Exponentiality for the construct of affine sets

@Applied General Topology c Universidad Polit´ecnica de Valencia Volume 9, No. 1, 2008 pp. 21-32 Exponentiality for the construct of affine sets V. Claes Abstract. The topological construct SSET of affine sets over the two-point set S contains many interesting topological subconstructs such as TOP, the construct of topological spaces, and CL, the construct of closure spaces. For this category and its subconstructs cartesian closedness is studied. We first give a classification of the subconstructs of SSET according to their behaviour with respect to exponentiality. We formulate sufficient conditions implying that a subconstruct behaves similar to CL. On the other hand, we characterize a conglomerate of subconstructs with behaviour similar to TOP. Finally, we construct the cartesian closed topological hull of SSET. 2000 AMS Classification: 54A05, 54C35, 18D15 Keywords: topological construct, affine space, cartesian closed category, cartesian closed topological hull, exponential object 1. Introduction The lack of natural function spaces in a topological construct that is not cartesian closed, has long been recognized as an akward situation for various applications in homotopy theory and topological algebra and for use in infinite dimensional differential calculus. For references to the original sources one might consult [22, 11, 12, 18, 21]. Topological constructs like TOP or the larger construct CL and several others that are commonly used by topologists, however are not cartesian closed. For the construct TOP of topological spaces this problem is extensively studied in the literature, see for example [17, 8, 4, 5, 19]. For the subconstruct CL of closure spaces, the author studied cartesian closedness in [6]. To remedy these facts, topologists have applied various methods. Either they dealt with the local problem, the description of exponential objects and with the construction of cartesian closed subconstructs, or they looked for larger cartesian closed constructs. A topological construct is cartesian closed if and only if every object Xis exponential in the sense that the functor X× − preserves coproducts and quotients. The reason for TOP not being cartesian 22 V. Claes closed is that X×− does not always preserve quotients, except for corecompact X. For CL it is just the other way around, X× − generally does not preserve coproducts, except for Xbeing indiscrete. TOP as well as CL are fully embedded in the construct SSET of affine spaces and affine maps which is a host for many other subconstructs that are important to topologists [13, 14] (see next section for the exact definitions). In this paper we investigate the problem of cartesian closedness for SSET and we describe the exponential objects and deduce results for its subconstructs. We prove that for a non-indiscrete affine space X, the functor X× − does not preserve coproducts in SSET and we describe a conglomerate of subconstructs of SSET (to which CL belongs), in which this negative result goes through. On the other hand, we determine a large subconstruct of SSET in all topological subconstructs of which (like for instance TOP) the functor X×− does preserve coproducts. In the final section of the paper we describe the cartesian closed topological hull of SSET. Remark that, as was observed by E. Giuli [13], our definition of affine spaces and maps, as we recall it in the next section, only differs slightly from the normal Boolean Chu spaces and continuous maps, as introduced by V. Pratt to model concurrent computation. Objects in SSET have a structure containing constants. This assumption makes SSET into a well-fibred topological construct in the sense of [1], which has the property that cartesian closedness is equivalent to the existence of ”nice” function spaces. 