A refactored proof of conceptual completeness

A refactored proof of
conceptual completeness
Spencer Breiner & Steve Awodey
October 20, 2012
Abstract
In this note we rework the Makkai/Reyes proof of conceptual com-
pleteness for pretoposes by highlighting the role of factorization systems.
In categorical logic, we replace logical theories T, T ′ , . . . by certain structured
categories ET, FT ′, . . .. Under this identification, a structure-preserving functor
I : ET → F ′ T corresponds to an interpretation of T in T′ . Often, we can also
think of the codomain F as a context for semantics, in which I determines an
T-model. We say that ET classifies T.
Of particular interest are the structure-preserving functors M : ET → Sets
which are, essentially, classical models of T. More precisely, the category of T-
models is equivalent to Hom str(ET, Sets), the category of structure-preserving
functors and natural transformations. In what follows, “theory” is synonymous
with “structured category”, and “model” with “structure-preserving functor to
Sets”.
Greater specificity requires selecting a logical doctrine which is, roughly
speaking, the system of connectives and inference rules which determines the
type of logic in which we are interested. For example, some familiar doctrines
include (Boolean or intuitionistic) propositional, first-order and higher-order
logics.
Other doctrines arise as fragments of some ambient language. For exam-
ple, algebraic theories are allowed no relation symbols and only (universally
quantified) equational axioms. Regular logic is the fragment of first-order logic
generated by the connectives ⊤, ∧, ∃; of course, in a Boolean context, this is
already a complete set of connectives for first-order logic.
1

These logical doctrines often correspond to certain categorical constructions.
Finite products are sufficient to express pairing and equality, allowing us to
interpret algebraic theories. Quantifiers can be interpreted as adjoints to sub-
stitution, and in a topos we can use subobject classifiers and exponentials to
interpret higher-order constructions. Functors which preserve these structures
are interpretations or models of the theory.
We will be particularly interested in the doctrine of pretoposes. This is
almost, but not quite, a fragment of first-order logic. Pretoposes have all finite
limits and regular factorizations, providing interpretations for ⊤, ∧, ∃. To these
we add two new connectives/type constructors for disjoint sums and quotients
by equivalence relations.
Quotients and finite coproducts are definable in Sets. Consequently, if
T → T extends T by adding quotients and coproducts, then any T-model M
has a unique extension to a T-model M. This, together with completeness
of set models, suffices to ensure that the addition of sums and quotients is a
conservative extension of T.
The principle benefit of adding these additional types is that it allows for
a more robust notion of notion of interpretation E → F. Thinking of F as a
semantic context, we now allow ourselves to construct E-models by quotienting
definable equivalence relations.
The unique extension of T-models to T-models also shows that the category
of models cannot possibly distinguish between T and T. The conceptual com-
pleteness theorem of Makkai and Reyes says that this worry goes no further: a
pretopos is determined (up to equivalence) by its category of models.
Specifically, suppose that we have a pretopos functor I : E → F. Given a
model N : F → Sets, we can “restrict along I” to define an E-model I ∗ N =
N ◦ I : E → Sets.
Theorem (Conceptual Completeness). A pretopos functor I : E → F is an
2

equivalence of categories if and only if I ∗ : Mod(F) → Mod(E) is.
Of course, one direction is trivial; the interesting observation is that a se-
mantic equivalence implies an equivalence of theories.
We will begin by recalling the central framework of categorical logic, special-
ized to pretoposes. A crucial ingredient in Makkai & Reyes’ proof is the “method
of diagrams”, which we discuss in section 2. From there, we will show that the
natural ternary factorization system in Cat induces a closely related system
in Ptop and another (in the opposite direction) between categories of model.
These factorizations, in turn, organize the proof of conceptual completeness.
1 Categorical Logic
Notes:
Lex, Ptop, etc. are quasi-algebraic theories. Therefore, they are closed
under filtered colimits (since these commute with finite limits).
Description of logical description of pushouts in Ptop
2 The Method of Diagrams
Classically, the diagram of an E-model M is constructed as follows:
• Extend the language of E by adding a new constant ca for each element
a ∈ |M|.
• Extend the theory E by adding the axiom ϕ(ca) whenever M |= ϕ(a).
Now fix a pretopos E and a model M : E → Sets. We would like to define
a pretopos Th(M) which corresponds to the logical diagram described above.
The first important observation is that the slice category E/A classifies A-
elements. First note that we can interpret E in E/A by sending each object E
to the second projection [E] = p2 : E × A → A. Of special interest are the
3

