ON SMALL TYPES IN UNIVALENT FOUNDATIONS

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Introduction
We investigate predicative aspects of constructive univalent foundations.By predicative and constructive, we respectively mean that we do not assume Voevodsky's propositional resizing axioms [Voe11,Voe15] or excluded middle and choice.Most of our work is situated in our larger programme of developing domain theory constructively and predicatively in univalent foundations.In previous work [dJE21a], we showed how to give a constructive and predicative account of many familiar constructions and notions in domain theory, such as Scott's D ∞ model of untyped λ-calculus and the theory of continuous dcpos.The present work complements this and other existing work on predicative mathematics Another reason for being interested in predicativity is the fact that propositional resizing axioms fail in some models of univalent type theory.A notable example of such a model is Uemura's cubical assembly model [Uem19].What is particularly striking about Uemura's model is that it does support an impredicative universe U in the sense that if X is any type and Y : X → U, then Π x:X Y (x) is in U again even if X isn't, but that propositional resizing fails for this universe.On the model-theoretic side, we also highlight Swan's (unpublished) results [Swa19b,Swa19a] that show that propositional resizing axioms fail in certain presheaf (cubical) models of type theory.Interestingly, Swan's argument works by showing that the models violate certain collection principles if we assume Brouwerian continuity principles in the metatheory.
By contrast, we should mention that propositional resizing is validated in many models when a classical metatheory is assumed.For example, this is true for any type-theoretic model topos [Shu19,Proposition 11.3].In particular, Voevodsky's simplical sets model [KL21] validates excluded middle and hence propositional resizing.We note, however, that in other models it is possible for propositional resizing to hold and excluded middle to fail, as shown by [Shu15,Remark 11.24].
Another interesting aspect of impredicativity is that it is expected, by analogy to predicative and impredicative set theories, that adding resizing axioms significantly increases the proof-theoretic strength of univalent type theory [Shu19, Remark 1.2].
This paper concerns resizing axioms, meaning we ask a given type to be equivalent to one in some fixed universe U of "small" types.Voevodsky [Voe11] originally introduced resizing rules which add judgements and hence modify the syntax of the type theory to make the given type inhabit U, rather than only asking for an equivalent copy in U.It is not known whether Voevodsky's resizing rules are consistent with univalent type theory in the sense that no-one has constructed a model of univalent type theory extended with such resizing rules, or proved a contradiction in the system.It is also an open problem [CCHM18, Section 10] whether we have normalization for cubical type theory extended with resizing rules.In fact, as far as we know, this is an open problem for plain Martin-Löf Type Theory as well.
Lastly, one may have philosophical reservations regarding impredicativity.For example, some constructivists may accept predicative set theories like Aczel's CZF and Myhill's CST, but not Friedman's impredicative set theory IZF.Or, paraphrasing Shulman's narrative [Shu11], one can ask why propositions (or (−1)-types) should be treated differently, i.e. given that we have to take size seriously for n-types for n > −1, why not do the same for (−1)-types?1.2.Related work.Curi investigated the limits of predicative mathematics in CZF [AR10] in a series of papers [Cur10a,Cur10b,Cur15,Cur18,CR12].In particular, Curi shows (see [Cur10a,Theorem 4.4 and Corollary 4.11], [Cur10b, Lemma 1.1] and [Cur15, Theorem 2.5]) that CZF cannot prove that various nontrivial posets, including sup-lattices, dcpos and frames, are small.This result is obtained by exploiting that CZF is consistent with the anti-classical generalized uniformity principle (GUP) [vdB06,Theorem 4.3.5].Our related Theorem 4.23 is of a different nature in two ways.Firstly, our theorem is in the spirit of reverse constructive mathematics [Ish06]: Instead of showing that GUP implies that there are no non-trivial small dcpos, we show that the existence of a non-trivial small dcpo is equivalent to weak propositional resizing, and that the existence of a positive small dcpo is equivalent to full propositional resizing.Thus, if we wish to work with small dcpos, we are forced to assume resizing axioms.Secondly, we work in univalent foundations rather than CZF.This may seem a superficial difference, but a number of arguments in Curi's papers [Cur15,Cur18] crucially rely on set-theoretical notions and principles such as transitive set, set-induction, and the weak regular extension axiom (wREA), which cannot even be formulated in the underlying type theory of univalent foundations.Moreover, although Curi claims that the arguments of [Cur10a,Cur10b] can be adapted to some version of Martin-Löf Type Theory, it is presently not clear whether there is any model of univalent foundations which validates GUP.However, one of the reviewers suggested that Uemura's cubical assemblies model [Uem19] might validate it.In particular, the reviewer hinted that [Uem19, Proposition 21] may be seen as a uniformity principle.
Finally, the construction of set quotients using propositional truncations is due to Voevodsky and also appears in [Uni13, Section 6.10] and [RS15, Section 3.4].While Voevodsky assumed resizing rules for his construction, we investigate the inter-definability of propositional truncations and set quotients in the absence of propositional resizing axioms.1.3.Organization.Section 2: Foundations and size matters, including impredicativity, relation to excluded middle, univalence and closure under embedded retracts.Section 3: Inter-definability of set quotients and propositional truncations, and equivalence of small set quotients and set replacement.Section 4: Nontrivial and positive δ V -complete posets and reductions to impredicativity and excluded middle.Section 5: Predicative unavailability of Tarski's fixed point theorem and Pataraia's lemma, and suprema of ordinals.Section 6: Comparison of completeness with respect to families and with respect to subsets.Section 7: Conclusion and future work.

