Assemblies. Effective topos can be seen as a “universe” for constructive mathematics in a sense that Ef f validates (as a topos) exactly those formulas of higher-order logic that are realizable (for a suitable higher-order generalization of realizability). However, if one does not need the whole power of higher-order logic, one can restrict attention to a subcategory ofassemblies, which admits a somewhat easier definition.
Definition 4.15. An object (X,∼) is¬¬-separated if it satisfies
∀x, y:X(¬¬[x∼y]→[x∼y])
That is, there is a computable function φ such that for any n ∈ [x∼ x], m∈ [y ∼y], given that [x∼y] is non-empty,φ· ⟨n, m⟩terminates andφ· ⟨n, m⟩ ∈[x∼y].
Each¬¬-separated object has a simpler description:
Definition 4.16. Anassemblyis a pair (X, E) whereXis an underlyingcarrier set, and
E is therealizability relation, a functionX → P(ω), s.t. E(x) is non-empty for everyx. Definition 4.17. Amorphismof assemblies (X, EX) and (Y, YE) is a functionf :X→Y, such that there is a recursive function{n}thattracks (or realizes)f:
a∈EX(x) =⇒ {n}(a)∈EY(f(x))
One can easily find computable functions tracking identity maps and composites; thus assemblies can be assembled together into a categoryAsm.
When unambiguous, we denote an assembly by its carrier setX and its realizability relation byEX. Every assembly (X, EX) can be viewed as an object of the effective topos, by taking the relation ∼X to be
{
[x∼Xx] =EX(x)
[x∼Xy] =∅ ifx̸=y
Clearly, (X, EX) is ¬¬-separated, as witnessed by e.g. λx.p1x. The converse holds as
well.
Proposition 4.18. Every ¬¬-separated object is isomorphic to an assembly.
For this we would like to use the following auxiliary lemma.
Lemma 4.19. Every object (X,∼) of Ef f is isomorphic to an object (X′,∼)such that EX′(x)is non-empty for every x∈X′.
Proof. Put X′ = {x ∈ X | EX(x) ̸= ∅}. The realizability relation on X′ is then a
restriction of∼toX′.
The morphismF: (X′,∼)→(X,∼) is just the inclusion. The morphism in the other direction isG: (X,∼)→(X′,∼) defined asG(x, y) = [x∼y]. We leave it to the reader to verify that F◦G≃idX andG◦F ≃idX′.
Proof of proposition 4.18. If (X,∼) is ¬¬-separated, then (X,∼) is isomorphic to the assembly (X/R, E) where
• R={(x, y)|[x∼y]̸=∅}
• E([x]) =∪x′∈[x][x′ ∼x′]
Without loss of generality, we may suppose that all elements ofX“exist” (i.e. [x∼x] is non-empty for every x). Then we define f : X → X/R by f(x) = [x] = {y | [x ∼ y] ̸= ∅}. Then ifa ∈ EX(x) =⇒ a ∈ EX/R([x]) and a ∈ [x ∼x′] =⇒ tr(a,s(a)) ∈ EX/R([x]) =Ex/R([x′]). This functionf induces a morphism F : (X,∼)→(X/R, E) in
Ef f.
The morphism going in the other directionG: (X/R, E)→(X,∼) is given by
G([x], y) =
{
∅ if [y∼x] =∅ [y∼y] otherwise We can see thatGis a well-defined morphism:
(ST) If a∈G([x], y), then [x∼y] is non-empty and thusa∈EX/R([x]) anda∈[y∼y].
(REL) If a ∈ G([x], y) and b ∈ E([x]) and c ∈ [y ∼ y′], then p1⟨b, a⟩ ∈ G([x], y) and
tr(s(c), c)∈G([x], y′).
(TL) If a ∈ E([x]), then a ∈ [x′ ∼ x′] for some x′ s.t. [x ∼ x′] is non-empty; thus
a∈G([x], x′).
(SV) Let a ∈ G([x], y) = [y ∼ y] and b ∈ G([x], y′) = [y′ ∼ y′], then, because X is
¬¬-separated, there is a recursive functionϕsuch thatϕab∈[y∼y′] if [y ∼y′] is non-empty. However, becausea∈G([x], y) implies that [x∼y] is non-empty, and
b∈G([x], y′) implies [x∼y′] is non-empty, we know that [y∼y′] is non-empty.
Thus, assemblies are exactly¬¬-separated objects inEf f.
Discrete and modest sets. Another important subcategory ofEf f is a subcategory
Mod ,→Asm,→ Ef f ofmodest sets. A modest set is a special case of a discrete object. Definition 4.20. A discrete object is a quotient of a subobject ofN.
This definition may seem like a mouthful at first; we will therefore stick to the following characterization of discrete objects.
Proposition 4.21 ([29, Proposition 3.2.18]). An object(X′,∼′) is discrete iff it is iso- morphic to an object(X,∼)such that n∈[x∼x]∩[y∼y]impliesx=y.
From this point we will assume that all the discrete objects are given in the “canonical form” above.
