Journal of Pure and Applied Algebra 46 (1987) 35-47 35 North-Holland COPRODUCTS AND ULTRAFILTERS Reinhard BORGER Fachbereich Mathematik und lnformatik, Fernuniversitiit, D-5800 Hagen 1, Fed. Rep. Germany Communicated by G.M. Kelly Received 28 February 1985 Revised 1 November 1985 Introduction and preliminaries In this paper we study connections between ultrafilters and coproduct preserva- tion properties of set-valued functors. It turns out that the problem can be reduced to coproducts indexed by measurable cardinals, if all needed coproducts exist. In particular, preservation of countable coproducts implies preservation of arbitrary ones provided that there are no uncountable measurable cardinals. This result has been proved for endofunctors of Set by Trnkovfi [9] and for arbitrary functors it is contained in [2]. Furthermore, we see that the ultrafilter functor is terminal in the (meta-)category of all endofunctors (see [1]). Then we prove some generalisations for functors that are not set-valued. Finally, we show by a counterexample that our results cannot be generalized to colimits of chains. I am indebted to the referee for valuable suggestions, notably for the simplifica- tion of the proof of Theorem 4.1. We always assume the axiom of choice. For all set-theoretical terminology see [4]. Particularly, if a is an infinite cardinal, an ultrafilter is called a-complete iff it is closed under intersections of less than a members. A cardinal a is called measurable iff there is a non-principal a-complete ultrafilter on a. (By von Neumann's conven- tion, a is identified with the set of all ordinals < a.) In particular, the countable car- dinal ~0 is measurable by our definition. Note that any a+-closed ultrafilter on a cardinal a is principal (a + is the least cardinal >a). If a is a cardinal, then a coproduct indexed by a set of cardinality < a is called an a-coproduct. The image of a map f is denoted by im f. The cardinality of a set M is called #M. 1. Coproduct preservation properties Our further investigation will be based on 0022-4049/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland) 36 R. Bdrger Lemma 1.1. Let a, ,8, be regular cardinals with fl <_ a. Let o~ be a category, in which all a +-coproducts exist. Furthermore, let F : O~-* Set be a functor that preserves all fl-coproducts and let (A¢)¢<a be a family of ;Y{-objects. For any subset MCa define B M := H ~EMA¢, and let "~M : BM ~ Ba be the canonically induced morphism. Then the following assertions hold: (i) For any MC a, F(,,M) is injective; (ii) For any ~ e F(Ba), the collection ~:= {MCa[x6im(F(,,M)) } is a fl-complete ultrafilter on ct; (iii) The coproduct B a = H ~<~ A~ is preserved by F iff for all ~ ~ F(Ba) the ultra- filter q~, is principal. Proof. (i) follows immediately from the fact that B~ is the coproduct of B M and B.\M with the injections "M and "%\M- (ii) By the above argument, F(B,~) is the disjoint union of imF(,,M) and im F(.%\M), hence exactly one of the two sets M, a\M belongs to ~. As ~eF(Ba), we obtain 0~. If McNCa are sets with Me~, consider the induced map : BM'-~BN; then we have ~e im F(UM) = im F(,,NO*)Cim F(~N), hence Ne D=. Now we have to show that ~ is closed under ),-indexed intersections for all y < fl (in particular for ),=2). So let (M¢)~< r be a family of members of ~. Define Ny'=~ {M~ t(<y}, and N,7:= ~ {M~ I ~<r/} \M r, for all r/<y. Then (N,),<y+l is a partition of a, and hence Ba is the coproduct of all BN,, r/<y+ 1 (with injections "N,)- Now the maps F(..%), rl<_y form a coproduct sink in Set, hence (im F(..N,),I<y is a partition of a, therefore ~e im F(..N,) for a unique r/_< y. But for all/7 < (, N,r e ~. would contradict M,1 e ~.~. Thus we have (iii) For any ~<a we have B{~} =A~, and B o is the coproduct of the B{¢} with the injections ,,{¢1. By (i) all F(,,{~I) are injective, and for any ~o<~1<a we have im(F(,,{~0l)) n im(F(,,{~}))= 0, since the contrary would lead to the contradiction 0= {~0} n {~1} ff~x for some xeF(Ba). Thus the given coproduct is preserved iff we have U {im(F("{¢}))l~<(} =F(B,~). This holds iff all ~.