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Frölicher versus differential spaces: A Prelude to Cosmology Paul Cherenack∗ Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa. ABSTRACT: Differential spaces due to R. Sikorski and Frölicher spaces (called earlier smooth spaces and studied by A. Frölicher) are compared. The category of Frölicher spaces is known to be topological over sets and Cartesian closed under the usual set adjunction. We see that the category of differential spaces is topological over sets but not Cartesian closed under the usual set adjunction. The tangent and cotangent bundles of an arbitrary Frölicher space X are defined and these notions are seen to coincide with the usual ones if X is a smooth manifold. We show that one can define, in a fairly natural way, the notions metric, connection, Ricci tensor, Riemannian curvature tensor and finally Einstein tensor for an arbitrary Frölicher space. Keywords: Frölicher space, differential space, topological functor, Cartesian closed, tangent bundle, connection, Einstein tensor. AMS Classifications: 18F15, 57R55, 58D15. Introduction In physics an object is often known by various scalar fields such as temperature and pressure defined on it. Local estimates of temperature can be put together to obtain a global temperature function. Various scalar functions can be combined via a smooth function of several variables to determine other scalar fields. Thus, if I denotes the current in an electric line and R is resistance, then the function V = f (I, R) = IR determines the scalar field V denoting potential. Such ideas lead to the notion of differential space and its generalizations which have played some role in mathematical physics. See [8], [12] among the wide variety of papers here. Penrose and Rindler [10] use related concepts. To be precise we now define the notion of differential space and map between differential spaces. Let X be a set and F be a set of real valued functions on X called scalar fields or just scalars which define an initial topology T on X. Thus, as one can show, U ∈ T if and only if U is the union of ∗ This paper is dedicated with thanks to Bernhard Banaschewski on his seventieth birthday c 2001 Kluwer Academic Publishers. Printed in the Netherlands. correctdiffeo2.tex; 4/05/2001; 11:30; p.1 2 sets of the form f −1 (V ) where f is a finite (pointwise defined) product of elements in F and V is an open subset of the reals IR. DEFINITION 0.1. The pair (X, F) is a differential space (assumed to possess the topology T ) if − for each open covering {Ui }i∈K of X and function g : X → IR, if g|Ui = hi |Ui for some hi : X → IR ∈ F and each i ∈ K, then g ∈ F . Thus, if locally a function comes from F, it must be in F. This is a sheaf theoretic property. − if f1 , · · · , fn is a collection of functions in F and g(x1 , · · · , xn ) is a smooth real-valued function (i.e., infinitely differentiable), then g ◦ (f1 , · · · , fn ) is again in F. A map f : (X, FX ) → (Y, FY ) of differential spaces is a map f : X → Y such that f ∗ FY = {g ◦ f |g ∈ FY } ⊂ FX . The class of such maps makes differential spaces into a category DSP. A map in DSP will be called a differentiable map. In the definition of differential spaces one would like to think of the scalar fields in F as smooth maps. There is only partial control though, over this property. To motivate our subsequent definition of Frölicher space, we note heuristically that one often considers curves on a physical surface where scalar fields (e.g., temperature or pressure) vary in a smooth way as a function of position. If such curves, as maps from IR to the surface, are smooth, one expects of course their compositions with such scalar fields to be smooth maps IR → IR. These intuitive notions are refined, in the case of IRn , by the following result of Boman [1] which shows the important fact that the converse implication holds as well. THEOREM 0.1. Let f : IRn → IR be a function. Then, f is smooth on IRn if and only if f ◦ c : IR → IR is smooth for each smooth curve c : IR → IRn . These ideas motivate the notion of Frölicher space which has been used in papers [3] by the author in smooth homotopy. Traditionally, Frölicher spaces were called by Frölicher and Kriegl [4] smooth spaces. Because such smooth spaces − are determined by contours (inputs) into them and scalar fields (outputs) on them, − as point subsets of the plane, they need not be smooth, correctdiffeo2.tex; 4/05/2001; 11:30; p.2 3 − Frölicher has worked extensively with them, we now call them Frölicher spaces but call the maps smooth. Their definition, where M is the set of smooth maps from IR to IR follows. DEFINITION 0.2. The triple (X, F, C) is a Frölicher space (assumed to possess the initial topology T defined by F) if − F ◦ C = {f ◦ g|f ∈ F, g ∈ C} ⊂ M. − ΦC = {f : X → IR|f ◦ c ∈ M for all c ∈ C} = F. − ΓF = {c : IR → X|f ◦ c ∈ M for all f ∈ F} = C. The elements of C are called contours. Frölicher and Kriegl [4] is our main reference for Frölicher spaces. Example 1 : The most important Frölicher spaces are finite dimensional smooth manifolds where if X is such a manifold, then CX = {c : IR → X|c is smooth } and FX = {f : X → IR|f is smooth }. Although there may at first glance seem little connection between Frölicher and differential spaces, one can show: THEOREM 0.2. Let (X, C, F) be a Frölicher space. Then, (X, F) is a differential space. Proof. We use the notation in the definition of differential space. Let c : IR → X be a contour. As hi ◦ c is smooth, so is g|Ui ◦ c|c−1 (Ui ) . Since the sets c−1 (Ui ) for i ∈ K cover IR, g ◦ c must be smooth. But, since (X, C, F) is a Frölicher