identity 1A, which is terminal in E/A, and the diagonal ∆A : A → A × A, which
is a new global element 1 → [A] ∈ E/A.
Given a model Ma : E/A → Sets we can immediately define
E M = [E] Ma a = Ma(∆A) : 1 → [A] Ma (i.e., a ∈ A M ).
On the other hand, an element a ∈ A M allows us to define a functor Ma by
sending each σ : E → A to the fiber (σ M ) −1 (a). A little thought shows that
these constructions are mutually inverse.
Whatever Th(M) is, it should at least have one model M∗ : Th(M) → Sets,
where we interpret each new constant ca as a itself. Moreover, that element a
should induce an interpretation ã : E/A → Th(M) such that M∗ ◦ã = Ma. This
suggests that Th(M) might be defined as a colimit of slice categories.
More generally, each σ : A → B induces a pullback functor σ ∗ : E/B → E/A.
Because pullbacks preserve fibers, whenever a ∈ A M and σ M (a) = b,
Mb ∼ = Ma ◦ σ ∗ : E/B → Sets.
Thus pullbacks will be the transition morphisms in our colimit.
The index category is provided by the Grothendieck construction, which
allows us to re-encode M as a “category of elements” M over E. An object
of M is a pair 〈A, a〉 where A is a formula (i.e., A ∈ E) and a is an element
of the definable set A M . We write 〈a ∈ A M 〉 for such an object. A morphism
〈a ∈ A M 〉 → 〈b ∈ B M 〉 is an arrow σ : A → B with σ M (a) = b. Composition is
computed in E, so we have an obvious projection M → E.
Using the fact that M preserves finite limits, one can show that M is a
filtered category:
• For any two objects 〈a ∈ A M 〉, 〈b ∈ B M 〉 there is a span
〈a ∈ A M 〉
p1
〈a, b〉 ∈ (A × B) M
4
p2
M 〈b ∈ B 〉

• For any parallel arrows σ, τ : 〈a ∈ A M 〉 ⇒ 〈b ∈ B M 〉 there is an equalizing
arrow
〈a ∈ Eq(σ, τ)〉
M 〈a ∈ A 〉
σ
τ
〈b ∈ B M 〉.
Moreover, σ : 〈a ∈ A M 〉 → 〈b ∈ B M 〉 if and only if Mb ∼ = Ma ◦ σ ∗ . Since
(slices of) Ptop are cocomplete, this allows us to define Th(M) as a colimit of
pretoposes over Sets:
Mb
E/B
σ ∗
E/A
Sets
Ma
˜
ã
b
Th(M) ∼
M∗
= lim E/A
−→
M
If E = ET is a classifying category we can reframe this definition more tradi-
tionally. We extend the language of E by adding a new constant ca : A for each
A ∈ E and a ∈ A M . Th(M) extends E by parameterized sentences which are
true in M; given a formula ϕ ↣ A:
Th(M) ⊲ ϕ(ca) ⇐⇒ a ∈ ϕ M
This suggests another perspective on Th(M), as the category of definable
sets relative to the model M. Each object in Th(M) is the equivalence class
of a pair 〈b ∈ B M , σ : A → B〉. We can think of this as the definable set
(σ M ) −1 (b) ⊆ A M . Suppose 〈c ∈ C M , τ : A → C〉 is another such pair.
One can check that (τ M ) −1 (c) defines the same subset of A M if and only
if the pairs are identified in Th(M) (up to isomorphism). Thus Th(M) is the
category of definable sets over M. From this we can see that Th(M) is actually
two-valued: Sub(1 Th(M)) ∼ = Sub({∗}) = {⊤, ⊥}.
Before continuing, we pause to consider the models H : Th(M) → Sets. If
〈∗ ∈ 1 M 〉 is terminal in M, then MH = H ◦ ˜∗ : E → Sets is an ordinary
E-model. Similarly, set MaH = H ◦ ã and note, by the outer triange below, that
5