2. Exponential objects in SSET and in its subconstructs. An affine space Xover the two point set S={0,1}is a structured set, where the structure on the underlying set Xis a collection of subsets of X. The sets belonging to the structure are called the “open” sets of X. An affine map from X−→ Yis a function fsuch that inverse images of open sets are open. An affine space can be isomorphically described in a functional way: An affine space (over S) (X, A) consists of a set Xand a subset Aof the powerset SX. An affine map f: (X, A)→(Y, B) is a function fsuch that β◦f∈ A for all β∈ B. In this paper we will use the functional description. We will restrict ourselves to the affine spaces whose affine structure contains the constant functions 0and 1. As in [13], the corresponding construct of affine spaces and affine maps will be denoted by SSET. In that paper it was proved that SSET is a well-fibred topological construct. An object Xin a category with finite products is exponential if the functor X×− has a right adjoint. In a well-fibred topological construct X, this notion can be characterized as follows: Xis exponential in Xiff for each X-object Ythe set HomX(X, Y ) can be supplied with the structure of a X-object - a function space or a power object YXsuch that (1) the evaluation map ev: X×YX→Yis a X-morphism (2) for each X-object Zand each X-morphism f:X×Z→Y, the map f∗:Z→YXdefined by f∗(z)(x) = f(x, z) is a X-morphism. Exponentiality for the affine sets 23 It is well known that in the setting of a topological construct X, an object Xis exponential in Xiff X× − preserves final episinks [15], [16]. Moreover, small fibredness of Xensures that this is equivalent to the condition that X× − preserves quotients and coproducts. A well-fibred topological construct Xis said to be cartesian closed (or to have function spaces) if every object is exponential. Before characterizing the exponential objects in the subcategories of SSET, we first prove some useful results for subconstructs of SSET. We first recall the following result from [20]. Proposition 2.1. [20] Every topological subcategory Xof SSET is a bicoreflective subcategory of some full bireflective subcategory Yof SSET. The following propositions can be proved using techniques similar to those developed for TOP in [17]. Proposition 2.2. Every topological subcategory of SSET is closed under the formation of retracts in SSET. Proposition 2.3. Every non-trivial topological subcategory Xof SSET contains all complemented topological spaces. In order to investigate the interaction of products and coproducts in SSET and in its subconstructs, we first look at the construction of coproducts in SSET and in its subconstructs. Let (Xi,Ai)i∈Ibe a family of affine sets, then one can easily verify that the coproduct in SSET of these objects is given by (∐Xi,A={ ⊔ i∈Iαi:αi∈ Ai}) with ⊔ i∈Iαi:∐ i∈IXi→S: (xi, i)→αi(xi) Proposition 2.4. Let Xbe a non-trivial topological subconstruct of SSET and (Xi,Ai)i∈Ia family of X-objects. For every i∈Iand every αi∈ Ai, the function ⊔ j∈Iβjbelongs to the affine structure A∐Xion the coproduct ∐ i∈IXi whenever βi=αiand βj=0for all j6=ior βj=1for all j6=i. Proof. Let Ybe the bireflective subcategory of SSET such as in proposition 2.1 and let (Xi,Ai)i∈Ibe a family of X-objects. For r= (ri)i∈I∈Π i∈IXi, we define the function fr:∐ i∈IXi→Π i∈IXi×I: (xi, i)→((yj)j∈I, i) with yi=xiand yj=rjfor j6=i. Let Abe the initial affine structure for the source (fr:∐ i∈IXi→Π i∈I(Xi,Ai)×(I, SI))r∈Π i∈IXi. Then, A=A1∪ A2with 24 V. Claes A1={1J◦prI◦fr|J⊂I, r ∈Π i∈IXi} =∪ J⊂I{ ⊔ i∈Iβi|βi=1if i∈Jand βi=0if i /∈J} A2={αi◦prXi◦fr|i∈I, αi∈ Ai, r ∈Π i∈IXi} =∪ i∈I{ ⊔ j∈Iβj|βi=αi∈ Aiand βj=1if j6=i} ∪ ∪ i∈I{ ⊔ j∈Iβj|βi=αi∈ Aiand βj=0if j6=i} From the previous proposition follows that the categories Xand Ycontain the discrete affine sets, and in particular (I, SI). Since Yis closed under the formation of initial structures in SSET, it follows that ( ∐ i∈IXi,A) belongs to Y. For all i∈I, the map ji: (Xi,Ai)→(∐ i∈IXi,A) is affine. Hence, ( ∐ i∈IXi,A) is coarser than the coproduct ( ∐ i∈IXi,A∐YXi) in the category Y. Since Xis a bicoreflective subcategory of Y, this implies that A ⊂ A∐XXi=A∐YXi. Hence, every non-trivial subcategory of SSET for which the affine structures are closed under arbitrary suprema is closed under the formation of coproducts in SSET. We will now recall a