Cartesian Closed Topological Categories and Tensor Products

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1 Cartesian Closed Topological Categories and Tensor Products Gavin J. Seal October 21, 2003 Abstract The projective tensor product in a category of topological R-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the cartesian closedness of X is related to the monoidal closedness of the category of R-module objects in X. 1 Introduction Because of the importance of cartesian closed categories in the study of topological spaces, it seems useful to introduce a similar notion for algebraic topological structures such as topological groups or topological vector spaces. Monoidal closedness is only partially satisfying in this regard, as it may endow the function space with uninteresting structures (see Example 2.9). Thus, the need for a more restrictive definition of a tensor product. Recall that cartesian closed categories are a particular case of (symmetric) monoidal closed categories. The category Mod R of R-modules is a crucial example of a monoidal closed category which is not cartesian closed. However, its associated bifunctor (its tensor product according to the categorical terminology) is precisely the usual tensor product, which is defined by a universal property involving the cartesian product of Set. This leads to a definition of a tensor product (not necessarily associative) in a category of R-modules which exploits the universal property. If this bifunctor is such that B is left adjoint, the category is said to be tensor closed. 1

2 In this article, we consider categories Mod R (X) of R-module objects in a category X, where R is itself a ring object in X. The category X will be concrete over Set and should be thought of as a category such as Top, Unif, or App (the categories of topological, uniform or approach spaces respectively). For example if X = Top, then Mod R (Top) will designate the category of R-module objects in Top, i.e. the category of topological R-modules. Note that with this approach, we encompass topological abelian groups and topological vector spaces in the same theory. Our aim is to study the relation between the tensor closedness of Mod R (X) and the cartesian closedness of X. The required link is provided by the free R-module functor, which is left adjoint to the forgetful functor G : Mod R (X) X. In particular, it is shown that if the base category X is cartesian closed topological (see for example [1]), then Mod R (X) is tensor closed. In this case, the category Mod R (X) is also monoidal closed with respect to the same tensor product. After establishing the definitions of a tensor product as well as certain of its properties in Section 2, we study in Section 3 a convenient base X for a category of R-module objects in a category X; this base is the smallest initially closed subcategory of X such that Mod R (X) = Mod R (X). These definitions allow us to state our main Theorem in Section 4 which gives an explicit description of how the tensor closedness of Mod R (X) and the cartesian closedness of X are linked. 1.1 Notations. A concrete category (X, U) over Set will always be supposed to have finite concrete products. The category of group objects in X will be denoted by Gr(X) (see for example [6]). In the same way, Ab(X) will denote the category of abelian group objects in X, and given a (unitary) ring object R in X, Mod R (X) will designate the category of R-module objects in X. Furthermore, Gr will be the category of groups, i.e. Gr = Gr(Set), Ab the category of abelian groups and Mod R the category of R-modules. The following maps will be used to describe the algebraic structure of objects (note that because our interest lies primarily in R-modules, all group operations will be denoted additively): α X : X X X (x, y) x + y ν X : X X x x µ X : R X X (λ, x) λx 2

3 In general, we will be working within the following diagram: Mod R (X) G X V U Mod R G Set where G and G denote the respective forgetful functors. Usually we will not distinguish between morphisms and their underlying maps (we will often write f : X Y instead of Uf : UX UY ). In order to simplify notations further, we will not distinguish either between morphisms and their image by the algebraic structure forgetful functors G and G. Finally, we recall the following result (see [12] for a more general version) which implies in particular that the forgetful functors of the form G : Mod R (X) X send V -initial sources to U-initial sources. 1.2 Proposition. Let (X, U) be a concrete category over Set having finite concrete products. Consider a group B and S = (h i : B V A i ) I a V -structured source in Gr (respectively in Ab or Mod R ). Suppose that there exists a U-initial lift L = (f i : X X i ) I of GS = (Ghi : GB UG A i ) I. Then L can be seen as a V -initial lift of S in Gr(X) (respectively in Ab(X) or Mod R (X)). Proof. The maps h i are group homomorphisms, so we have h i α B = α V Ai (h i h i ) for i I. As the right-hand side of this equality can be seen as the underlying map of an X-morphism and L is initial, we get that α B is an X-morphism. In the same way we check that ν B and µ B are X-morphisms. 2 Tensor Products Taking the definition of a tensor product in Mod R over Set as a model, we get the following definition. Note that the projective tensor product for normed or Banach vector spaces (see for example [11], Section V.20) is defined in a similar way. The notion of a universal bimorphism (see [3]) is related but does not take into account the structure of the product A B. 2.1 Definition. Let (X, U) be a concrete category over Set, and R a commutative ring object in X. The category A = Mod R (X) has tensor products if there exists a bifunctor : A A A such that: 3

