Asperti A.Categories,types,and structures.1991

12. Examples of Internal Models

1 is the function that takes (an equivalence class in R1R0 of) a triple < T, F, G > in (the equivalence class in R1 of) the triple < λtλm .T(pm)( 0Ft ), 0(F), 0(G) >.

Note that λtλm .T(pm)( 0Ft )) is a morphism from the retraction 0F to the retraction 0G; indeed,

0G ˚ λtλm .T(pm)( 0Ft ) ˚ 0F =

= λz. 0G(λm .T(pm)( 0Fz) )

= λz. (λxλt.G(pt)(x(pt)) ) (λm .T(pm)( 0Fz) ) = λz. λt.G(pt)( (λm .T(pm)( 0Fz) ) (pt))

= λz. λt.G(pt)( (T(pt)( 0Fz) )

G(N) ˚ λt.StN ˚ M = projG(N) ˚ S ˚ M = G(N) ˚ projG(N) ˚ S ˚ M = λt.StN . Moreover,

∆M,G-1( ∆M,G( < T, λx.M, G > ) ) =

=∆M,G-1( < λtλm .T(pm)t , M, 0(G) > )

=< λmλt.T(pm)t, λx.M, G >

=< λm.T(pm), λx.M, G >

and clearly T and λm.T(pm) are equivalent in T R0.

As for the converse, note first that if S = 0(G) ˚ S, then

St(pm) = 0(G)(St)(pm) = (λm.G(pm)(St(pm)))(pm) = G(pm)(St(pm)) = Stm Thus: ∆M,G( ∆M,G-1( < S, M, 0(G) > ) ) =

=∆M,G( < λmλt.Stm, λx.M, G > )

=< λtλm .St(pm) , M, 0G >

=< λtλm .Stm , M, 0G >

=< S, M, 0G >

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Define then

∆ ( < M,G, < T, λx.M, G > > ) = < M, G, < λtλm .T(pm)t , M, 0G > >

and

∆-1( < M,G, < S, M, 0G > > ) = <M,G, < λmλt.Stm, λx.M, G > >.

12.1.2 Remark It is possible to give an elegant categorical characterisation of the models of

object of RETA: more formally, that RETA is internal to RETA w.r.t. PERA. We have thus the following facts:

1. (AA < A in a CCC C and RETA internal to RETA with respect to PERA ) imply A |= λβp .

2. A |= λβp implies

( AA < A in PERA and RETA internal to RETA w.r.t. PERA ). The reader should complete the details for exercise.

12.2 PER inside ω-Set

This example continues our presentation of PER an an internal category M of ω-Set (see examples 7.2.7 and 7.3.9.2, where it is shown that the construction gives also an internal model.)

Let M* be defined as c* in 11.4.2. We next define an internal right adjoint to the constant functor K : M → M*, that by def. 11.4.4 will complete the construction of a model for λ2.

Again, we shall take advantage of the wise blend of set-theory and computability on which ω-Set is based, in order to avoid the most formal details.

The following lemma motivates the definition of below.

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12.2.3 Definition

1.o : [Mo → Mo] → Mo, is given by o(F) = ∩A MoF(A),

2.1: [M o → M 1] → M 1 is defined as follows. If m |_ <τ,F,G>ω-Set[M o,M 1], set

1(<τ,F,G>) = <{m.0} o(F)→ ο(G) , o(F), o(G)> .

12.2.4 Proposition. in definition 12.2.3 is well defined. In particular, 1 is realized by any number p such that , for all m , p.m = m.0 .