space, g ∈ F . Suppose that c is again a contour. Since fi ◦c, i = 1, · · · , n, is smooth, the composite (f1 , · · · , fn ) ◦ c is smooth. But, then g ◦ (f1 , · · · , fn ) ◦ c is smooth. This implies that g ◦ (f1 , · · · , fn ) belongs to F. DEFINITION 0.3. A map of Frölicher spaces or just smooth map is a map arising from the underlying map of differential spaces. The resulting category of Frölicher spaces is denoted FRL. It follows that on identifying a Frölicher space with its corresponding differential space that one obtains: PROPOSITION 0.1. The category FRL is a full subcategory of DSP. Let C be a category and U : C → SET S be a faithful functor to the category of sets. Here, the category C will be either DSP or FRL and U will be the forgetful functor. Let S = {hi |B → U (Ai )}i∈I be a collection of set maps. Suppose that B = U (A). Then, A will be called an initial object for S if correctdiffeo2.tex; 4/05/2001; 11:30; p.3 4 1. there are (unique) maps {gi : A → Ai }i∈I lying in C such that U (gi ) = hi for i ∈ I and 2. given an object A0 and maps {ki |A0 → Ai }i∈I such that U (ki ) = hi for i ∈ I, there is a necessarily unique map α : A0 → A in C such that U (α) is the identity map on B. Dually, one can define the notion of final object (for U and a set of maps S0 = {hi |U (Ai ) → B}i∈I ). If any set S has an initial object, then every set S0 has a final object, U is said to be a topological functor and C is called topological over SET S. For more detail, see Brümmer [2]. Now, for the forgetful functor U : FRL → SET S, Frölicher and Kriegl [4] have shown that FRL is topological. They also show that FRL is Cartesian closed. We will show to start with (Section 1) that, for the forgetful functor U : DSP → SET S, the category DSP is topological. If C is topological, it has all limits and colimits. One merely lifts the limit in SET S to C. We will examine these limits and colimits and also initial subobjects in FRL and DSP. We will see (Section 2) that initial subobjects in FRL compared to DSP better reflect any singularities which the subobject possesses. Furthermore, in FRL, tangent cones can be defined for Frölicher spaces and tangent cones can be more appropriate than tangent spaces for the study of singularities in a Frölicher space. For instance, in algebraic geometry, tangent cones are used in “blowing up” singularities. See [5]. In Section 3 we compare the products in FRL with those of DSP. This enables us to see in Section 5 that unlike FRL, if DSP is Cartesian closed (which seems unlikely), the Cartesian closedness does not arise from that of sets. In Section 4 we show that colimits of objects in FRL are the same irrespective of whether taken in DSP or FRL. The Cartesian closedness of FRL enables us to construct in Section 6 the tangent bundle, cotangent bundle and associated type (p, q)-tensor bundles over an object in FRL in a natural way. We can also construct corresponding cone bundles in FRL. Finally, in our last section, we show how to write down the Einstein tensor for an arbitrary Frölicher differential space. Although much of the theory is abstract, it leads to many classical type questions. The author would like to thank the referee for useful advice. 1. The category DSP is topological over sets We will show that, for the forgetful functor U : DSP → SET S, the category of differential spaces is topological over sets. correctdiffeo2.tex; 4/05/2001; 11:30; p.4 5 ∗ is a collection of maps X → IR, then F ∗ generates a collection If FX X ∗ of maps X → IR such that (X, ΣF ∗ ) is a of functions denoted ΣFX X 0 ) is a differential space with F ∗ ⊂ F 0 , differential space and if (X, FX X X ∗ ⊂ F 0 . More precisely, one can show, see [11] , that ΣF ∗ is then ΣFX X X the set of all g : X → IR such that, for each P ∈ X and on some open ∗ ), set U with P ∈ U (for the initial topology given by FX g|U = h ◦ (f1 |U , · · · , fn |U ) ∗ and n ∈ IN . for some smooth function h : IRn → IR, f1 , · · · , fn ∈ FX Let now {hi : Y → Xi }i∈I be a collection of set maps where Xi is a differential space with structure functions FXi . Set FY∗ = {g : Y → IR|g = f ◦ hi with f ∈ FXi and i ∈ I} . Then, we show: LEMMA 1.1. The differential space (Y, ΣFY∗ ) is the initial object for the collection of maps {hi : Y → Xi }i∈I . Proof. Clearly, the maps hi : (Y, ΣFY∗ ) → (Xi , FXi ), where i ∈ I, are maps of differential spaces. Let (Y, F 0 ) be a differential space such that hi : (Y, F 0 ) → (Xi , FXi ) is a map of differential spaces for each i ∈ I. Then, for f ∈ FXi , f ◦ hi ∈ F 0 for each i ∈ I. But then FY∗ ⊂ F 0 and, from the above discussion, ΣFY∗ ⊂ F 0 . As we mentioned in the introduction, from the lemma it follows that DSP must thus also have final objects and be topological for the forgetful functor U : DSP → SET S. 2. Initial subobjects and tangents in FRL and DSP Initial subobjects of an object in FRL exist. Extracting from the more general construction in Frölicher and Kriegl [4], let (X, CX , FX ) be a Frölicher space and A a subset of X. Then, the inclusion iA : A → X places an initial structure on A where the resulting Frölicher space is (A, CA , FA ) with − CA = {c : IR → A|iA ◦ c ∈ CX }. − FA = ΦΓ {f ◦ iA |f ∈ FX }. correctdiffeo2.tex; 4/05/2001; 11:30; p.5 6 With this structure A is called a Frölicher subspace of X. Here the description of CA is most useful. Initial here means that if iA : (A, C 1 , F 1 ) → (X, CX , FX ) is a map of Frölicher spaces, then the identity map IA : (A, C 1 , F 1 ) → (A, CA , FA ) defines a smooth map. Examples 2 : We use the above notation. | denote the rationals. Then, C − Let X = IR and A = Q A consists of | the constant maps and FA thus