general method to construct hereditary bicoreflective subcategories of SSET [9], [10] ,[13]. In order to define a subconstruct of SSET we put an algebra structure on S={0,1}. Recall that an algebra structure on the set Sis a class of operations Ω = {ωi:Sni→S|i∈I} of arbitrary arities. Hence the niare arbitrary cardinal numbers, and there is no condition on the size of the indexing system I. We assume that Ω contains the constant operations. For every set X, by point-wise extension, the powerset SXcarries an algebra structure. We denote by SSET(Ω) the subconstruct of SSET consisting of those affine sets (X, A) for which Ais a Ω-subalgebra of the function algebra SX. The objects in SSET(Ω) are called affine sets over the algebra (S, Ω). For example the category CL of closure spaces and the category TOP of topological spaces can be obtained this way. The construct obtained this way, by considering for Ω the class containing the constant operations and the complementation¯: S→Sdefined by¯(0) = 1,¯(1) = 0 will be denoted by SSET(C). Lemma 2.5. SSET(Ω) is a subcategory of SSET(C)whenever Ωcontains an operation ωT:ST→Sthat satisfies the following condition: ∃(xt)t∈T,(yt)t∈T∈STsuch that ωT((xt)t∈T) = 0,ωT((yt)t∈T) = 1 and xt= 0 implies yt= 0 for all t∈T. Exponentiality for the affine sets 25 Proof. Let (X, A) be an SSET(Ω)-object. For α∈ A, define the serie functions (ft:X→S)t∈Tas follows. ft=   0if xt= 0 1if yt= 1 αif xt= 1, yt= 0 Since Acontains all constant functions, we have that ft∈ A for all t∈T. Then, Acontains the function ωT◦Π t∈Tft:X→S:x→ωT(ft(x))t∈T. If α(x) = 1, then ωT◦Π t∈Tft(x) = ωT((xt)t∈T) = 0 =¯◦α(x). If α(x) = 0, then ωT◦Π t∈Tft(x) = ωT((yt)t∈T) = 1 =¯◦α(x). We can conclude that¯◦α=ωT◦Π t∈Tft∈ A and thus (X, A) is an SSET(C)-object.  Lemma 2.6. If Xis a non-trivial topological subcategory of SSET and D2 is the two point discrete space, then for every non-indiscrete object (X, A)the following holds: (X, A)×D2∈X⇒(X, A) is not an exponential object in X Proof. For a non-constant function α∈ A,0⊔αis an element of AX⊔X, while it is not contained in AX×D2. The previous negative result has important consequences with respect to exponential objects in SSET and to cartesian closedness of topological subconstructs. Corollary 2.7. If Xis a topological subconstruct of SSET which is finitely productive in SSET, then the class of exponential objects in Xcoincides with the class of indiscrete spaces. We now characterize a conglomerate of subconstructs of SSET, which are not finitely productive in SSET, in which the class of exponential objects also coincides with the class of indiscrete objects. Proposition 2.8. For every category SSET(Ω) that is not a subcategory of SSET(min : S2→S, max : S2→S)the exponential objects are exactly the indiscrete affine sets. Proof. Suppose that SSET(Ω) has a non-indiscrete exponential object (X, A). Then we have that (X, A)⊔(X, A) is isomorphic to (X, A)×D2. By proposition 2.4, it follows that 0⊔αbelongs to AX×D2for every α∈ A. The product AX×D2is the smallest Ω-subalgebra of SX×D2containing B={α◦prX|α∈ A} ∪ { prD2, prD2: (x, a)→a}. Hence, 0⊔α=ωα((γ◦prX)γ∈A, prD2, prD2) with ωα:SA∪S→Sa composition of algebraic operations of Ω. For a non-constant function α∈ A, there exists x, y ∈Xsuch that α(x) = 1 and α(y) = 0. Define (fγ:S×S→S)γ∈A as follows: •If γ(x) = γ(y), put fγthe constant function with value γ(x). •If γ(x) = 1 and γ(y) = 0, put fγ=pr1:S×S→S: (a, b)→a 26 V. Claes •If γ(x) = 0 and γ(y) = 1, put fγ=¯ ◦pr1:S×S→S: (a, b)→a Then, for b∈Swe have: •ωα◦( Π γ∈Afγ×pr2×pr2)(0, b) = ωα((γ(y))γ∈A, b,¯ b) = ωα((γ◦prX)γ∈A, prD2, prD2)(y, b) = 0⊔α(y, b) = 0 = min(0, b). •ωα◦( Π γ∈Afγ×pr2×pr2)(1, b) = ωα((γ(x))γ∈A, b,¯ b) = ωα((γ◦prX)γ∈A, prD2, prD2)(x, b) = 0⊔α(x, b) = b= min(1, b). This implies that min(a, b) = ωα◦( Π γ∈Afγ×pr2×pr2)(a, b), which means that the operation min : S×S→Scan be written in terms of the operation ωα, the complementation¯ and the constant functions. If SSET(Ω) is a subcategory of SSET(C), it now follows that SSET(Ω) is a subcategory of SSET(min : S2→S). For the categories SSET(Ω) that are not embedded