Proof. By definition, if F : Mo→Mo, then o(F) is in Mo. o is realized by (any index of) any total recursive function. By lemma 12.2.2, 1 is well defined as its definition does not depend on the choice of the realizer m for <F,τ,G>. Therefore we only need to show that 1 is realized by p. Namely, that if

1.m1,m2 |__ <τ,F,G>

2.n1 o(F) n2

3.A Mo

and thus (p.m1.n1) G(A) (p.m2.n2) by (4), (5) and the definition of (F(A)→G(A)) Mo. ♦

12.2.5 Remark. It should be clear that lemma 12.2.2 is a very simple but crucial lemma. Note first that the morphisms in ω-Set[Mo,M1] are described as triples, <τ,F,G>, where τ: Mo →A MoQ(F(A)→G(A)) is such that τ(A) Q(F(A)→G(A)) and F, G: Mo → Mo give the source and target of τ(A). This is a sound description, since M* = ( [Mo→Mo], [M1→M1],...) may be viewed as the internal category of functors from the discrete category, whose object of objects is Mo, to M. Thus [Mo→M1] or ω-Set[Mo,M1] are internal natural transformations. By lemma 12.2.2, now, there is a uniform n which realizes τ(A) for all A. In a sense these internal natural transformations are “almost” constant maps and only depend on the source and target objects.

We need now to prove that there exists an internal natural isomorphism ∆, which gives the adjunction between and K. We can use the simplicity of the intended universe and perform the construction directly. As in the proof of the internal Cartesian closure of M, the point is to show that the functional dependencies, usually implicit in the external world (or given by “indexes”: recall a,b |_ Λa,b), can be turned into internal constructions. Once more, this will be straightforward to check within ω-Set, as the realizers for the natural isomorphism and its inverse will not depend on their “indexes.”

Let K, with components ko and k1, be the internal constant functor in definition 11.4.3.

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12.2.6Definition. For m |__ < τ,ko(B),G> ω-Set[Mo,M1] , set

∆B,G (< τ,ko(B),G>) = < {m.0}Β→ ο(G) ,Β, o(G)> M1 . and ∆−1B,G ( <{m}Β→ ο(G) ,Β, o(G)> )

=< λX Mo.{m}Β→G(X) ,ko(B),G> ω-Set[Mo,M1].

12.2.7Proposition. ∆B,G and ∆−1B,G in 12.2.6 are well defined, for all B Mo. and G:

Mo→Mo (= [Mo → Mo]). Moreover, they are realized by p and k (respectively), such that p.m = m.0 and k.m.n = m .

Proof: By lemma 12.2.2, ∆B,G does not depend on the choice of the particular realizer m. As for ∆−1B,G, note that, if m(Β→ ο(G))m, then n,q ( nBq m.n (∩A MoG(A)) n.q), by definition of ο, and hence m.n G(A) n.q, for all A Mo. Therefore, m (Β → G(A)) m for all A Mo and ∆−1B,G too is well defined. By definition, they are uniformly realized by p and k as above. In particular, p and k compute ∆B,G and ∆−1B,G for all B and G. ♦

12.3 PL-Categories Inside Their Groethendiek Completion

This example show another way for obtaining an internal model from a PL-category, different from the internalization process of chapter 11.

12.3.1 Definition. Given any E-indexed category G, define the category ∫G, the Grothendieck

Let (E, G, Ω) be an external model (that is an indexed category with the additional requirements on functor G in 12.2.1); then the Grothendieck completion ∫G assumes a particularly simple form Objects: (e, σ)∫G iff σ ObjG(e) = E[e, Ω]

(hence we can identify objects of ∫G with arrows σ: e→Ω) Morphisms: (α,f)∫G[σ:e→Ω,τ:e'→Ω] iff

α E[e,e'] and f G(e)[σ,G(α)(τ)] = G(e)[σ,τ ˚ α].

The point is that ∫G contains an internal category which, in a sense, internalizes the external model (E, G, Ω).

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Warning We are here forcing our terminology, since the category ∫G does not need to have all finite limits, at least not in general, and hence we could not speak of “internal categories in ∫G.” However, all the needed pullback diagrams exist in ∫G, as pointed out below.

The situation for COMP is more delicate, since ∫G does not have all finite limits. However, the

The terminal object is given by tG(t), the terminal object of the local category G(t) (for t terminal in E). As for products, they can be defined by using the Cartesian structure of the local categories:

In order to define f G(Ω×Ω)[ tG(Ω×Ω), x0→fst], we can use again the Cartesian closed structure of G(Ω×Ω) and the fact that, by lemma 11.2.3, one has

x0 = x0 ° id = x0 ° <fst,snd> = fst ×G(Ω×Ω) snd.

