Algebraic cocompleteness and finitary functors
aa r X i v : . [ c s . L O ] F e b ALGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS
JI ˇR´I AD ´AMEK ∗ Department of Mathematics,Czech Technical University in Prague,Czech Republicand Institute of Theoretical Computer Science,Technical University Braunschweig,Germany e-mail address : [email protected]
Abstract.
A number of categories is presented that are algebraically complete and co-complete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. Example:assuming GCH, the category
Set ≤ λ of sets of power at most λ has that propery, whenever λ is an uncountable regular cardinal.For all finitary (and, more generally, all precontinuous) set functors the initial algebraand terminal coalgebra are proved to carry a canonical partial order with the same idealCPO-completion. And they also both carry a canonical ultrametric with the same Cauchycompletion. Finally, all endofunctors of the category Set ≤ λ are finitary if λ has countablecofinality and there are no measurable cardinals µ ≤ λ . Introduction
The importance of terminal coalgebras for an endofunctor F was clearly demonstratedby Rutten [17]: for state-based systems whose state-object lie in a category K and whosedynamics are described by F , the terminal coalgebra νF collects behaviors of individualstates. And given a system A the unique coalgebra homomorphism from A to νF assignsto every state its behavior. However, not every endofunctor possesses a terminal coalgebra.Analogously, an initial algebra µF need not exist.Freyd [14] introduced the concept of an algebraically complete category: this meansthat every endofunctor has an initial algebra. More than a decade prior to Freyd’s lectureTrnkov´a proved that the category Set ≤ℵ of countable sets and mappings is algebraicallycomplete [21], Thm.2. This has inspired Koubek and the author to prove that for everycardinal λ the category Set ≤ λ Key words and phrases: initial algebra, terminal coalgebra, algebraically complete category, finitaryfunctor. ∗ Supported by the Grant Agency of the Czech Republic under the Grant 19-00902S.
Preprint submitted toLogical Methods in Computer Science © J. Adámek CC (cid:13) Creative Commons
J. AD ´AMEK of sets of power at most λ and K - Vec ≤ λ of vector spaces of dimension at most λ , for any field K , are algebraically complete, [5],Example 14. The algebraic completeness of the category of classes and maps has beenproved in [9].We dualize the above concept, and call a category algebraically cocomplete if everyendofunctor has a terminal coalgebra. Assuming the generalized continuum hypothesis(GCH), we prove below that for every uncountable, regular cardinal λ the category Set ≤ λ is algebraically complete and cocomplete. (That is, every endofunctor F has both µF and νF .) In contrast, the category of countable sets is not algebraically cocomplete (Example2.15). And the algebraic cocompleteness of the category Set ≤ℵ is proved to be equivalent to the continuum hypothesis (Corollary 2.16).Further examples of algebraically complete and cocomplete categories, assuming GCH,are K - Vec ≤ λ for regular infinite cardinals λ > | K | and Nom ≤ λ the category of nominal sets of cardinality at most λ , for regular cardinals λ > ℵ . And if G is a group with 2 | G | < λ , then the category G - Set ≤ λ of G -sets (sets with an action of G ) of cardinality at most λ is algebraically complete andcocomplete.If we work in the category of cpo ’s as our base category, then Smyth and Plotkin provedin [18] that the initial algebra coincides with the terminal coalgebra for all locally continuousendofiunctors. That is, the underlying objects are equal, and the structure maps are inverseto each other. Is there a connection between initial algebras µF and terminal coalgebras νF for set functors F , too? We concentrate on precontinuous set functors which is ageneralization of finitary set functors encompassing also all continuous functors (preservinglimits of ω op -chains) and closed under arbitrary products, subfunctors, coproducts andcomposition. Each precontinuous functor has a terminal coalgebra νF which carries acanonical partial order, and an initial algebra that, as a subposet of νF , has the same ideal cpo -completion whenever F ∅ 6 = ∅ . Consquently, assuming GCH, if the initial algebra isuncountable, it has the same cardinality as the terminal coalgebra.And, analogously, assuming GCH the initial algebra and terminal coalgebra of a precon-tinuous functor with F ∅ 6 = ∅ carry a canonical ultrametric such that the Cauchy completionsof µF and νF coincide. This complements the result of Barr [12] that for every finitary,continuous set functor with F ∅ 6 = ∅ the metric space νF is the Cauchy completion of µF .Finitary functors are also a topic of the last section devoted to non-regular cardinals: if λ is a cardinal of countable cofinality, we prove that every endofunctor of Set ≤ λ is finitary,assuming that no measurable cardinal is smaller or equal to λ . For example for λ = ℵ ω .Surprisingly, Set ≤ℵ ω is not algebraically complete. Related Work
This paper extends results of the paper [4] presented at the conferenceCALCO 2019. The result about the ideal cpo -completion of the initial algebra and the lastsection on non-regular cardinals are new. In [4] a proof was presented that assuming GCH,a set functor F having µF of uncountable regular cardinality has νF of the same cardinality.Unfortunately, the proof was not correct. We present in Corollary 4.14 a different proofassuming F is precontinuous. LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 3
Acknowledgement
The author is grateful to the referees for a number of very useful com-ments. And to Stefan Milius and Lary Moss for valuable discussions about precontinuousfunctors. 2.
Algebraically Cocomplete Categories
For a number of categories K we prove that the full subcategory K ≤ λ on objects of powerat most λ is algebraically cocomplete. Power is a cardinal defined below. Definition 2.1.
An object is called connected if it is non-initial and is not a coproduct oftwo non-initial objects. An object has power λ if it is a coproduct of λ connected objectsand λ is the least cardinal possible. Example 2.2. In Set , connected objects are the singleton sets, and power has its usualmeaning. In the category K - Vec of vector spaces over a field K the connected spaces arethose of dimension one, and power means dimension. In the category Set S of many-sortedsets the connected objects are those with precisely one element (in all sorts together), andthe power of X = ( X s ) s ∈ S is simply | ` s ∈ S X s | . Definition 2.3.
A category K is said to have width w ( K ) if it has coproducts, every objectis a coproduct of connected objects, and w ( K ) is the smallest cardinal β such that(a) K has at most β connected objects up to isomorphism, and(b) for all cardinals α ≥ β there exist at most α morphisms from a connected object to anobject of power α . Example 2.4. (1)
Set has width 1. More generally,
Set S has width | S | . Indeed, in Example 2.2 wehave seen that the number of connected objects up to isomorphism is | S | , and (b) clearlyholds.(2) K - Vec has width | K | + ℵ . Indeed, the only connected object, up to isomorphism,is K . For a space X of dimension α the number of morphisms from K to X is | X | . If K isinfinite, then α ≥ | K | implies | X | = α (and | K | = | K | + ℵ ). For K finite, the least cardinal λ such that | X | ≤ α holds for every α -dimensional spaces X with α ≥ λ is ℵ (= | K | + ℵ ).(3) G - Set , the category of sets with an action of the group G , has width at most 2 | G | for infinite groups, and at most ℵ for finite ones, see our next lemma. Recall that objectsare pairs ( X, · ) where X is a set and · is a function from G × X to X (notation: gx for g ∈ G and x ∈ X ) such that h ( gx ) = ( hg ) x for h, g ∈ G and x ∈ X , and ex = x for x ∈ X ( e neutral in G ) . Morphisms are the equivariant functions: those preserving the unary operation g. − forevery g ∈ G .(4) The category Nom of nominal sets has width ℵ , see 2.6 below. Recall that fora given countably infinite set A , the group S f ( A ) of finite permutations consists of allcomposites of transpositions. A nominal set is a set X together with an action of the group S f ( A ) on it (notation: πx for π ∈ S f ( A ) and x ∈ X ) such that every element x ∈ X has a J. AD ´AMEK finite support. This means a finite subset A ⊆ A such that for every finite permutation wehave: π ( a ) = a for all a ∈ A implies πx = x. Morphisms are the equivariant functions.
Lemma 2.5.
For every group G the category G - Set of sets with an action of G has width w ( G - Set ) ≤ | G | + ℵ . Proof. (1) We first observe an important example of a G -set given by any equivalencerelation ∼ on G which is equivariant , i.e., fulfilling g ∼ g ′ ⇒ hg ∼ hg ′ for all g, g ′ , h ∈ G .