consists of all functions. Since Q then has the discrete topology, we then call it a discrete Frölicher space. It is reasonably clear that there are examples of fractal curves which are Frölicher subsets of IR2 and having the discrete Frölicher structure. − Let C be the curve and subset of IR2 defined in polar coordinates by r = 1θ , θ ∈ (0, ∞) but also containing (0, 0). A straight forward computation of the arc length l(θ) of the curve r = 1θ from θ0 to θ (θ > θ0 ) shows that l(θ) ≥ ln θθ0 . Since any smooth curve c : [0, 1] → IR2 must have finite arc length, it follows that there is no smooth curve connecting ( θ10 , θ0 ) to (0, 0). Although C is a connected subspace of IR2 in the usual topology, a contour into C thus maps either to (0, 0) or to the complement of (0, 0). Hence, a scalar on C can take any value at (0, 0) irrespective of values at other point. The Frölicher subspace (A, CA , FA ) thus is disconnected topologically. Before we present other examples we define the tangent space at a point of a differential space according to the usual definition and an alternate definition of tangent at a point of a Frölicher space. This last definition follows a common way for defining tangents in the theory of differentiable manifolds. DEFINITION 2.1. 1. Let (X, F) be a differential space. A tangent vector V at P ∈ X is a derivation V : F → IR at P , i.e. a linear map such that V (f g) = f (P )V (g) + g(P )V (f ). The set of such vectors form the tangent space T XP of X at P . 2. Let (X, F) be a differential space and P ∈ X. Let c ∈ ΓF and suppose that c(a) = P and f ∈ F. Suppose that Vc is the derivation correctdiffeo2.tex; 4/05/2001; 11:30; p.6 7 defined by setting Vc (f ) = lim t→a f ◦ c(t) − f ◦ c(a) . t−a Set T CXP = {Vc |c ∈ ΓF, c(a) = P }. It is easy to see that T CXP ⊂ T XP . We call T CXP the tangent cone to X at P . In fact, as examples below show, T CXP need not be a vector space. Examples 3 : 1. The rationals as a Frölicher subspace of IR have trivial tangent spaces equal to their tangent cones: Since contours must have | . The constant values, the tangent cones must be trivial. Let q ∈ Q function f : IR → IR such that f (q) = 1 and f (r) = 0 if r 6= q 2 belongs to FQ | . Since f = f , one can show that, for any 0 derivation D at q, D(f ) = 0. Let g ∈ FQ | and g = f g. Then, D(g 0 ) = f (q)D(g). Since g(q)g 0 = (g 0 )2 , D(g 0 ) = 0 and thus D(g) = 0. Hence, the tangent space at q is trivial. 2. Except at (0, 0) the Frölicher curve C above has a one-dimensional tangent space(= tangent cone). At (0, 0) the tangent space(= tangent cone) is trivial(as in 1.). 3. Let B be the Frölicher subspace of IR2 defined by xy = 0. The tangent cone to B agrees with the tangent space and is 1-dimensional except at (0, 0) where the tangent cone is B and the tangent space is IR2 . A scalar on B is a function f : B → IR such that f is smooth on the x and y-axes, respectively. 4. Let V be an algebraic subvariety of IRn defined by polynomial equations f1 , · · · , fm and P = (a1 , · · · , an ) ∈ V . Let pi be the degree of the first non-vanishing homogeneous term in fi regarded as a polynomial in the xj − aj for i = 1, · · · , m and j = 1, · · · , n. Set p = min {pi }i=1,···,m . If g is a polynomial in the polynomial ring R generated by the xj − aj for j = 1, · · · , n, let (g)k denote the term of degree k in the same variables. Consider the ideal I generated by (f1 )p , · · · , (fm )p in R. The tangent cone to V at P is then the zero set of I in IRn . One can show that the tangent space to V at P is the zero set of (f1 )1 , · · · , (fm )1 . Thus, if m = 1 and f1 = x2 + y 2 , then T CV(0,0) = {(0, 0)} and T V(0,0) = IR2 . Let now X be a differential space, A ⊂ X and FA∗ = {f ◦ iA |f ∈ FX } where iA : A → X is the inclusion. Then A with scalars Σ(FA∗ ) is called a differential subspace of X. Examples 4 : We consider examples given above in DSP. correctdiffeo2.tex; 4/05/2001; 11:30; p.7 8 | → IR which are | as a differential subspace of IR has scalars f : Q 1. Q locally in the usual topology the restrictions of locally smooth | is the functions on IR. Thus, the tangent space to a point q ∈ Q same as the tangent space when q is regarded as a point of IR and one dimensional. 2. The curve C in Examples 2, as a differential subspace of IR2 , is connected. Scalars at (0, 0) are the restrictions locally of smooth functions on IR2 and hence cannot take on arbitrary values at (0, 0). For f ∈ FC , let ∂ f (0 + h, 0) − f (0, 0) lim |C,(0,0) f = . ∂x h h→0,(h,0)∈C Since f is the restriction locally of an element in FIR2 , this ∂ definition makes sense. One readily shows that ∂x |C,(0,0) is a ∂ derivation on FC . The derivation ∂y |C,(0,0) is defined similarly. As ∂ ∂ ∂ ∂x |C,(0,0) x|C = 1, ∂x |C,(0,0) y|C = 0, ∂y |C,(0,0) y|C = 1 and ∂ ∂y |C,(0,0) x|C = 0, the tangent space of C at (0, 0) is at least 2-dimensional. To each derivation D on FC , one can associate a derivation D on FIR2 by setting D(g) = D(g|C ). As D must have ∂ ∂ |(0,0) + b ∂y |(0,0) , it follows that the form D = a ∂x ∂ ∂ D = a ∂x |C,(0,0) + b ∂y |C,(0,0) . Hence, the tangent space of C at (0, 0), as a differential space, is 2-dimensional. 