Foundations and Small Types
We work with a subset of the type theory described in [Uni13] and we mostly adopt the terminological and notational conventions of [Uni13].We include + (binary sum), Π (dependent products), Σ (dependent sum), Id (identity type), and inductive types, including 0 (empty type), 1 (type with exactly one element : 1), N (natural numbers).We assume a universe U 0 and two operations: for every universe U, a successor universe U + with U : U + , and for every two universes U and V another universe U V such that for any universe U, we have U 0 U ≡ U and U U + ≡ U + .Moreover, (−) (−) is idempotent, commutative, associative, and (−) + distributes over (−) (−).We write U 1 :≡ U + 0 , U 2 :≡ U + 1 , . . .and so on.If X : U and Y : V, then X + Y : U V and if X : U and Y : X → V, then the types Σ x:X Y (x) and Π x:X Y (x) live in the universe U V; finally, if X : U and x, y : X, then Id X (x, y) : U.The type of natural numbers N is assumed to be in U 0 and we postulate that we have copies 0 U and 1 U in every universe U.This has the useful consequence that while we do not assume cumulativity of universes, embeddings that lift types to higher universes are definable.For example, the map (−) × 1 V takes a type in any universe U to an equivalent type in the higher universe U V. We assume function extensionality and propositional extensionality tacitly, and univalence explicitly when needed.Finally, we use a single higher inductive type: the propositional truncation of a type X is denoted by X and we write ∃ x:X Y (x) for Σ x:X Y (x) .Apart from Section 3, we assume throughout that every universe is closed under propositional truncations, meaning that if X : U then X : U as well.
2.1.The Notion of a Small Type.We introduce the fundamental notion of a type being U-small with respect to some type universe U, and specify the impredicativity axioms under consideration (Section 2.2).We also note the relation to excluded middle (Section 2.2) and univalence (Section 2.3).Finally, in Section 2.4 we establish our main technical result on small types, namely that being small is closed under retracts.
Definition 2.1 (Smallness, [E + 22, UF.Size]).A type X in any universe is said to be U-small if it is equivalent to a type in the universe U.That is, X is U-small :≡ Σ Y :U (Y X).
Definition 2.2 (Local smallness, [Rij17]).A type X is said to be locally U-small if the type (x = y) is U-small for every x, y : X.
(i) Every U-small type is locally U-small.
(ii) The type Ω U of propositions in a universe U lives in U + , but is locally U-small by propositional extensionality.
2.2.Impredicativity and Excluded Middle.We consider various impredicativity axioms and their relation to (weak) excluded middle.The definitions and propositions below may be found in [Esc19, Section 3.36], so proofs are omitted here.
(i) By Propositional-Resizing U ,V we mean the assertion that every proposition P in a universe U is V-small.(ii) We write Ω-Resizing U ,V for the assertion that the type Ω U is V-small.(iii) The type of all ¬¬-stable propositions in a universe U is denoted by Ω ¬¬ U , where a proposition P is ¬¬-stable if ¬¬P implies P .By Ω ¬¬ -Resizing U ,V we mean the assertion that the type Ω ¬¬ U is V-small.(iv) For the particular case of a single universe, we write Ω-Resizing U and Ω ¬¬ -Resizing U for the respective assertions that Ω U is U-small and Ω ¬¬ U is U-small.Proposition 2.5.
(i) The principle Ω-Resizing U ,V implies Propositional-Resizing U ,V for every two universes U and V. (ii) The conjunction of Propositional-Resizing U ,V and Propositional-Resizing V,U implies Ω-Resizing U ,V + for every two universes U and V.
It is possible to define a weaker variation of propositional resizing for the ¬¬-stable propositions only (and derive similar connections), but we don't need it in this paper.
(i) Excluded middle in a universe U asserts that for every proposition P in U either P or ¬P holds.(ii) Weak excluded middle in a universe U asserts that for every proposition P in U either ¬P or ¬¬P holds.
We note that weak excluded middle says precisely that ¬¬-stable propositions are decidable and is equivalent to de Morgan's Law.
Proposition 2.7.Excluded middle implies impredicativity.Specifically, 2.3.Smallness and Univalence.With univalence we can prove that the statements Propositional-Resizing U ,V and Ω-Resizing U ,V are subsingletons.More generally, univalence allows us to prove that the statement that X is V-small is a proposition, which is needed at the end of Section 4.4.
Proposition 2.8 [Esc19, has-size-is-subsingleton]. If V and U V are univalent universes, then X is V-small is a proposition for every X : U.
The converse also holds in the following form.
Proposition 2.9.The type X is U-small is a proposition for every X : U if and only if the universe U is univalent.
2.4.Small Types and Retracts.We show our main technical result on small types here, namely that being small is closed under retracts.