~ are principal. [] Lemma 1.1 enables us to prove the following surprising theorem, which was proved by Trnkovfi [9] for endofunctors of Set. Theorem 1.2. Let a be an uncountable cardinal, and let o~ be a category, in which all a-coproducts exist. Let F: or'--, Set be a functor. Then the following statements are equivalent: (i) F preserves all a-coproducts; (ii) F preserves all fl+-coproducts for all measurable cardinals fl<a. Coproducts and ultrafilters 37 Proof. (i) = (ii) is trivial. (ii) = (i). Assume (ii) is true and (i) is false. Application of (ii) to fl= 1~0 yields that Fpreserves all countable coproducts. Now there is a smallest a such that F does not preserve all fl+-coproducts. (Then, of course, F preserves fl-coproducts.) Let (A¢)¢< B be a family, whose coproduct is not preserved by F. By Lemma 1.1(iii), there is an ~eF(II¢<#A~) such that the ultrafilter ~, (as above) is not principal. On the other hand, by Lemma 1. l(ii), ~ is fl-complete. Hence fl is measurable, which contradicts (ii). [] Application of Theorem 1.2 to contravariant hom-functors from the category of unital rings to Set gives the following result: If R is an unital ring and a is a non- Ulam-measurable cardinal, then every unital ring homomorphism from a product of a rings to R depends on only one component provided that this is true for count- able products. This and related results can be found in [3]. Theorem 1.2 reduces the question of a-coproduct preservation to the case where a =fl+ and fl is measurable. But the assumption that g0 is the only measurable car- dinal is relatively consistent with ZFC (= Zermelo-Fraenkel set theory + axiom of choice). On the other hand, uncountable measurable cardinals have to be very large [8, 10], and adding the assumption of their existence leads to a theory strictly stronger than ZFC. If ~0 is the only measurable cardinal, then Theorem 1.2 says that preservation of countable coproducts implies preservation of arbitrary ones. Proposition 1.3. Let a be an infinite cardinal, and let ~5" be a category with a- coproducts, let F, G : ~'--, Set be the functors, and let (o : F~ G be a natural trans- formation. If G preserves a-coproducts and if F preserves finite coproducts, then F preserves a-coproducts. Proof. For any family (A¢)¢<# of ~'-objects with fl<a and for x~F(L[~<aA), the ultrafilter ~ is contained in the ultrafilter @,(x)~), which is constructed like ~, but with F being replaced by G. As both are ultrafilters, ~ = @~,(x)(~) is u-complete by Lemma 1.1, and this proves that the coproduct is preserved by F. [] 2. The ultrafilter functor In this section we want to apply Lemma 1.1 and Theorem 1.2 to endofunctors of Set. Every set is a disjoint union (= Set-coproduct) of singletons, and we can apply Lemma 1.1 for this canonical coproduct. Then the assignment ~--* ~, even becomes natural in a suitable way. In order to show this, we have to define the ultrafilter functor T: Set-,Set: For every set Xlet T(X) be the set of all ultrafilters on Xand for any set map/:X~ Yand any ~ T(X) define T(/)(~):= {BC rl/-l[B] ~.~}. Obviously T is a functor. 38 R. B6rger Theorem 2.1. (i) T preserves finite coproducts. (ii) If F: Set ~ Set /s an endofunctor preserving finite coproducts, then there exists a unique natural transformation x : F--* T. (iii) lf F and x are as in (ii) and if fl is a cardinal, then F preserves all fl-coproducts iff for all sets X and all ~ ~ F(X) the ultrafilter x(X)(x) is fl-complete. Proof. (i) is trivial. (ii) Let X be a set, and for any subset A CX let "A : A ---'X be the inclusion map. Applying Lemma 1.1(ii) to the representation of X as a coproduct of singletons yields that x(X)(x) := {A CX [~imF(,~,4)} is an ultrafilter on X. x :F~ T turns out to be a natural transformation. Now we have to show the uniqueness of x. Let x': F~ T be an arbitrary natural transformation. Then it suffices to show that x(X)(~)Cx'(X)(~) (since both are ultrafilters) for all sets X and all ~ e F(X). For any A ~ x(X)(~) we get ~, = F(,, A )(a) for some aeF(A), hence ,~,~1[A] =A ¢x'(A)(,~) and x' (X)(a:) = (x' (X)) o (F(a A ))(a) = ( T(~ A )) o (x'(A))(a), yields A e x'(X)(,~). (iii) The 'only if' part is a direct translation of Lemma 1.1 (ii) to this special case, and the 'if'-part follows from Lemma 1. l(iii). [] In particular Theorem 2.1 means that the functor that assigns to every set X the set of fl-complete ultrafilters on X is terminal in the meta-category of fl-coproduct preserving endofunctors.