3. Products in FRL and DSP spaces Suppose that one is given a collection {(Xi , Fi )}i∈I of differential Q spaces. Let i∈I Xi be the or a collection {(Xi , Ci , Fi )}i∈I of Frölicher Q set product of the sets {Xi }i∈I and πj : i∈I Xi → X for j ∈ I denote Q j the projection map. The initial structure on P = i∈I Xi in both DSP and FRL is generated by the set ∗ FP = [ i∈I {fi ◦ πi |fi ∈ Fi , i ∈ I} . In smooth spaces one obtains − CP = {c : IR → P| if c(t) = (ci (t))i∈I , then ci ∈ Ci }. ∗ − FP = ΦΓFP correctdiffeo2.tex; 4/05/2001; 11:30; p.8 9 Here, the requirement that each component of c is a smooth map is most useful. We now provide an example that shows that finite products in FRL and DSP need not be the same. IN denote the Frölicher space and thus Example 5 : Let IR⊕ differential space whose underlying set is IRIN and whose Frölicher space structure is generated by the set C 0 is the set {(xi (t))i∈IN ∈ CIRIN | except for finitely many i, xi (t) is identically 0}. IN → IR defined by setting It is clear that the function l : IR⊕ l((xi )i∈IN ) = ∞ X xi i=1 IN . Let IRIN denote a second copy of IRIN with is a scalar on IR⊕ ⊕⊕ ⊕ coordinates (yi )i∈IN . Then, we show: LEMMA 3.1. The function k(xi , yj ) = P∞ i=1 xi yi is IN × IRIN for the Frölicher space product structure 1. a scalar on IR⊕ ⊕⊕ but 2. not for the differential space product structure where the scalars are of the form f (g1 , · · · , gn , h1 , · · · , hm ) with f : IRn+m → IR a smooth function, g1 , · · · , gn ∈ FIRIN and h1 , · · · , hm ∈ FIRIN . ⊕ ⊕⊕ IN × IRIN has the form Proof. Since every contour c : IR → IR⊕ ⊕⊕ c(t) = (xi (t), yj (t))i,j∈IN where all but finitely many xi (t) and yj (t) are identically 0, the first statement is clear. To see the last assertion, suppose that f (g1 , · · · , gn , h1 , · · · , hm ) = ∞ X xi yi . (1) i=1 One can write down power series expansions at (0, 0, · · ·): gl = ∞ X ail xi + · · · ∞ X bjm yj + · · · i=1 and hk = j=1 for l = 1, · · · n and k = 1, · · · m. Here, one lets xi = yi = 0 for i > M and then lets M → ∞. Substituting into f and equating the result to the right hand side of expression (1), since we can assume that each gl correctdiffeo2.tex; 4/05/2001; 11:30; p.9 10 can pair up with at most one hm and vice-versa, one can assume n = m. From the resulting equation, equating coefficients, one also obtains a matrix equation AB = I (2) where A is the ∞ × n matrix c1 a11 · · · c1 an1 c2 a12 · · · c2 an2 , .. .. .. . . . the cp are the appropriate coefficients in the expansion of f at (0, · · · , 0), B is the n × ∞ matrix b11 b12 · · · .. .. .. . . . bn1 bn2 · · · and I is the ∞ × ∞ identity matrix 1 0 0 ... 0 1 0 ... 0 0 1 .. . ··· ··· ··· . ... Since the invertibility of the n × n matrix consisting of the first n rows of A implies that all columns in B after the first n columns are zero, it is clear that equation (2) has no solution. Hence, k(xi , yj ) is IN × IRIN . not a scalar for the differential structure on IR⊕ ⊕⊕ Let IR1 denote the reals but with the smooth structure where FIR1 = ΦΓ(FIR ∪ {|t|}). p Clearly, |x| ∈ FIR1 . It seems likely that x2 + y 2 is a scalar for the Frölicher product IR1 × IR1 but not the differential product IR1 × IR1 . No proof of this fact has been found. 4. Coequalizers and coproducts in FRL and DSP Let {Xi }i∈I be a set of differential or Frölicher spaces with Fi the set of S S scalars on Xi (i ∈ I), b i∈I Xi the disjoint union and iXi : Xi → b i∈I Xi the inclusion map. Place the differential or Frölicher final structure on correctdiffeo2.tex; 4/05/2001; 11:30; p.10 11 S b i∈I Xi given by the set {iXi }i∈I . The resulting differential or Frölicher space is the coproduct of {Xi }i∈I in the respective category, denoted ` by i∈I Xi and F` i∈I Xi = ( f: a i∈I Xi → IR| for each i ∈ I, f |Xi ∈ Fi ) is the set of scalars for the coproduct. See [4], [11]. Thus, from the above description, it follows that the inclusion of FRL into DSP is coproduct preserving. Examples 6 : In Examples 2.1 (the setting is the category FRL) | can be identified with a countable union of points. 1. Q 2. C can be identified with a coproduct of IR and a point. A similar analysis holds for coequalizers. Let f, g : X → Y be two differentiable maps between differential or Frölicher spaces and Q = Y / ' where ' is the smallest equivalence relation with the property that y ' y 0 if y = f (x) and y 0 = g(x) for some x ∈ X. Furthermore, suppose that q : Y → Q is the set quotient map and Q acquires a final structure in differential spaces or Frölicher spaces via this quotient map. In both differential spaces, as it is easy to see, and Frölicher spaces (see [4]) FQ = {f : Q → IR|f ◦ q ∈ FY }.Thus, the inclusion functor of FRL into DSP preserves coequalizers and, because every colimit can be formed from coproducts and coequalizers, the inclusion functor preserves colimits. Now, for a differential space X̂ = (X, FX ) let Υ(X̂) = (X, ΓFX , ΦΓFX ) be the associated Frölicher space. One can prove: LEMMA 4.1. The association of a Frölicher space Υ(X̂) to a differential space X̂induces a functor DSP → FRL. Proof. Let f : X̂ → Ŷ be a map of differential spaces. We need to show that f is a smooth map of the associated Frölicher spaces. The rest of the proof is transparent. Let c ∈ ΓFX . We must then show that f ◦ c ∈ ΓFY . But, f ◦ c ∈ ΓFY if and only if h ◦ f ◦ c is smooth for all h ∈ FY . But, h ◦ f ∈ FX and this is thus the case. For a differential space X̂, the identity set map IX : Υ(X̂) → X̂ is a map of differential spaces which one can use to satisfy the solution set condition in Freyd’s Adjoint Functor Theorem (see [6]). One thus obtains: correctdiffeo2.tex; 4/05/2001; 11:30; p.11 12 PROPOSITION 4.1. The inclusion functor FRL → DSP has a right adjoint. It is, in fact, easy to show that the right adjoint is the functor Υ. Note that, for differential spaces, tangent cones are defined to be the tangent cones of the associated Frölicher spaces. 