A~i (ai : c → J~xi .φiKM )i∈I

18 A~i A~i in (c ↓ FM ) deﬁnes a unique arrow (ai)i∈I : c → limi∈I J~xi .φiKM ≃ FM (limi∈I {~xi .φi}). Now, for A~ any arrow b : c → J~x .φKM equipped with a cone ([θi]: b → ai)i∈I in (c ↓ FM ), there is a canonical A~ A~i arrow [θ]T : {~x .φ} → limi∈I {~xi .φi} in CT providing a factorization of the [θi]T’s, whence JθKM also factorizes each JθiKM , thus providing a universal factorization of the cone in (c ↓ FM ). For any ~ ′ ′ Ai u : c → c, it is easy to see that the composite (ai)i∈I ◦ u : c → FM (limi∈I {~xi .φi}) is the limit of ′ the ai ◦ u’s in (c ↓ FM ) by the uniqueness of the factorization of a cone through the limit. M As a consequence, the ﬁbred category R = (iC ↓ FM ) is cartesian. Its objects are the pairs A~ (c,a), where c is an object of C and a is a c-indexed basic element a : c → J~x .φKM , while an arrow (c1,a1) → (c2,a2) is a pair (u, [θ]T) consisting of an arrow u : c1 → c2 in C and an arrow [θ]T in CT such that a2 ◦ u = JθKM ◦ a1: u c1 c2

a1 a2

JθK ~a1 M ~a2 J~x1 .φ1KM J~x2 .φ2KM In order to complete our generalization of the construction of section 2, we need to equip the M category R with a Grothendieck topology representing the analogue of the antecedents topology. This is not straightforward since, in the general context, there are no vertical covers a priori. Indeed, one would be tempted to deﬁne in each ﬁber (c ↓ FM ) a topology of c-indexed antecedents, made of families of triangles

c b JθiKM ◦b=a

Jθ K A~i i M A~ J~xi .φiKM J~x .φKM

A~i A~ A~ for ([θi]: {~xi .φi}→{~x .φ})i∈I a family in BT({~x .φ}) and b ranging over all the antecedents of a along the arrows JθiKi∈I . However this would not work, since, though FM sends the family A~i A~ ([θi]: {~xi .φi}→{~x .φ})i∈I to an epimorphic family in E, there is no reason for a c-indexed basic element a to have a c-indexed antecedent along any of the JθiKM ’s. In order to take the horizontal components into account, we need to leave the domain of ‘antecedent’ generalized elements vary −1 among the objects of C, and require them to cover the ﬁbers JθiKM (a) of a along the arrows JθiKM :

−1 `i∈I JθiKM (a) c y a

` JθiKM i∈I A~i A~ ` J~xi .φiKM J~x .φKM i∈I

−1 (Note that, as in the set-based setting, some JθiKM (a) may be empty, though they jointly cover c). In light of the characterization of the antecedents topology in the set-based setting in terms of lifted topologies, provided by Corollary 2.5, we are thus led to defining the antecedents topology M on R as follows: ant M Definition 3.2. The antecedents topology JM on R is the Grothendieck topology on (iC ↓ FM ) induced by the (fM ,JT)-topology on (1E ↓ FM ) (in the sense of Definition 2.3); that is, it has as a basis the collection of families

(uij ,[θi]T) (dij ,bij ) (c,a))i∈I, j∈Ji

A~ A~i A~ where a : c → J~x .φKM is a basic generalized element, ([θi]T : {~xi .φi}→{~x .φ})i∈I is a family A~ in BT({~x .φ}), (uij : dij → c)j∈Ji is a family of arrows in C (for each i ∈ I) and (bij : dij → A~i J~xi .φiKM )j∈Ji is a family of arrows (for each i ∈ I) making the diagrams

uij dij c

bij a

Jθ K A~i i M A~ J~xi .φiKM J~x .φKM

19 commutative and such that the family of arrows

−1 (bij : dij → hJθiKM i (a))j∈Ji f as in the following diagram is epimorphic :

u dij ij b fij

−1 πi JθiKM (a) c bij y a

Jθ K A~i i M A~ J~xi .φiKM J~x .φKM

ant Remarks. (a) The covering families in the definition of JM are indexed by a dependent sum, with first a set indexing a basic cover of JT and for each term of this cover, a basic covering family. To obtain a more conventional presentation, one can equivalently use a single indexing set J and require a family of squares ({θi},ui)i∈I to induce an epimorphism h(bi,ui)ii∈I and have (θi)i∈I in JT. In particular, for a presentation as above, take J = ì∈I Ji and impose {θ(i,j)} := {θi}.