Definition 4.22. A modest set is a discrete assembly.
Example 4.23. A natural numbers objectN is a modest set.
It has been show by Van Oosten [28] that discrete objects can be intuitively understood as “discrete spaces”. We will explain this analogy in proposition4.33.
The notion of a discrete object can also be generalized to an arbitrary slice.
Definition 4.24. A mapf : (Y,∼)→(X,∼) is discrete if it is a quotient of a subobject of the natural numbers object in the slice toposEf f /(X,∼).
Once again, we would prefer to use a characterization of discrete maps in more concrete terms.
Proposition 4.25. If a mapF: (Y,∼)→(X,∼)is discrete, then the following proposi- tion holds in realizability logic:
F(y, x)∧F(y′, x)∧[[y∼y]∩[y′∼y′]]→[y∼y′]
In other words, every fiber F−1(x)is discrete.
The converse of the proposition above holds for assemblies; see [29, Proposition 3.4.4]. The full subcategory D ,→ Ef f of discrete objects is reflective ([29, Proposition 3.2.19]), and we will describe this reflection in geometrical terms in section4.8.
Uniform objects.
Definition 4.26. An object (X,∼) isuniformif it is covered by an object in the image of∇, that is, if there is an epimorphism∇Y →(X,∼).
We would also make use of the following characterization:
Definition 4.27. [29, Proposition 2.4.7] An object is uniform if it is isomorphic to an object (X,∼) for which there is a ψ∈∩x∈X[x∼x].
Example 4.28. As a somewhat trivial example, any object ∇(X) is uniform with ψ
being, e.g. 0.
Example 4.29. The canonical example of a uniform object is a powersetP(N). To see this, we first note that the realizability relation on P(N) can be rewritten as
P ⇔Q:= [∀n:N.(P(n)→ {n})∧ ∀n:N.(P(n)↔Q(n))]
just by unfolding the definitions. We can “bundle up” the strictness ofPin the definition of P(N), that is, we can consider predicatesP :N→ P(ω), such that for all x∈P(n),
p1x = n. Then P(N) is isomorphic to an object (X,∼), where X contains only such
predicatesN→ P(ω) that satisfies the aforementioned property, and
P ∼Q:= [∀n:N.(P(n)↔Q(n))]
Then we can see that (X,∼) is uniform, and λx.x ∈ ∩P∈X[P ∼P]. In general, every powerset object is uniform, see [29, Proposition 3.2.6].
A small digression. The name uniform comes from theuniformity principle for the realizability interpretation of second-order arithmetic: ∀X∃x.ϕ(X, x) → ∃x∀X.ϕ(X, x), which is non-classical. In fact, the following uniformity principle holds in the topos logic ofEf f for all formulas ϕand for all uniform (X,∼) and discrete (Y,≈):
U P :=∀x:X∃y:Y.ϕ(x, y)→ ∃y:Y∀x:X.ϕ(x, y)
Note that for a uniform object (X,∼), the following statement are equivalent in realiz- ability logic:
{
[∀x:X.([x∼x]→ϕ(x,y¯))]↔[∀x:X.ϕ(x,y¯)] [∃x:X.([x∼x]∧ϕ(x,y¯))]↔ ∃x:X.(ϕ(x,y¯))
HenceU P can be translated into realizability logic in the following way:
∀x:X.∃y:Y.([y≈y]∧ϕ(x, y))→ ∃y:Y.([y≈y]∧ ∀x:X.ϕ(x, y))
To see thatU P is realized, suppose thatn realizes the antecedent. Thus,p1(n) gives us
the realizer that is shared between all the witnesses for the existential clause for every
x:X. Generally, a witness forxmight be different than a witness for x′ ̸=x, but since (Y,≈) is discrete, different elements cannot share a realizer. We can conclude that there is a single y ∈Y that serves as a witness for everyx∈ X in the antecedent of U P. It then follows straightforwardly thatnrealizes the consequent.
Uniform maps. A generalization of the previous notion is that of auniform map. Definition 4.30. A mapF : (Y,∼)→(X,≈) isuniform if there is a setZ, an epimor- phism Q:∇(Z)→ ∇Γ(X,≈), and an epimorphismH :A→(Y,∼) over (X,≈), where
A is a pullback
A ∇(Z)
(X,≈) ∇Γ(X,≈)
P Q
A (Y,∼) (X,≈)
H
P
F
We will not go into the details on the motivation and theory behind uniform maps, but we will just mention that they provide a generalization of the uniformity principle to slices ofEf f, specifically uniform epimorphisms are (strong) orthogonal to discrete maps [29, Theorem 3.4.9].
The following characterization of uniform maps will also come in handy.
Proposition 4.31 ([29, Proposition 3.4.6]). A map F : (Y,∼)→(X,≈) is uniform iff there are recursive functionsα, βsuch that for ally∈Y,x∈X,n∈[x≈x],m∈F(y, x)
there exists an y′∈Y and {
α(n)∈F(y′, x)
β(n, m)∈[y∼y′]