Definition 2.10 (Sections and retractions).A section is a map s : X → Y together with a left inverse r : Y → X, i.e. the maps satisfy r • s ∼ id.We call r the retraction and say that X is a retract of Y .
We extend the notion of a small type to functions as follows.
Definition 2.11 (Smallness (for maps), [E + 22, UF.Size]).A map f : X → Y is said be V-small if every fibre is V-small.
Theorem 2.13.Every section into a V-small type is V-small.In particular, its domain is V-small.Hence, the V-small types are closed under retracts.
Proof.We show that the domain is V-small from which it follows that the section is V-small by Lemma 2.12(ii).So suppose we have a section s : Remark 2.14.In [dJE21b] we had a weaker version of Theorem 2.13 where we included the additional assumption that the section was an embedding.(Note that if every section is an embedding, then every type is a set [Shu16, Remark 3.11(2)], but that all sections into sets are embeddings [Esc19, lc-maps-into-sets-are-embeddings].)We are grateful to the anonymous reviewer who proposed the above strengthening.

Set Quotients, Propositional Truncations and Set Replacement
We investigate the inter-definability and interaction of type universe levels of propositional truncations and set quotients in the absence of propositional resizing axioms.In particular, we will see that it is not so important if the set quotient or propositional truncation lives in a higher universe.What is paramount instead is whether the universal property applies to types in arbitrary universes.However, in some cases, like in Section 5.2, it is relevant whether set quotients are small and we show this to be equivalent to a set replacement principle in Section 3.4.
We start by recalling (the universal property of) the propositional truncation, which, borrowing terminology from category theory, we could also call the subsingleton reflection or propositional reflection.
Some sources, e.g.[Uni13], also demand that the diagram above commutes definitionally: for every x : X, we have f (x) ≡ f (|x|).Having definitional equalities has some interesting consequences, such as being able to prove function extensionality [KECA17, Section 8].We do not require definitional equalities, but notice that we do have f (x) = f (|x|) (up to an identification) for every x : X, as P is a subsingleton.In particular it follows using function extensionality that f is the unique factorization.
Notice that if a propositional truncation exists, then it is unique up to unique equivalence.
Remark 3.2.Some remarks regarding universes are in order: (i) In Definition 3.1, the subsingleton P may live in an arbitrary universe, regardless of the universe in which X sits.The importance of this will be revisited throughout this section and in Example 3.4 in particular.(ii) In Definition 3.1, we haven't specified in what universe X should be.When adding propositional truncations as higher inductive types, one typically assumes that X : U if X : U, and indeed this is what we do in most of this paper.In this section, however, we will be more general and instead assume that X : F (U) where F is a (meta)function on universes, so that the above case is obtained by taking F to be the identity.We will also consider F (U) = U 1 U in the final subsection.
While in general propositional truncations may fail to exist in intensional Martin-Löf Type Theory, it is possible to construct a propositional truncation of some types in specific cases [EX15, Section 3.1].A particular example [KECA17, Corollary 4.4] is for a type X with a weakly constant (viz.any of its values are equal) endofunction f : the propositional truncation of X can be constructed as Σ x:X (x = f (x)), the type of fixed points of f .
We review an approach by Voevodsky, who used resizing rules, to constructing propositional truncations in general in the next section.
NB.We do not assume the availability of propositional truncations in this section.
Definition 3.3 (Voevodsky propositional truncation, X v ).The Voevodsky propositional truncation X v of a type X : U is defined as Because of function extensionality, one can show that X v is indeed a proposition for every type X.Moreover, we have a map Observe that X v : U + , so using the notation from Remark 3.2, we have F (U) = U + .However, as we will argue for set quotients, it does not matter so much where the truncated proposition lives; it is much more important that we can eliminate into subsingletons in arbitrary universes, i.e. that − v satisfies the right universal property.Given X : U and a map f : X → P to a proposition P : U with i : is-subsingleton(P ), we have a map X v → P given as Φ → Φ(P, i, f ).However, if the proposition P lives in some other universe V, then we seem to be completely stuck.To clarify this, we consider the example of functoriality.
Example 3.4.If we have a map f : X → Y with X : U and Y : U, then we get a lifting simply by precomposition, i.e. we define But obviously, we also want functoriality for maps f : X → Y with X : U and Y : V, but this is impossible with the above definition of |f | v , because for X v we are considering propositions in U, while for Y v we are considering propositions in V.
In particular, even if the types X : U and Y : V are equivalent, then it does not seem possible to construct an equivalence between X v and Y v .This issue also comes up if one tries to prove that the map Proposition 3.5 [KECA17, Theorem 3.8].If our type theory has propositional truncations with X : U whenever X : U, then X v is U-small.
Proof.We will show that X and X v are logically equivalent (i.e.we have maps in both directions), which suffices, because both types are subsingletons.We obtain a map X → X v by applying the universal property of X to the map |−| v : X → X v .Observe that it is essential that the universal property allows for elimination into subsingletons in universes other than U, as X v : U + .For the function in the other direction, simply note that X : U, so that we can construct X v → X as Φ → Φ( X , i, |−|) where i witnesses that X is a subsingleton.Thus, as is folklore in the univalent foundations community, we can view higher inductive types as specific resizing axioms.But note that the converse to the above proposition does not appear to hold, because even if X v is U-small, then it still wouldn't have the appropriate universal property.This is because the definition of X v is a dependent product over propositions in U only, which now includes X v , but still misses propositions in other universes.In the presence of resizing axioms, we could obtain the full universal property, because we would have (equivalent copies of) all propositions in a single universe: Proposition 3.6 (see e.g.[Esc19, Section 36.5]).If Propositional-Resizing U ,U 0 holds for every universe U, then the Voevodsky proposition truncation satisfies the full universal property with respect to all types in all universes.