this is equivalent to an element aH ∈ A MH .
(−)×A
MH
E Th(M)
˜∗
ã
H
E/A
MaH
Sets
From this we can define a family of functions hA : A M → A MH by setting
hA(a) = aH. If σ M (b) = a then ˜ b ◦ σ ∗ = ã. Consequently
H ◦ ã = H ◦ ˜ b ◦ σ ∗ = MbH ◦ σ∗ = MaH .
This ensures that hA ◦ σ M = σ MH ◦ hB, so that h is a natural transformation.
Thus the Th(M)-model H defines a model homomorphism h : M → MH. In
fact, this construction is reversible: h is sufficient to construct H.
Moreover, a natural transformation θ : H → K induces a homomorphism
MH → MK which commutes with the maps from M. This leaves us with the
following:
Proposition 2.1. Th(M) classifies E-models under M. If H : Th(M) → Sets,
H ◦ ˜∗ defines a model MH and the assignment h(a) = c H a defines an E-model
homomorphism h : M → MH.
3 Connecting syntax and semantics
In this section we will show that certain semantic properties of interpretations
I : E → F naturally correspond dual properties for the functors I∗ : Mod(E) →
Mod(F). Specifically, we will prove the theorem below (definitions to follow).
Theorem 3.1.
(a) I is conservative if and only if I ∗ is supercovering.
(b) I is full on subobjects if and only if I ∗ stabilizes subobjects.
(c) I is subcovering if and only if I ∗ is faithful.
6

3.2 Proof of (a)
Lemma 3.3. For a pretopos functor I : E → F, the following are equivalent:
(i) I is conservative.
(ii) I is injective on subobjects.
(iii) I is faithful.
Proof. First of all, notice that pretoposes are balanced: epi + mono ⇒ iso. Any
monic M ↣ A induces an equivalence relation M + M ⇒ A + A. The quotient
A ⊕ A is two copies of A glued along M, and M is the equalizer of A ⇒ A ⊕ A.
M
M
This means that every monic in a pretopos is an equalizer; one easily shows that
an epic equalizer must be an isomorphism.
The implication (i)⇒(ii) is immediate. To see that (ii)⇒(ii), suppose that
I is injective on subobjects and that f = g : A ⇒ B ∈ C. Then Eq(f, g) A,
which implies that
But then f = g, so I is faithful.
Eq(If, Ig) = I(Eq(f, g)) IA.
Lastly, suppose that I is faithful. Faithful functors reflect monomorphisms:
if If : IA ↣ IB and g, h : C ⇒ A, then
(f ◦ g = f ◦ h) ⇒ (If ◦ Ig = If ◦ Ih) ⇒ (Ig = Ih) ⇒ (g = h).
Essentially the same argument shows that I reflects epis as well. If If is an
isomorphism then it is both epic and monic. By reflection, f is also monic and
epic so, because E is balanced, f is an isomorphism.
Naïvely we expect that the semantic property which is dual to conservativity
should bear a family resemblence to essential surjectivity. However, the follow-
ing example shows that conservative interpretations do not, in general, induce
essentially surjective I ∗ .
7

