A Categorical Construction of Ultrafilters

A CA TEGORICAL CONSTR UCTION OF UL TRAFIL TERS DANIEL LITT, ZACHAR Y ABEL, AND SCOTT D. KOMINERS De dic ate d to Pr ofessor Elem ´ er Elad R osinger Abstract. Ultrafilters are useful mathematical ob jects having applications in nonstandard analysis, Ramsey theory , Bo olean algebra, top ology , and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms of the inv erse limit of an inv erse family of finite parti- tions; this is an elemen tary and intuitiv e presentation of a consequence of the profiniteness of Stone spaces. W e then apply this construction to answ er a question of Rosinger in the negative. 1. Introduction It is well-kno wn that the category Stone of Stone spaces with contin uous maps is categorically equiv alent to the pro-completion of the category FinSet of finite sets (see [3, p. 236]). W e illuminate this equiv alence in the context of spaces of ultrafilters, in an elementary setting whic h do es not require top ological metho ds. In particular, we giv e an elementary construction of ultrafilter spaces as an inv erse limit, without resorting to Stone spaces or to the correspondence betw een maxi- mal ideals and ultrafilters. W e then give a brief application of this construction, answ ering a question of Rosinger [4] in the negativ e. 2. Ul trafil ters Definition. Let S b e a set. An ultr afilter on S is a subset U of 2 S , the p ow er set of S , such that: (1) ∅ 6∈ U , (2) A ∈ U , A ⊂ B = ⇒ B ∈ U , (3) A ∈ U , B ∈ U = ⇒ A ∩ B ∈ U , (4) A 6∈ U = ⇒ S \ A ∈ U . W e sa y that an ultrafilter is fr e e if it contains no finite sets. It has b een sho wn (see, for example, [1, F orm 63], [2, pp. 145–146]) that Theorem 1 (F ree Ultrafilter Theorem) . If S is infinite, then ther e exists a fr e e ultr afilter on S . The standard proof of Theorem 1, giv en in [2, pp. 145–146], considers, more gen- erally , filters on S , i.e. subsets of 2 S satisfying Prop erties (1)–(3) from Definition 2. It prov es via Zorn’s lemma that given an y filter F , there exists an ultrafilter U ⊃ F . T aking F to b e the c ofinite filter (the collection of all sets whose complements are finite) giv es Theorem 1. 2000 Mathematics Subje ct Classific ation. 54D80 (16B50). Key wor ds and phr ases. Ultrafilters, inverse limits, finite partitions. 1 2 D. LITT, Z. ABEL, AND S. D. KOMINERS No w, let Sets denote the category of sets and let FP ( S ) ⊂ Sets denote the set of finite partitions of a set S . Let FPS ( S ) ⊂ Sets denote the set of finite partitions of subsets of S with the partial ordering defined as follo ws: ∆ 0 ≤ ∆ if and only if for all D 0 ∈ ∆ 0 , there exists a unique D ∈ ∆ such that D 0 ⊂ D , i.e. ∆ 0 is a subset of a (p ossibly trivial) refinemen t of ∆. This turns FPS ( S ) into an inv erse family with morphisms { ψ ∆ 0 , ∆ : ∆ 0 ≤ ∆ } , where ψ ∆ 0 , ∆ is defined b y ψ ∆ 0 , ∆ : A ∈ ∆ 0 7→ B ∈ ∆ s.t. A ⊂ B . The follo wing prop erty of ultrafilters will b e useful: Lemma 2. L et U b e an ultr afilter on S , and let ∆ ∈ FP ( S ) . Then ther e exists a unique D ∈ ∆ such that D ∈ U . Pr o of. Assume to the contrary that no suc h D exists. Then S \ D ∈ U for each D ∈ ∆. Hence their in tersection, \ D ∈ ∆ S \ D = ∅ , is in U by Prop ert y 3 of Definition 2, which contradicts Property 1 of Definition 2; that is, the empty set cannot b e in U . No w, assume that D , D 0 ∈ ∆ are both in U . Then D ∩ D 0 = ∅ ∈ U —again, a con tradiction.  