Then the quotient set G/ ∼ is a G -set (of equivalence classes [ g ]) w.r.t. the action g [ h ] = [ gh ].This G -set is clearly connected.(2) Let ( X, · ) be a G -set. For every element x ∈ X we obtain a subobject of ( X, · ) onthe set Gx = { gx ; g ∈ G } (the orbit of x ) . The equivalence on G given by g ∼ g ′ iff gx = g ′ x is equivariant, and the G -sets Gx and G/ ∼ are isomorphic. Moreover, two orbits are disjointor equal: given gx = hy , then x = ( g − h ) y , thus, Gx = Gy .(3) Every object ( X, · ) is a coproduct of at most | X | connected objects. Indeed, let X be a choice class of the equivalence ≡ given by x ≡ y ⇔ Gx = Gy, then ( X, · ) is a coproduct of the orbits of x for x ∈ X .(4) The number of connected objects, up to isomorphism, is at most 2 | G | + ℵ . Indeed, itfollows from the above that the connected objects are represented by precisely all G/ ∼ where ∼ is an equivariant equivalence relation. If | G | = β then we have at most β β equivalencerelations. For β infinite, this is equal to 2 | G | , for β finite, this is smaller than 2 | G | + ℵ .(5) The number of morphisms from G/ ∼ to an object ( X, · ) is at most | X | . Let α ≥ | G | + ℵ . If the power of ( X, · ) is α , than the cardinality of X is at most α sincethe cardinality of X in (3) is at most α . Consequently, there exist at most α morphisms p : G/ ∼→ ( X, . ) . Indeed, every morphism p is determined by the value x = p ([ e ]) since p ([ g ]) = p ( g [ e ]) = g · x holds for all [ g ] ∈ G/ ∼ . Corollary 2.6.
The category
Nom of nominal sets has width ℵ . The proof is completely analogous: in (2) each orbit S f ( A ) (cid:14) ∼ ≃ S f ( A ) x is a nominalset. And the number of all such orbits up to isomorphism is ℵ , see Lemma A1 in [8]. In(5) we have | X | ≤ α · ℵ = α for all α ≥ ℵ .The following lemma is based on ideas of V. Trnkov´a [20]: LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 5
Lemma 2.7.
A commutative square A a ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ a ❆❆❆❆❆❆❆❆ B b ❆❆❆❆❆❆❆❆ B b ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ B in A is an absolute pullback, i.e., a pullback preserved by all functors with domain A , pro-vided that (1) b and b are split monomorphisms, and (2) there exist morphisms ¯ b and ¯ a : A a ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ a ❆❆❆❆❆❆❆❆ B b ❆❆❆❆❆❆❆❆ B b ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ ¯ a ` ` ❆❆❆❆❆❆❆❆ B ¯ b ` ` ❆❆❆❆❆❆❆❆ satisfying ¯ b b = id , ¯ a a = id , and a ¯ a = ¯ b b (2.1) Proof.
The first square above is a pullback since given a commutative square b c = b c for c i : C → B i there exists a unique c with c i = a i · c ( i = 1 , a is split monic.Put c = ¯ a · c . Then c = a c follows from b being monic: b c = b ¯ b b c ¯ b b = id= b ¯ b b c b c = b c = b a ¯ a c ¯ b b = a ¯ a = b a c c = ¯ a c And c = a c follows from b being monic: b c = b c = b a c c = a c = b a c b a = b a For every functor F with domain A the image of the given square satisfies the analogousconditions: F b and F b are split monomorphisms and F ¯ b , F ¯ a verify (2.1). Thus, theimage is a pullback, too. Corollary 2.8 (See [20]) . Every set functor preserves nonempty finite intersections.
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Indeed, if A above is nonempty, choose an element t ∈ A and define ¯ b and ¯ a by¯ b ( x ) = ( y if b ( y ) = xa ( t ) if x / ∈ b [ B ]and ¯ a ( x ) = ( y if a ( y ) = xt if x / ∈ a [ A ]It is easy to see that (2.1) holds. Remark 2.9. (a) We recall that the cofinality of an infinite cardinal λ is the smallestcardinal µ such that λ is a join of a µ -chain of smaller cardinals. An infinite cardinal iscalled regular if it equals its cofinality. The first non-regular cardinal is ℵ ω .(b) Recall that for a set X of infinite cardinality λ a collection of subsets is called almostdisjoint if the intersection of arbitrary two distinct members has cardinality smaller than λ .Tarski [19] proved that for every set X of infinite cardinality λ (not necessarily regular)there exists an almost disjoint collection Y i ⊆ X ( i ∈ I ) with | I | > λ. (c) Given an element t ∈ X there exists an almost disjoint collection Y i as above with t ∈ Y i for all i ∈ I . Indeed, take any almost disjoint collection ( Y i ) i ∈ I and use Y i ∪ { t } instead (for all i ∈ I ). Moreover, we can assume | Y i | = λ for all i : see [13], Thm. 2.8. Definition 2.10.
Let K be a category of width w ( K ). For every infinite cardinal λ > w ( K )we denote by K ≤ λ the full subcategory of K on objects of power at most λ . Proposition 2.11.
Let F be an endofunctor of K ≤ λ , λ regular. Given an object X = ` i ∈ I X i ,with all X i connected, every morphism b : B → F X with B of power less than λ factorizesthrough F c for some subcoproduct c : C → X where C = ` j ∈ J X j and | J | < λ : F C
F c (cid:15) (cid:15) B = = ④④④④ b / / F X
Proof. (1) It is sufficient to prove this in case X has power precisely λ (otherwise put c = id X ).And we can also assume that B is connected. In the general case we have, by Definition2.1, a coproduct of connected objects B = ` k ∈ K B k with | K | < λ , and we find for each k a coproduct injection c k : C k → X corresponding to the k -th component of b , then put C = ` k ∈ K C k (which has power less than λ = λ since each C k does and | K | < λ ) and put c = [ c k ] : C → X . LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 7
Since λ > w ( K ), there are less than λ connected objects up to isomorphism. Thus, as λ is regular, in the coproduct of λ connected objects representing X , at least one compomemtmust appear λ times. Thus X has the form X = a λ R + X where both R and X have power less than λ : X is the coproduct of all components thatappear less than λ times as X i for some i ∈ I and R is the coproduct of all the othercomponents where isomorphic copies are not repeated.(2) By Remark 2.9 we can choose t ∈ λ and an almost disjoint collection of sets M k ⊆ λ , k ∈ K , with t ∈ M k , | M k | = λ and | K | > λ . Denote for every M ⊆ λ by Y M the coproduct Y M = G M R + X and for every k ∈ K let b k : Y M k → X be the coproduct injection.Consider the following square of coproduct injections for any pair k , l ∈ K : Y M l ∩ M k a k z z ✉✉✉✉✉✉✉✉✉ a l $ $ ❍❍❍❍❍❍❍❍❍ Y M k b k $ $ ■■■■■■■■■■ Y M l b l z z ✉✉✉✉✉✉✉✉✉✉ X This is an absolute pullback. Indeed, it obviously commutes. And b k and b l are splitmonomorphisms: define ¯ b k : X → Y M k as identity on the summand X whereas the i -th copy of R is sent to copy i if i ∈ M k , andto copy t else. Then ¯ b k b k = id . Analogously for b l . Next define ¯ a l : Y M l → Y M k ∩ M l as identity on the summand X whereas the i -th copy of R is sent to copy i if i ∈ M k , andto copy t else. Then clearly ¯ a k a l = id and a k ¯ a l = ¯ b k b l . Thus, the above square is an absolute pullback by Lemma 2.7.(3) We are ready to prove that for a connected object B every morphism b : B → F X has the required factorization. For every k ∈ K since | M k | = λ we have an isomorphism y k : X → Y M k which composed with b k : Y M k → X yields an endomorphism z k = b k · y k : X → X .
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The following morphisms B b −−→ F X
F z k −−−→ F X ( k ∈ K )are not pairwise distinct because | K | > λ , whereas | K ( B, F X ) | ≤ λ . Indeed, the latterfollows since F X has at most λ components (since F is an endofunctor of K ≤ λ ) so that (b)in Definition 2.3 implies that K ( B, F X ) has cardinality at most λ . Choose k = l in K with F z k · b = F z l · b . (2.2)Compare the pullbacks Z of z k and z l and Y M k ∩ M l of b k and b l : Z p (cid:15) (cid:15) ✤✤✤✤ p k } } ④④④④④④④④④④④④ p l ! ! ❇❇❇❇❇❇❇❇❇❇❇❇ X y k (cid:15) (cid:15) Y M k ∩ M l a k ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ a l ❇❇❇❇❇❇❇❇❇❇❇ X y l (cid:15) (cid:15) Y M k b k ! ! ❈❈❈❈❈❈❈❈❈❈❈ Y M l b l } } ⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤ X Since y k and y l are isomorphisms, the connecting morphism p between the above pullbacksis an isomorphism, too. We know that | M k ∩ M l | < λ since M k , M l are members of ouralmost disjoint family, thus the coproduct injection c : Y M k ∩ M l → X has less than λ summands, as required. And, due to (2), the pullback of z k and z l is absolute.The equality (2.2) thus implies that b factorizes through F p k . Since clearly p k = c · p , thisimplies that b factorizes through F c , as required.