Consider the (single-sorted) theories of countably-many and continuum-
many distinct constants, respectively:
E = Ptop{ai = aj ⊢ ⊥ | i = j ∈ ω}.
F = Ptop{ai = aj ⊢ ⊥|i = j ∈ 2 ω }.
Any countable subset of S ⊂ 2 ω induces an interpretation IS : E → F. IS∗
acts by sending each F-model N to its reduct N ↾ S; this is the forgetful functor
which omits constants outside of S. It is easy to check that IS is conservative,
but it cannot possibly be essentially surjective: there are no countable models
of F. However, the following weaker property holds:
Definition 3.4. We call I ∗ supercovering if for any E-model M, any element
a ∈ A M and any R ↣ A with a ∈ R M , there exists a F-model N and a
homomorphism h : M → I ∗ N such that h(a) ∈ R I∗ N .
Lemma 3.5. I : E → F is conservative if and only if I ∗ is supercovering.
Proof. First suppose that I ∗ is supercovering, and that R A ∈ E. By
completeness we can find a E-model M and an element a0 ∈ A M such that
a0 ∈ R M . By the assumption, this gives a F-model N and h : M → I ∗ N with
h(a0) ∈ R I∗ N = IR N . Then IR IA by soundness, so that I is injective on
subobjects. By the previous lemma, it follows that I is conservative.
On the other hand, suppose that I is not supercovering. There are some
M0, a0 ∈ A M0 and a0 ∈ R M0 such that, for any homomorphism h : M0 → I ∗ N,
h(a0) ∈ R I∗ N . Consider the following pushout in Ptop:
E/A
IA
F/IA
ã0
Th(M0)
F/IA ⊕ Th(M0).
Call this aggregate theory T and note that a T-model consists of (i) an F-model
N with an element a1 ∈ IA N together with (ii) a homomorphism h : M0 →
E/A
M ′ ∈ E such that (iii) M ′ = I ∗ N and h(a0) = a1.
8

By assumption, every T-model satisfies IR(a1) and so, by completeness, T
proves this sentence. The derivation is finite, so only finitely many of the axioms
involved arise from Th(M0), say {⊢ ϕi(cbi )} with bi ∈ B M0
i . By weakening we
may assume that all the formulas share the same constants cb0 : B, and also
that all depend on the constant ca0 : ϕi ↣ A × B. Translating these axioms to
F and noting that a1 = h(a0) = ca0 in T:, this tells us that
F/(a1 : IA) ⊲ Iϕi(a1, cb0 ) ⊢ IR(a1)
i
The constant cb0 does not appear on the right-hand side of the derivation, so
we can replace it by an existential quantifier. Setting ɛ = ∃y : B.
ϕi(ca0 i , y),
and replacing a1 by a free variable, this leaves us with
F ⊲
Iɛ(x) ⊢x:A IR(x)
Now notice that ɛ(a0) is satisfied in M (b0 ∈ B M0 witnesses the existential)
while a0 ∈ R M0 by assumption. But then
E ⊲
and this shows that I is not conservative.
3.6 Proof of (b)
ɛ(x) ⊢x:A R(x) ,
Suppose that A ∈ E and S ≤ IA ∈ F, so that for any F-model N, S N ⊆ IA N =
A I∗ N . Therefore, if we have N0, N1 ∈ Mod(F) and an E-homomorphism h :
I ∗ N0 → I ∗ N1 we can compare S N1 to the image hA(S N0 ).
Definition 3.7. Given I : E → F we say that I ∗ stabilizes a subobject S ≤ IA
if
N0
hA S 1
N
⊆ S ⊆ A I∗N1 for any h : I ∗ N0 → I ∗ N1. I ∗ is stabilizing if for any A ∈ E and S ≤ IA ∈ F,
I ∗ stabilizes S.
9