3.2.Set Quotients from Propositional Truncations.In this section we assume to have propositional truncations with X : F (U) when X : U for some (meta)function F on universes.We will be mainly interested in F (U) = U and F (U) = U 1 U for the reasons explained below.We prove that we can construct set quotients using propositional truncations.The construction is due to Voevodsky and also appears in [Uni13, Section 6.10] and [RS15, Section 3.4].However, while Voevodsky assumed propositional resizing rules in his construction, the point of this section is to show that resizing is not needed to prove the universal property of the set quotient, provided propositional truncations are available.Our proof follows our earlier Agda development [Esc18] (see also [Esc19, Section 3.37]) and is fully formalized [dJE21c].
(1) The image of a function f : X → Y is defined as im(f ) :≡ Σ y:Y ∃ x:X (f (x) = y).
(2) A function f : X → Y is a surjection if for every y : Y , there exists some x : X such that f (x) = y.
(3) The corestriction of a function f : X → Y is the function f : X → im(f ) given by x → (f (x), |x, refl|).
Remark 3.8.Note that if X : U and Y : V and f : X → Y , then im(f ) : V F (U V), because Σ x:X (f (x) = y) : U V and − takes types in W to subsingletons in F (W).In case F is the identity, then we obtain the simpler im(f ) : U V.
Proof.By definition of the corestriction.
Lemma 3.10 (Image induction, [E + 22, UF.ImageAndSurjection]).For a surjective map f : X → Y , the following induction principle holds: for every prop-valued P : Y → W, with W an arbitrary universe, if P (f (x)) holds for every x : X, then P (y) holds for every y : Y .In the other direction, for any map f : X → Y , if the above induction principle holds for the specific family P (y) :≡ ∃ x:X (f (x) = y), then f is a surjection.
Proof.Suppose that f : X → Y is a surjection, let P : Y → W be subsingleton-valued and assume that P (f (x)) holds for every x : X.Now let y : Y be arbitrary.We are to prove that P (y) holds.Since f is a surjection, we have ∃ x:X (f (x) = y).But P (y) is a subsingleton, so we may assume that we have a specific x : X with f (x) = y.But then P (y) must hold, because P (f (x)) does by assumption.
For the other direction, notice that if P (y) :≡ ∃ x:X (f (x) = y), then P (f (x)) clearly holds for every x : X.So by assuming that the induction principle applies, we get that P (y) holds for every y : Y , which says exactly that f is a surjection.3.2.2.Set Quotients.We now construct set quotients using images and specialize image induction to the set quotient.
Definition 3.11 (Equivalence relation).An equivalence relation on a type X is a binary type family ≈ : X → X → V such that it is (i) subsingleton-valued, i.e. x ≈ y is a subsingleton for every x, y : X; (ii) reflexive, i.e. x ≈ x for every x : X; (iii) symmetric, i.e. x ≈ y implies y ≈ x for every x, y : X; (iv) transitive, i.e. the conjunction of x ≈ y and y ≈ z implies x ≈ z for every x, y, z : X.
Definition 3.12 (Set quotient, X/≈).We define the set quotient of X by ≈ to be the type X/≈ :≡ im(e ≈ ) where and p is the witness that ≈ is subsingleton-valued.
Of course, we should prove that X/≈ really is the quotient of X by ≈ by proving a suitable universal property.The following definition and lemmas indeed build up to this.For the remainder of this section, we will fix a type X : U with an equivalence relation ≈ : X → X → V.
Remark 3.13.By Remark 3.8, and because Ω V : V + , we have X/≈ : T F (T ) with T :≡ V + U.In the particular case that F is the identity, we obtain the simpler X/≈ : V + U.
Lemma 3.14.The quotient X/≈ is a set.Proof.Observe that (X/≈) ≡ im(e ≈ ) is a subtype of X → Ω V (as pr 1 : X/≈ → (X → Ω V ) is an embedding), that X → Ω V is a set (by function extensionality) and that subtypes of sets are sets.Definition 3.15 (η).The map η : X → X/≈ is defined to be the corestriction of e ≈ .
Although, in general, the type X/≈ lives in another universe than X (see Remark 3.13), we can still prove the following induction principle for (subsingleton-valued) families into arbitrary universes.Lemma 3.16 (Set quotient induction).For every subsingleton-valued P : X/≈ → W, with W any universe, if P (η(x)) holds for every x : X, then P (x ) holds for every x : X/≈.
Proof.The map η is surjective by Lemma 3.9, so that Lemma 3.10 yields the desired result.
Definition 3.17 (Respect equivalence relation).A map f : X → A respects the equivalence relation ≈ if x ≈ y implies f (x) = f (y) for every x, y : X.
