In this section we recall Van Oosten’spath object construction [28] inEf f, in which for every objectX we associate an objectPX of “paths” inX, makingPX into an internal groupoid. Van Oosten uses a slightly different notion of a path than the one employed in this thesis, in particular the path object functor X7→PX is not an exponent. The main motivation for Van Oosten’s work is to find a way to understand the effective topos in topological term. Specifically, Van Oosten’s path object is constructed in a such a way, that the terminology behind discrete objects and discrete reflection obtain an intuitive geometrical meaning. Specifically, a discrete object is an object containing only non- trivial Van Oosten’s paths, and a discrete reflection ofX is an object of path-connected components of X.
n-paths and discrete reflection. We start by defining the intervals inEf f.
Definition 4.32. For eachn∈Nwe define ann-intervalIn to be an assembly with the underlying set{0, . . . , n} and the realizability relationEIn(i) ={i, i+ 1}.
Byn-path in an object (X,∼) we mean a mapIn→(X,∼). As it was noted in [28], an interesting property of discrete objects is that they contain no non-trivial paths. In fact, we have the following proposition:
Proposition 4.33. An object (K,∼) is discrete iff there are no non-constant n-paths p:In→K,n≥1.
Proof. Suppose thatK doesn’t have non-constantn-paths. Letn∈[k0∼k0]∩[k1∼k1]
for k0, k1 ∈K. Then consider a mapp:I1→K induced by the set-level map p(i) =ki
and tracked byλx.n. Since this map is constant, we have to concludek0=k1.
For the other direction, suppose we have a mapP :In→K. By the totality ofP, we
have totP ∈ ∩ i∈In([i∼i]→ ∪ ki∈KP(i, ki)). Because 1∈[0∼0]∩[1∼1], we have t:=totP·1∈P(0, k0)∩P(1, k1)
for somek0, k1∈K. By the stability ofP, we havestP(t)∈[k0∼k0]∩[k1∼k1]. Hence,
sinceKis discrete,k0=k1. Similarly,stP(totP·i)∈[ki−1∼ki−1]∩[ki∼ki], and by the
same argument we have that all ki’s are equal. We can now show that P is isomorphic to the constant map rk0 : In → K, rk0(i, y) = [y ∼ k0]. Let n ∈ P(i, y). Then, by
single-valuedness, svP(n, t)∈[y∼ki] = [y∼k0] =rk0(i, y). Thus,P is a constant path in K.
It was also shown in [28] that discrete reflection ofXcan be seen as the collection ofn- path-connected components ofX. For an assemblyX the construction can be replicated as follows: the discrete reflectionXdofX is obtained by quotientingXby the equivalence relation∼P defined as
x∼P y if there is ann-pathp:In →X, with p(0) =xandp′(m) =y
ThusXd={[x]|x∈X} andy∈[x] iff there is a sequence of realizers a0, . . . , an−1 and
elementsx1, . . . , xn−1ofX such thata0∈E(x)∩E(x1), a1∈E(x1)∩E(x2), . . . , an−1∈
E(xn−1)∩E(y). Intuitively, a class [x]∈ Xd contains all points ofX that are n-path-
connected tox. The realizability relation onXd is defined as a⊩Xd[x] ⇐⇒ a⊩X y for somey∈[x]
The map [–] :X →Xd is tracked byλx.x.
Proposition 4.34. The constructionX7→Xd restricts to a functor Asm→Mod , which makes Mod a reflexive subcategory of Asm.
Proof. Given a morphismf :X →Y (tracked byf), we obtain a morphismfd:Xd→Yd
defined as
fd([x]) = [f(x)]
and realized byf. One can check that (−)dis indeed a functor.
To establish the reflection we have to verify two things: thatXd is indeed discrete, and that the arrow [–] is universal.
For the first part, suppose that [x],[y]∈Xd, [x] ̸= [y]. Then, by definition,x̸∈[y], and there is no path betweenxandy inX; in particular, there is no 1-pathp′:I1→X,
such thatsp′=xandtp′=y. Thus,Xd is modest.