5. The non-Cartesian closedness of DSP The category of topological spaces is not Cartesian closed; so, in algebraic topology, where Cartesian closedness is useful, the category of compactly generated spaces was considered as an alternative (see [6]). The rationals form a non-compactly generated space and in the | , IR) discussion below we try to see why hom-sets such as HomDSP (Q should provide examples where the Cartesian closedness of DSP fails. We will show that two possible ways to begin constructing Cartesian | plays a role) fail and that the set mappings closedness for DSP (where Q defining sets and FRL to be Cartesian closed do not work for the category of differential spaces. − Let X be a differential space and X̂ the associated smooth space. The identity map I : X̂ → X is a map of differential spaces which induces a map I : HomDSP (X, IR) → HomDSP (X̂, IR) = HomF RL (X̂, IR). The set HomFRL (X̂, IR), since FRL is Cartesian closed, can be given an FRL structure and, via I, one can put an initial structure on G = HomDSP (X, IR) in DSP. Will G with this initial structure be under set adjunction a Cartesian closed object in DSP? In other words, is there a natural bijection θ : HomDSP (Z, G) → HomDSP (Z × X, IR) | be viewed as a differential where θ(f )(z, x) = f (z)(x)? Let X = Q | , IR) consists | subspace of IR. Since Q̂ is discrete, Ĝ = HomDSP (Q̂ | → IR. Also, I : G → Ĝ is the inclusion map. The of all maps Q̂ scalars FG on G are generated by functions of the form f ◦ I where f ∈ FĜ . However, f ∈ FĜ if and only if f ∈ HomDSP ( a | q∈Q q, IR) ∼ = Y ˆ IR = IR, | q∈Q correctdiffeo2.tex; 4/05/2001; 11:30; p.12 13 ˆ Let Z = IR. A map l : IR → where f is identified with (f (q)) ∈ IR. G sending t to lt is smooth if and only if I ◦ l is smooth and using our earlier statement about the functions on G if and only if I(lt ) = (lt (q))q∈Q | (3) is smooth for each t in each coordinate. However, the corresponding | → IR in differential spaces is differentiable if and only map ˆl : IR×Q ˆ if l(t, q) = lt (q) is smooth in an IR2 -neighborhood of each point | . Clearly, however, smoothness of the expression in (t, q) ∈ IR × Q (3) just above does not imply the smoothness required for ˆl. Thus, Cartesian closedness in DSP cannot be obtained by putting an initial structure on G. − We now attempt to make G, as above, into a differential space by identifying G with the set of maps f : IR − I → IR where I ranges over the discrete subsets of IR consisting exclusively of irrationals. This identification is possible since every f ∈ G is smooth in a neighborhood of each rational. Let I be a discrete subset of IR consisting exclusively of irrationals. Then, set HI = {f : IR → IR|f is smooth except at point in I} . Notice that every map in HI restricts to a unique map in G and that every map in G that extends to a map in HI does so uniquely. One can thus view G as a direct limit in sets of the HI . Let Ie = {(i, j)|i, j ∈ I, i < j and no l ∈ I is strictly between i and j} . It is easy to see that HI , since FRL is Cartesian closed, can be endowed with the Frölicher structure of the Frölicher space HomFRL ( a IRj , IR) j∈Ie where IRj is a copy of IR. Since DSP is topological over sets, G can be viewed as a differential space and the direct limit of the HI in DSP, where I < I 0 if I ⊂ I 0 . We then ask if G with this structure is a Cartesian object in DSP. Since coproducts and coequalizers are the same taken over Frölicher spaces in DSP, as we have seen earlier, G will also be an Frölicher space. Now take a path c : IR → G where c(a) ∈ HI in DSP. A scalar g : G → IR is essentially a collection of scalars gI : HI → IR such that if I ⊂ I 0 , then the restriction of gI 0 to HI is gI . This implies that if c(t) ∈ HI for t in an open interval O containing a, then the associated map correctdiffeo2.tex; 4/05/2001; 11:30; p.13 14 ĉ : O × Q → IR extends to a map O × IR → IR which is smooth except possibly on lines y = i, i ∈ I. However, without writing down the detail, it is not difficult, for a rational number q, to find a sequence rn of irrational numbers converging to q and a sequence of numbers an converging to a such that c(an ) is not extendable to a smooth function at rn for each n ∈ IN . Then, the associated map ĉ will not be smooth at (a, q) which is impossible if one wishes G to be a Cartesian object under the map making sets Cartesian closed. IN defined earlier. Suppose that − Consider now the Frölicher space IR⊕ DSP is Cartesian closed under the usual set adjunction and K deIN , IR) notes the differential space with underlying set HomDSP (IR⊕ and arising from the Cartesian closedness of DSP. Let K be the associated Frölicher space and g : IR → K a smooth map. Since IN → IR g : IR → K is also a differentiable map, the map ĝ : IR × IR⊕ defined by setting ĝ(t, x) = g(t)(x) is also a differentiable map (using the assumed Cartesian closedness of DSP). Suppose that IN taken in DSP is also the product in FRL. the product IR × IR⊕ Then, ĝ is a smooth map. Hence, using the Cartesian closedness of FRL, g must also define a smooth map g : IR → L where L IN , IR) denotes the Frölicher space with underlying set HomFRL (IR⊕ arising from the Cartesian closedness of FRL. Thus, every contour of K is a contour of L. IN → IR is Let g : IR → L be a contour of L. Then ĝ : IR × IR⊕ IN a smooth map with the product in FRL. Again, since IR × IR⊕ is also the product in differential spaces, g : IR → K must be a differential map and induce a smooth map g : IR → K. Thus, L and K must have the same contours and are thus equal. Finally, if DSP is Cartesian closed we have, using the natural bijection from sets: IN IN HomDSP (IR⊕ × IR⊕ , IR) I N ∼ = HomFRL (IR , K) ⊕ ∼ HomDSP (IRIN , K) = ⊕ IN ∼ , L) = HomFRL (IR⊕ IN ∼ = HomF RL (IR × IRIN , IR) ⊕ ⊕ where the products are taken in the appropriate categories. Since IN with itself is different in DSP and