Observe that respecting an equivalence relation is property rather than data, when the codomain A of the map f : X → A is a set.
Lemma 3.18.The map η : X → X/≈ respects the equivalence relation ≈ and the set quotient is effective, i.e. for every x, y : X, we have x ≈ y if and only if η(x) = η(y).
Proof.By definition of the image and function extensionality, we have for every x, y : X that η(x) = η(y) holds if and only if holds.If ( * ) holds, then so does x ≈ y by reflexivity and symmetry of the equivalence relation.Conversely, if x ≈ y and z : X is such that x ≈ z, then y ≈ z by symmetry and transitivity; and similarly if z : X is such that y ≈ z.Hence, ( * ) holds if and only if x ≈ y holds.Thus, η(x) = η(y) if and only if x ≈ y, as desired.
The universal property of the set quotient states that the map η : X → X/≈ is the universal function to a set preserving the equivalence relation.We can prove it using only Lemma 3.16 and Lemma 3.18, without the need to inspect the definition of the quotient.
Theorem 3.19 (Universal property of the set quotient).For every set A : W in any universe W and function f : X → A respecting the equivalence relation, there is a unique function f : Proof [dJE21c].Let A : W be a set and f : X → A respect the equivalence relation.
The following auxiliary type family over X/≈ will be at the heart of our proof: Claim.The type B(x ) is a subsingleton for every x : X/≈.
Proof of claim.By function extensionality, the type expressing that B(x ) is a subsingleton for every x : X/≈ is itself a subsingleton.So by set quotient induction, it suffices to prove that B(η(x)) is a subsingleton for every x : X.So assume that we have (a, p), (b, q) : B(η(x)).
It suffices to show that a = b.The elements p and q witness respectively.By Lemma 3.18 and the fact that f respects the equivalence relation, we obtain f (x) = a and f (x) = b and hence the desired a = b.
We then define the (nondependent) function f : X/≈ → A as pr 1 • k.We proceed by showing that f • η = f .By function extensionality, it suffices to prove that f (η(x)) = f (x) for every x : X.But notice that: Finally, we wish to show that f is the unique such function, so suppose that g : X/≈ → A is another function such that g • η = f .By function extensionality, it suffices to prove that g(x ) = f (x ) for every x : X/≈, which is a subsingleton as A is a set.Hence, set quotient induction tells us that it is enough to show that g(η(x)) = f (η(x)) for every x : X, but this holds as both sides of the equation are equal to f (x).
Remark 3.20 (cf.[Esc19, Section 3.21]).In univalent foundations, some attention is needed in phrasing unique existence, so we pause to discuss the phrasing of Theorem 3.19 here.Typically, if we wish to express unique existence of an element x : X satisfying P (x) for some type family P : U → V, then we should phrase it as is-singleton(Σ x:X P (x)), where is-singleton(Y ) :≡ Y × is-subsingleton(Y ).That is, we require that there is a unique pair (x, p) : Σ x:X P (x).This becomes important when the type family P is not subsingleton-valued.However, if P is subsingleton-valued, then it is equivalent to the traditional formulation of unique existence: i.e. that there is an x : X with P (x) such that every y : X with P (y) is equal to x.This happens to be the situation in Theorem 3.19, because of function extensionality and the fact that A is a set.
We stress that although the set quotient increases universe levels, see Remark 3.13, it does satisfy the appropriate universal property, so that resizing is not needed.
Having small set quotients is closely related to propositional resizing, as we show now.
Proposition 3.21.Suppose that − does not increase universe levels, i.e. in the notation of Remark 3.2, the function F is the identity.
(i) If Ω-Resizing V,U holds for universes U and V, then the set quotient X/≈ is U-small for any type X : U and any V-valued equivalence relation.(ii) Conversely, if the set quotient 2/≈ is U-small for every V-valued equivalence relation on 2, then Propositional-Resizing V,U holds. Proof.
(i) If we have Ω-Resizing V,U , then Ω V is U-small, so that X/≈ ≡ im(e ≈ ) is U-small too when X : U and ≈ is V-valued.(ii) Let P : V be any proposition and consider the V-valued equivalence relation x ≈ P y :≡ (x = y) ∨ P on 2. Notice that (2/≈ P ) is a subsingleton ⇐⇒ P holds, so if 2/≈ P is U-small, then so is the type is-subsingleton(2/≈ P ) and therefore P .
3.3.Propositional Truncations from Set Quotients.The converse, constructing propositional truncations from set quotients, is more straightforward, although we must pay some attention to the universes involved in order to get an exact match.
Definition 3.22 (Existence of set quotients).We say that set quotients exist if for every type X and equivalence relation ≈ on X, we have a set X/≈ with a map η : X → X/≈ that respects the equivalence relation such that the universal property set out in Theorem 3.19 is satisfied.
Theorem 3.23.Any set quotient satisfies the induction principle of Lemma 3.16, i.e. the induction principle is implied by the universal property of the set quotient.