in SSET(C), it follows from lemma 2.5 that: (1) ωα((γ(x))γ∈A,0,0) = 0, because ωα((γ(x))γ∈A,0,1) = 0⊔α(x, 0) = 0 (2) ωα((0)γ∈A,1,0) = 0, because ωα((γ(y))γ∈A,1,0) = 0⊔α(y, 1) = 0 (3) ωα((0)γ∈A,0,0) = 0 By defining fγ=pr1:S×S→S: (a, b)→aif γ(x) = 1 and otherwise fγ=0, we have that ωα◦( Π γ∈Afγ×pr2×0)(a, b) = min(a, b). Indeed, for b∈Swe have •ωα◦( Π γ∈Afγ×pr2×0)(0, b) = ωα((0)γ∈A, b, 0) = 0 •ωα◦( Π γ∈Afγ×pr2×0)(1, b) = ωα((γ(x))γ∈A, b, 0) = b Where the last equation follows from previous observation (1) and the fact that ωα((γ(x))γ∈A,1,0) = 0 ⊔α(x, 1) = 1 So in each category SSET(Ω) which has an non-indiscrete exponential object, Ais closed under finite minima for every object (X, A). It can be proved in a similar way that Ais closed under finite maxima. One can easily prove that the indiscrete affine sets are exponential. So we can conclude that the exponential objects of SSET(Ω) are exactly the indiscrete affine sets.  Thus for the categories SSET,CL and SSET(C) the exponential objects are exactly the indiscrete objects. From the proof of previous proposition follows that in all the categories SSET(Ω) which are not embedded in SSET(min : S2→S, max : S2→S), the functor X× − does not preserve coproducts for non-indiscrete objects X. In SSET(min : S2→S, max : S2→S) itself, the functor X× − preserves coproducts for some non-indiscrete objects X. In the following proposition these objects are characterized. Exponentiality for the affine sets 27 Proposition 2.9. In the construct SSET(min : S2→S, max : S2→S), we have that the functor (X, A)×− preserves coproducts if and only if Ais a finite set. Proof. It can be easily verified that for (X, A), with Aa finite set, (X, A)× ∐ i∈I(Yi,Bi) is isomorphic to ∐ i∈I(X, A)×(Yi,Bi) for every collection SSET(min : S2→S, max : S2→S)-objects (Yi,Bi)i∈I. Suppose now that the functor (X, A)× − preserves coproducts, then we have that (X, A)×(A, SA) is isomorphic to ∐ α∈A(X, A). We consider the function ⊔ α∈Aα:∐ α∈A(X, A)→S: (x, α)→α(x) Since SSET(min : S2→S, max : S2→S) is a bicoreflective subcategory of SSET, this function ⊔ α∈Aαbelongs to A∐ α∈A(X,A)and thus ⊔ α∈Aαbelongs to AX×(A,SA). The product AX×(A,SA)is the smallest subalgebra containing B={α◦prX|α∈ A} ∪ { f◦prA|f∈SA}. Hence, there exists a finite set Iand for every i∈Ithere exist αi∈ A, fi∈SAsuch that ⊔ α∈Aα= max i∈Imin(αi◦prX, fi◦prA). For β∈ A, set Iβ={i∈I|fi(β) = 1}. For every x∈X, we have: β(x) = ⊔ α∈Aα(x, β) = max i∈Imin(αi(x), fi(β)) = max i∈Iβ αi(x). Since the set Iis finite, this implies that Aalso shall be finite.  3. Subconstructs in which the functor X× − preserves coproducts The categories considered in the previous section fail to be cartesian closed because the functor X×− does not preserve coproducts. From the last proposition, it follows that the condtion, SSET(Ω) is a subcategory of SSET(min : S2→S, max : S2→S), is not a sufficient condition such that the functor X× − preserves coproducts for all objects X. In this section we formulate a sufficient condition. It is known [17] that in TOP and in all its topological subconstructs the functor X× − preserves coproducts for all objects X. We generalize these results to a larger subconstruct of SSET. Definition 3.1. Let Dbe the full subcategory of SSET with objects all affine sets (X, A)that satisfy the following two conditions. (D1) α∈ A, β ∈ A ⇒ min(α, β)∈ A (D2) {αi|i∈I} ⊂ A and min(αi, αj) = 0for each i6=j⇒max i∈Iαi∈ A It is clear that TOP is a subcategory of this category D. For a collection B ⊂ SXwe can define a D-structure Aon Xas follows. Let Cconsist of all finite minima of elements of B ∪ {0,1}. By adding to Cthe maxima of collections functions (αi)i∈Iof Cthat satisfy the condition min(αi, αj) = 0for each i6=j, we get a D-structure A. Moreover, Ais the smallest D-structure on Xcontaining B.Bis called the subbase of Aand Cthe base of A. 28 V. Claes Proposition 3.2. D is a topological category. Proof. For a family of functions (fi:X→(Xi,Ai))i∈Iwith (Xi,Ai)∈Dthe D-structure Agenerated by the subbase B={αi◦fi|αi∈ Ai, i ∈I}is the unique initial structure on Xfor the given source.  