Remark 2.12.
The assumption that λ is a regular cardinal has only been used in the aboveproof in Step (1), where we claimed that one component is repeated λ times in X . If ourcategory has finitely many connected objects up to isomorphism, then the above propositionalso holds for non-regular infinite cardinals. Remark 2.13. (a) If an infinite cardinal λ has cofinality n , then λ n > λ , see [15], Corollary 1.6.4.(b) Every ordinal α is considered as the set of all smaller ordinals. In particular ℵ isthe set of all natural numbers and ℵ the set of all countable ordinals.(c) Recall the Continuum Hypothesis (CH) stating that the cardinal successor of ℵ is 2 ℵ , and the General Continuum Hypothesis (GCH) which states that for every infinite cardinal λ the cardinal successor is 2 λ .(d) Under GCH every infinite regular cardinal λ fulfils λ n = λ for all cardinals 1 ≤ n < λ . LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 9
See Theorem 1.6.17 in [15].
Theorem 2.14.
Assuming GCH, let K be a cocomplete and cowellpowered category of width w ( K ) . Then K ≤ λ is algebraically cocomplete for all uncountable regular cardinals λ > w ( K ) .Proof. Let F be an endofunctor of K ≤ λ . Form a collection a i : A i → F A i ( i ∈ I ) representingall coalgebras of F on objects of power less than λ (up to isomorphism of coalgebras). Wehave | I | ≤ λ . Indeed, for every cardinal n < λ let I n ⊆ I be the subset of all i with A i having n components.Given i ∈ I n , for every component b : B → A i of A i we know, since λ > | K | , that there areat most λ morphisms from B to F A i (recalling that F A i has at most λ components). Thusthere are at most n · λ = λ morphisms from A i to F A i . And the number of objects A i with n components is at most w ( K ) n < λ n = λ (see Remark 2.13). Thus, there are at most λ indexes in I n . Since I = S n<λ I n , this proves | I | ≤ λ = λ .Consequently A = ` i ∈ I A i is an object of K ≤ λ , and we have the coalgebra structure α : A → F A of a coproduct of ( A i , α i ) in Coalg F . Let e : ( A, α ) → ( T, τ ) be the widepushout of all homomorphisms in
Coalg F with domains ( A, α ) carried by epimorphisms of K . Since K is cocomplete and cowellpowered, and since the forgetful functor from Coalg F to K creates colimits, this means that we form the corresponding pushout in K and get aunique coalgebra structure τ : T → F T making e a homomorphism: ` A i α / / e (cid:15) (cid:15) F ( ` A i ) F e (cid:15) (cid:15) T τ / / F T
We are going to prove that (
T, τ ) is a terminal coalgebra.(1) Firstly, for every coalgebra β : B → F B with B having power less than λ thereexists a unique homomorphism into ( T, τ ). Indeed, the existence is clear: compose theisomorphism that exists from (
B, β ) to some ( A i , α i ), the i -th coproduct injection to ( A, α )and the above homomorphism e . To prove uniqueness, observe that by definition of ( T, τ ),this coalgebra has no nontrivial quotient: every homomorphism with domain (
T, τ ) whoseunderlying morphism is epic in K is invertible. Given homomorphisms u , v : ( B, β ) → ( T, τ ) B β / / v (cid:15) (cid:15) u (cid:15) (cid:15) F B
F u (cid:15) (cid:15)
F v (cid:15) (cid:15) T τ / / q (cid:15) (cid:15) F T
F q (cid:15) (cid:15) Q / / ❴❴❴ F Q form their coequalizer q : T → Q in K . Then Q carries the structure of a coalgebra making q a homomorphism. Thus, q is invertible, proving u = v . (2) Next, consider an arbitrary coalgebra β : B → F B . Express B = ` i ∈ I B i where B i are connected and assume | I | = λ (the case | I | < λ has just been handled). For every set J ⊆ I we denote by u J : ` i ∈ J B i → B the subcoproduct. In case | J | < λ we are going toprove that there exists a set J ⊆ J ′ ⊆ I with | J ′ | < λ such that the summand u J ′ : B J ′ = a i ∈ J ′ B i → B carries a subcoalgebra. That is, there exists β J ′ : B J ′ → F B J ′ for which u J ′ is a homomor-phism. Indeed, we put J ′ = [ n<ω J n for the following ω -chain of sets J n ⊆ I with | J n | < λ . First J = J , and given J n , apply Proposition 2.11 to β.u J n : B J n → F B . We conclude that this morphismfactorizes through
F b J n +1 for some subset J n +1 ⊆ I of power less that λ : B J n β n / / ❴❴❴ u Jn (cid:15) (cid:15) F B J n +1 F u Jn +1 (cid:15) (cid:15) B β / / F B
Thus, for the union J ′ = S J n we get | J ′ | < λ because λ is uncountable and reg-ular, therefore | ` n<ω J n | < λ . And u J ′ carries the following subcoalgebra β J ′ : ` j ∈ J ′ B j → F (cid:0) ` j ∈ J ′ B j (cid:1) of ( B, β ): Given j ∈ J ′ let n be the least number with j ∈ J n . Denote by w : B j → ` i ∈ J n B i and v : ` i ∈ J n +1 B i → ` j ∈ J ′ B j the coproduct injections. Then the j -th compo-nent of β ′ is the following composite B j w −→ a i ∈ J n B i β n −→ F (cid:0) a i ∈ J n +1 B i (cid:1) F v −−→ F (cid:0) a i ∈ J ′ B (cid:1) To prove that the square below ` i ∈ J ′ B i β J ′ / / u J ′ (cid:15) (cid:15) F (cid:0) ` i ∈ J ′ B i (cid:1) F u J ′ (cid:15) (cid:15) ` i ∈ I B i β / / F (cid:0) ` i ∈ I B i (cid:1) commutes, consider the components for j ∈ J ′ separately. The upper passage yields, since u J ′ · v = u J n +1 : ` i ∈ L n B i → ` i ∈ I B i , the result F u J ′ · ( F v · β n · w ) = F u J n +1 · β n · w = β · u J n · w . The lower passage yields the same.
LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 11 (3) We conclude that every coalgebra β : B → F B is a colimit, in
Coalg F , of coalgebrason objects of power less than λ . Indeed, this is non-trivial only for objects B ∈ K ≤ λ where B = ` i ∈ I B i with B i indecomposable and | I | = λ . Then for every set J ′ ⊆ I with | J ′ | < λ forwhich B J ′ carries a subcoalgebra ( B J ′ , β J ′ ) of ( B, β ) we take this as an object of our diagram.And morphisms from ( B J ′ , β J ′ ) to ( B J ′′ , β J ′′ ) (where again J ′′ ⊆ I fulfils | J ′′ | < λ ), are thosecoproduct injections u : B J ′ → B J ′′ for J ′ ⊆ J ′′ that are coalgebra homomorphisms. Sinceevery subset J ⊆ I with | J | < λ is contained in J ′ for some object of this diagram, it easilyfollows that the cocone of coproduct injections from ( B J ′ , β J ′ ) to ( B, β ) is a colimit of theabove diagram.(4) Thus, from Part (1) we conclude that there exists a unique homomorphism from(
B, β ) to (
T, τ ). Example 2.15.
The category
Set ≤ λ of sets of power at most λ is, under GCH, algebraicallycocomplete for all uncountable regular cardinals λ .However, the category Set ≤ℵ of countable sets is not algebraically cocomplete. Indeed,the restriction P f of the finite power-set functor to it does not have a terminal coalgebra.Assuming the contrary, let τ : T → P f T be a (countable) terminal coalgebra for therestricted functor. For every subset A ⊆ N denote by X A the following countable, finitelybranching graph: 0 / / / / / / · · · / / i / / = = ④④④④④④④④④ · · · ( i ∈ A )It is obtained from the infinite path on N by taking an extra successor for i ∈ N iff i ∈ A .Thus X A is a coalgebra for P f . The unique homomorphism h A : X A → ( T, τ ) takes 0 to anelement h A (0) ∈ T . The desired contradiction is achieved by proving for all subsets A , A ′ of N that if h A (0) = h A ′ (0), then A = A ′ – but then | T | ≥ | N | .From h A (0) = h A ′ (0) we derive A ⊆ A ′ ; by symmetry, A = A ′ follows. Since h A and h A ′ are coalgebra homomorphisms, the relation R ⊆ X A × X A ′ of all pairs ( x, x ′ ) with h A ( x ) = h A ′ ( x ′ ) is a graph bisimulation. By assumption 0 R
0, and for every i ∈ A we finda leaf of A with distance i + 1 from 0. Hence, there is a leaf of A ′ with the same distance.This proves A ⊆ A ′ . Corollary 2.16.