First a lemma. Note that any subobject S ≤ IA defines and ideal in
SubE(A):
Γ = R ∈ Sub(A) | IR ≤ S ≤ IA .
S belongs to the essential image of I just in case this ideal is principle.
Lemma 3.8. If Γ is not principle and F/IA is consistent then:
(i) The theory T = F/(a1 : IA) ∪ {IR(a1) ⊢ ⊥}R∈Γ ∪ {⊢ S(a1)} is consistent.
(ii) Suppose that M is an E-model and a0 ∈ A M0 such that a0 ∈ R M0 for each
R ∈ Γ. Then the theory
is consistent.
T ′ M = F/(a1 : IA) ⊕ Th(M) ∪ {S(a1) ⊢ ⊥}
E/A
Proof. If Γ is not principle then for each R ∈ Γ we have IR S. By complete-
ness, we can find NR ∈ Mod(F) and aR ∈ S I∗ N such that aR ∈ R I∗ N = IR N .
Using compactness we can see that T is consistent. For any finite subset
{Ri} ⊆ Γ we must have
Indeed, the element a Ri
F ⊲ S(x) ⊢x:IA IRi(x).
i
witnesses S ≤
i Ri. This witness gives a model of
the finite subset of axioms F/IA ∪ {⊢ S(a1)} ∪ {IRi(a1) ⊢ ⊥}. Since any finite
subset has a model, the full theory T does as well.
For the proof of (ii), suppose that M models E and that a0 ∈ R M for R ∈ Γ.
Consider the pushout
E/A
IA
F/IA
ã0
Th(M)
F/IA ⊕ Th(M).
We extend this pushout theory by the axiom {S(a1) ⊢ ⊥} to obtain T ′ M .
10
E/A

We suppose that T ′ M
is inconsistent and restrict attention to finite sub-
set of axioms from Th(M), say {⊢ ϕi(cbi )}, which are necessary to derive
a contradiction. As in the previous lemma, we may weaken these formu-
las to involve the same constant cb : B and also to include the constant
ca0 . As before we replace the formulas ϕi by an existentially quantified meet:
ɛ(x : A) = ∃y : B.
ϕi(x, y).
ing:
i
With this notation, the assumption of inconsistency amounts to the follow-
F/(a1 : IA) ⊲
Iɛ(a1) ⊢ S(a1) .
The last implication means that ɛ belongs to Γ, and therefore a0 ∈ ɛ M . On the
other hand,
ϕi(ca0 , cb) ∈ Th(M)
for each i, and therefore M satisfies
i ϕi(a0, b). Then b is a witness show-
ing that M satisfies ɛ(a0). This contradiction demonstrates that T ′ M
consistent.
must be
Theorem 3.9. I is full on subobjects if and only if I ∗ stabilizes all subobjects.
More specifically, S ≤ IA belongs to the essential image of I if and only if I ∗
stabilizes S.
Proof. If IR ∼ = S then it is easy to see that I ∗ must stabilize S. Indeed, for any
h : I ∗ N0 → I ∗ N1 and a ∈ S N0
Then I ∗ stabilizes S.
hA(a) = hR(a) ∈ R I∗ N1 = IR N1 ∼ = S N1 .
On the other hand, suppose that S is not in the essential image of I, so that
the ideal Γ ⊆ SubE(A) from the previous lemma is not principle. We will show
that the consistency of the theories defined in the lemma imply that I ∗ does
not stabilize S.
11

Since T is consistent we may find an F-model N0 and an element a0 ∈ S N0
such that a0 ∈ IR N0 for each R ∈ Γ. This means that I ∗ N0 and a0 satisfy the
assumptions in the second part of the lemma, so that TI ∗ N0
is consistent.
A model of the latter theory consists of another F-model N1 together with
an E-model homomorphism h : I ∗ N0 → I ∗ N1 such that h(a0) ∈ S N2 . Then
a0 ∈ S N1 witnesses the fact that I ∗ does not stabilize S.
3.10 Proof of (c)
Definition 3.11. We say that I : E → F subcovers B ∈ F if there is an object
A ∈ E, a subobject S ≤ IA ∈ F and a regular epimorphism q : S ↠ B. I is
subcovering if it subcovers every B ∈ F.
Fix an object B ∈ F. Given an object A ∈ E, a partial function IA ⇀ B is
a two-place relation ϕ ↣ IA × B such that
F/(〈b, b ′ 〉 : B × B) ⊲ ϕ(a, b) ∧ ϕ(a, b ′ ) ⊢a:IA b = b ′ .
Categorically speaking, ϕ is a partial function just in case the projection ϕ ↣
IA × B → IA is monic.
We sometimes write ϕ : IA ≥ S → B to indicate that S = (∃b : B.ϕ) is the
domain of definition for ϕ. Now define
Φ = {ϕ ↣ IA × B | A ∈ E, ϕ : IA ⇀ B}.
Lemma 3.12. If I does not subcover B and F/B is consistent then:
(i) the theory T = F/(b:B) ∪ {∃a:IA. ϕ(a, b) ⊢ ⊥ | ϕ ∈ Φ} is consistent.
(ii) for any T-model 〈N, b0〉 the following theory TN is consistent:
Th 1 (N) ⊕
F⊕Th(I E
∗N) Th 2
(N) ∪ {c 1 b0 = c2b0 ⊢ ⊥}
12

