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Topological *-autonomous categories, revisited

Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing object.


Introduction
This paper is an updated version of [Barr, 2006]. In the course of preparing some lectures on the subject, I discovered to my great chagrin that that paper was badly flawed. Several of the arguments had gaps or flaws. In the process of repairing them, I discovered that the main results were not only correct, but that better results were available. This updated paper is the result. [Mackey, 1945] introduced the category of pairs of vector spaces, equipped with a bilinear pairing into the ground field. It is likely that he viewed this abstract duality as a replacement for the topology. See also [Mackey, 1946], the review of the latter paper by Dieudonné as well as Dieudonné's review of [Arens, 1947], for a clear expression of this point of view. In this paper we fully answer this question. [Barr, 2000] showed that the full subcategory of the category of (real or complex) topological vector spaces that consists of the Mackey spaces (defined in 2.7 below)is *autonomous and equivalent to both the full subcategory of weakly topologized topological vector spaces and to the full subcategory of topological vector spaces topologized with the strong, or Mackey topology. This means, first, that those subcategories can, in principle at least, be studied without taking the topology into consideration. Second it implies that both of those categories are * -autonomous.
André Joyal has raised the question whether there was a similar result for vector spaces over the field Q p of p-adic rationals. This was mentioned in the original paper, but oddly that situation was not explored. Thinking about this question, I realized that there is a useful general theorem that answers this question for any locally compact field and also for locally compact abelian groups.
The results in this paper support the following conclusion. Let K be a spherically complete field (defined below) and |K| its underlying discrete field. Then the following five categories are equivalent: 1. chu(K-Vect, |K|) (Section 3) 2. The category V w (K) of topological K-spaces topologized with the weak topology for all their continuous linear functionals into K.
3. The category V s (K) of topological K-spaces topologized with the strong topology (see Section 2) for all their continuous linear functionals into K.
4. The category V w (|K|) of topological |K|-spaces topologized with the weak topology for all their continuous linear functionals into |K|.
5. The category V s (|K|) of topological |K|-spaces topologized with the strong topology (see Section 2) for all their continuous linear functionals into |K|.
A normed field is spherically complete if any family of closed balls with the finite intersection property has non-empty intersection. A locally compact field is spherically complete (so this answers Joyal's question since Q p , along with its finite extensions, is locally compact) and spherically complete is known to be strictly strong than complete. [Barr, 2006, Section 2] is a result on adjoint functors that is interesting and possibly new. The argument given there is flawed. Although the result is bypassed in the current paper, the result seemed interesting enough to give a full proof of. This appears as an appendix to this paper.
1.1. Terminology. We assume that all topological objects are Hausdorff. As we will see, each of the categories contains an object K with special properties. It will be convenient to call a morphism V G G K a functional on V . In the case of abelian groups, the word "character" would be more appropriate, but it is convenient to have one word. In a similar vein, we may refer to a mapping of topological abelian groups as "linear" to mean additive. We will be dealing with topological objects in categories of topological vector spaces and abelian groups. If V is such an object, we will denote by |V | the underlying vector space or group.
If K is a topological field, we will say that a vector space is linearly discrete if it is a categorical sum of copies of the field.
2. The strong and weak topologies 2.1. Blanket assumptions. Throughout this section, we make the following assumptions.

1.
A is an additive equational closed symmetric monoidal category and T is the category of topological A-algebras.
2. K is a uniformly complete object of T .
3. there is a neighbourhood U of 0 in K such that (a) U contains no non-zero subobject; (b) whenever ϕ : T G G K is such that ϕ −1 (U) is open, then ϕ is continuous.