The category
Set ≤ℵ is algebraically cocomplete iff the continuum hypoth-esis holds. Indeed, if CH holds, then the proof of Theorem 2.14 applies: we do not need the fullstrentgh of GCH in that proof. All we need is the statement at the beginning of the proofthat for every n < ℵ there are at most ℵ coalgebras of power n , which follows from CH.Conversely, if ℵ < ℵ , then the restriction of P f to Set ≤ℵ does not have a terminalcoalgebra, the argument is as in the preceding example.3. Algebraically Complete Categories
Remark 3.1. (a) Let K be a category with an initial object 0 and with colimits of i -chains for all limitordinals i ≤ λ . Recall from [1] the initial-algebra chain of an endofunctor F : its objects F i i ≤ λ and its connecting morphisms w ij : F i → F j i ≤ j ≤ λ are defined by transfinite recursion as follows: F ,F i +1 F ( F i , and F j colim i We recall here some facts from [10]. Let λ be an infinite regular cardinal.(a) An object A of a category K is called λ -presentable if its hom-functor K ( A, − )preserves λ -filtered colimits. And A is called λ -generated if K ( A, − ) preserves λ -filteredcolimits of diagrams where all connecting morphisms are monic.(b) A category K is called locally λ -presentable if it is cocomplete and has a stronggenerator consisting of λ -presentable objects G i for i ∈ I . That is, every subobject m : A → B such that all morphisms G i → B factorize through m is invertible.(c) Every such category has a factorization system (strong epi, mono). An object is λ -generated iff it is a strong quotient of a λ -presentable one.(d) In the case λ = ℵ we speak about locally finitely presentable categories . Definition 3.3 (See [8]) . A strictly locally λ -presentable category is a locallly λ -presentablecategory in which every morphism b : B → A with B λ -presentable has a factorization b = b ′ · f · b for some morphisms b ′ : B ′ → A and f : A → B ′ with B ′ also λ -presentable. Examples 3.4 (See [8]) . (a) The categories Set , K - Vec and G - Set , where G is a finite group, are strictly locallyfinitely presentable.(b) Nom is strictly locally ℵ -presentable.(c) Set S is strictly locally λ -presentable for λ > | S | .(d) Given an infinite group G , the category G - Set is strictly locally λ -presentable if λ > | G | . Definition 3.5. A category K has strict width w ( K ) if it has width w ( K ), its coproductinjections are monic, and every connected object is finitely presentable. Example 3.6. (1) The category Set S has strict width | S | + ℵ , since connected objects (see Exam-ple 2.2) are finitely presentable.(2) K - Vec has strict width | K | + ℵ : the only connected object K is finitely presentable.(3) G - Set has strict width at most 2 | G | + ℵ . Indeed, it follows from the proof ofLemma 2.5 that the only connected objects are the quotients of G , and they are easily seento be finitely presentable.(4) Nom has strict width ℵ , the argument is as in (3). Lemma 3.7. If a category has strict width w ( K ) , then for every regular cardinal λ ≥ w ( K ) its λ -presentable objects are precisely those of power less than λ . Proof. If X is λ -presentable and X = ` i ∈ I X i with connected objects X i , then in case card I <λ we have nothing to prove. And if card I ≥ λ , form the λ -filtered diagram of all coproducts ` j ∈ J X j where J ranges over subsets of I with card J < λ . Its colimit ois X . Since K ( X, − )preserves this colimit, there exists a factorization of id X through one of the colimit injections v : ` j ∈ J X j → X . Now v is monic (by the definition of strict width) and split epic, hence itis an isomorphism. Thus, X ≃ ` j ∈ J X j has power at most card J < λ .Conversely, if X has power less than λ , then it is λ -presentable because every coproductof less than λ objects which are λ -presentable is λ -presentable. Remark 3.8. (a) In a strictly locally λ -presentable category every λ -generated object is λ -presentable.This was proved for λ = ℵ in [8], Remark 3.8, the general case is analogous.(b) In every locally λ -presentable category K all hom-functors of λ -presentable objectscollectively reflect λ -filtered colimits. That is, given a λ -filtered diagram D with objects D i ( i ∈ I ), then a cocone c i : D i → C of D is a colimit iff for every λ -presentable object Y the following holds:(i) every morphism f : Y → C factorizes through some c i and(ii) given two such factorizations u , v : Y → C , c i · u = c i · v , there exists a connectingmorphism d ij : D i → D j of D with d ij · u = d ij · v .This is stated as an exercise in [10], and proved for λ = ℵ in [8], Lemma 2.7.(c) Let D be a full subcategory of K representing all λ -presentable objects up to isomor-phism. It follows from (b) that every object K ∈ K is a colimit of the λ -filtered diagramof all morphisms a : A → K with A ∈ D . More precisely, colimit of the forgetful functor ofthe slice category, D : D /K → K with the canonical cocone given by all a ’s. Theorem 3.9. Let α be a regular cardinal such that K is a strictly locally α -presentablecategory with a strict width. Then K ≤ λ is algebraically complete for every cardinal λ ≥ max( α, w ( K )) .Proof. Following Remark 3.1, it is sufficient to prove that K ≤ λ has colimits of i -chains forall ordinals i ≤ λ , and every endofunctor of K ≤ λ preserves colimits of λ -chains.(1) We first prove that K ≤ λ is closed in K under strong quotients e : B → C . Thus,assuming that B has power at most λ , we prove the same about C . For every i ∈ I the component e i : B i → C of e factorizes, by Proposition 2.11 through the subcoproduct ` j ∈ J ( i ) C j for some J ( i ) ⊆ J of power less than λ . The union K = S i ∈ I J ( i ) has power atmost λ = λ , and e factorizes through the coproduct injection v k : ` j ∈ K C j → C . Since e isa strong epimorphism, so is v K . But being a coproduct injection, v K is also monic. Thus v K is an isomorphism, proving that C = ` j ∈ K C j has power at most card K ≤ λ .(2) K ≤ λ has for every limit ordinal i ≤ λ colimits of i -chains ( B j ) j
A . The functor B : λ → D (cid:14) X given by i ( B i , b i ) is cofinal, i.e., for every object ( A, a )of D (cid:14) X (a) there exists a morphism of D (cid:14) X into some ( B i , b i ) and (b) given a pair ofmorphisms u , v : ( A, a ) → ( B i , b i ) there exists j ≥ i with u and v merged by the connectingmorphism b ij : B i → B j of our chain. Indeed, since A is α -presentable and λ ≥ α , themorphism a : A → colim i<λ B i factorizes through b i for some i < λ . And since u , v above fulfil b i · u = b i · v (= a ), some connecting morphism b ij also merges u and v .Consequently, in order to prove that F preserves the colimit X = colim B i , it is sufficientto verify that it preserves the colimit of D , where D : D (cid:14) X → K ≤ λ is the codomainrestriction of D above: since B : λ → D (cid:14) X is cofinal, the colimits of the diagrams F.D and( F B i ) i<λ coincide. We apply Remark 3.8(b) with α in place of λ , and verify the conditions(i) and (ii) for the cocone F a : F A → F X of F.D (in K ). Thus F X = colim F D in K whichimplies F X = colim F D in K ≤ λ .Ad (i) Given a morphism f : Y → F X with Y α -presentable, then Y has by Lemma 3.7power less than α , thus, by Proposition 2.9 there exists a coproduct injection c : C → X with C α -presentable such that f factorizes through F c (which is a member of our cocone).Ad (ii) Let u , v : Y → F A , with A α -presentable, fulfil F a · u = F a · v . We are to finda connecting morphism h : ( A, a ) → ( B, b ) in D (cid:14) X with F h · u = F h · v . By the strictness of K , since A is α -presentable, for a : A → B thereexist morphisms b : B → X and f : X → B with B α -presentable and a = b · f · a . It is sufficient to put h = f · a : A → B . Then F a · u = F b · v implies F h · u = F h · v , as desired. Example 3.10. (1) The category Set ≤ℵ of countable sets is algebraically complete, but not alge-braically cocomplete (Example 2.15).