Proof [dJ22a].Suppose that P : X/≈ → W is a proposition-valued type-family over the set quotient X/≈ and that we have ρ : Π x:X P (η(x)).We write S :≡ Σ x :X/≈ P (x ) and define the map f : X → S by f (x) :≡ (η(x), ρ(x)).Note that f respects the equivalence relation since η does and P is proposition-valued.Moreover, S is a set, because subtypes of sets are sets and the quotient X/≈ is a set by assumption.Hence, by the universal property, f induces a map f : X/≈ → S such that f • η = f .We claim that f is a section of pr 1 : S → X/≈.Note that this would finish the proof, because if we have e : Π x :X/≈ pr 1 f (x ) = x , then we obtain P (x ) for every x by transporting pr 2 f (x ) along e(x ).But f must be a section of pr 1 , because we can take both pr 1 • f and id for the dashed map in the commutative diagram so pr 1 • f and id must be equal by the universal property of the set quotient.
Theorem 3.24.If set quotients exist, then every type has a propositional truncation.
Proof [dJ22a].Let X : U be any type and consider the U 0 -valued equivalence relation x ≈ 1 y :≡ 1.To see that X/≈ 1 is a subsingleton, note that by set quotient induction it suffices to prove η(x) = η(y) for every x, y : X.But x ≈ 1 y for every x, y : X, and η respects the equivalence relation, so this is indeed the case.Now if P : V is any subsingleton and f : X → P is any map, then f respects the equivalence relation ≈ 1 on X, simply because P is a subsingleton.Thus, by the universal property of the quotient, we obtain the desired map f : X/≈ 1 → P and hence, X/≈ 1 has the universal property of the propositional truncation.
Remark 3.25.Because the set quotients constructed using the propositional truncation live in higher universes, we embark on a careful comparison of universes.Suppose that propositional truncations of types X : U exist and that X : F (U). Then by Remark 3.13, the set quotient X/≈ 1 in the proof above lives in the universe (U 1 U) F (U 1 U).
In particular, if F is the identity and the propositional truncation of X : U lives in U, then the quotient X/≈ 1 lives in U 1 U, which simplifies to U whenever U is at least U 1 .In other words, the universes of X and X/≈ 1 match up for types X in every universe, except the first universe U 0 .
If we always wish to have X/≈ 1 in the same universe as X , then we can achieve this by assuming F (V) :≡ U 1 V, which says that the propositional truncations stay in the same universe, except when the type is in the first universe U 0 in which case the truncation will be in the second universe U 1 .
Proof.If we have set quotients, then we have propositional truncations by Theorem 3.24 which we can use to construct effective set quotients following Section 3.2.But any two set quotients of a type by an equivalence relation must be equivalent, so the original set quotients are effective too.
3.4.Set Replacement.In this section, we return to our running assumption that universes are closed under propositional truncations, i.e. the function F above is assumed to be the identity.We study the equivalence of a set replacement principle and the existence of small set quotients.These principles will find application in Section 5.2.Definition 3.27 (Set replacement, [E + 22, UF.Size]).The set replacement principle asserts that the image of a map f : In particular, if U and V are the same, then the image is U-small.The name "set replacement" is inspired by [BBC + 22, Section 2.19], but is different in two ways: In [BBC + 22], replacement is not restricted to maps into sets, and the universe parameters U and V are taken to be the same.Rijke [Rij17] shows that the replacement of [BBC + 22] is provable in the presence of a univalent universe closed under pushouts.
We show that set replacement is logically equivalent to having small set quotients, where the latter means that the quotient of a type X : U by a V-valued equivalence relation lives in U V.
Definition 3.28 (Existence of small set quotients).We say that small set quotients exist if set quotients exists in the sense of Definition 3.22, and moreover, the quotient X/≈ of a type X : U by a V-valued equivalence relation lives in U V.
Note that we would get small set quotients if we added set quotients as a primitive higher inductive type.Also, if one assumes Ω-Resizing V , then the construction of set quotients in Section 3.2.2yields a quotient X/≈ in U V when X : U and ≈ is a V-valued equivalence relation on X.
Theorem 3.29.Set replacement is logically equivalent to the existence of small set quotients.
Proof [dJ22a, dJ22b].Suppose set replacement is true and that a type X : U and a V-valued equivalence relation ≈ are given.Using the construction laid out in Section 3.2.2,we construct a set quotient X/≈ in U V + as the image of a map X → (X → Ω V ).But by propositional extensionality Ω V is locally V-small and by function extensionality so is X → Ω V .Hence, X/≈ is (U V)-small by set replacement, so X/≈ is equivalent to a type Y : U V. It is then straightforward to show that Y satisfies the properties of the set quotient as well, finishing the proof of one implication.
Conversely, let f : X → Y be a map from a U-small type to a locally V-small set.Since X is U-small, we have X : U such that X X.And because Y is locally V-small, we have a V-valued binary relation = V on Y such that (y = V y ) (y = y ) for every y, y : Y .We now define the V-valued equivalence relation ≈ on X by (x ≈ x ) :≡ (f (x) = V f (x )), where f is the composite X X f − → Y .By assumption, the quotient X /≈ lives in U V. But it is straightforward to work out that im(f ) is equivalent to this quotient.Hence, im(f ) is (U V)-small, as desired.
The left-to-right implication of the theorem above is similar to [Rij17, Corollary 5.1], but our theorem generalizes the universe parameters and restricts to maps into sets.The latter is the reason why the converse also holds.