4 i) for all objects A, B of A there exists a bilinear X-morphism ι : A B A B; ii) for every bilinear X-morphism f : A B C there exists a unique A-morphism f : A B C with ι f = f. Thus, the following diagram commutes in X: f A B C ι f A B The object A B is called the tensor product of A and B in A and ι is the canonical bilinear map. If the underlying set V (A B) of A B is the usual algebraic tensor product, the tensor product is said to be concrete. Recall that a category (X, U) is well-fibred if each fibre is a set and the fibre of a singleton or of the empty set contains a unique element. From now on, we will mostly be considering well-fibred topological categories, although this is often done more for convenience than necessity. 2.2 Proposition. Let X be a well-fibred topological category over Set and R a commutative ring object in X. Then Mod R (X) has concrete tensor products. Proof. If A and B are objects in A = Mod R (X), we define the concrete tensor product of A and B in the following way. Let A B be the tensor product of V A and V B in Mod R. Consider the V -structured source S = (f i : A B V C i ) I indexed by all pairs (f i, C i ) with C i Ob(A) and f i : A B V C i Mor(Mod R ) such that there exists g i : A B C i Mor(X) making the following diagram commute in Set: g i A B C i ι f i A B Let (f i : T (A, B) C i ) I denote the U-initial lift of GS. In particular, the map ι : A B V T (A, B) is an X-morphism (where A, B, C Ob(A)) if and only if g i = f i ι : A B C i are X-morphisms for all i, so that ι is naturally an X-morphism. Furthermore, the targets in S are all objects of 4

5 V A and the maps f i are linear, so it follows by Proposition 1.2 that T (A, B) is an object of A. Finally, if f : A B C is a bilinear X-morphism, we have that f : T (A, B) C is an X-morphism by definition. Therefore we can write T (A, B) as A B Ob(A) since it is indeed a (concrete) tensor product of A and B in A. 2.3 Remark. From now on, well-fibred topological categories over Set will be endowed with the concrete tensor product just described. 2.4 Proposition. Let X be a well-fibred topological category over Set, R a commutative ring object in X and A, B objects of A = Mod R (X). Then i) A B = B A; ii) R A = A; iii) A {0} = {0} = {0} A, where {0} is a terminal object in A. Proof. The proof is straightforward, so we omit the details. 2.5 Remark. Although it would seem that associativity does not follow from the definition, at the present time the author does not know of an example of a tensor product (in the above sense) that is not associative. Because of the importance of cartesian closed categories, and since the tensor product is related to the cartesian product by its definition, we introduce the notion of tensor closed categories. 2.6 Definition. A category A = Mod R (X) which has tensor products is called tensor closed provided that for each A-object B the functor ( B) : A A is left adjoint. Thus, for every object C in A there exist an object C B in A and a morphism ev : C B B C such that for each object A and morphism f : A B C there exists a unique morphism ˆf : A C B with f = ev ( ˆf id B ). The morphism ev is called an evaluation map. The following result states that, as in the case of cartesian closed categories, the object C B can be chosen to be the space of Mod R (X)-morphisms from B to C. In other words, Mod R (X) has canonical function spaces. 5

6 2.7 Theorem. Let X be a well-fibred topological category over Set, R a commutative ring object in X and A = Mod R (X). Then the following statements are equivalent: a) A is tensor closed; b) for any objects A, B, C of A, there exists an object H(B, C) of A such that an R-linear map f : A B C is an A-morphism if and only if ˆf : A H(B, C) is an A-morphism, where f(x y) = ˆf(x)(y). The object H(B, C) will be denoted by Hom A (B, C). Proof. The proof is almost identical to the corresponding proof in the context of cartesian closed categories (see [1]), so we omit the details. 2.8 Remark. If A is tensor closed and its tensor product is associative, it does not follow that it is a monoidal closed category. Indeed, the tensor product may not satisfy all the axioms for a monoidal category (see [6], Chapter VII). However, this situation does not occur when the tensor product is concrete. 2.9 Example. We will see further on that if X is a well-fibred cartesian closed topological category, then Mod R (X) is tensor closed. However, the following example is interesting. Let A, B and C be topological R-modules. By taking the final R-module topology on the set A B with respect to ι : A B A B, where A B is endowed with the topology of separate continuity, and hom A (B, C) with the topology of pointwise convergence, it is known that an R-linear map f : A B C is continuous if and only if ˆf : A homa (B, C) is. In fact, Mod R (Top) with these topologies for the tensor product and function space is monoidal closed. However, if A B is the usual cartesian product of A and B, the canonical map ι : A B A B is not necessarily continuous (see for example [9], Section III.5), thus with these structures Mod R (Top) is not tensor closed. 3 A Topological Hull Recall that the target of the forgetful functor G : Gr(Top) Top can be restricted to the category of completely regular spaces (not necessarily T 1 ), and we get a functor G : Gr(Top) CReg or even G : Gr(CReg) 6