(2) For every uncountable regular cardinal λ the category Set ≤ λ is algebraically com-plete (by Theorem 3.9) and, assuming (GCH), algebraically cocomplete (by Theorem 2.14).The former was already proved in [5], Example 14, using an entirely different method.(3) For non-regular uncountable cardinals λ the category Set ≤ λ need not be alge-braically cocomplete, see Example 5.1. Example 3.11. Let λ be a regular uncountable cardinal. The following categories arealgebraically complete and, assuming (GCH), algebraically cocomplete:(a) Set S ≤ λ whenever λ > | S | ,(b) K - Vec ≤ λ whenever λ > | K | ,(c) Nom ≤ λ , and(d) G - Set ≤ λ for groups G with λ > | G | .This follows from Theorems 2.14 and 3.9.4. Precontinuous Set Functors and ultrametrics In the preceding parts we have presented categories in which every endofunctor F has aninitial algebra µF and a terminal coalgebra νF . The category of sets is, of course, not oneof them. However, for set functors we know that µF exists whenever νF does, see [22]. Weobserve below that µF , as a coalgebra, is a subcoalgebra of νF . Moreover, we introducea wide class of set funcors we call precontinuous and for them we will see, assuming GCH,that νF carries a canonical ultrametrics such that, whenever F ∅ 6 = ∅ , (1) the ultrametric subspace µF has the same Cauchy completion as νF and(2) the coalgebra structure τ : νF → F ( νF ) is the unique continuous extension of ι − , theinverted algebra structure ι : F ( µF ) → µF .Thus one can say that the terminal coalgebra is determined, via its ultrametric, by theinitial algebra.For finitary set functors which also preserve limits of ω op -sequence Barr proved more: νF is a complete space which is the Cauchy completion of µF , see [12]. Proposition 4.1 ( µF as a subcoalgebra of νF ) . If a set functor F has a terminal coalgebra,then it also has an initial algebra carried by a subset µF ⊆ νF such that the inclusion map m : µF ֒ → νF is the unique coalgebra homomorphism µF ι − / / m (cid:15) (cid:15) F ( µF ) F m (cid:15) (cid:15) νF τ / / F ( νF ) Proof. (1) Assume first that F preserves monomorphisms. Form the unique cocone of theinitial-algebra chain (see Remark 3.1) with codomain νF , m i : F i → νF ( i ∈ Ord)satisfying the recursive rule m i +1 ≡ F ( F i F m i −−−−→ F ( νF ) τ − −−−→ νF ( i ∈ Ord) . Easy transfinite induction verifies that m i is monic for every i . Since νF has only a setof subobjects, there exists an ordinal λ such that all m i with i ≥ λ represent the same LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 17 subobject. Thus the commutative triangle below W λ w λ,λ +1 / / m λ ! ! ❉❉❉❉❉❉❉❉ F W λm λ +1 { { ①①①①①①①① νF implies that w λ,λ +1 is invertible. Consequently, the following algebra F ( F λ w − λ,λ +1 −−−−−→ F λ m λ : F λ → νF put µF = m λ [ F λ ⊆ νF . Choose an isomorphism r : µF → F λ m = m λ · r : µF → νF is the inclusion map.Then there exists a unique algebra structure ι : F ( µF ) → µF for which r is an isomorphismof algebras: r : ( µF, ι ) ∼ −−→ ( F λ , w − λ,λ +1 ) . The following commutative diagram µF ι − / / r (cid:15) (cid:15) F ( µF ) F r (cid:15) (cid:15) F λ w λ,λ +1 / / m λ (cid:15) (cid:15) F ( F λ m λ +1 x x qqqqqqqqqqqqqqqq F m λ (cid:15) (cid:15) νF τ / / F ( νF )proves that m = m λ · r is the unique coalgebra homomorphism, as required.(2) Let F be arbitrary. We can assume F ∅ 6 = ∅ . Our proposition holds for the functor G in Remark 3.1(e). Since F and G agree an all nonempty sets and F ∅ 6 = ∅ 6 = G ∅ , theyhave the same terminal coalgebras. Since the initial algebra of G is, as we have just seen,obtained via the initial-algebra chain, and F has from ω onwards the same initial-algebrachain, F and G have the same initial algebras. Thus, our proposition holds for F too.Barr [12] calls a functor continuous if it preserves limits of ω op -chains. Below we call acone of an ω op -chain a prelimit if every other cone has a most one factorization through it.That is, a prelimit is a collectively monic cone. Definition 4.2. A set functor is called precontinuous if it preserves nonempty prelimits of ω op -chains. Example 4.3. (1) All finitary set functors are precontinuous. To verify this, we can restrictourselves to set functors F preserving finite intersections and inclusion and distinct from C ,the constant functor of value ∅ . In fact, the case C is trivial, and for every other functoruse G of Remark 3.1(e): since F is fnitary, so is G . Let a n : A n +1 → A n be an ω op -chain with a prelimit b n : B → A n . Given distinctelements x, x ′ of F B , we are to find n with F b n ( x ) = F b n ( x ′ ) . Since F is finitary and preserves inclusion, there is a finite subset C ⊆ B with x, x ′ lying in F C . The cone ( b n ) is collectively monic, thus there exists n such that the restriction of b n to C is monic. F preserves (finite intersections, thus) monomorphisms, hence F b n has thedesired propery: its restriction to F C is monic.(2) All continuous set functors are of course precontinuous. Example: polynomialfunctors for infinitary signatures.(3) Composites, products and coproducts of precontinuous functors are clearly precon-tinuous.(4) Subfunctors of a precontinuous functor F are precontinuous. Indeed, let µ : G → F be a natural mono-transformation. And let a n : A n +1 → A n be an ω op -chain with a prelimit b n : B → A n . Since the cone ( F b n ) is collectively monic, so is (( F b n ) · µ B ). By naturalitythe last cone is ( µ A n · Gb n ). Thus also ( Gb n ) is collectively monic, as required.(5) The functor D of discrete probability distributions is precontinuous. Recall thata discrete probability distribution on a set X is given by a function p : X → [0 , 1] such P x ∈ X p ( x ) = 1. (Thus all p ( x ) but countably many are 0.) This extends to a probabilitydistribution on X by µ ( M ) = P m ∈ M p ( m ) for all M ⊆ X .The functor D assing to a set theset of its discrete probability distributions. Given a function f : X → Y , then Df assignsto a distribution µ on X the distribution M µ ( f − ( M )) for all M ⊆ Y . For an argumentwhy D preserves prelimits of ω op -chains see [23], Example 15.(6) The functor F X = ( DX ) A is precontinous: it is a composite of D and the continuousfunctor ( − ) A . Its coalgebras are probabilistic labelled transition systems with the set A ofactions. Remark 4.4. Restricting ourselves to nonempty prelimits in the above definition is needed:finitary functors do not preserve prelimits of ω op -chains in general. Consider the functor C , taking the empty set to 2 and all other sets to 1.The following proof is based on ideas of Worrell [23]. Recall the connecting maps v i,j of the terminal-coalgebra chain from Remark 3.1(d). Theorem 4.5. For every precontinuous set functor F the terminal-coalgebra chain con-verges in δ ≥ ω steps and the connecting map v δ,ω : νF → F ω is monic.Proof. (1) We can restrict ourselves to precontinuous set functors preserving finite intersec-tions (thus preserving monomorpihsms) and inclusion and distinct from C , the constantfunctor of value ∅ . In fact, the theorem is trivial for C , and for every other set functor F all ordinals i fulfil F i = ∅ by [23], Lemma 6. Thus the functor G of Remark 3.1(e) has thesame terminal-coalgebra chain as F . And it is precontinuous, since F is.(2) The connecting morphisms v ω,n : F ω → F n n < ω from a prelimit that F preserves: the cone of F v ω,n = v ω +1 ,n +1 is collectively monic. Thus the factorizingmorphism v ω +1 ,ω of that cone is monic: recall from Remark 3.1 that is it defined by thecomposites v ω +1 ,ω · v ω,n +1 = F v ω,n for all n < ω . LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 19 It follows that for every infinite ordinal i the connecting map v i,ω is monic. Let us verifyit by transfinite induction. We have seen that this holds for ω + 1. If this holds for i it holdsalso for i + 1: We know that F v i,ω = v i +1 ,ω +1 is monic, hence so is v i +1 ,ω = v ω +1 ,ω · v i,ω +1 ,as required. Limit steps are trivial: a limit cone of a chain of monomorphisms has moniclimit projections. And soince i > ω , the limit does not change when the first ω steps aredeleted from the chain.Since the set F ω v i , ω stops after δ for some ordinal δ , as claimed. Remark 4.6. (1) For a precontinuous functor F the initial algebra exists and has theform µF = F i i ≥ δ (Remark 3.1(b)). Thus we have the subobject m : F i → F δ m = v δ +1 ,δ · F m · w i,i +1 . (2) All the examples of the precontinuous functors in 4.3 preserve nonempty countableintersetctions. (For finitary functors the verification is analogous to the proof of Proposition10 in [23].) It then follows that we can choose δ = ω + ω. Indeed, the limit defining F ω + ω v ω + i,ω for i < ω .Thus F preserves that limit which means that the terminal-coalgebra chain converges in ω + ω steps.