Largeness of Complete Posets
A well-known result of Freyd in classical mathematics says that every complete small category is a preorder [Fre64, Exercise D of Chapter 3].In other words, complete categories are necessarily large and only complete preorders can be small, at least impredicatively.Predicatively, by contrast, we show that many weakly complete posets (including directed complete posets, bounded complete posets and sup-lattices) are necessarily large.We capture these structures by a technical notion of a δ V -complete poset in Section 4.1.In Section 4.2 we define when such structures are nontrivial and introduce the constructively stronger notion of positivity.Section 4.3 and Section 4.4 contain the two fundamental technical lemmas and the main theorems, respectively.Finally, we consider alternative formulations of being nontrivial and positive that ensure that these notions are properties rather than data and shows how the main theorems remain valid, assuming univalence.4.1.δ V -complete Posets.We start by introducing a class of weakly complete posets that we call δ V -complete posets.The notion of a δ V -complete poset is a technical and auxiliary notion sufficient to make our main theorems go through.The important point is that many familiar structures (dcpos, bounded complete posets, sup-lattices) are δ V -complete posets (see Examples 4.3).Definition 4.1 (δ V -complete poset, δ x,y,P , δ x,y,P ).A poset is a type X with a subsingletonvalued binary relation on X that is reflexive, transitive and antisymmetric.It is not necessary to require X to be a set, as this follows from the other requirements.A poset (X, ) is δ V -complete for a universe V if for every pair of elements x, y : X with x y and every subsingleton P in V, the family δ x,y,P : 1 + P → X inl( ) → x; inr(p) → y; has a supremum δ x,y,P in X.
Remark 4.2 (Classically, every poset is δ V -complete).Consider a poset (X, ) and a pair of elements x y.If P : V is a decidable proposition, then we can define the supremum of δ x,y,P by case analysis on whether P holds or not.For if it holds, then the supremum is y, and if it does not, then the supremum is x.Hence, if excluded middle holds in V, then the family δ x,y,P has a supremum for every P : V. Thus, if excluded middle holds in V, then every poset (with carrier in any universe) is δ V -complete.
The above remark naturally leads us to ask whether the converse also holds, i.e. if every poset is δ V -complete, does excluded middle in V hold?As far as we know, we can only get weak excluded middle in V, as we will later see in Proposition 4.6.This proposition also shows that in the absence of excluded middle, the notion of δ V -completeness isn't trivial.For now, we focus on the fact that, also constructively and predicatively, there are many examples of δ V -complete posets.(i) There is a locally small δ V -complete poset with decidable equality that is nontrivial in an unspecified way if and only if weak excluded middle in V holds.(ii) There is a locally small δ V -complete poset with decidable equality that is positive in an unspecified way if and only if excluded middle in V holds.

Maximal Points and Fixed Points
As is well known, in impredicative mathematics, a poset has suprema of all subsets if and only if it has infima of all subsets.Perhaps counter-intuitively, this "duality" theorem can be proved predicatively.However, in the absence of impredicativity, it is not possible to fulfil its hypotheses when trying to apply it, because there are no nontrivial examples.
To explain this, we first have to make the statement of the duality theorem precise.A single universe formulation is "every V-small V-sup-lattice has all infima of families indexed by types in V".The usual proof, adapted from subsets to families, shows that this formulation is predicatively provable, but in our predicative setting Theorem 4.23 tells us that there are no nontrivial examples to apply it to.
It is natural to wonder whether the single universe formulation can be generalized to locally small V-sup-lattices (with necessarily large carriers), resulting in a predicatively useful result.However, as one of the anonymous reviewers pointed out that this generalization gives rise to a false statement and suggested the ordinals as a counterexample in a set-theoretic setting: it is a class with suprema for all subsets but has no greatest element.This led us to prove (Section 5.2) in our type-theoretic context that the locally small, but large type of ordinals in a univalent universe V is a V-sup-lattice.But this is not a V-inf-lattice, because the unique family indexed by the empty type does not have a greatest lower bound since the type of ordinals has no greatest element.
Similarly, consider a generalized formulation of Tarski's theorem [Tar55] that allows for multiple universes, i.e. we define Tarski's-Theorem V,U ,T as the assertion that every monotone endofunction on a V-sup-lattice with carrier in a universe U and order taking values in a universe T has a greatest fixed point.Then Tarski's-Theorem V,V,V corresponds to the original formulation and, moreover, is provable predicatively, but not useful predicatively because Theorem 4.24 shows that its hypotheses can only be fulfilled for trivial posets.On the other hand, Tarski's-Theorem V,V + ,V is provably false because the identity map on the V-sup-lattice of ordinals in V is a counterexample.Analogous considerations can be made for a lemma due to Pataraia [Pat97,Esc03] saying that every dcpo has a greatest monotone inflationary endofunction.

A Predicative Counterexample.
Because the type of ordinals in V is not V-small even impredicatively, the above does not rule out the possibility that a V-sup-lattice X has all V-infima provided X is V-small impredicatively.To address this, we present an example of a V-sup-lattice, parameterized by a proposition, that is V-small impredicatively, but predicatively does not necessarily have a maximal element.In particular, it need not have a greatest element or all V-infima.monotone.Moreover, if y ≺ f (x), then β ↓ y ≺ α ↓ x, so that we get x ≺ x with y = f (x ), and f is thus a simulation.