(b) As in the set-based case, the JθiKM ’s are only jointly epimorphic, so a generalized element A~ a : c → J~x .φKM may have no antecedent along a chosen θi, that is, the corresponding set Ji may be empty. However, antecedent families live over subfamilies of JT-covering families, as shown by Proposition 3.3 below. M ant Proposition 3.3. There is a comorphism of sites (R ,JM ) → (CT,JT) sending a basic general- A~ A~ ized element a : c → J~x .φKM to the underlying sort {~x .φ}.

ant Proof. This follows immediately from the characterization of JM as the Grothendieck topology on (iC ↓ FM ) induced by the (fM ,JT)-topology on (1E ↓ FM ). Remark. The comorphism of sites of Proposition 3.3 is not a fibration, unlike its topos-theoretic ∗ extension provided by the canonical projection functor (1E ↓ fM ) → Sh(CT,JT). This explains why the antecedent topology is not a lifted topology but rather a topology induced on a smaller subcategory by a lifted topology existing at the topos level, and hence that its description is more involved than that for its topos-theoretic extension. This is an illustration of the importance of developing invariant constructions at the topos-theoretic level and of investigating only later how such notions can be described at the level of sites. ant Since JM is the Grothendieck topology on (iC ↓ FM ) induced by the (fM ,JT)-lifted topology M on (1E ↓ FM ), the canonical projection functor πM : R → C to C is a comorphism of sites M ant (R ,JM ) → (C,J). We are now ready to define the over-topos at M in the general setting: Definition 3.4. The T-over-topos of E at M is defined as

M ant uM E[M] ≃ Sh(R ,JM ) −→ E M ant where uM is the geometric morphism induced by the comorphism of sites πM : (R ,JM ) → (C,J).

Remarks. (a) The inverse image of uM is the pre-composition sending c to HomC(πM (−),c) (since πM is continuous by Theorem 4.44 [3]), while the direct image is the restriction to sheaves of the right Kan extension along πM . (b) In the set-based case, choosing as a presentation site the category {∗} with the trivial topology on it, there is only one ﬁber, and the antecedents topology on R M of Deﬁnition 1.1 is completely concentrated in it.

Theorem 3.5. The E-topos uM : E[M] →E satisﬁes the universal property of the T-over-topos at M: that is, for any E-topos g : G→E there is an equivalence of categories

∗ GeomE [g, E[M]] ≃ T[G]/g M natural in g.

20 Proof. The proof naturally generalizes that for a set-based model. In one direction, suppose that f is a geometric morphism over E from g to uM :

f G E[M]

g uM E

ant This deﬁnes a JM -continuous, cartesian functor

∗ M f R −→ G. By Theorem 2.9, it also induces a morphism of cartesian stacks over E

f M −→ G/g∗ whose components fc ∗ (c ↓ FM ) −→ G/g c

A~ (for each c in C) are cartesian functors sending a given basic generalized element a : c → J~x .φKM ∗ ˜ ∗ to a certain arrow fc(a): N(c,a) → g c in G. We want to associate with f a morphism f : N → g M A~ A~ in T[G]. For each {~x .φ} in CT, the category (iC ↓ J~x .φKM ) of basic generalized elements is small; we can thus deﬁne

N{~xA~ .φ} := colim N(c,a) ≃ colim a N(c,a) (i ↓{~xA~ .φ}) c∈C C A~ a∈HomE (c,J~x .φKM ) and, by using the universal property of the colimit, an arrow f{~xA~.φ} as in the following diagram (where the ja’s are the legs of the colimit cocone):

fc(a) ∗ N(c,a) g c

g∗(a) ja f ~ {~xA.φ} A~ N{~xA~ .φ} J~x .φKg∗M

Let us ﬁrst check this yields a JT-continuous cartesian functor N : (CT,JT) → G, showing how the above assignment on objects naturally extends to arrows. A~1 A~2 Given an arrow [θ]T : { ~x1 .φ1}→{ ~x2 .φ2} in CT, composing with JθKM allows one to ~ A1 associate with any generalized element a : c → J ~x1 .φKM a generalized element JθKM ◦ a : c → ~ A2 J ~x2 .φKM . So by the functoriality of fc we have:

fc([θ]T) N(c,a) N(c,JθKM ◦a)

fc(a) fc(JθKM ◦a) g∗(c)