Proposition 3.3. D is a bicoreflective subcategory of SSET Proof. For an affine set (X, A) the bicoreflection is given by 1X: (X, A′)→(X, A) with A′the D-structure generated by the subbase A. Remark that Dis not a hereditary subcategory of SSET as follows from the next example. Example 3.4. Let X={0,1,2,3}and A={0,1,1{1,3},1{2,3},1{3}}, then (X, A) is a D-object. Then (Y, A|Y) = ({0,1,2},{0,1,1{1},1{2}}) is the SSET-subspace of (X, A) with underlying set {0,1,2}. min(1{1},1{2}) = 0 and max(1{1},1{2}) = 1{1,2}/∈ A|Y, so (Y, A|Y) does not belong to the category D. Hence, there is no algebraic structure Ω on Ssuch that Dhas the form SSET(Ω). Proposition 3.5. If SSET(Ω) is a subcategory of D, then it is a subcategory of TOP or a subcategory of SSET(C). Proof. For an arbitrary set I, take ∞/∈Iand define for every i∈Ithe function αi:I∪ {∞} → Swith αi(i) = 1 and αi(x) = 0 for x6=i. Let Abe the smallest Ω-subalgebra of SI∪{∞} containing {αi|i∈I}. Then, (I∪ {∞},A) is an SSET(Ω)-object. Since SSET(Ω) is a subcategory of D, we have that max i∈Iαibelongs to A. Hence, max i∈Iαi=ωI(αi)i∈I, with ωI:SI→Sa composition of algebraic operations of Ω. This gives the following information about ωI: • ∀j∈I:ωI(αi(j))i∈I= max(αi(j))i∈I= 1 •ωI(0)i∈I=ωI(αi(∞)i∈I) = max(αi(∞)i∈I) = max(0)i∈I= 0 Now two cases can arise: (1) ∀x6= (0)i∈I∈SI:ωI(x) = 1. In this case ωI= max i∈I (2) There exists a x6= (0)i∈I∈SIsuch that ωI(x) = 0. Choose j∈I such that prj(x)6= 0 and define (yi)i∈I∈SIwith yj= 1 and yi= 0 for i6=j. Then we have ωI(x) = 0, ωI(yi)i∈I=ωI(αi(j))i∈I= 1 and pri(x) = 0 implies yi= 0. It then follows from lemma 2.5 that SSET(Ω) is a subcategory of SSET(C). We can conclude that either for every set I,ωI= max i∈Iand thus SSET(Ω) is a subcategory of TOP or SSET(Ω) is a subcategory of SSET(C).  Exponentiality for the affine sets 29 It can be verified that coproducts are universal in D, i.e. coproducts are preserved under pullbacks along arbitrary morphisms. From proposition 2.4 and the condition (D2) follows that Dand its subconstructs are closed under the formation of coproducts in SSET. Combining this with 2.2, the following theorem can be proved. Theorem 3.6. In every topological subcategory of Dcoproducts are preserved by the functor X× −. Corollary 3.7. The exponential objects of a subcategory of Dare the objects Xfor which the functor X× − preserves quotients. 4. Cartesian closed topological hull of SSET In section 2, we proved that for finitely productive subcategories of SSET, products do not distribute over coproducts. If we want to work in a cartesian closed construct in which products are formed similarly to the ones in SSET, we have to consider a larger scope. We will look for cartesian closed topological constructs that are larger than SSET and in which SSET is finally densely embedded. We know from [7] that quotients in SSET are productive. In fact we have the same situation as for the category CL [6]. For CL a cartesian closed extension was constructed in [6] using the method presented by J. Ad´amek and J. Reiterman in [2]. We first look if this method is also applicable to SSET. Definition 4.1. For affine sets (X, A)and (Y, B)we consider the collection of functions N={Γβ|β∈ B} on Hom(X, Y )with Γβ: Hom(X, Y )→Sdefined by Γβ(f) = 1 iff β◦f=1. Analogous to CL, we can prove the following result for this structure on the Hom-sets of SSET. Proposition 4.2. If M⊆Hom(X, Y )is a subset endowed with an affine structure Msuch that the evaluation map ev:(X, A)×(M, M)→(Y, B)is an affine map, then the following conditions hold: (1) N|M⊆ M (2) ev:X×(M, N|M)→Yis affine. This shows that SSET is a type of category as considered in 4.3 of [2]. Consider the following superconstruct Kof SSET. Objects of Kare triples (X, A,A) where Xis a set, Ais a cover of Xsuch that U′⊆U, U ∈A⇒U′∈A and Ais an affine structure on Xwhich is A-final in the sense that (i: (U, A|U)→(X, A))U∈Ais final in SSET. The members of Aare called generating sets. A morphism in K, f: (X, A,A)→(Y, B,B) is a function that is affine (with respect to (X, A) and (Y, B)) and preserves the generating sets: U∈A⇒f(U)∈B.