(3) Recall the morphism ¯ u from Remark 3.1(f): Lemma 4.7. Every precontinuous functor fulfils ¯ u = v δ,ω · m · w ω,i . Proof. We prove that the squares defining ¯ u in 3.1(f) commute when we substitute theright-hand side of our equation for ¯ u . That is, we prove v ω,n · [ v δ,ω · m · w ω,i ] · w n,ω = F n ! : F n → F n v δ,n · m · w n,i = F n ! : F n → F n n = 0 is clear. Assuming the equation holds for n weprove it for n + 1. The previous remark yields v δ,n +1 · m · w n +1 ,i = v δ,n +1 · v δ +1 ,δ · F m · w i,i +1 · w n +1 ,i . This simplifies to v δ +1 ,n +1 · F m · w n +1 ,i +1 = F v δ,n · F m · F w n,i which is equal to F n +1 !: apply F to the induction hypothesis. Remark 4.8. For every precontinuous functor νF is a canonical subset of F ω v δ,ω .And this endows νF with a canonical ultrametric, as our next lemma explains. Recall thata metric d is called an ultrametric if for all elements x , y , z the triangle inequality can bestrengthened to d ( x, z ) ≤ max( d ( x, y ) , d ( y, z )). Lemma 4.9. Every limit L of an ω op -chain in Set carries a complete ultrametric: assign to x = y in L the distance − n where n is the least natural number such that the correspondinglimit projection separates x and y . Proof. Let l n : L → A n ( n ∈ N ) be a limit cone of an ω op -chain a n : A n +1 → A n ( n ∈ N ).For the above function d ( x, y ) = 2 − n where l n ( x ) = l n ( y ) and n is the least such number we see that d is symmetric. It satisfiesthe ultrametric inequality d ( x, z ) ≤ max (cid:0) d ( x, y ) , d ( y, z ) (cid:1) for all x, y, z ∈ L . This is obvious if the three elements are not pairwise distinct. If they are, the inequalityfollows from the fact that if l n separates two elements, then so do all l m with m ≥ n .It remains to prove that the space ( L, d ) is complete. Given a Cauchy sequence x r ∈ L ( r ∈ N ), then given k ∈ N there exists r ( k ) ∈ N with d ( x r ( k ) , x n ) < − k for every n ≥ r ( k ) . Choose r ( k )’s to form an increasing sequence. Then d ( x r ( k ) , x r ( k +1) ) < − k , i.e., l k ( x r ( k ) ) = l k ( x r ( k +1) ). Therefore, the elements y k = l k ( x r ( k ) ) are compatible: we have a k +1 ( y k +1 ) = y k for all k ∈ N . Consequently, there exists a unique y ∈ L with l k ( y ) = y k for all k ∈ N . Thatis, d ( y, x r ( k ) ) < − k . Thus, y is the desired limit: y = lim k →∞ x r ( k ) implies y = lim n →∞ x n . We conclude that for a finitary set functor both νF and µF carry a canonical ultra-metric: νF as a subspace of F ω v ω + ω,ω : νF → F ω 1, and µF as a subspace of νF via m . Given t = s in νF we have d ( t, s ) = 2 − n for the least n ∈ N with v ω + ω,n ( t ) = v ω + ω,n ( s ).The isomorphism τ : νF ∼ −−→ F ( νF ) then makes F ( νF ) also a canonical ultrametric space,analogously for F ( µF ). Notation 4.10. Given a precontinuous set functor F with F ∅ 6 = ∅ , choose an element p : 1 → F ∅ . This defines the following morphisms for every n ∈ N : e n ≡ F n F n p −−−−→ F n +1 w n +1 ,ω −−−−−−→ F ω ¯ u −−→ F ω ε n = e n · v ω,n : F ω → F ω . Observation 4.11. (a) For every n ∈ N we have a commutative square below F n F n p / / F n +1 F n +1 ! (cid:15) (cid:15) F n F n +1 v n +1 ,n o o This is obvious for n = 0. The induction step just applies F to the given square.(b) v ω,n · e n = id F n . LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 21 Indeed, in the following diagram F n F n p / / F n +1 w n +1 ,ω / / F n +1 ! ( ( PPPPPPPPPPPP F ω ¯ u / / F ω v ω,n (cid:15) (cid:15) v ω,n +1 w w ♦♦♦♦♦♦♦♦♦♦♦♦ F n +1 v n +1 ,n ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ F n F n u , see Remark 3.1(f), the left-handone does by (a), and the lower right-hand triangle is clear.(c) v ω,n · ε n = v ω,n .This follows from (b): precompose it with v ω,n .(d) ε n · ε n +1 = ε n . Indeed, we have ε n · ε n +1 = ¯ u · w n +1 ,n · F n p · v ω,n · ε n +1 def. of ε n = ¯ u · w n +1 ,n · F n p · ( v n +1 · v ω,n +1 ) · ε n +1 = ¯ u · w n +1 ,n · F n p · v n +1 · v ω,n +1 · ε n +1 by (c)= ¯ u · w n +1 ,n · F n p · v ω,n = ε n . Theorem 4.12. For a precontinuous set functor F with F ∅ 6 = ∅ the Cauchy completions ofthe ultrametric spaces µF and νF coincide. And the algebra structure ι of µF determinesthe coalgebra structure τ of νF as the unique continuous extension of ι − .Proof. Assume first that F preserves inclusion (Remark 3.1).(a) We prove that the subset¯ u = v δ,ω · m · w ω,i : µF → F ω F ω 1, thus, the complete space F ω u [ µF ] and v δ,ω [ νF ].For every x ∈ F ω ε n ( x ) lies in the image of e n · v ω,n which, in view of thedefinition of e n , is a subset of the image of ¯ u . And we have x = lim n →∞ ε n ( x )because Observation 4.11 (c) yields v ω,n ( x ) = v ω,n ( ε n ( x )), thus d (cid:0) x, ε n ( x ) (cid:1) < − n for all n ∈ N . (b) The continuous map ι − has at most one continuous extension to νF . And τ issuch an extension: it is not only continuous, it is an isometry. And it extends ι − byProposition 4.1: µF ι − / / _(cid:127) m (cid:15) (cid:15) F ( µF ) _(cid:127) F m (cid:15) (cid:15) νF τ / / F ( νF )Since F m is an inclusion map, τ is an extension of ι − . Once we have established (a) and (b) for inclusion-preserving finitary functors, it holdsfor all finitary set functors F . Indeed, there exists an inclusion-preserving set functor G thatagrees with F on all nonempty sets and functions, and fulfils G ∅ 6 = ∅ , see Remark 3.1(e).Consequently, the coalgebras for F and G coincide. And the initial-algebra chains coincidefrom ω onwards, in particular, we can assume F i G i 0, that is, F and G have the sameinitial algebra. Example 4.13. For the finite power-set functor P f the initial algebra can be described as µ P f = all finite extensional trees,(where trees are considered up to isomorphism). Recall that a tree is called extensional if for every node x the maximum subtrees of x are pairwise non-isomorphic. And it iscalled strongly extensional if it has no nontrivial tree bisimulation. (A tree bisimulation isa bisimulation R on the tree which (a) relates the root with itself, (b) relates only verticesof the same height.) For finite trees these two concepts are equivalent. Worrell proved in[23] that the terminal coalgebra can be described as follows: ν P f = all finitely branching strongly extensional treeswith the coalgebra structure inverse to tree tupling. Whereas the terminal-coalgebra chainyields P ωf P ωf t = s the distance d ( t, s ) = 2 − n , where n is the leastnumber with ∂ n t = ∂ n s . Here ∂ n t is the extensional tree obtained from t by cutting it atlevel n and forming the extensional quotient of the resulting tree. Corollary 4.14. Assuming GCH, let F be a precontinuous set functor whose initial algebrahas an uncountable regular cardinality. Then the terminal coalgebra has the same cardinality.Shortly, µF ≃ νF . Indeed, let λ = | µF | . The Cauchy completion of µF has power at most λ ω = λ (see Remark 2.13(d)), thus, | νF | ≤ λ by the above theorem. And | νF | ≥ λ follows fromProposition 4.1. Remark 4.15. The above corollary does not extend to ℵ : The finitary set functor F X = X × Example 4.16. Let Σ be a (possibly infinitary) signature with at least one nullary symbol.We choose one, ⊥∈ Σ . Recall that a Σ -tree is an ordered tree labelled in Σ so that a