We can thus deﬁne N θ T : N A~ → N A~ as the arrow determined by the universal [ ] { ~x1 1 .φ1} { ~x2 2 .φ2} property of the colimit by the requirement that all the diagrams of the form

fc([θ]T) N(c,a) N(c,JθKM ◦a)

j ja JθKM ◦a N[θ]T N A~ N A~ { ~x1 1 .φ1} { ~x2 2 .φ2} should commute. ∗ M The fact that N is cartesian follows at once from the fact that the functor f : R → G is, in light of the deﬁnition of N in terms of colimits and of the stability of these under pullback. So it remains to prove its JT-continuity. A~i A~ A~ Let ([θi]T : {~xi .φi}→{~x .φ})i∈I be a family in BT({~x .φ}); we want to show that the ~ ~ family N[θi]T : N Ai → N{~xA.φ}i∈I is epimorphic. First, we notice that this condition can be {~xi .φi}

21 conveniently phrased in terms of basic generalized elements, as follows: for any basic generalized element w : e → N{~xA~ .φ} there are an epimorphic family {uk : ek → e | k ∈ K} lying in C and for each k ∈ K an element ik ∈ I and a basic generalized element wk : ek → N A~ such that ik { ~xik .φik } the diagram uk ek e

wk w N[θ ]T ik N A~ N A~ ik {~x .φ} { ~xik .φik } commutes. Since colimits yield epimorphic families in a topos, in light of the deﬁnition of N we can further rewrite this condition as follows: for any basic generalized element w : e → N(c,a), A~ where a : c → J~x .φKM , there are an epimorphic family {uk : ek → e | k ∈ K} lying in C and for ~ Aik each k ∈ K an element ik ∈ I, an object (dk,bk) of the category (iC ↓ J ~xik .φik KM ) and an arrow wk : ek → N(dk,bk) such that the diagram

uk ek e

wk w N[θ ]T ik N(dk,bk) N(c,a) commutes. A~ Now, given a generalized element a : c → J~x .φKM , we may obtain a covering family for the ant antecedent topology JM (with respect to our original JT-cover) by covering each of the ﬁbers −1 −1 JθiKM (a) by basic generalized elements bij : dij → JθiKM (a): f u dij ij b fij

−1 πi JθiKM (a) c bij y ∗ a JθiKM (a)

Jθ K A~i i M A~ J~xi .φiKM J~x .φKM

∗ ant ∗ The functor f being JM -continuous, f sends this family to an epimorphic family

f(uij , [θi]T): N d ,b → N c,a ( ij ij ) ( )i∈I, j∈Ji in G; but this clearly implies our thesis. Conversely, let f : N → g∗M be a morphism of T-models in G. We can regard f as a CT-indexed A~ A~ family {f{~xA~ .φ} : J~x .φKN → J~x .φKg∗M } of morphisms in G (subject to the naturality conditions). A~ For each basic generalized element a : c → J~x .φKM , we deﬁne N(c,a) as the following pullback:

∗ N c,a g c ( ) y g∗(a)

f ~ A~ {~xA.φ} A~ J~x .φKN J~x .φKg∗M

M Given a morphism (u, [θ]T) : (c1,a1) → (c2,a2) in R , we have by the naturality of f (seen as a morphism of JT-continuous cartesian functors) the following commutative square:

f ~a {~x 1 .φ } ~a1 1 1 ~a1 J~x1 .φ1KN J~x1 .φ1Kg∗M

JθKN JθKg∗M f ~a {~x 2 .φ } ~a2 2 2 ~a2 J~x2 .φ2KN J~x2 .φ2Kg∗M

We can thus deﬁne N(u,[θ]M) : N(c1,a1) → N(c2,a2) as the unique arrow making the following diagram (where the front and back faces are pullback) commute:

22 N g∗(c ) (c1,a1) 1 ∗ N(u,[θ]M) g (u) a1 ∗ N(c2,a2) g (c2) y f ~a {~x 1 .φ } 1 1 ~a1 ~a1 a2 J~x1 .φ1KN J~x1 .φ1Kg∗M JθKg∗M

f ~a JθKN {~x 2 .φ } ~a2 2 2 ~a2 J~x2 .φ2KN J~x2 .φ2Kg∗M This yields a functor M N(−) R −→ G. ant We want to show that this functor is cartesian and JM -continuous. The fact that it is cartesian follows easily from the fact that the functor FN : CT → G corresponding to the model N is, in light M of the construction of finite limits in the category R provided by Proposition 2.8. ant Concerning JM -continuity, we shall consider the extension of N(−) to the category (1E ↓ FM ) and show its continuity with respect to the extended topology, as the latter admits a more natural characterization as the (fM ,JT)-lifted topology; recall that this topology has a basis consisting of multicomposites of covering families of horizontal arrows for the fibration rfM and of covering families of horizontal arrows for the fibration tfM , so continuity can be checked separately with respect to each of these families. Let us start by showing the continuity with respect to the covering families of horizontal arrows for tfM ; this can be checked without any problems in terms of the site of definition (C,J) for E. A~ For a J-covering family (ui : ci → c)i∈I and a generalized element a : c → J~x .φKM , consider the following diagram: ∗ N(ci,a◦ui) g ci

N ∗ (ui,1 ~ ) g ui {~xA.φ} ∗ N(c,a) g c y g∗(a)

A~ A~ J~x .φKN J~x .φKg∗M f ~ {~xA.φ} The lower square and the outer rectangle are pullbacks, whence by the pullback lemma the upper square is also a pullback. Since (C,J) is a site of deﬁnition for E, g∗ sends J-covering families to epimorphic families; so we have a pullback of epimorphisms

N g∗c ` (ci,a◦ui) ` i i∈I i∈I y hN i ∗ (ui,1 ~ ) i∈I hg u i {~xA.φ} i i∈I ∗ N(c,a) g c ensuring that N(−) sends the horizontal covering family

A~ A~ ~ (ui, 1{~xA.φ}) : (ci, ({~x .φ},a ◦ ui)) → (c, ({~x .φ},a))i∈I to an epimorphic family. Now we turn to the proof of the continuity of N(−) with respect to the covering families of horizontal arrows for the ﬁbration rfM . A~ T Given a family [θi]: {~xi.φi}→{~x.φ}i∈I in B and a generalized element a : c → J~x .φKM , we want to prove that N(−) sends the family

−1 ∗ (πi, [θi]) : (JθiKM (a), ({~xi.φi}, JθiKM (a))) → (c, ({~x.φ},a))i∈I M in R to an epimorphic family. Consider, for each i ∈ I, the following diagram:

23 ∗ −1 −1 ∗ N(JθiK (a),JθiK (a)) g (JθiK (a)) ∗ N(πi,[θi]) g (πi) ∗ JθiK (a)) ∗ N(c,a) g (c) y f ~ { ~x Ai .φ } A~i i i A~i a J~xi .φiKN J~xi .φiKg∗M JθiKg∗M

JθiKN f ~ A~ {~xA.φ} A~ J~x .φKN J~x .φKg∗M

The front square is a pullback by deﬁnition of N(c,a), the back square is a pullback by deﬁnition ∗ −1 ∗ of N(JθiK (a),JθiK (a)) and the right-hand lateral one is a pullback since g is cartesian. So the composite of the back square with the right-hand lateral square is a pullback, and hence by the pullback lemma the left-hand lateral square is also a pullback (as the front square is a pullback). Our thesis thus follows from the fact that, since N is a T-model by our hypotheses, the family

A~i A~ JθiKN : J~xi .φiKN → J~x .φKN i∈I is epimorphic. Finally, proving that the functors deﬁned above are mutually quasi-inverses is a straightforward exercise which we leave to the reader. To conclude this section, we remark that our construction of the over-topos is functorial; that is, a morphism of T-models naturally induces a canonical morphism between the corresponding T- over-toposes. Let f : M1 → M2 be a morphism in T[E]. This is a same as a natural transformation