The second part amounts to filling in a dotted morphismg in the following diagram, where Y is a modest set:
X
Y Xd
f
[–]
g
We may simply put g([x]) =f(x). To see that this is well-defined, suppose thaty∈[x], i.e. there is an m-pathp′ :Im→X connectingxand y. Then there is aa0 ∈EX(x)∩
EX(p′(1)). Given a realizerf forf, we havef·a0⊩Y f(x), f(p′(1)). BecauseY is modest, f(x) =f(p′(1)). We can then apply this reasoning toa1∈EX(p′(1))∩EX(p′(2)) to get f(p′(1)) =f(p′(2)), etc. By induction we obtainf(x) =f(y).
Finally, if a ⊩Xd [x], then a ⊩X y for some y ∈ [x], and f ·a ⊩Y f(y) = f(x) =
g([x]).
Path object. All then-paths inX can be organized into a single object (path object); for that, some paths are identified via anorder and endpoint preserving map.
Definition 4.35. A mapσ : In → Im is order and endpoint preserving iff it is order preserving and satisfiesσ(0) = 0 andσ(n) =m.
We can now define a path objectP(X,∼) for a given object (X,∼) in Ef f.
Definition 4.36 (Van Oosten’s path object). The underlying set ofP(X,∼) is the set of all pairs (n, f) withn≥1 andf being a morphismf :In →(X,∼). The realizability relation is defined as follows. For (n, f),(m, g) ∈ P(X,∼) we have ⟨a, s, b⟩ ∈ [(n, f) ∼ (m, g)] if
• a∈E(X,∼)In(f) • b∈E(X,∼)Im(g)
• Either there is an order and endpoint preserving function σ : In → Im such that
s ∈ [f ∼ gσ] in (X,∼)In; or, there is an order and endpoint preserving function
σ:Im→In such thats∈[g∼f σ] in (X,∼)Im.
Proposition 4.37 ([28, Proposition 2.6]). 1. The construction of P(X,∼)extends to an endofunctorP:Ef f → Ef f, which preserves finite limits.
2. The object P(X,∼)comes with well-defined maps (a) s, t:P(X,∼)→(X,∼)(source and target maps) (b) c: (X,∼)→P(X,∼)(constant map)
(c) ∗:P(X,∼)×(X,∼)P(X,∼)→P(X,∼)(composition of paths)
(d) ˜·:P(X,∼)→P(X,∼)(path reversal)
Proposition 4.38([28, Proposition 2.7]).There is a a morphismL:P(X,∼)→PP(X,∼
)“contracting a path onto its endpoint”, i.e. satisfying internallys(L(p)) =p,t(L(p)) =
c(t(p)).
The two propositions above imply that Ef f possesses the structure of a (nice) path object category [6, 12], which in turn gives rise to a model of type theory in which the identity type of X is interpreted as PX. We will compare the model with ours in the upcoming chapter.
Notes
In this chapter we recalled the definition of the effective topos, some of its properties and various classes of objects from the theory of realizability toposes. The topic of categorical realizability is vast and this purpose of this chapter is to make the thesis self-contained. We can recommend lecture notes [38,3] to a reader interested in basic categorical realizability. The book [29] is a grand reference for realizability triposes and toposes.
In the next chapter we apply the construction of chapter3to the effective topos. We also discuss the differences and similarities to the approach of Van Oosten.
Chapter 5
Model category structure on
Ef f
f
This chapter is devoted to the study of the model category structure onEf f, as presented in chapter3. First, we describe the interval object, given by∇2≃I1. The we study the
relation between the notions of fibrant and contractible objects (types) on the one hand, and notions of uniform maps and discrete maps inEf f on the other hand. We show that inAsm, contractible maps/trivial fibrations are exactly uniform maps. We also show that for fibrant objects the notions of uniformity and contractibility coincide. Next, we show that discrete objects are fibrant, and that discrete maps with discrete bases are fibrations. We discuss importance of the discrete reflection and use it to show that the homotopy category of fibrant assemblies is equivalent to the category of modest sets. Finally, we contrast our approach with Van Oosten’s notion of homotopy based on the path object construction.