Theory and Applications of Categories, Vol. 24, No. 14, 2010, pp. 378–393.

ON ∗-AUTONOMOUS CATEGORIES OF TOPOLOGICAL MODULES

MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL

Abstract. Let R be a commutative ring whose complete ring of quotients is Rinjective. We show that the category of topological R-modules contains a full subcategory that is ∗-autonomous using R itself as dualizing object. In order to do this, we develop a new variation on the category chu(D, R), where D is the category of discrete R-modules: the high wide subcategory, which we show equivalent to the category of reflexive topological modules.

1. Introduction

Recall that a ∗-autonomous category C is a category that has an internal hom that we will denote − −∘ − and an object ⊥, called a dualizing object, with respect to which the canonical map C [figure] (C −∘ ⊥) −∘ ⊥ is an isomorphism for every object C. See, for example, [Barr (1999)] for more details. More generally, in a symmetric closed category (which will usually be monoidal), say that the object C is ⊥-reflexive if the canonical map C [figure] (C −∘ ⊥) −∘ ⊥ is an isomorphism. We can then ask if the full subcategory of reflexive objects is closed under the internal hom. If it is, then that subcategory will be ∗-autonomous. This question of whether the internal hom of two reflexive objects is still reflexive is too general to hope for a full answer. However, we will be tackling a special case here.

The study of ∗-autonomous category, and of duality in general, has almost invariably been carried out under the hypothesis that the dualizing object is injective. The Hahn-Banach theorem, central to the study of topological vector spaces, is just the assertion that the ground field (R or C) is injective in the category of locally convex spaces. The injectivity of the circle group R/Z in locally compact abelian groups is central to the duality theory of that category. And it is the extension of that injectivity to the category of subobjects of arbitrary products of locally compact abelian groups that allows the construction of ∗-autonomous categories of such groups, see [Barr & Kleisli (2001)]. What is different in this paper is the that the dualizing object is not assumed to be injective.

ON ∗-AUTONOMOUS CATEGORIES OF TOPOLOGICAL MODULES  379

Throughout this paper we will suppose, unless explicitly mentioned otherwise, that R is a commutative ring with unit whose complete ring of quotients (see Section 2) is Rinjective. A topological R-module is R-cogenerated if it can be embedded, algebraically and topologically, in a power of the ring R. Throughout this paper when R appears in a topological context, it will bear the discrete topology.

The first and third authors would like to thank NSERC of Canada for its support of this research.

We would all like to thank McGill and Concordia Universities for partial support of the second author’s visits to Montreal.

Received by the editors 2009-10-23 and, in revised form, 2010-09-20.

Transmitted by F. William Lawvere. Published on 2010-09-21.

2000 Mathematics Subject Classification: 16D90, 18D10.

Key words and phrases: ∗-autonomous, chu category, high and wide subcategory.

\begin{matrix}c \\ ◯ \end{matrix} Michael Barr, John F. Kennison, and R. Raphael, 2010. Permission to copy for private use granted.

If M is such a module, its dual M^{∗} is the set of all continuous homomorphisms to R, topologized as a subspace of R^{||M||}, where ||M|| is the set underlying M and R is, as always, topologized discretely. It turns out that the canonical homomorphism M [figure] M^{∗∗} is always a topological embedding. When it is an isomorphism, we say that M is reflexive. If M and N are two R-cogenerated topological R-modules, we define M −∘ N as the set of continuous homomorphisms M [figure] N topologized as a subspace of all functions M [figure] N, that is as a subspace of N^{||M||}. We should point out that we are not claiming, at this point, that −∘ is the internal hom of a closed category, although it will follow when we have finally proved the next result, that it is.

The main purpose of this paper is to prove the following (Section 5).

1.1. Theorem. When M and N are reflexive modules as described above, then so is M −∘ N. The category of such reflexive modules is ∗-autonomous.

We can do even better when R is self-injective (that is, injective as module over itself). Given any topological R-module M, there is a strongest and a weakest topology on the module underlying M that has the same continuous homomorphisms to R as M does. Call these extremal topologies weak and strong.

1.2. Theorem. [See Section 6.] When R is self-injective the full subcategories of weak and strong topological R-modules are equivalent to each other and each is ∗-autonomous.

The connection between the two theorems is that the category in the first theorem turns out to be the category of weak topological R-modules.

The complete ring of quotients of any ring R without nilpotents is R-injective, but there are many other examples. See Section 2 below for a fuller discussion.

Fields (and arbitrary products of fields) are self-injective, but there are many other such rings. For example, when K is a field and p ∈ K[x] is any non-zero polynomial, the residue ring R = K[x]/(p) will always be self-injective, [Nicholson & Yousif (2003), 1.5 (3)]. But if p has repeated factors (in a splitting field of K), R will even have nilpotents.

The main tool used here is the “Chu construction”, see [Barr (1999)]. However, in addition to having to restrict to the full subcategory of separated extensional Chu objects, it turns out that a further restriction is needed in the absence of injectivity of the dualizing object. This led to the introduction of the “high” and “wide” subcategories of chu objects, see 4.8.

2. The ring R

All objects we study in this note are modules over a commutative ring R of which we make one further assumption: that the complete ring of quotients of R be R-injective (for which it is necessary and sufficient that the complete ring of quotients be self-injective).