7 CReg; remark furthermore that this last functor has a left adjoint F : CReg Gr(CReg) = Gr(Top) such that the insertion of generators η X : X G F X is an embedding (indeed, Markov s proof in [7] can be reproduced to yield this result without the T 1 condition). As CReg is topological over Set, this proves that the category of completely regular spaces is the smallest initially closed subcategory of Top containing the image of G. Because we will need a canonical full subcategory X of X to establish the relation between tensor closedness of Mod R (X) and the cartesian closedness of X, we generalize this result to a more abstract situation (note that the cartesian closed hull of Top is distinct from that of CReg). This will put forth the importance of having an embedding η X : X G F X, and will show that the topologicity of X can be seen as a consequence of this. 3.1 Setting. In this section, we will be considering the following situation. Let (A, V ) be a topological category over a category B, and (X, U) a concrete category over Y. Suppose that G : A X is a functor sending V -initial sources to U-initial sources, and (η, ɛ) : F G : B Y is an adjoint situation such that the following diagram commutes: A G X V U B G Y Then by Wyler s Taut Lift Theorem [12] the adjoint situation (η, ɛ) : F G : B Y can be lifted along V and U to (η, ɛ) : F G : A X. Note that because of the more abstract setting studied in this Section, all the (faithful) functors will be mentioned explicitly. Suppose now that (B, G) is concrete. We require that for every map f : Y UG A, there exists a unique U-initial lift f : X G A. Let M denote a class of morphisms containing all the universal arrows η Y : Y GF Y, where Y runs through the objects of Y. We define the following full subcategories of X: the objects of X are the X Ob(X) such that the universal arrow η X : X G F X is U-initial. the objects of X are the X Ob(X) such that there exists a U-initial morphism f : X G A whose underlying morphism f : UX UG A is in M; 7

8 the objects of X are the X Ob(X) such that there exists a U-initial morphism f : X G A; It is clear that X X X. The following proposition shows that these inclusions are in fact equalities. 3.2 Proposition. Let f : X G A be a U-initial morphism. η X : X G F X is also a U-initial morphism. Then the universal arrow Proof. Let g : UY UX be a Y-morphism such that η UX g : UY GF UX can be lifted to an X-morphism. Since F X is a free object over X, there exists an A-morphism f : F X A such that G f η X = f. So Uf g = UG f η UX g : UY UG A can be lifted to an X-morphism. By initiality of f we can conclude that g can also be lifted to an X-morphism and η X is U-initial. Another way of characterizing the objects of X is given by the following proposition. 3.3 Proposition. Let β : Y G F X be the U-initial lift of η UX : UX GF UX. Then G F Y = G F X, so that β = η Y and η Y is U-initial. Proof. Note that the underlying morphism of η Y and η X is the same. The initiality of β and the universality of η X : X G F X yield the identity-carried arrow G F X G F Y, and the universality of η Y yields the identity-carried arrow G F Y G F X. This shows that G F X and G F Y are isomorphic. Finally, id GF UX : GF UX UG F X has a unique U-initial lift, so we have G F Y = G F X. The following proposition demonstrates the importance of the initiality of the morphisms η X. It also gives an alternate description of the morphisms of X. 3.4 Proposition. Let X, Y be objects of X. A map f : UX UY can be lifted to an X-morphism if and only if F f : V F X V F Y can be lifted to an A- morphism. Proof. If there is an arrow f : X Y, then F f : F X F Y is a V -lift of F f. On the other hand, if g : F X F Y is a V -lift of F f, then 8