M1 (CT,JT) f E

M2 whose components f ~ A~ {~xA.φ} A~ J~x .φKM1 J~x .φKM2 are indexed by the objects of CT. This induces an indexed functor α : M1 ⇒ M2 assigning to each object c of C the functor αc=(c↓f) (c ↓ FM1 ) (c ↓ FM2 )

A~ A~ sending a basic global element a : c → J~x .φKM1 to f{~xA~ .φ} ◦ a : c → J~x .φKM2 . This clearly yields a comorphism of sites ant ant ((iC ↓ FM1 ),JM1 ) → ((iC ↓ FM2 ),JM2 ) over C, which thus induces a geometric morphism

E[α]: E[M1] →E[M2] over E between the associated over-toposes.

4 The T-over-topos as a bilimit

In this section we explore the 2-dimensional aspects of the construction of the T-over-topos, exhibiting it as a bilimit in the bicategory of Grothendieck toposes; we shall also discuss its relation with the notion of totally connected topos. We will see that, when M is a model of T in Set, hence a point fM : Set → Set[T] of Set[T] over Set, this yields a totally connected topos over Set, which coincides with the colocalization of Set[T] at M studied in [4][Theorem C3.6.19]; indeed, in this case the over-topos uM : Set[M] → Set admits a Set-point sM : Set → Set[M] providing a factorization of fM : Set → Set[T] as sM : Set → Set[M] followed by the geometric morphism fM ◦ uM : Set[M] → Set[T], which satisfies the universal property of the colocalization of Set[T] at M. Note, however, that, for a section s : S→E of a S-topos p : E → S, the colocalization of E at s, as defined in [4] differs in general from the over-topos of S at s, since the universal property of the former provides an equivalence between GeomS (F,us) and GeomS (F, E)/(s ◦ g) (where E is regarded as a S-topos via p), while the universal property of the latter provides and equivalence between GeomS (F,us) and Geom(F, E)/(s ◦ g).

24 Indeed, we can geometrically deﬁne the over-topos of F at a geometric morphism f : F →E as a F-topos uf : F[f] → F with a morphism ξf : F[f] → E satisfying the following universal property: for any F-topos g : G→F, there is an equivalence

GeomF (F,uf ) ≃ Geom(F, E)/(f ◦ g) natural in g, induced by composition with ξf . Note that the condition p ◦ s = 1 is not sufficient in general to ensure that ξs is a morphism over S (i.e., that pαs : p ◦ ξs → p ◦ s ◦ us =∼ us, where αs : ξs → s ◦ us is the canonical geometric transformation, is an isomorphism), which explains why the colocalization of E at s is not in general equivalent to the over-topos of S at s. Of course, the above definition can be readily generalized to yield a notion of over-topos relative to an arbitrary base topos; if we do so and regard s as a morphism of S-toposes then the S-over-topos of S at s coincides precisely with the colocalization of E at s. Next, we recall some classical notions about bilimits in the bicategory of Grothendieck toposes; the reader may refer to [4][B4.1] and [6] for more details. Definition 4.1. Let K be a bicategory and f : X → Z, g : Y → Z two 1-cells with common codomain. − A bipullback of f, g is an invertible 2-cell

∗ f,g g f X ×Z Y Y

f ∗g α g ≃ XZ f

such that for any 0-cell W and any invertible 2-cell

Wt Y s β g ≃ XZ f

there is an arrow uβ : W → X ×Z Y , unique up to unique invertible 2-cell, and a pair of invertible 2-cells αs, αt such that:

W t αt u ≃ Wt Y β α s g∗f s β g = ≃ f,g ≃ X ×Z Y Y s XZ f ∗g α g f ≃ XZ f

In other words, the bipullback induces for each W an equivalence in Cat

f,g K[W,f],K[W,g] K[W, X ×Z Y ] ≃ K[W, X] ×K[W,Z] K[W, Y ]. In the sequel, the universal 2-cell in the bipullback will be considered implicit, and we shall use the same notation as for a pullback in a 1-category. − A bicomma object is a 2-cell p (f ↓ g) g Y

pf λf,g g XZ f such that for any other 2-cell Wt Y

s µ g XZ f

25 there exists a 1-cell uµ : W → (f ↓ g), unique up to a unique invertible 2-cell, and a pair of invertible 2-cells αs, αt such that:

W t αt u ≃ Wt Y µ αs ∗ f,g g f s µ g = ≃ X ×Z Y Y s XZ ∗ λf,g g f f g XZ f

In other words, we have an equivalence in Cat

K[W, (f ↓ g)] ≃ K[W, f] ↓ K[W, g].