Recall from Definition 3.28 what it means to have small set quotients.If these are available, then the type of ordinals has all small suprema.Theorem 5.8 (Extending [Uni13, Lemma 10.3.22]).Assuming small set quotients, the large ordinal Ord V has suprema of families indexed by types in V.
Proof [dJE22].Given α : I → Ord V , define α as the quotient of Σ i:I α i by the V-valued equivalence relation ≈ where (i, x) ≈ (j, y) if and only if α i ↓ x and α j ↓ y are isomorphic as ordinals.By our assumption, the quotient α lives in V. Next, [Uni13, Lemma 10.3.22]tells us that (α, ≺) with is an ordinal that is an upper bound of α.So we show that α is a lower bound of upper bounds of α.To this end, suppose that β : Ord V is such that α i β for every i : I.In light of Lemma 5.7, this assumption yields two things: (1) for every i : I and x : α i there exists a unique b x i : β such that α i ↓ x = β ↓ b x i ; (2) for every i : I, a simulation f i : α i → β such that for every x : α i , we have f i (x) = b x i .We are to prove that α β.We start by defining Observe that f respects ≈, for if (i, x) ≈ (j, y), then by univalence, so b x i = b y j by uniqueness of b x i .Thus, f induces a map f : α → β satisfying the equality f ([(i, x)]) = f (i, x) for every (i, x) : Σ j:J α j .
It remains to prove that f is a simulation.Because the defining properties of a simulation are propositions, we can use set quotient induction and it suffices to prove the following two things: i , then there exists j : I and y : α j such that α i ↓ y ≺ α j ↓ x and b y j = b.For (I), observe that if α i ↓ x ≺ α j ↓ y, then β ↓ b x i ≺ β ↓ b y j , from which b x i ≺ b y j follows using Lemma 5.5.For (II), suppose that b ≺ b x i .Because f i (see item (2) above) is a simulation, there exists y : α i with y ≺ x and f i (y) = b.By Lemma 5.5, we get α i ↓ y ≺ α i ↓ x.Moreover, b y i = f i (y) = b, finishing the proof of (II).
In Section 3.4 we saw that set replacement is equivalent to the existence of small set quotients, so the following result immediately follows from the theorem above.But the point is that an alternative construction without set quotients is available, if set replacement is assumed.
Theorem 5.9.Assuming set replacement, the large ordinal Ord V has suprema of families indexed by types in V. U :≡ U 1 and V :≡ U 0 , so that V U ≡ U holds.Thus, our V-families-based approach generalizes the traditional subset-based approach.

Conclusion
Firstly, we have shown, constructively and predicatively, that nontrivial dcpos, bounded complete posets and sup-lattices are all necessarily large and necessarily lack decidable equality.We did so by deriving a weak impredicativity axiom or weak excluded middle from the assumption that such nontrivial structures are small or have decidable equality, respectively.Strengthening nontriviality to the (classically equivalent) positivity condition, we derived a strong impredicativity axiom and full excluded middle.
Secondly, we showed that Tarski's greatest fixed point theorem cannot be applied in nontrivial instances in our predicative setting, while generalizations of Tarski's theorem that allow for large structures are provably false.Specifically, we showed that the ordinal of ordinals in a univalent universe does not have a maximal element, but does have small suprema in the presence of small set quotients, or equivalently, set replacement.More generally, we investigated the inter-definability and interaction of type universes of propositional truncations and set quotients in the absence of propositional resizing axioms.In particular, we showed that in the presence of propositional truncations, but without assuming propositional resizing, it is possible to construct set quotients that happen to live in higher type universes but that do satisfy the appropriate universal properties with respect to sets in arbitrary type universes.
Finally, we clarified, in our predicative setting, the relation between the traditional definition of a lattice that requires completeness with respect to subsets and our definition that asks for completeness with respect to small families.
In future work, it would be interesting to study the predicative validity of Pataraia's theorem and Tarski's least fixed point theorem.Curi [Cur15,Cur18] develops predicative versions of Tarski's fixed point theorem in some extensions of CZF.It is not clear whether these arguments could be adapted to univalent foundations, because they rely on the settheoretical principles discussed in the introduction.The availability of such fixed-point theorems might be useful for application to inductive sets [Acz77], which we might otherwise introduce in univalent foundations using higher inductive types [Uni13].In another direction, we have developed a notion of apartness [BV11] for continuous dcpos [GHK + 03] that is related to the strictly-below relation introduced in this paper.Namely, if x y are elements of a continuous dcpo, then x is strictly below y if x is apart from y.In [dJ21], we give a constructive analysis of the Scott topology [GHK + 03] using this notion of apartness.
Definition 3.5].Hence, [Shu16, Theorem 5.3] tells us that f can be split as Y r − → A s − → Y for some maps s and r .Now X and A are equivalent as witnessed by the maps x → r (s(x)) and a → r(s (a)).Finally, we recall from the proof of [Shu16, Theorem 5.3] that A