9 UG g η UX = η UY f, so by initiality of η Y X-morphism. we can conclude that f is an 3.5 Proposition. The category X is topological, and it is the smallest initially closed subcategory of X containing G A. Proof. Let (f i : Y UX i ) I be a source in Y, where the X i are objects of X. We have that (F f i : F Y V F X i ) I is a source in B for which there exists a V -initial lift (g i : A F X i ) I. Furthermore, there exists a U-initial lift β : X G A of η Y : Y GF Y = UG A, so X is an object of X. Since the η Xi : X i G F X i are initial morphisms and UG g i Uβ = GF f i η Y = η UXi f i we get a source (f i : X X i) I in X lifting (f i : Y UX i ) I. Let us prove that it is U-initial. Let f : UZ UX be a map such that f i f : UZ UX i can be lifted to an X-morphism for all i. This implies that F f i F f = F (f i f) : V F Z V F X i can be lifted to an A-morphism for all i, so we get an A-morphism h : F Z A over F f : V F Z V F X. Because β : X G A is initial and UG h η UZ = Uβ f, we conclude that f : UZ UX can be lifted to an X-morphism. Finally, it is clear that if X is an initially closed subcategory of X containing G A, then it must contain the objects X Ob(X) such that η X : X G F X is initial (this is the U-initial lift of η UX : UX UG F X). 3.6 Examples. 1. The category CReg of completely regular spaces is the smallest initially closed subcategory of Top containing the image by G of topological groups, abelian topological groups or topological R-modules if R is Tychonoff (for the construction of the free functor such that the insertion of generators is an embedding, see [7], Theorem 1 for groups and [2], Theorem for R-modules; in both cases, the proofs can be reproduced without the T 1 hypothesis on the base space). 2. In the same way, we can see that there is no proper initially closed subcategory of Unif containing the image by G of uniform groups (see [8], Section 5). 9

10 3. It can be proved (see [10]) that if X is a cartesian closed topological category over Set and A = Gr(X) (or A = Mod R (X)), then X = X. In other words, there is no proper initially closed subcategory of X containing the image by the forgetful functor of the category of group objects (respectively R-module objects) in X. 4 Closed Categories In this Section, we investigate how the cartesian closedness of a category X may be related to the tensor closedness of Mod R (X). Our main result shows that these two notions are in fact closely linked. 4.1 Lemma. Let X be a well-fibred topological cartesian closed category, X an object of X, R a ring object in X, and A an R-module object in X. Then the internal hom-object Hom X (X, A) is an R-module object in X. Proof. We set H = Hom X (X, A). As X is cartesian closed, the map α H : H H H is an X-morphism if and only if the map ev (α H id X ) : H H X A is an X-morphism (where ev : H X A is an evaluation map relatively to the cartesian closure of X). Moreover, the diagonal map X : X X X, the middle permutation map ρ : H H X X H X H X and the evaluation ev : H X A are all X-morphisms. Since ev (α H id X ) = α A (ev ev) ρ (id H H X ) we conclude that α H is an X-morphism. We also note that ev (ν H id X ) = ν A ev, from which we deduce that ν H is an X-morphism. Finally, we have that ev (µ H id X ) = µ A (id R ev), so H is indeed an R-module in X. 4.2 Proposition. Let X be a well-fibred topological cartesian closed category, and R a commutative ring object in X. Then A = Mod R (X) is tensor closed. Proof. Let A, B, C be objects in A, and H(A, B) the V -initial lift of the inclusion of hom A (A, B) into Hom X (A, B). By the preceding lemma and the fact that the inclusion is linear, we get that H(A, B) is an object of A. An R-linear map f : A B C is an A-morphism if and only if the bilinear map f : A B C is an X-morphism. Since X is well-fibred cartesian closed, this is true if and only if ˆf : A homx (B, C) is an 10

11 X-morphism, which is the case if and only if ˆf : A hom A (B, C) is an A-morphism. By Theorem 2.7, we are done. 4.3 Proposition. Let X be a well-fibred topological category, and R a commutative ring object in X. Suppose that A = Mod R (X) is tensor closed. Then for every objects Y, Z in X there exists an object Hom X (Y, Z) in X such that if ˆf : X Hom X (Y, Z) is an X-morphism, then f : X Y Z is an X-morphism, where f(x, y) = ˆf(x)(y). Proof. Let Y, Z Ob(X). Recall that hom X (Y, Z) = hom X (Y, Z) by definition of X. Any X-morphism f : X Y yields an A-morphism F f : F X F Y, so we have a map φ : hom X (Y, Z) hom A (F Y, F Z). The U-initial lift φ : H Hom A (F Y, F Z) gives us an object H = Hom X (Y, Z) of X. Suppose now that we have an arrow ˆf : X Hom X (Y, Z). This gives us an X-morphism g = φ ˆf : X Hom A (F Y, F Z), which in turn yields the A- morphism ĝ : F X Hom A (F Y, F Z) by the universal property of F X. Since A is tensor closed, we get an A-morphism g : F X F Y F Z and by definition of the tensor product, an X-morphism g : F X F Y F Z. Moreover, the inclusion map η Z : Z G F Z is U-initial and g (η X η Y )(x, y) = g(x, y) = g(x y) = ĝ(x)(y) = φ ˆf(x)(y) = ˆf(x)(y) Z for all (x, y) X Y. By setting f = g (η X η Y ), we get an X-morphism f : X Y Z such that f(x, y) = ˆf(x)(y). 4.4 Remark. Note that the condition ˆf : X hom X (Y, Z) is an X- morphism implies that f : X Y Z is an X-morphism is equivalent to the evaluation map ev : hom X (Y, Z) Y Z is an X-morphism. 4.5 Theorem. Let X be a well-fibred topological category, R a commutative ring object in X, and A = Mod R (X). The following are equivalent: a) X is cartesian closed; 11