Deﬁnition 4.2. Let K be a bicategory and X a 0-cell. Then a bipower with 2 of X is a 2-cell

∂0

2 X λX X

∂1 where ∂0 and ∂1 are called respectively the universal domain and universal codomain of X, such that for any 2-cell f Yµ X g

2 there exists a 1-cell uµ : W → X , unique up to a unique invertible 2-cell, and a pair of invertible 2-cells αf , αg such that:

f αf ∂0 ≃ uµ 2 Yµ X = Y X λX X αg g ≃ ∂1 g

In other words, we have an equivalence of categories

Cat[2, K[Y,X]] ≃ K[Y,X2].

Remark. Bipullbacks, bicommas and bipowers are related as follows: the bipower of X is the bicomma of the identity 1-cell with itself

∂ X2 0 X

∂1 λX X X and the bicomma of two arrows f,g can be constructed from bipowers and pullbacks as:

g,∂0 2 (f ↓ g) Z ×X X Z y y

f,∂1 2 2 ∂0 Y ×X X X X y ∂1 λX Y X X

Remark. Whenever they exist, bilimits are unique up to unique equivalence in K.

26 There are other shapes of finite bilimits that can be defined, and that exist in the bicategory of Grothendieck toposes, but we only need the above ones in this paper. The following fact is standard, and its proof can be found either in [4] or in [6]: Proposition 4.3. The bicategory of Grothendieck toposes is finitely bicomplete; in particular it has bipullbacks, bicommas and bipowers with 2.

2 Proposition 4.4. The universal codomain ∂1 : Set[T] → Set[T] is the over-topos of Set[T] at the universal domain UT, that is, we have a geometric equivalence

2 Set[T] ≃ Set[T][UT] and an invertible 2-cell:

≃ Set[T]2 Set[T][UT]

≃ u ∂1 UT Set[T]

Proof. This is a consequence of the universal property of the T-over-topos and the way in which a T-model M in E corresponds to a geometric morphism gM : E → Set[T] by the universal property of classifying toposes. Indeed, any object of GeomE [gM , ∂1] is an invertible 2-cell

f E Set[T]2

gM ≃ ∂1 Set[T] which produces a 2-cell: ∗ ∂0 f

∗ E λSet[T]f Set[T]

∗ ∂1 f≃gM But this is exactly the same as a morphism of T-models f : N → M in E. T 2 Corollary 4.5. For any geometric theory , (CT,JUT ) is a small, cartesian, subcanonical site for Set[T]2. Proof. It suﬃces to apply the construction of the T-over-topos provided by Theorem 3.5 in the particular case E = Sh(CT,JT), M = UT and (C,J) = (CT,JT). Note that iC is simply the Yoneda [embedding yT : CT ֒→ Sh(CT,JT), and also FM coincides with yT, whence the site for Set[T][UT 2 provided by Deﬁnition 3.4 is (yT ↓ yT) ≃CT, that is, the arrow category of CT. Proposition 4.6. Let E be a Grothendieck topos and M in T[E], with E[M] the associated T-over- topos. Then the following square E[M] Set[T]2

uM ∂1 g E M Set[T] is a bipullback, where the top morphism is the name of the universal geometric transformation from gM ◦ uM to the canonical morphism E[M] → Set[T]. Proof. This is just unravelling of universal properties. Suppose we have a square:

f G Set[T]2

g ∂1 g E M Set[T]

27 This yields a morphism of T-models f in T[G] such that ∂1f = gM ◦ g, that is, an object of T[G]/g∗M deﬁning a unique morphism f as in the following diagram:

G f f

1 E[M] M Set[T]2 g uM ∂1 Set T E gM [ ]

Hence by the uniqueness up to unique equivalence of the bipullback, the T-over-topos coincides with the bipullback. Remark. Equivalently, we can deﬁne E[M] →E as a comma topos:

u E[M] M E

λM Set[T] Set[T]