FRL, it the product of IR⊕ is clearly impossible to have all these bijections. Thus DSP is not Cartesian closed under the canonical set adjunction. correctdiffeo2.tex; 4/05/2001; 11:30; p.14 15 IN taken in DSP is not the product in FRL, If the product IR × IR⊕ following similar arguments, one sees that DSP is not Cartesian closed under the canonical set adjunction. − In a Cartesian closed category, products commute with coequalizers and it would be useful to show that DSP is not Cartesian closed using this fact. 6. Tangent and cotangent bundles on a Frölicher space Let X and Y be Frölicher spaces. Then, (X, Y ) will denote the Frölicher space whose underlying set is HomFRL (X, Y ) and with structure arising from the Cartesian closedness of FRL. We then write FX = (X, IR) to denote FX with its associated Frölicher structure. Using the notation of Definition 2.1, the sets DX = {D : FX → IR|D is a derivation at some P ∈ X} and DX,C = {Vc ∈ DX |c ∈ CX , c(0) = P and P ∈ X} can be viewed as Frölicher subspaces of (FX , IR) and the product X × DX is a Frölicher space. We are now able to make the following definition: DEFINITION 6.1. The tangent bundle T X (resp., tangent cone bundle TCX) on X is the Frölicher subspace of X × DX (resp., X ×DX,C )consisting of all (P, D) such that D is a derivation at P . The projection map π : T X → X (resp., Π : T CX → X) is the smooth map sending (P, D) to P . A vector field on X is (most properly) a section of Π or (more generally) π. Remarks : The notion of tangent bundle can clearly be extended to differential spaces. In fact, for differential spaces, there are other, not quite so canonical ways to define tangent bundles. See [9]. However, | , it is clear that the tangent bundle of a looking at the rationals Q differential space need not coincide with the tangent bundle of the associated Frölicher space. The tangent cone bundle of a differential space can be defined as the tangent cone bundle of the associated Frölicher space. This may be useful in some circumstances, but, as | , information might be lost in certain cases. in the case of Q Example 7 : Let X = IRn . Then, there is a bijection δ : T X → IRn × IRn correctdiffeo2.tex; 4/05/2001; 11:30; p.15 16 P where if (P, D) ∈ T X and D = ni=1 di ei , using the notation in Nakahara [7], then δ(P, D) = (P, (di )). A map ζ : IR → T X is a contour if and only if, in ζ(t) = (c(t), n X di (t)ei,c(t) ), i=1 P c(t) is a contour of X and ni=1 di (t)ei,c(t) is a contour of DX where P ∂ ei,c(t) denotes the partial ∂x taken at c(t). But, ni=1 di (t)ei,c(t) is i smooth if and only if the map µ : IR × FX → IR sending (t, f ) to Pn d (t)e i (f )(c(t)) is smooth. Taking f = xi , one sees that di (t) i i=1 must be a smooth function of t for i = 1, · · · , n. The map µ is smooth if and only if, for any smooth map IR → FX sending s → fs , di (t) and ei (fs )(c(t)), for i = 1, · · · , n are smooth in s and t. But, ˆ ei (fs )(c(t)) = ∂ f (s,c(t)) and fˆ(s, t) = fs (t) is smooth in s and t. It ∂xi follows that ζ is smooth if and only if c(t) and di (t), i = 1, · · · , n are smooth. In that event, the bijection δ is a smooth isomorphism. Let X be an n-dimensional smooth manifold and U an open coordinate neighborhood of X isomorphic as a smooth manifold to IRn . One can extend the above arguments to show that π −1 (U ) is isomorphic to IRn × IRn as a Frölicher space. It is then clear that the tangent bundle defined for X as a smooth manifold is smooth isomorphic to TX when X is viewed as a Frölicher space. Let E be a Frölicher space and π : E → X be a smooth map such that π −1 (x) is a given vector space or a vector space aside from the fact that the addition is only partially defined, for each x ∈ X. Then, by LIN(E, R) we mean the Frölicher subspace of (E, IR) consisting of all f such that f restricted to π −1 (x) satisfies f (ca) = cf (a)(c ∈ IR) and f (a + b) = f (a) + f (b), whenever a + b is defined, for each x ∈ X. Define a relation ' on LIN = X × LIN(T X, IR) by setting (P, f ) ' (Q, g) if and only if P = Q and, for some neighborhood U of P, f |π−1 (U ) = g|π−1 (U ) . DEFINITION 6.2. The cotangent bundle T ∗ X to a Frölicher space is LIN/ '. The projection map π10 : T ∗ X → X is the smooth map induced from the projection of LIN onto its first factor. Replacing T X by T CX, one obtains the notion of cotangent cone bundle, denoted T ∗ CX and where the projection is Π10 . A section s of π10 or Π01 will be called a (smooth) 1-form provided that, for each point P ∈ X, P has a neighborhood U such that s(x) = [(x, s0 )] for some fixed s0 ∈ LIN(T X, IR) and all x ∈ U . Thus, s is represented locally by a global section. correctdiffeo2.tex; 4/05/2001; 11:30; p.16 17 Remarks : Notice that the fiber (π10 )−1 (P ) above P has an induced vector space structure. One commonly writes T ∗ XP = (π10 )−1 (P ). We would prefer to omit the condition given on the section s but it is not clear whether this is possible and, in practice, one needs to work at P ∈ X with an element of LIN(T X, IR) restricted to a suitable neighborhood of P . Example 8 : Let X = IRn and q : LIN → T ∗ X be the quotient map. A map τ : IR → T ∗ X is smooth if and only if f ◦ τ is smooth for each smooth map f : T ∗ X → IR. But f is smooth if and only if f ◦ q is smooth on LIN. Let [(P, L)] be an equivalence class in T ∗ X and Pn L|π−1 (P ) = i=1 ai,P dxi |P . There is then a bijection υ : T ∗ X → IRn × IRn sending [(P, L)] to (P, (ai,P )). We wish to show that this bijection is a smooth isomorphism. Let c : IR → T ∗ X be a contour and write υ(t) = υ ◦ c(t) = (Pt , (ai,Pt (t))). Using the fact that q is a quotient, one sees that there is a smooth map ψ : T ∗ X → X such that ψ ◦ q = λ1 where λ1 is projection onto the first factor of LIN. As λ1 (Pt , Lt ) = Pt is smooth in t and ψ = P1 ◦ υ = π10 , where P1 is the projection of IRn × IRn onto the first of two factors, υ is smooth in its first coordinate. Define Θi : LIN → IR P by setting Θi (P, L) = ai,P where L|π−1 (P ) = ni=1 ai,P dxi |P . Again, as q is a set quotient, there is a map ρi : T ∗ X → IR such that ρi ◦ q = Θi . We wish first to show that Θi is smooth. Let d : IR → LIN be a smooth map and d(t) = (Pt , Lt ). Then, Lt is smooth in t if and only if the associated map L̂ : IR × T X → IR defined by L̂(t, v) = Lt (v) is smooth. Identifying T X with IRn × IRn , as above, one can write L̂(t, P, w) = A(t, P ) · w where A(t, P ) is a matrix with entries which are smooth functions of t and P , w is written as a column vector and · is matrix multiplication. ∂ . Then, with w viewed as a column vector, Let w = ∂x i ai,Pt (t) = Θi ◦ d(t) = A(t, Pt ) · w . Hence, Θi and thus also the ρi are smooth. Since ρi = Pn+i ◦ υ, where Pn+i is projection on the n + i-th coordinate of IRn × IRn , υ must be smooth. Conversely, suppose that the ai,PP t (t) for i = 1, · · · , n and Pt are smooth functions of t. Let β(t) = (Pt , ni=1 ai,Pt (t)dxi |Pt ). Then the second component of β is smooth since the induced map sending (t, (Pt , bi ) to P aiPt bi , arising from the Cartesian closedness of FRL and using the identification of T X with IRn × IRn , is clearly smooth. As β is then smooth, so is q ◦ β. Hence, υ −1 is smooth and the conclusion sought follows. correctdiffeo2.tex; 4/05/2001; 11:30; p.17 18 Let E1 , · · · , En be Frölicher spaces and πi : Ei → X be a smooth map for i = 1, · · · , n such that πi−1 (P ) is a given vector space or a vector space aside from the fact that the addition is only partially defined, for each P ∈ X. Let Π∗ : E1 ⊕E2 · · ·⊕En → X be the pullback of π1 ×π2 × · · · × πn : E1 × E2 × · · · × En → X n along the diagonal map δ : X → X n which is clearly smooth. The set E1 ⊕ E2 · · · ⊕ En is usually called the Whitney product of E1 , E2 , · · · , En . By MULTILIN(E1 , · · · , En ; IR) we mean the Frölicher subspace of (E1 ⊕ E2 · · · ⊕ En , IR) consisting of all f such that 1. f (a1 , · · · , cai , · · · , an ) = cf (a1 , · · · , ai , · · · , an ) for c ∈ IR, (a1 , · · · , an ) ∈ (π ∗ )−1 (P ) and P ∈ X. 2. f ((a1 , · · · , ai−1 , bi , ai+1 , · · · , an ) + (a1 , · · · , ai−1 , ci , ai+1 , · · · , an )) = f (a1 , · · · , ai−1 , bi , ai+1 , · · · , an ) + f (a1 , · · · , ai−1 , ci , ai+1 , · · · , an ) for (a1 , · · · , ai−1 , bi , ai+1 , · · · , an ), (a1 , · · · , ai−1 , ci , ai+1 , · · · , an ) in (π ∗ )−1 (P ) and P ∈ X but only if (a1 , · · · , ai−1 , bi , ai+1 , · · · , an ) + (a1 , · · · , ai−1 , ci , ai+1 , · · · , an ) is defined. Define a relation ≈ on MULTILIN(E1 , · · · , En ) = X × MULTILIN(E1 , · · · , En ; IR) by setting (P, f ) ≈ (Q, g) if and only if P = Q and there is a neighborhood U of P such that f |(Π∗ )−1 (U ) = g|(Π∗ )−1 (U ) . DEFINITION 6.3. The higher level tensor bundle T pq X of type (p, q) is equal to the Frölicher space MULTILIN(T ∗ X, · · · , T ∗ X , T X, · · · , T X )/ ≈ | {z p } | {z q } correctdiffeo2.tex; 4/05/2001; 11:30; p.18 19 where T ∗ X appears in the first p factors and T X appears in the last q factors. Replacing T X by T CX and T ∗ X by T ∗ CX, one obtains the tensor cone bundle. The projections π qp and Πpq are the evident ones. A section s of π pq or Πpq is called a (p, q)-tensor field if, for each P ∈ X, there is a neighborhood U of P such that s(Q) = [(Q, K)] for some K ∈ MULTILIN(T ∗ X, · · · , T ∗ X, T X, · · · , T X; IR) and all Q ∈ U . A pointwise cone Riemannian metric g is a (0, 2)-tensor field such that, if gP is the restriction of g to (Π02 )−1 (P ), then 1. gP (U, V ) = gP (V, U ). 2. The induced fiber preserving smooth map (g∗ ) : T CX → LIN(T CX, IR), where (g∗ )P (X)(Y ) = gP (X, Y ), X, Y ∈ π −1 (P ) and P ∈ X, is a smooth isomorphism onto its image. A pointwise Riemannian metric is defined by taking π 02 instead of Π02 , T X instead of T CX and requiring, in addition, (g∗ )P to be bijective. Remarks : In the case of finite dimensional smooth manifolds, condition (2) of Definition 6.3 is usually replaced by the condition: gP (U, V ) = 0 for all U ∈ T XP implies V = 0. However, a Frölicher space may have infinite dimensional tangent spaces and this necessitates the revised requirement. Since the fibers of π : T X → X need not be finite dimensional, T 10 need not be the same as T X. Thus, we distinguish between ordinary and higher level tensor bundles, which involve taking duals. Application of g∗ in tensor analysis is lowering of indices. Let X be an Frölicher space and c : I = [0, 1] → X be a smooth map. Suppose that c(a) = P . Then, P has a neighborhood U such that g(x) = [(x, g ∗ )] for each x ∈ U . As g ∗ ∈ LIN(T X, T X; IR) and d : IR → T X × T X defined by setting d(t) = (c0 (t), c0 (t)) is smooth, g ∗ (c0 (t), c0 (t)) is smooth for t in a neighborhood of a. Letting g P = gP |π−1 (P )×π−1 (P ) if g = [(P, gP )], g c(t) (c0 (t), c0 (t)) = g ∗ (c0 (t), c0 (t)) is a real R valued smooth function for t near a and hence for all t ∈ I. Thus, c ds can be defined as Z 1q 0 g c(t) (c0 (t), c0 (t))dt. correctdiffeo2.tex; 4/05/2001; 11:30; p.19 20 Example 9 : Let X ⊂ IR2 be the union of the x and y axes. Let α : IR → IR be a smooth function such that α(t) = 0 for |t| ≤ 81 and α(t) = t for t ≥ 21 . Define c : IR → X by setting c(t) = (α(t), α(−t)). The curve c(t) starts at (0, 1) and passing through (0, 0) proceeds to (1, 0). Define the Riemannian metric in the usual way except at (0, 0). Thus, g = dx along the x axis and g = dy along the y axis. At (0, 0), let g = dx + dy. Then, Z c ds = − Z 0 α‘(−y)dy + −1 Z 1 α‘(x)dx. 0 It is now clear that the notion of geodesic can be extended to Frölicher spaces. 