Lemma.
Suppose there is an embedding T G G i∈I T i and there is a morphism ϕ : T G G K. Then there is a finite subset J ⊆ I and a commutative diagram Proof. Since ϕ −1 (U) is a neighbourhood of 0 in T , it must be the meet with T of a neighbourhood of 0 in i∈I T i . From the definition of the product topology, we must have a finite subset J ⊆ I and neighbourhoods U j of 0 in T j such that But the latter is a subobject and therefore must be 0. Now let topologized as a subspace of j∈J T 0 and ϕ 0 be the induced map. It is immediate that ϕ −1 0 (U) ⊇ j∈J U j which is a neighbourhood of 0 in the induced topology and hence ϕ 0 is continuous. Finally, since K is complete, we can replace T 0 by its closure in j∈J T j .
2.3. Theorem. Suppose S is a full subcategory of T that is closed under finite products and closed subobjects and that K ∈ S satisfies the assumptions in 2.1. If V is the closure of S under all products and all subobjects and K is injective in S, then it is also injective in V . Proof. It is sufficient to show that if V ⊆ i∈I S i with each S i ∈ S, then every morphism V G G K extends to the product. But the object V 0 constructed in the preceding lemma is a closed suboject of j∈J S j so that V 0 ∈ S and the fact that K is injective in S completes the proof.

Definition. A bijective morphism
Of course, a bijective morphism induces an injection so the only issue is whether the induced map is a surjection.
2.5. Proposition. A finite product of weak isomorphisms is a weak isomorphism.
Proof. Assume that J is a finite set and for each j ∈ J, V j G G V ′ j is a weak isomorphism. Then since finite products are the same as finite sums in an additive category, we have 2.6. Theorem. Assume the conditions of Theorem 2.3 and also suppose that for every object of S, and therefore of V , there are enough functionals to separate points. Then for every object V of V , there are weak isomorphisms τ V G G V G G σV with the property that σV has the coarsest topology that has the same functionals as V and τ V has the finest topology that has same functionals as V .
Proof. The argument for σ is standard. Simply retopologize V as a subspace of K Hom(V,K) .
Let {V i G G V } range over the isomorphism classes of weak isomorphisms to V . We define τ V as the pullback in The bottom map is the diagonal and is a topological embedding so that the top map is also a topological embedding. We must show that every functional on τ V is continuous on V . Let ϕ be a functional on τ V . From injectivity, it extends to a functional ψ on V i . By Lemma 2.2, there is a finite subset J ⊆ I and a functional ψ 0 on j∈J V j such that ψ The dashed arrow exists because of Proposition 2.5, which completes the proof. 2.7. Remark. We will call the topologies on σV and τ V the weak and strong topologies, respectively. They are the coarsest and finest topology that have the same underlying A structure and the same functionals as V . The strong topology is also called the Mackey topology.
so that the topology on W ′′ is at least as fine as that of W ′ . Then we have a commutative diagram Apply Hom(−, K) and use the injectivity of K to get: The bottom arrow is a bijection and the left hand arrow is a surjection, which implies that the top arrow is a surjection.
Proof. For σ, this is easy. If f : W G G V is a morphism, the induced σf : σW G G σV will be continuous if and only if its composite with every functional on V is a functional on W , which obviously holds.
To see that τ is a functor, suppose f : It is a weak isomorphism by the preceding proposition. Thus we get τ W is also a weak isomorphism so that τ V is one of the factors in the computation of τ V ′ and then τ V ′ G G τ V is a continuous bijection, while the other direction is evident.

Proposition. For any
Proof. It is easiest to assume that the underlying objects |V | = |σV | = |τ V | and similarly for V ′ . Then for any f : Thus the two composition of the two maps below and give the identity in each direction.
Let V w ⊆ V and V s ⊆ V denote the full subcategories of weak and strong objects, respectively. Then as an immediate corollary to the preceding, we have:

Chu and chu
Now we add to the assumptions on A that it be a symmetric monoidal closed category in which the underlying set of A −• B is Hom (A, B). We denote by E and M the classes of surjections and injections, respectively. We briefly review the categories Chu(A, K) and chu(A, K). See [Barr, 1998] for details. The first has a objects pairs (A, X) of objects of A equipped with a "pairing" commutes. This says that f a, y = a, gy for all a ∈ A and y ∈ Y . This can be enriched over A by internalizing this definition as follows. Note first that the map The category Chu(A, K) is complete (and, of course, cocomplete). The limit of a diagram is calculated using the limit of the first coordinate and the colimit of the second. The full subcategory chu(A, K) ⊆ Chu(A, K) consists of those objects (A, X) for which the two transposes of A ⊗ X G G K are injective homomorphisms. When A G G G G X −• K, the pair is called separated and when X G G G G A −• K, it is called extensional. In the general case, one must choose a factorization system (E , M ) and assume that the arrows in E are epic and that M is stable under −•, but here these conditions are clear. Let us denote by Chu s (A, K) the full subcategory of separated pairs and by Chu e (A, K) the full subcategory of extensional pairs.
The inclusion Chu s (A, K) G G Chu(A, K) has a left adjoint S and the inclusion Chu e (A, K) G G Chu(A, K) has a right adjoint E. Moreover, S takes an extensional pair into an extensional one and E does the dual. In addition, when (A, X) and (B, Y ) are separated and extensional, (A, X) −• (B, Y ) is separated but not necessarily extensional and, dually, (A, X) ⊗ (B, Y ) is extensional, but not necessarily separated. Thus we must apply the reflector to the internal hom and the coreflector to the tensor, but everything works out and chu(A, K) is also * -autonomous. See [Barr, 1998] for details.
In the chu category, one sees immediately that in a map (f, g) : (A, X) G G (B, Y ), f and g determine each other uniquely. So a map could just as well be described as an f : A G G B such that x.ỹ ∈ X for every y ∈ Y . Hereỹ : B G G K is the evaluation at y ∈ Y of the exponential transpose Y G G B −• K. Although the situation in the category of abelian groups is as described, in the case of vector spaces over a field, the hom and tensor of two separated extensional pairs turns out to be separated and extensional already ( [Barr, 1996]).

The main theorem
4.1. Theorem. Assume the hypotheses of Theorem 2.6 and also assume that the canonical map I G G K −• K is an isomorphism. Then the categories of weak spaces and strong spaces are equivalent to each other and to chu(A, K) and are thus * -autonomous.
Proof. The first claim is just Theorem 7.7. Now define F : V G G chu by F (V ) = (|V |, Hom(V, K)) with evaluation as pairing. We first define the right adjoint R of F . Let R(A, X) be the object A, topologized as a subobject of K X . Since it is already inside a power of K, it has the weak topology. Let f : |V | G G A be a homomorphism such that for all x ∈ X,x.f ∈ Hom(V, K). This just means that the composite V G G R(A, X) G G K X πx G G K is continuous for all x ∈ X, exactly what is required for the map into R(A, X) to be continuous. The uniqueness of f is clear and this establishes the right adjunction.
We next claim that F R ∼ = Id. That is equivalent to showing that Hom(R(A, X), K) = X. Suppose ϕ : R(A, X) G G K is a functional. By injectivity, it extends to a ψ : K X G G K. It follows from 2.2, there is a finite set of elements x 1 , . . . , x n ∈ X and morphisms θ 1 , . . . , θ n such that ψ factors as K X G G K n (θ 1 ,...,θn) G G K. Applied to R(A, X), this means that ϕ(a) = θ 1 x 1 , a +· · ·+ θ n x n , a . But the θ i ∈ I and the tensor products are over I so that the pairing is a homomorphism A ⊗ I X G G K. This means that ϕ(a) = θ 1 x 1 + · · · θ n x n , a and θ n x 1 + · · · + θ n x n ∈ X.
Finally, we claim that RF = S, the left adjoint of the inclusion V w ⊆ V . If V ∈ V , then RF V = R(|V |, Hom(V, K)) which is just V with the weak topology it inherits from K Hom(V,K) , exactly the definition of SV . It follows that F |V w is an equivalence.
Since V w and V s are equivalent to a * -autonomous category, they are * -autonomous.
The fact that the categories of weak and Mackey spaces are equivalent was shown, for the case of B (Banach) spaces in [Dunford & Schwartz, 1958, Theorem 15, p. 422]. Presumably, the general case has also been long known, but I am not aware of a reference.