Proposition 4.7. Let be M be in T[E] and g : G→E be a geometric morphism. Then the following square is a pullback: G[g∗M] E[M] y

G E

Proof. This is a simple application of the bipullback lemma: in the diagram below

G[g∗M] E[M] Set[T][UT]

ug∗M uM uUT g G E M Set[T] both the right and outer squares are bipullbacks, hence the left one is so. We now turn to the geometric interpretation of the over-topos construction. Deﬁnition 4.8 (Theorem C3.6.16 [4]). A geometric morphism f is said to be totally connected if the following equivalent conditions are fulﬁlled:

∗ − f has cartesian left adjoint f!; − f has a right adjoint in the 2-category of Grothendieck toposes; − f has a terminal section. Morally, a topos is totally connected if it has a terminal point, dually to local toposes who have an initial point. In particular, the universal codomain morphism ∂1 of a topos is totally connected (as the universal domain morphism ∂0 is local). Totally connected geometric morphisms are stable under pullback (cf. Lemma C3.6.18 [4]), so, by Proposition 4.6, the T-over-topos E[M] → E is totally connected; indeed, its terminal point is the identity morphism 1M . In particular, all the ﬁbers of this morphisms are also totally connected, while its sections

E[M] s uM E E just are the name of some homomorphism N → M in E.

28 Let us now analyse the T-over-topos construction in the particular case T = O, the (one-sorted) theory of objects. This yields a ‘thickening’ of the notion of slice topos; indeed, for any object E in a topos E, the “totally connected component” of E in E

E[E] Set[O]2 y uE ∂1 E E Set[O] classiﬁes generalized elements F → E of the object E, as opposed to the usual ´etale topos

E/E Set[O•] y E b E E Set[O] which is constructed from the universal fibration of the classifier of pointed objects over the classifier of objects. Indeed, whilst the objects of E : E/E → E are generalized elements of E, that is, morphisms F → E, E actually classifies overb E global elements of E in the sense that its sections b E/E s E b E E correspond to global elements 1 → E. In particular, its fibers at points are discrete (as E only has a set of global elements). Note that any point of E[E] defines by composition with uE a point p of E over which it lies. By the universal property of the pullback, it thus yields a section s of the totally connected geometric morphism at the corresponding fiber:

Set[p∗E] E[E] y s up∗(E) uE

Set Set p E

Note that s defines a generalized element X → p∗(E) of the stalk of E at p, equivalently a X- indexed family of global elements of p∗(E); this should be compared with the étale case, where points of the étale topos are mere global elements of the stalk. Finally, we observe that the two constructions are related as in the following diagram, where the upper square is a pullback as the bottom and front square are:

O E/E y Set[ •]

2 E E[E] Set[O] b y uE ∂1 E Set[O] E

Acknowledgements: We thank Laurent Laﬀorgue for his careful reading of a preliminary version of this paper.

References

[1] M. Artin, A. Grothendieck and J. L. Verdier, Théorie des topos et cohomologie étale des schémas - (SGA 4), Séminaire de Géométrie Algébrique du Bois-Marie, année 1963-64; second edition published as Lecture Notes in Math., vols 269, 270 and 305 (Springer-Verlag, 1972). [2] O. Caramello, Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’ (Oxford University Press, 2017).

29 [3] O. Caramello, Denseness conditions, morphisms and equivalences of toposes, arxiv:math.CT/1906.08737v3 (2020). [4] P. T. Johnstone, Sketches of an Elephant: a topos theory compendium. Vols. 1 and 2, vols. 43 and 44 of Oxford Logic Guides (Oxford University Press, 2002). [5] J. Giraud, Classifying topos, in Toposes, Algebraic Geometry and Logic, pp. 43-56 (Springer, 1972). [6] S. Zoghaib, Th´eorie homotopique de Morel-Voevodsky et applications, 2020, available at http://www.normalesup.org/ zoghaib/math/dea.pdf.

Olivia Caramello Dipartimento di Scienza e Alta Tecnologia, Universita` degli Studi dell’Insubria, via Val- leggio 11, 22100 Como, Italy. E-mail address: [email protected]

Institut des Hautes Etudes´ Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France. E-mail address: [email protected]

Axel Osmond IRIF, 8 Place Aurelie´ Nemours, 75013 Paris, France. E-mail address: [email protected]