7. Derivation of Einstein tensor We will show how one can formally define the Einstein tensor by looking at the tensor bundles of a Frölicher space. On singular spaces it would be better to look at the tangent cone bundle but the definitions require more care. Our references are [7],[13]. DEFINITION 7.1. Let A be an Frölicher space, f ∈ FA and X [A] be the set of smooth sections of the projection π : T A → A,i.e., the smooth vector fields of A. The covariant derivative ∇X (f ) of f or the directional derivative X[f ] of f is the element in FA defined by setting ∇X (f )(a) = X[f ]a = q(X[a])(f ) where, if X(a) = (a, Da ), then q(X(a)) = Da . A smooth map ∇ : X [A] × X [A] → X [A] is a covariant derivative or connection if 1. ∇ is bilinear. 2. ∇f X Y = f ∇X Y for f ∈ FA . 3. ∇X (f Y ) = X[f ]Y + f ∇X Y (the Leibniz rule). 4. ∇ is induced from an smooth map ∇0 : X [A] × T A → T A, mapping fiber to fiber, on setting (∇X )Y (P ) = ∇0X (Y (P )). correctdiffeo2.tex; 4/05/2001; 11:30; p.20 21 Remark: The last condition in our definition of connection is normally implicit in the theory of connections on a finite dimensional smooth manifold. Let F (0,1) be the collection of smooth (0, 1)-tensor fields or, in other words, smooth 1-forms. Define h : F (0,1) → LIN(T A; IR) by letting h(ω)(X(Q)) = ω P (X(Q)), for Q in a neighborhood of P , where ω(P ) = [(P, ω P )] and ω P defines ω in a neighborhood of P . Here, we write X(P ) merely to indicate that X(P ) ∈ T XP . As h ◦ q(P, ω P )(X(Q)) = ω P (X(Q)), for Q in a neighborhood of P , h ◦ q and hence h is smooth. In order for ∇Y , where Y is a smooth vector field, to satisfy the Leibniz rule ∇Y (h(ω)(X)) = ∇Y h(ω)(X) + h(ω)(∇0Y X) (4) for X ∈ T A. Using condition (4) in the definition of connection (Definition 7.1), ∇Y h(ω) : T A → IR, defined by equation (4) directly above, is smooth. Thus, (P, ∇Y h(ω)) ∈ A × LIN(T A; IR) and hence the assignment P → [(P, ∇Y h(ω))] defines a smooth 1-form ∇Y ω. In a similar way, one defines h : F (0,2) → MULTILIN(T A, T A; IR). The relation ∇Z (h(g)(X, Y )) = (∇Z h(g))(X, Y ) + h(g)(∇0Z X, Y ) + h(g)(X, ∇0Z (Y )) implies that the map ∇Z h(g) : T A × T A → IR is smooth and finally one lets ∇Z g be the map sending P to [(P, ∇Z h(g))]. We note, for the following definition, that the set of vector fields X [A] on a Frölicher space A has a Lie product. Let X, Y ∈ X [A], X(P ) = (P, DPX ) and Y (P ) = (P, DPY ). Since multiplication on IR is smooth, the map A → T A defined by setting [X, Y ](P ) = (P, DPX DPY − DPY DPX ) is smooth and we have our Lie product. DEFINITION 7.2. 1. The covariant derivative ∇ is a metric connection if and only if ∇X g = 0 for each X ∈ X [A]. 2. The smooth map T : X [A] × X [A] → X [A] defined by setting T (X, Y ) = ∇X Y − ∇Y X − [X, Y ] is called the torsion tensor. correctdiffeo2.tex; 4/05/2001; 11:30; p.21 22 3. The smooth map R : X [A]×X [A]×X [A] → X [A] defined by setting R(X, Y, Z) = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z is called the Riemannian curvature tensor. Let A be a smooth finite dimensional manifold. Let LIN(X [A]; X (A)) be the set of smooth linear maps X [A] → X [A] induced from pointwise linear maps of the tangent bundle. One can show that there is a trace function TR : LIN(X [A]; X [A]) → FA , additive and invariant under coordinate change. The Riemannian curvature tensor R : X [A] × X [A] × X (A) → X [A] induces, because of the Cartesian closedness of FRL, a map R : X [A] × X [A] → LIN(X [A]; X [A]). The map TR ◦ R : X [A] × X [A] → FA is a contraction of R and one calls RIC = TR ◦ R the Ricci tensor. Let g be a Riemannian metric on a smooth space. DEFINITION 7.3. 1. Suppose that R : X [A] × X [A] → LIN(X [A]; X [A]) is the map induced from R (using the Cartesian closedness of FRL and condition 4 of Definition 7.1) and one possesses a trace function TR : LIN(X [A]; X [A]) → FA . Then, the Ricci tensor RIC : X [A] × X [A] → FA is defined by setting RIC = TR ◦ R. 2. Because of the Cartesian closedness of FRL, the Ricci tensor RIC(X, Y ) induces a smooth linear map [ : X [A] → LIN(X [A], FA ). RIC 3. If the smooth map g∗∗ : X [A] → LIN(X [A], FA ) (defined by setting g∗∗ (Q)(X)(Y ) = h(X(Q), Y (Q)) where g = [(Q, h)] for Q in a neighborhood of P ) is a smooth isomorphism, then g is called a global Riemannian metric. 4. For a global Riemannian metric, the scalar curvature R on a smooth space A is defined by setting −1 [ ◦ RIC). R = TR(g∗∗ 5. Finally, for a global Riemannian metric, G = RIC − 21 Rg is called the Einstein tensor. Remark: In the case of finite dimensional smooth manifolds, a pointwise Riemannian metric is a global Riemannian metric. The same assertion is unclear for general smooth spaces. correctdiffeo2.tex; 4/05/2001; 11:30; p.22 23 In conclusion, one sees that categorical language surrounding the category FRL can fix many of the terms in mathematical physics in a natural way and provide some clues to the direction of future endeavors. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. J. Boman, Differentiability of a Function and of its Compositions with Functions of One Variable, Math. Scand. 20(1967),249-268. G. C. L. Brümmer, Topological Categories, Topology and its Applications 18(1984),27-41. P. Cherenack, Smooth Homotopy, Bolyai Society Mathematical Studies: Topology with Applications,Szekszárd (Hungary),1993, pp. 47-70,4. A. Frölicher and A. Kriegl, Linear Spaces and Differentiation Theory. 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