Examples.
Example 1. Vector spaces over a locally compact field. For the second example, let K be a locally compact field. Locally compact fields have been classified, see [Pontrjagin, 1968] or [Weil, 1967]. Besides the discrete fields and the real and complex numbers, they come in two varieties. The first are finite algebraic extensions of the field Q p which is the completion of the rational field in the p-adic norm. The second are finite algebraic extensions of the field S p , which is the completion in t-adic norm of the field Z p {t} of Laurent series over the field of p elements. Notice that all these locally compact fields are normed.
We take for S the category of normed linear K-spaces, except in the case that K is discrete, we require also that the spaces have the discrete norm. We know that K is injective in the discrete case. The injectivity of K in the real or complex case is just the Hahn-Banach theorem, which has been generalized ultrametric fields according to the following, found in [Robert, 2000].

Theorem. [Ingelton]
Let K be a spherically complete ultrametric field. E a Knormed space and v a subspace of E. For every bounded linear functional ϕ defined on V , there exists a bounded linear functional ψ defined on E whose restriction to V is ϕ and such that ||ϕ|| = ||ψ||.
An ultrametric is a metric for which the ultratriangle inequality, ||x + y|| ≤ ||x|| ∨ ||y||, holds. This is obviously true for p-adic and t-adic norms. Spherically complete means that the meet of any descending sequence of closed balls is non-empty. This is known to be satisfied by locally compact ultrametric spaces.
Regardless of the topology on a field K (assuming it is topological field), K is its own endomorphism ring.
Notice that if K is non-discrete, then what we have established is that both V s and V w are equivalent to chu(Vect-|K|, |K|). But exactly the same considerations show that the same is true if we ignore the topology on K and use the discrete norm. The category S will now be the category of discrete finite-dimensional |K|-vector spaces. Its product and subobject closure will consist of spaces that are mostly not discrete, but there are still full subcategories of weakly and strongly topologized spaces within this category and they are also equivalent to chu(Vect-|K|, |K|).
Thus, these categories really do not depend on the topologies. Another interpretation is that this demonstrates that, for these spaces, the space of functionals replaces the topology, which was arguably Mackey's original intention.
Example 2. Locally compact abelian groups. For the abelian groups, we take for V the category of those abelian that a subgroups (with the induced topology) of products of locally compact abelian groups. The object K in this case is the circle group R/Z. A simple representation of this group is as the closed interval [−1/2, 1/2] with the endpoints identified and addition mod 1. The group is compact. Let U be the open interval (−1/3, 1/3). It is easy to see that any non-zero point in that interval, added to itself sufficiently often, eventually escapes that neighborhood so that U contains no non-zero subgroup. It is well-known that the endomorphism group of the circle is Z. If Then the {U i } form a neighborhood base in K and one readily sees that f −1 (U i ) ⊆ T i which implies that f is continuous.
We take for S the category of locally compact abelian groups. The fact that K is an injective follows directly from the Pontrjagin duality theorem. A result [Glicksberg, 1962, Theorem 1.1] says that every locally compact group is strongly topologized. Thus both categories of weakly topologized and strongly topologized groups that are subobjects of products of locally compact abelian groups are equivalent to chu(Ab, |K|) and thus are *-autonomous.
We can ask if the same trick of replacing K = R/Z by |K|, as in the first example, can work. It doesn't appear so. While Hom(K, K) = Z, the endomorphism ring of |K| has cardinality 2 c and is non-commutative, so we cannot draw no useful inference about maps from |K| n G G |K|, even for finite n.
Example 3. Modules over a self injective cogenerator. If we examine the considerations that are used in vector spaces over a field, it is clear that what is used is that a field is both an injective module over itself and a cogenerator in the category of vector spaces. Then if K is a such a commutative ring, we can let T be the category of topological K-modules, S be the full subcategory of submodules of finite powers of K with the discrete topology and V the limit closure of S. Then chu(Mod K , K) is equivalent to each of the categories V s and V w of topological K-modules that are strongly and weakly topologized, respectively, with respect to their continuous linear functionals into K. We now show that there is a class of commutative rings with that property. Let k be a field and K = k[x]/(x n ). When n = 2, this is called the ring of dual numbers over k.