nodewith a label σ ∈ Σ n has precisely n successor nodes. We consider these trees again up toisomorphism. A Σ-tree is called well-founded if every branch of it is finite.By Theorem II.3.7 of [11] the polynomial functor H Σ X = ` Σ n × X n has the initialalgebra µH Σ = all well-founded Σ-treesand the terminal coalgebra νH Σ = all Σ-trees. LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 23 Moreover, H Σ preserves limits of ω op -chains, hence, νH Σ = H ω Σ 1. The complete ultrametricon νH Σ assigns to Σ-trees t = s the distance d ( t, s ) = 2 − n , where n is the least numberwith ∂ n t = ∂ n s . Here ∂ n t is the Σ-tree obtained cutting t at height n and relabelling allleaves of height n by ⊥ . We conclude that νH Σ is the Cauchy completion of µH Σ . Consequently, if the set of all well-founded trees has a regular uncountable cardinality, thenthis is also the cardinality of the set of all Σ-trees.The preceding example only works if Σ = ∅ . In case Σ = ∅ we have H Σ ∅ = ∅ andTheorem 4.7 does not apply.5. Precontinuous Set Functors and CPO’s Analogously to the preceding section we prove that given a precontinuous set functor F with F ∅ 6 = ∅ both νF and µF carry a canonical partial order with a common ideal cpo -completion.Again, the coalgebra structure τ : νF → F ( νF ) is the unique continuous extension of ι − . Notation 5.1. Given a precontinuous set functor F with F ∅ 6 = ∅ choose an element p : 1 → F ∅ . This defines a partial order ⊑ on the set F ω t = s in F ω 1, put t ⊑ s if t = ε n ( s ) for some n ∈ N (see Notation 4.10). Theorem 5.2 ([1, Theorem 3.3]) . ( F ω , ⊑ ) is a cpo , i.e., every directed subset has a join.Moreover, every strictly increasing ω -chain in F ω has a unique upper bound. The theorem in [1] is formulated for continuous functors for which the statement isthat νF is a cpo . However, the proof remains the same for precontinuous functors if weformulate the result as above. Example 5.3. (1) For P f , given strongly extensional trees t = s , then t ⊑ s iff t = ∂ n s for some n .(2) The functor F X = Σ × X + 1 has a terminal coalgebra νF = F ω ∗ + Σ ω . Forwords t = s we have t ⊑ s iff t is a prefix of s. (3) More generally, given a signature Σ, for Σ-trees t = s we have t ⊑ s iff t is a cuttingof s , i.e., t = ∂ n s for some n (see Example 4.16.We conclude that for a precontinuous set functor F with F ∅ 6 = ∅ both νF and µF carrya canonical partial order: νF as a subposet of F ω v δ,ω and µF as a subposet of νF via m . For t = s in νF we have t ⊑ s iff v δ,ω ( t ) = v δ,ω · ε n ( s ) for some n ∈ N . This partial order depends on the choice of an element p of F ∅ . Remark 5.4. Recall the concept of ideal completion of a poset P. This is a cpo ¯ P containing P as a subposet such that for every monotone function f : P → Q , where Q is a cpo , thereexists a unique continuous extension f : ¯ P → Q . Theorem 5.5. For a finitary set functor F with F ∅ 6 = ∅ the ideal completions of the posets µF and νF coincide. And the algebra structure ι of µF determines the coalgebra structure τ of νF as the unique continuous extension of ι − .Proof. As in the proof of Theorem 4.12, we can assume that F is preserves inclusion. Weprove that F ω µF and νF .(1) Given x ∈ F ω 1, we find an ω -sequence in µF with join x . In fact, the sequence ε n ( x ) lies in the image of ¯ u : µF ֒ → F ω ε n ). This is an ω -sequence:for every n ∈ N we have ε n ( x ) ⊑ ε n +1 ( x ) ⊑ x . Recall from Theorem 5.2 that strict ω -sequences have unique upper bounds in F ω 1, thus, x = G n ∈ N ε n ( x ) . (2) Let D ⊆ F ω x = F D . If x / ∈ D , then D is cofinal with thesequence ε n ( x ), n ∈ N . Indeed, every d ∈ D fulfils d ⊑ x , that is, d = ε n ( x ) for some n .The rest follows from (1).(3) F ω µF . Indeed, let Q be a cpo and f : µF → Q acontinuous function. Extend it to F ω x ∈ F ω − ¯ u [ µF ]¯ f ( x ) = [ n ∈ N f (cid:0) ε n ( x ) (cid:1) . This map is continuous due to (2), and the extension is unique due to(1).(4) F ω νF . The argument is analogous.(5) The proof that τ is the unique continuous extension of ι − is completely analogousto the metric case in Theorem 4.12.6. Unexpected Finitary Endofunctors We know from Theorem 3.9 that all endofunctors of Set ≤ λ preserve colimits of λ -chains.In particular, all endofunctors of Set ≤ ω (the category of countable sets) are finitary. Nev-ertheless, this category is not algebraically cocomplete, as shown in Example 2.15. In thissection we turn to singular cardinals, e.g. ℵ ω = W n<ω ℵ n with countable cofinalities (see Re-mark 2.13). The category Set ≤ℵ ω is also not algebraically cocomplete, as we demonstratein the next example. We are going nonetheless to prove that all endofunctors of Set ≤ℵ ω arefinitary, i.e., they preserve all existing filtered colimits. Below we use ideas of Section 4.6 of[11]. Here is a surprisingly simple functor which is finitary but has no terminal coalgebra: Example 6.1. The endofunctor F X = ℵ ω × X of Set ≤ℵ ω (a coproduct of ℵ ω copies of Id) does not have a terminal coalgebra. We use thefact that (cid:0) ℵ ω (cid:1) ω > ℵ ω LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 25 see Remark 2.13(b). The proof that νF does not exist in Set ≤ℵ ω is similar to that ofExample 2.15. Suppose that τ : T → ℵ ω × T is a terminal coalgebra. For every function f : N → ℵ ω define a coalgebra ˆ f : N → ℵ ω × N byˆ f ( n ) = (cid:0) f ( n ) , n + 1 (cid:1) for n ∈ N . We obtain a unique homomorphism h f : N h f (cid:15) (cid:15) ˆ f / / ℵ ω × N id × h f (cid:15) (cid:15) T τ / / ℵ ω × T We claim that for all pairs of functions f , g : N → ℵ ω we have h f (0) = h g (0) implies f = g . Indeed, since τ · h f (0) = τ · h g (0), the square above yields (cid:0) f (0) , h f (1) (cid:1) = (cid:0) g (0) , h g (1) (cid:1) , in other words h f (1) = h g (1) and f (0) = g (0) . The first of these equations yields, via the above square, h f (2) = h g (2) and f (1) = g (1) , etc. This is the desired contradiction: since for all functions f : N → ℵ ω the elements h f (0)of T are pairwise distinct, we get | T | ≥ |ℵ N ω | > ℵ ω . Remark 6.2. (a) Recall that a filter on a set X is a collection of nonempty subsets of X closed upwardsand closed under finite intersections. The principle filter is one containing a singleton set.Maximum filters are called ultrafilters and are characterized by the property that for every M ⊆ X either M or its complement is a member.(b) Recall further that a cardinal α is measurable if there exists a nonprinciple ultrafilteron α closed under intersections of less than α members. Every measurable cardinal isinaccessible, see e.g. [15], Lemma 5.27.2. Therefore, the assumption that no measurablecardinal exists is consistent with ZFC set theory. This also implies that every measurablecardinal is larger than ℵ ω .(c) To say that a cardinal α is not measurable is equivalent to saying that for everynonprinciple ultrafilter on α contains members U , U , U , . . . such that T k<ω U k is no mem-ber. Theorem 6.3. Let λ be a cardinal of cofinality ω such that no measurable cardinal is smalleror equal to λ (e.g. λ = ℵ ω ). Then every endofunctor of Set ≤ λ preserves filtered colimits.Proof. (1) Set ≤ λ has colimits of λ -chains and every endofunctor preserves them. Indeed, let a chain with objects A i and connecting morphisms a i,j ( i ≤ j < λ ) be givenin Set ≤ λ . If a i : A i → A ( i < λ ) is a colimit in Set , then this is also a colimit in Set ≤ λ because | A | ≤ X i<λ | A i | ≤ λ = λ . For every endofunctor F we prove that ( F a i ) i<λ is also a colimit cocone in Set (thusin Set ≤ λ ). This is equivalent to proving that the cocone has properties (i) and (ii) ofRemark 3.8. That is(i) F A is the union of the images of F a i ,and(ii) given i < λ , every pair of elements of F A i that F a i merges is also merged by F a ij forsome connecting morphism a i,j : A i → A j .For (i) use Proposition 2.11 which, by Remark 2.12, applies to nonregular cardinals:given b ∈ F A there exists a subset c : C ֒ → A with | C | < λ such that b ∈ F c [ F C ]. It followsthat the subset fulfils c ⊆ a i for some i < λ , hence b ∈ F a i [ F A i ].For (ii), let x , x ∈ F A i fulfil F a i ( x ) = F a i ( x ). As above, there exists a subset c : C ֒ → A i with | C | < λ such that x , x ∈ F c [ F C ]. Since | C × C | < λ , we can find anordinal j with i ≤ j < λ such that every pair in C × C merged by a i is also merged by a i,j .In other words ker a i · c ⊆ ker a i,j · c . Consequently, a i · c factorizes through a ij · c : C _(cid:127) c (cid:15) (cid:15) A i a i,j / / a i (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ A ja j ~ ~ ⑦⑦⑦⑦⑦⑦⑦ A f > > ⑦⑦⑦⑦ From a ij · c = f · a i · c we derive, given y k ∈ F C with x k = F c ( y k ), that F a i,j · ( x ) = F a i,j (cid:0) F c ( y ) (cid:1) = F f (cid:0) F a i ( x ) (cid:1) and analogously for x . Thus, F a i ( x ) = F a i ( x ) implies F a i,j ( x ) = F a i,j ( x ), as desired.