Define then f = Λ(fst) where fst G(Ω×Ω)[fst×snd, fst]. SND is defined analogously.

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It remains to give the pairing isomorphism <,>; note first of all that the required pullbacks (see

definition 7.3.6, and also appendix A at the end of chapter 7) are given by the following data:

X = [,]0×∫G[,]0 ˚ <id×fst, id×snd> : Ω×(Ω×Ω)→Ω ∏X : X→c1×c1 (that is X → [,]0 ×∫G [,]0)

∏X = ( <id×fst, id×snd>, id )

ρ : X → c0×c0×c0 (that is X → tG(Ω×Ω×Ω)) ρ = (id, !)

Y = [,]0 ˚ id×x0 : Ω×(Ω×Ω)→Ω ∏Y : Y→c1 (that is Y → [,]0 ) ∏Y = ( id×x0, id)

ρ' : Y → tG(Ω×Ω×Ω) ρ' = (id, !)

Observe now that, as we are dealing with objects of G(Ω×(Ω×Ω)), by interpreting all products and exponentials locally in G(Ω×(Ω×Ω)) ), one has

X = (fst → fst ˚ snd)×(fst → snd ˚ snd) Y = fst → ((fst ˚ snd)×(snd ˚ snd))

and hence X Y, since G(Ω×(Ω×Ω)) is Cartesian closed.

The isomorphism <,>: X→Y (in ∫G ) is then given by <,> = (id,f), where f is the

isomorphism between X and Y in G(Ω×(Ω×Ω)). The required equations are now easily verified.

Take then evalfst,snd G(Ω×Ω)[(fst→snd) × fst, snd] and set

f = Λ(evalfst,snd) G(Ω×Ω)[tG(Ω×Ω), ((fst→snd) × fst) → snd].

Before giving the isomorphism Λ, we need to express the pullback diagrams of theorem 7.3.7 (see also appendix A). They can be instantiated as

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12. Examples of Internal Models

where:

The internal Cartesian closed category Γ Cat(∫G) is actually an internal model when (E, G, Ω) is

an external one. In order to define the required adjunction, we need first to show that we can indeed construct the internal category Γ*; that is, the exponents c0c0 and c1c0 exist in ∫G (as for the limits, not all the exponents exist in ∫G).

12.3.2 Lemma

i. For any object e of E, c0tG(e) exists in ∫G and it is given by tG(Ωe). ii.For any object e of E, c1tG(e) exists in ∫G and it is given by ([,]0 ˚ eval).

Proof Set σ = tG(e).

i. Note first that in ∫G, for any e' and τ, every arrow (β,g):τ→tG(e') has g = !. We must then show that, for c0σ = tG(Ωe), the following diagram commutes, for any object τ and arrow (α,!):

which is immediate.

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12.Examples of Internal Models

ii.Let τ: e'→Ω be any object of ∫G; we are looking for the unique Λ(α, f) such that the following diagram commutes:

previous lemma, let us look for an arrow h G(ΩΩ×Ω)[tG(ΩΩ)˚ fst, ( 0 ˚ fst) → eval]. Then the f we need will then be ∆(h). Observe now that projeval G(ΩΩ×Ω)[ 0 ˚ fst, eval]; this immediately gives the required h in the following diagram:

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12. Examples of Internal Models

In conclusion, the Grothendieck completion provides yet another example of the general categorical construction of model in chapter 11. Moreover, it gives further insight into the external versus internal approach by giving a setting where the external models may be viewed as internal constructions.

References The examples in this chapter may be found, in part, in the many papers mentioned at the end of the previous chapter. Since Moggi's hint for the small completeness of the PER construction, in particular, these models have been thoroughly explored by many authors. The three sections above develop the ideas in Amadio and Longo (1986) and in Longo and Moggi (1988), and follow Asperti and Martini (1989), respectively. By this, the first two sections extend to higher order the understanding of simple types as partial equivalence relations and retractions in Scott (1976). The consistency of λβηp is shown in Berardi (1988).

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