Proof. For any partial function ϕ : IA ≥ S → B, we can find a model Nϕ and
an element bϕ such that bϕ does not belong to the image of ϕ. Otherwise, by
completeness, ϕ : IA ≥ S → B would exhibit a subcover of B.
If T were inconsistent, we could specialize to a finite subset of axioms
{∃ai : IAi. ϕ(ai, b) ⊢ ⊥} involved in the contradiction. The assumption of
inconsistency amounts to the following:
F/(b : B) ⊲
∃ai :IAi. ϕi(ai, b).
i
However,
i ϕi defines a partial function I(
i Ai) ⇀ B and the element b
i ϕi
defined in the last paragraph ensures that this formula cannot hold in F/B.
Now we need to unwrap the theory defined in (ii), relative to a T model
〈N, b0〉. A model of the inner theory F ⊕ Th(I
E
∗N) consists of an F-model N ′
together with an E-model homomorphism h : I∗N → I∗N ′ . It is easy to see
that I induces a functor IN : Th(I ∗ N) → Th(N) so that the following diagram
commutes:
F
E
F
˜∗
˜∗
Th(I ∗ N)
IN
Th(N).
This gives us a map F ⊕Th(I E
∗N) → Th(N) which we push out against itself
to form the cokernel pair
Th 1 (N) ⊕
F⊕Th E
2 (I∗N) Th(N).
Here we have two copies of Th(N) and we distinguish between objects in the
first and objects in the second by adding superscripts: for any ϕ ↣ D and
d ∈ D N
ϕ 1 (c 1 d) ∈ Th 1 (N) ϕ 2 (c 2 d) ∈ Th 2 (N).
A model of this cokernel pair consists of two F-model homomorphisms h1 :
N → N1 and h2 : N → N2. Because these h agree over F we have N1 = N ′ =
13

N2. Because they agree over Th(I ∗ N) we know that I ∗ h1 = I ∗ h2 : I ∗ N → N ′ .
In particular, for a ∈ IA N , c 1 a = c 2 a.
Finally, we obtain T ′ N by adding the axiom (c1 b0 = c2 b0
⊢ ⊥); we need to
see that this theory is consistent. If it were not then we could find some finite
subset of axioms
{⊢ ϕ 1 i (cai, c 1 di )} ∪ {⊢ ψ2 j (caj , c 2 dj )} ∪ {c1 b0 = c2 b0
⊢ ⊥}
which are by themselves sufficient to derive a contradiction. Here ai : IAi while
di : Di where D is not in the image of I.
Without loss of generality we may weaken these formulas in order to combine
the a’s and d’s into a common context IA × D, and we also weaken to include
the constant cb0 . Furthermore, we may expand our collection so that Th1 (N)
and Th 2 (N) provide the same set of formulas: {ϕi} = {ψj}. Letting ɛ(a, b) =
∃z :D. ϕ(a, b, z), this leaves us with
i
F/ 〈a, b〉 : IA × B ⊲
ɛ(a, b 1 ) ∧ ɛ(a, b 2 ) ⊢ b 1 = b 2
.
This latter sequent says that ɛ defines a partial map IA ⇀ B and therefore
belongs to Φ. On the other hand, ɛ(ca, cb0 ) belongs to Th(N). This means that
b0 lies in the image of ɛ N , and this runs contrary to the assumption that 〈N, b0〉
satisfies the axiom (∃a : IA. ɛ(a, b0) ⊢ ⊥). Therefore we conclude that T ′ N is
consistent whenever I fails to subcover B.
Theorem 3.13. I is subcovering if and only if I ∗ is faithful. More specifically,
I subcovers B just in case, for any F-model homomorphisms g, h : N0 → N1,
I ∗ g = I ∗ h implies that gB = hB.
Proof. As usual, showing that the syntactic property implies the model-theoretic
one is easy. Suppose that N0, N1 ∈ Mod(F) and gB = hB : B N0 ⇒ B N1 . If I
subcovers B, there is S ≤ IA with q : S ↠ F . Since q is epi,
gB = hB
⇒
gB ◦ q N1 = hB ◦ q N1
⇒
14
q N0 ◦ gS = q N0
◦ hS
⇒
gS = hS
.