(2) Set ≤ λ is closed under existing filtered colimits in Set . It is sufficient to prove thisfor existing directed colimits due to Theorem 5 of [10]. Let D : ( I, ≤ ) → Set ≤ λ be a directeddiagram with a colimit a i : A i → A ( i ∈ I ) in Set ≤ λ . We prove properties (i) and (ii) ofRemark 3.8.Ad(i): for the union m : A ′ ֒ → A of images of all a i we are to prove A ′ = A . Factorize a i = m · a ′ i for a ′ i : A i → A ′ ( i ∈ I ), then ( a ′ i ) i ∈ I is clearly a cocone of D . Thus, we have f : A → A ′ with a ′ i = f.d i ( i ∈ I ). From ( m.f ) .a i = a i for all i ∈ I we deduce m.f = id,thus, A ′ = A .Ad(ii): given i ∈ I and elements x , x ′ ∈ A merged by a i , we prove that they are mergedby some connecting map a i,k : A i → A k of D ( k ∈ i ). Without loss of generality we assumethat i is the least element of I (if not, restrict D to the upper set of i which yields a directed LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 27 diagram with the same colimit). For every k ≥ i put x k = a i,k ( x ) and define, given j ∈ J ,a map b j : A j → A + { t } in an element y ∈ A j as follows b j ( y ) = ( t if a j,k ( x j ) = a j,k ( y ) for some k ≥ ja j ( y ) elseSince ( I, ≤ ) is directed, it is easy to see that this yields a cocone of D . We have a unique f : A → A + { t } with b i = f.a i ( i ∈ I ). Clearly b i ( x ) = t , therefore, a i ( x ) = a i ( x ′ ) implies b i ( x ′ ) = t . This proves a i,k ( x ) = a i,k ( x ′ ) for some k ≥ i , as desired.(3) To prove the theorem, it is sufficient to verify for every endofunctor F of Set ≤ λ and every object X that every element a ∈ F X in the image of F y for some finite subset y : Y ֒ → X . Indeed, since filtered colimits are, due to (2), formed as in Set , the fact that F preserves them is then proved completely analogously to (1) above.First, F preserves colimits of ω -chains: given an ω -chain D : ω → Set ≤ λ with objects D i for i < ω and given cardinals λ < λ < λ . . . with λ = W n<ω λ n , we define a λ -chain¯ D : λ → Set ≤ λ by assigning to i < λ the value ¯ D i = D and to every i with λ n ≤ i < λ n +1 the value ¯ D i = D n +1 . (Analogously for the connecting morphisms). Then D is cofinal in¯ D (and F D cofinal in F ¯ D ), thus, they have the same colimits. Since F preserves colim ¯ D ,it also preserves colim D .(4) For every element a ∈ F X we are going to prove that there exists a finite subset y : Y ֒ → X with a ∈ F y [ F Y ].Denote by F the collection of all subsets y : Y ֒ → X for which a ∈ F y [ F Y ]. We are toprove that F contains a finite member. Put Z = \ Y ∈ F Y with inclusion z : Z ֒ → X . (a) First, suppose Z ∈ F . We prove that Z is finite, which concludes the proof.In case Z is infinite, we derive a contradiction. Express Z as a union of a strictlyincreasing ω -chain Z = S n<ω Z n . Obviously Z n / ∈ F (due to Z n $ Z ). But then F doesnot preserve the union z = S z n of the corresponding inclusion maps z n : Z n ֒ → X since a ∈ F z [ F Z ] but a / ∈ F z n [ F Z n ], a contradiction.(b) Suppose Z / ∈ F . We prove in part (c) that F contains sets V , V with V ∩ V = Z .Since for the inclusion maps v i : V i → X we have a ∈ F v i [ F V i ] but a / ∈ F z [ F Z ], we see that F does not preserve the intersection z = v ∩ v . Thus, Z = ∅ , see Corollary 2.8. Then wechoose t ∈ X and prove { t } ∈ F , concluding the proof. Let h : X → X be defined by h ( x ) = ( x x ∈ V t else . That is, since V and V are disjoint, for the constant function k : X → X of value t we have hv = v and hv = kv . Then we compute, given a i ∈ F V i with a = F v i ( a i ): F k ( a ) = F ( kv )( a ) = F ( hv )( a ) = F h ( a ) as well as a = F v ( a ) = F ( hv )( a ) = F h ( a ) . Consequently, F k ( a ) = a , and since for the inclusion y : { t } → X we have the followingcommutative triangle { t } y (cid:15) (cid:15) X ! ; ; ✈✈✈✈✈✈✈✈✈ k / / X this yields a ∈ F y [ F { t } ], as required.(c) Assuming that V , V ∈ F implies V ∩ V = Z , we derive a contradiction. Put X = X − Z and define a filter F on X by Y ∈ F iff Z ∪ Y ∈ F . F is closed under super-sets (since F is) and does not contain ∅ (since Z / ∈ F ). We verifythat it is closed under finite intersection. Given Y , Y ∈ F , then by our assumption above:( Z ∪ Y ) ∩ ( Z ∪ Y ) = Z . This implies that the sets V i = Z ∪ Y i are not disjoint. Thus, F preserves the intersection v ∩ v , see Corollary 2.8. Consequently, a lies in the F -image of v ∩ v , i.e., V ∩ V ∈ F .Since Y ∩ Y = Z ∪ ( V ∩ V ), this yields Y ∩ Y ∈ F .By the Maximality Principle the filter F is contained in an ultrafilter U . This ultrafilteris nonprincipal: for every x ∈ X we know (from our assumption Z / ∈ F ) that { x } / ∈ F ,hence, { x } / ∈ U . Since card X is not measurable, there exists by Remark 6.2(c) a collection U n ∈ U ( n < ω ) with \ n<ω U n / ∈ U . We define an ω -chain S k ⊆ X − Z ( k < ω ) of sets that are not members of U by the followingrecursion: S = \ n<ω U n . Given S k , put S k +1 = S k ∪ ( X − U k ) . Assuming S k / ∈ U then, since we know that X − U k / ∈ U , we get S k +1 / ∈ U by the fact that U is an ultrafilter.We observe that X = [ k<ω S k . Indeed, every element x ∈ X either lies in S or in its complement S n<ω ( X − U n ). In thelatter case we have x ∈ X − U k ⊆ S k for some k . We achieved the desired contradiction: F does not preserve the ω -chain colimit X = colim k<ω ( Z ∪ S k ). Indeed, Z ∪ S k / ∈ F , thus, for itsembedding s k : Z ∪ S k ֒ → X we have a / ∈ F s k [ F ( Z ∪ S k )]. LGEBRAIC COCOMPLETENESS AND FINITARY FUNCTORS 29 The argument used in point (1) of the above proof is analogous to that of Theorem 3.10of [8]. 7. Conclusions and Open problems We have presented a number of categories that are algebraically complete and cocomplete,i.e., every endofunctor has a terminal coalgebra and an initial algebra. Examples include(for sufficiently large regular cardinals λ ) the category Set ≤ λ of sets of power at most λ , Nom ≤ λ of nominal sets of power at most λ , K - Vec ≤ λ of vector spaces of dimension atmost λ , and G - Set ≤ λ of G -sets (where G is a group) of power at most λ .All these results assumed the General Continuum Hypothesis. It is an open questionwhat could be proved without this assumption.We have introduced the concept of a precontinuous set functor encompassing all finitaryones, all continuous ones and composites, products and coproducts of those. For thesefunctors F with F ∅ 6 = ∅ we have presented a sharper result: both µF and νF carry acanonical partial ordering and these two posets have the same conservative cpo -completion.Moreover, by inverting the algebra structure of µF we obtain the coalgebra structure of νF as the unique continuous extension. In place of posets and cpo ’s, we have also got the sameresult for ultrametric spaces and their Cauchy completions.For cardinals λ of countable cofinality we have proved that the category Set ≤ λ is notalgebraically complete, but every endofunctor is finitary. The proof used the set-theoreticalassumption that no cardinal smaller or equal to λ is measurable. 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