12 b) A is tensor closed and the iso-carried morphism φ : F (X Y ) F (X) F (Y ) λ (x,y) (x, y) λ (x,y) (x y) (x,y) X Y is an A-isomorphism. Proof. (x,y) X Y a) = b). The fact that A is tensor closed follows from Proposition 4.2. We now outline the proof that φ is an isomorphism. We first mention that given an X-morphism f : X Y Z, it is possible to construct a (unique) X- morphism f : F ab X F ab Y F ab Z that extends f and such that f (x, ) : F ab Y F ab Z and f (, y) : F ab X F ab Z are group homomorphisms (where F ab denotes the free abelian group functor). This construction can be completed with standard techniques similar to those used in the construction of a free group in a cartesian closed category. We also recall that the free R-module over an object X in a cartesian closed topological category can be obtained by taking a quotient of F ab (R F ab X) (see [10]). Thus, by applying the extension described above twice, we obtain an X- morphism F ab (R F ab X) F ab (R F ab Y ) F ab (R F ab (X Y )). Finally, after taking quotients appropriately we get the required morphism F (X) F (Y ) F (X Y ). b) = a). By Proposition 4.3 and Remark 4.4, we only have to prove that if f : X Y Z is an X-morphism then ˆf : X hom X (Y, Z) is an X-morphism. Thus, let f : X Y Z be an X-morphism and consider the arrow g = F f φ 1 : F X F Y F Z. As A is tensor closed, this yields an arrow ĝ η X : X Hom A (F Y, F Z). Furthermore, ĝ η X (x)(y) = ĝ(x)(y) = F f φ 1 (x y) = F f(x, y) = f(x, y) Z, so the restriction of ĝ η X (x) to Y is a map from Y to Z, which is the restriction of an A-morphism F Y F Z. This means that ĝ η X can be seen as a map ˆf : X hom X (Y, Z), and by initiality of Hom X (Y, Z) Hom A (F Y, F Z), we can conclude that ˆf is an X-morphism. 4.6 Proposition. Let X be a well-fibred topological cartesian closed category over Set. Then A = Mod R (X) is a symmetric monoidal closed category. It follows that the free module functor F : X Mod R (X) is a morphism of monoidal closed categories. 12

13 Proof. We can prove that (A B) C = A (B C) by using the isomorphism F (A B) = F A F B and the fact that the tensor product preserve quotients (recall that it is left adjoint). Another (more classical) way to prove this is to define a trifunctor with a universal property and use the cartesian closure of X. The other conditions for (A, ) to be a symmetric monoidal category (see [4], Section 6) follow from Proposition 2.4 and the fact that the underlying set of a tensor product is the usual tensor product of R-modules. Finally, it is monoidal closed because it is tensor closed. The last statement follows easily. 4.7 Remark. Theorem 4.5 states that the category of R-module objects (for example abelian group or vector space objects) in any cartesian closed topological category is convenient to work in when considering tensor products. More surprisingly, it states that if tensor closedness is required, there is no other choice as long as the ismorphism between F (X Y ) and F X F Y holds. Furthermore, the theorem suggests that an interesting category to study topological vector spaces or abelian topological groups would be the category QTop(R) of c-spaces. Indeed, if X = Top we have that CReg = X and the cartesian closed hull of X is precisely QTop(R) (see [5]). In this sense, the category of c-spaces is the monoidal closed hull of the category of topological vector spaces. Acknowledgements The author wishes to thank Claude-Alain Faure who suggested the original idea of this paper, as well as Walter Tholen and Gábor Lukács for their support and many fruitful discussions. References [1] J. Adámek, H. Herrlich, and G. E. Strecker. Abstract and Concrete Categories. The Joy of Cats. Pure and Applied Mathematics. John Wiley, New York, [2] V. I. Arnautov, S. T. Glavatsky, and A. V. Mikhalev. Introduction to the Theory of Topological Rings and Modules. Number 197 in Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York,

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