3. The Inverse Limit W e require one additional definition, whic h is cen tral to our categorical approach to ultrafilters: Definition. The inverse limit of an inv erse family ( X i , f ij ) in a category C is the univ ersal ob ject X (unique up to a unique isomorphism) equipped with arro ws π i : X → X i with π j = f ij ◦ π i . That is, X is suc h that for any Y ∈ Ob( C ) and collection of maps u i : Y → X i suc h that u j = f ij ◦ u i for all f ij , there exists a unique u : Y → X suc h that the diagram Y u         u i                             u j   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 X π i   ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ π j @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ X i f ij / / X j comm utes for all f ij . A CA TEGORICAL CONSTRUCTION OF UL TRAFIL TERS 3 F or an inv erse family ( X i , f ij ) in Sets , the in verse limit can b e explicitly con- structed as (1) lim ← − X i = n ( a i ) ∈ Y X i : a j = f ij ( a i ) for all f ij o , whic h may , in some cases, b e empty . 4. Our Ca tegorical Constr uction W e may now giv e categorical interpretations of b oth the set free ultrafilters and the set of all ultrafilters o ver a set I . In particular, consider the function Big : FP ( I ) → FPS ( I ) given by Big : ∆ 7→ { D ∈ ∆ : D is infinite } . Then w e hav e the following theorem: Theorem 3. The set of fr e e ultr afilters on I is in c anonic al bije ction with lim ← − ∆ ∈ FP ( I ) Big(∆) . F urthermor e, the set of al l ultr afilters on I is in c anonic al bije ction with lim ← − ∆ ∈ FP ( I ) ∆ . Pr o of. W e pro ve the second claim ab ov e; the first follo ws analogously . W e claim that eac h ultrafilter induces a unique elemen t of the in verse limit by the mapping Φ : U 7→ (∆ ∩ U ) ∆ ∈ FP ( I ) ∈ lim ← − ∆ ∈ FP ( I ) ∆ , where lim ← − ∆ ∈ FP ( I ) ∆ j Y ∆ ∈ FP ( I ) ∆ . W e first chec k that any element of the image of the abov e map is in the in verse limit, as claimed. First, note that for all ∆ ∈ FP ( I ), ∆ ∩ U is a singleton by Lemma 2, so Φ( U ) is indeed an elemen t of Q ∆ ∈ FP ( I ) ∆. T o see that Φ( U ) is in lim ← − ∆ ∈ FP ( I ) ∆, w e chec k that Φ( U ) satisfies the conditions of the construction in Equation (1) of Section 3. In particular, we hav e that for all ψ ∆ 0 , ∆ with ∆ 0 ≤ ∆, U ∩ ∆ 0 ⊆ ψ ∆ 0 , ∆ ( U ∩ ∆ 0 ) , so ψ ∆ 0 , ∆ ( U ∩ ∆ 0 ) ∈ U b y Prop ert y 2 of Definition 2. But, b y definition, ψ ∆ 0 , ∆ ( U ∩ ∆ 0 ) ∈ ∆, so U ∩ ∆ = ψ ∆ 0 , ∆ ( U ∩ ∆ 0 ), as desired (as eac h set con tains a single elemen t). So w e hav e that Φ( U ) ∈ lim ← − ∆ ∈ FP ( I ) ∆ . W e claim that Φ is the desired canonical bijection. T o see that this map is injectiv e, consider ultrafilters U , U 0 with Φ( U ) = Φ( U 0 ). Note that for eac h A ∈ U , w e may take the partition ∆ A = { A, I \ A } ; then, as Φ( U ) = Φ( U 0 ), we must hav e U 0 ∩ ∆ A = A . Thus, A ∈ U 0 , so U ⊆ U 0 . The reverse inclusion follows identically , so U = U 0 . W e no w show that Φ is surjective. Choose a tuple ( a ∆ ) ∈ lim ← − ∆; w e claim that the set U = { a ∆ : ∆ ∈ FP ( I ) } 4 D. LITT, Z. ABEL, AND S. D. KOMINERS is an ultrafilter and that Φ( U ) = ( a ∆ ). T o chec k that U is an ultrafilter, we verify the four definitional prop erties. (1) ∅ 6∈ U : The empty set is not an element of any partition ∆. (2) A ∈ U, A ⊂ B = ⇒ B ∈ U : Consider the partitions ∆ 1 = { B , I \ B } and ∆ 2 = { A, B \ A, I \ B } . Noting that A = a ∆ 2 , we hav e ∆ 2 ≤ ∆ 1 and thus a ∆ 1 = ψ ∆ 2 , ∆ 1 ( A ) = B . So B ∈ U , as desired. (3) A ∈ U, B ∈ U = ⇒ A ∩ B ∈ U : Consider the partions ∆ 1 = { A, I \ A } , ∆ 2 = { B , I \ B } , ∆ 3 = { A ∩ B , A \ B , B \ A, I \ ( A ∪ B ) } . W e ha v e that ∆ 3 ≤ ∆ 1 , ∆ 2 , so ψ ∆ 3 , ∆ 1 ( a ∆ 3 ) = A, ψ ∆ 3 , ∆ 2 ( a ∆ 3 ) = B . But then a ∆ 3 = A ∩ B , so A ∩ B ∈ U . (4) A 6∈ U = ⇒ I \ A ∈ U : Let ∆ = { A, I \ A } . Then at least one of A, I \ A (namely , a ∆ ) is in U . Since it is not A b y assumption, it must b e I \ A . Clearly , Φ( U ) = ( a ∆ ), b y construction, so Φ is bijectiv e. An iden tical pro of gives the first claim, as w e nev er use the cardinalit y of the sets in volv ed. That is, the restriction b y Big guarantees that all the elemen ts of eac h ( a ∆ ) are infinite; there is alwa ys at least one infinite element in any finite partition of an infinite set (on finite sets, the in verse limit will indeed b e empt y , as there are no free ultrafilters on finite sets), by the pigeonhole principle.  5. A Concrete Example As an application of our results on ultrafilters, we note an interesting corollary: Theorem 4. Consider a function f : I → X , wher e I is an infinite indexing set. F or ∆ ∈ FP ( X ) , let ∆( f ) denote the set ∆( f ) := { D ∈ ∆ : f − 1 ( D ) is infinite } . Then lim ← − ∆ ∈ FP ∆( f ) 6 = ∅ . Pr o of. F or ∆ ∈ FP ( X ), let f − 1 (∆) = { f − 1 ( D ) : D ∈ ∆ } . Note that ev ery partition in FP ( I ) admits a represen tation in this fashion. Then an y free ultrafilter U on I gives an element of the inv erse limit ab ov e, e.g. ( D ∈ ∆ : f − 1 ( D ) ∈ U ) ∆ ∈ FP ( X ) , whic h is an elemen t of the inv erse limit precisely by the argumen t in Section 4, ab o ve.  Corollary 5. L et X b e a set and T : X → X b e a function. F or ∆ ∈ FP ( X ) , let ∆( x ) := { D ∈ ∆ : { n ∈ N : T n ( x ) ∈ D } is infinite } . Then for e ach x ∈ X , we have lim ← − ∆ ∈ FP ( X ) ∆( x ) 6 = ∅ . Pr o of. Fixing x , we may take I = N and f : n 7→ T n ( x ) in Theorem 4. The result follo ws immediately .  Corollary 5 negativ ely answ ers the conjecture Rosinger p osed in [4]. A CA TEGORICAL CONSTRUCTION OF UL TRAFIL TERS 5 A cknowledgements The authors are extremely grateful to Professor Elem´ er Elad Rosinger for bring- ing their attention to the problem and for his helpful comments and suggestions on the work. They would also lik e to ackno wledge Brett Harrison for his excellent suggestions on earlier drafts of this pap er. References [1] P . How ard and J. E. Rubin: Conse quenc es of the Axiom of Choic e , Mathematical Surveys and Monographs 59 . American Mathematical So ciety , 1991. [2] J. S. Pin to and R. F. Hoskins: Infinitesimal Methods for Mathematic al A nalysis . Horwood Publishing Ltd., 2004. [3] P . T. Johnstone: Stone Sp aces . Cambridge Universit y Press, 1982. [4] E. E. Rosinger: A fixed p oint conjecture. . Dep ar tment of Ma thema tics, Har v ard University E-mail addr ess : dalitt@fas.harvard.edu Dep ar tment of Ma thema tics, Har v ard University E-mail addr ess : zabel@fas.harvard.edu Dep ar tment of Ma thema tics, Har v ard University c/o 8520 Burning Tree Ro ad, Bethesda, MD, 20817 E-mail addr ess : kominers@fas.harvard.edu E-mail addr ess : skominers@gmail.com