Since S ≤ IA, this implies that gIA = hIA and therefore (I ∗ g)A = (I ∗ h)A.
This means that I ∗ is faithful.
For the converse we apply the previous lemma. Suppose that I does not
subcover B. As T is consistent we can find a model 〈N0, b0〉. Similarly, we
can find a model of T ′ . As discussed above, such a model consists of a pair
N0
of F-model homomorphisms g, h : N0 → N1 such that I ∗ g = I ∗ h. The fact
that 〈g, h〉 satisfies the additional axiom (c 1 b0 = c2 b0 ) means that g(b0) = h(b0)
and, consequently gB = hB. It follows that whenever I ∗ g = I ∗ h implies that
gB = hB, then I must subcover B.
4 Conceptual Completeness
With the model-theoretic characterizations of these syntactic properties, we are
in a good position to proof conceptual completeness.
Theorem 4.1. Suppose that I : E → F is a pretopos functor.
(i) If I ∗ is essentially surjective, then I is faithful.
(ii) If I ∗ is e.s.o. and full, then I is full.
(iii) If I ∗ is an equivalence of categories, then I is essentially surjective.
(iv) If I ∗ is an equivalence of categories, then so is I.
Proof. If I ∗ is essentially surjective, then it is certainly supercovering. For any
E-model M there is an isomorphism h : M ∼
−→ I ∗ N. Clearly if a ∈ R M then
h(a) ∈ R I∗ N , so this suffices. Therefore, by theorem 3.1(i), I is conservative
and, by lemma 3.3, faithful as well.
Now suppose that I ∗ is full, so that any morphism h : I ∗ N0 → I ∗ N1 has a
lift h : N0 → N1 with I ∗ h = h. For any S ≤ IA ∈ F, this means
hA(S N0 ) = (I ∗ h)A(S N0 ) = hIA(S N0 ) = hS(S N0 ) ⊆ S N1 .
15

Then I ∗ stabilizes every S ≤ IA so by 3.1(ii), I is full on subobjects.
Now suppose that f : IA → IA ′ ∈ F. We can represent f as a graph
Γf ↣ IA×IA ′ . Because I is full on subobjects, Γf has a preimage R ↣ A×A ′ .
If I ∗ is also essentially surjective then I will be conservative. From conservativity
and the fact that Γf was a provably function relation one can easily show that
R must be provably functional. Then I sends the composite R ↣ A × A ′ → A ′
to f, and I is full.
Now assume that I ∗ is additionally faithful, so that it is an equivalence of
categories. By 3.1(iii), this means that I is finitely covering. For any B ∈ F
there is a subquotient IA ≥ S ↠ B. Now form the kernel pair
S × S
B
IA × IA
S
IA
Since I is full on subobjects, both S ∼ = IR and S × S
B
∼ = IK are in the essential
image of I. As I is full, the projections lift to maps K ⇒ R ∈ E. Because I is
conservative and the kernel pair is an equivalence relation, so is K ↣ R × R.
Then we may form the quotient Q ∼ = R/K ∈ E and IQ ∼ = B. Hence I is
B
essentially surjective, and an equivalence of categories.
16
.