Proposition. K is self injective.
We base this proof on the following well-known fact: 5.3. Lemma. Let k be a commutative ring, K is a k-algebra, Q an injective k-module, and P a flat right K-module then Hom k (K, Q) is an injective K-module.
The K-module structure on the Hom set is given by (rf )(a) = f (ar) for r ∈ K and a ∈ P .
Proof. Suppose A G G G G B is an injective homomorphism of K-modules. Then we have Hom R (B, Hom k (P, Q)) Hom R (A, Hom k (P, Q)) G G Hom R (A, Hom k (P, Q)) Hom k (P ⊗ R A, Q) ∼ = and the flatness of P , combined with the injectivity of Q force the bottom arrow to be a surjection.
Proof of 5.2. From the lemma it follows that Hom k (K, k) is a K-injective. We claim that, as K-modules, Hom k (K, k) ∼ = K. To see this, we map f : K G G Hom k (K, k). Since these are vector spaces over k, we begin with a k-linear map and show it is K-linear. A k-basis for K is given by 1, x, . . . , x n−1 . We define f (x i ) : K G G k for 0 ≤≤ n − 1 by f (x i )(x j ) = δ i+j,n (the Kronecker δ). For this to be K-linear, we must show that Clearly, the f (x i ), for 0 ≤ i ≤ n are linearly independent and so f is an isomorphism.

Proposition.
K is a cogenerator in the category of K-modules.
Proof. Using the injectivity, it suffices to show that every cyclic module can be embedded into K. Suppose M is a cyclic module with generator m. Let i be the first power for which x i m = 0. I claim that m, xm, . . . , x i−1 m are linearly independent over k. If not, suppose that λ 0 m + λ 1 xm + · · · λ i−1 x i−1 m = 0 and not all coefficients 0. Let λ j be the first non-zero coefficient, so that λ j x j + · · · + λ i−1 x i−1 m = 0. Multiply this by x i−j−1 and use that x l m = 0 for l ≥ i to get λ j x i−1 m = 0. But by assumption, x i−1 m = 0 so that this would imply that λ j = 0, contrary to hypothesis. Thus there is a k-linear map f : M G G K given by f (x j m) = x n−i+j . Since the x j are linearly independent, this is k-linear and then it is clearly K-linear.

Interpretation of the dual of an internal hom
These remarks are especially relevant to the vector spaces, although they are appropriate to the other examples. The fact that (U −• V ) * ∼ = U −• V * can be interpreted that the dual of U −• V is a subspace of V −• U, namely those linear transformations of finite rank. An element of the form u ⊗ v * acts as a linear transformation by the formula (u ⊗ v * )(v) = v, v * u. This is a transformation of row rank 1. Sums of these elements is similarly an element of finite rank.
This observation generalizes the fact that in the category of finite dimensional vector spaces, we have that (U −• V ) * ∼ = V −• U (such a category is called a compact *autonomous category). In fact, Halmos avoids the complications of the definition of tensor products in that case by defining U ⊗V as the dual of the space of bilinear forms on U ⊕V , which is quite clearly equivalent to the dual of U −• V * ∼ = V −• U * ( [Halmos, 1958, Page 40]). (Incidentally, it might be somewhat pedantic to point out that Halmos's definition makes no sense since U ⊕ V is a vector space in its own right and a bilinear form on a vector space is absurd. It would have been better to use the equivalent form above or to define Bilin(U, V ).) Since linear transformations of finite rank are probably not of much interest in the theory of topological vector spaces, this may explain why the internal hom was not pursued. 7. Appendix: Some generalities on adjoints.
In an earlier version of this paper, the proof of 2.13 was based on some formal results of adjoints. The argument got greatly simplified and these results were not needed, largely because of the concreteness of the categories involved. Still, it seemed worthwhile to include these formal results. which implies that the back adjunction F RB ′ G G B ′ is also an isomorphism, which is possible only if R is full and faithful. The reverse implication is just the dual.