An ideal I of a commutative ring R is called dense if whenever 0 ≠ r ∈ R, then rI ≠ 0. The complete ring of quotients Q of R is characterized by the fact that it is an essential extension of R and every homomorphism from a dense ideal to Q can be extended to a homomorphism R [figure] Q. Details are found in [Lambek (1986), Sections 2.3 and 4.3].

It is worth going into a bit more detail about the reference [Lambek (1986)]. In section 2.3, the complete ring of quotients for commutative rings is constructed, while in 4.3 the construction is carried out in the non-commutative case, where the definition of “dense” is more complicated. Unfortunately, all the discussion of injectivity is carried out in the latter section and it is not easy to work out reasonable conditions under which the complete ring of quotients is injective. Here is one simple case, although far from the only one.

2.1. Example. Let K be a field. Each of the rings K[x]/(x)^{2} and K[x, y]/(x, y)^{2}, can be readily seen to be its own complete rings of quotients (they have no proper dense ideals), but the first is and the second is not self-injective. In the second case the ideal (x, y) contains every proper ideal (and is therefore large, as defined in the third paragraph below), but is not dense.

Let A be an R-module. An element a ∈ A is called a weak torsion element if there is a dense ideal I ⊆ R with aI = 0. We say that A is weak torsion module if every element of A is and that A is weak torsion free if it contains no non-zero weak torsion elements.

An R-module is said to be R-cogenerated if it can be embedded into a power of R. A topological R-module is called R-cogenerated if it can be embedded algebraically and topologically into a power of R, with R topologized discretely. Among other things, this implies that the topology is generated by (translates of) the open submodules.

An ideal I ⊆ R is called large if its intersection with every non-zero ideal is non-zero. An obvious Zorn’s lemma argument shows that if every map from a large ideal of R to a module Q can be extended to a map R [figure] Q, then this is true for all ideals and so Q is R-injective. A dense ideal is characterized by the fact that its product with every non-zero ideal is non-zero. If, for example, there are no nilpotents, then when the product of two ideals is non-zero, so is their intersection and then the complete ring of quotients is injective. But while this condition is sufficient, it is far from necessary as the example K[x]/(x)^{2} makes clear.

3. The category C

Let C denote the category of R-cogenerated topological R-modules. When C is an object of C, we let |C| denote the discrete module underlying C and let ||C|| denote the discrete set underlying C. If C and C′ are objects of C, we let hom(C, C′) denote the R-module of continuous R-linear homomorphisms and Hom(C, C′) denote the set ||hom(C, C′)||.

We denote by C^{⋆} the module hom(C, R) topologized as a subspace of R^{||C||}.

3.1. Proposition. For C ∈ C, the canonical map C [figure] R^{||C^{⋆}||} is a topological embedding.

Proof. Let C [figure] R^{S} define the topology on C. Then for each s ∈ S, the composite C [figure] R^{S} [figure: p_{s}] R, where p_{s} is the projection, defines a homomorphism C [figure] R. Thus there is a function S [figure] Hom(R, C) that leads to the commutative triangle

[figure: C; R^{S}; C; R^{Hom(C,R)}; R^{S}; R^{Hom(C,R)}]

and the diagonal map, being an initial factor of a topological embedding, is one itself.

For later use, we prove a theorem of a purely topological nature. It is true for arbitrary topological R-modules, not just for objects of C. Compare the proof of Theorem 3.1 in [Barr (2006)].

3.2. Theorem. Suppose A ⊆ ∏_{s∈S} A_{s} is an embedding of topological modules and D is a discrete module. Then every continuous map A [figure] D factors through a quotient of A of the form A/(A ∩_{(∏s∈S−S_{0})})A_{s} where S_{0} is a finite subset of S.

Proof. If f : A [figure] D is given, then ker f must be open. A subbasic open neighbourhood of 0 in the product consists of a set of the form A ∩ (U × ∏_{s≠s_{1}} A_{s}) with U open in A_{s_{1}}. In particular, every subbasic open neighbourhood at 0 includes a set of the form A∩(∏_{s≠s_{1}} A_{s}) and every basic neighbourhood of 0, being a finite intersection of subbasic

sets, contains a set of the form A ∩ (∏_{s∉S_{0}} A_{s}) which implies that for some choice of a finite set S_{0}, f vanishes on A ∩ (∏_{s∈S_{0}} A_{s}).

In fact, by analyzing this argument we find that hom(−, D) commutes with filtered colimits (a product is a filtered colimit of the finite products) whenever D contains a neighbourhood of 0 that contains no non-zero submodule. However, we require only the form stated here.

3.3. Corollary. For any family {A_{s}} of topological modules and any discrete object D, we have that the canonical map ∑ hom(A_{s}, D) [figure] hom(∏ A_{s}, D) is an isomorphism.

Crucial to this paper is the following theorem, which is proved in [Barr, et. al. (2009), Corollary 3.8]. It is understood that R always carries the discrete topology. Incidentally, this is the only place that the injectivity condition is used.

3.4. Theorem. Let C [figure] C′ be an algebraic and topological inclusion between objects of C. Then the cokernel of hom(C′, R) [figure] hom(C, R) is weak torsion.

4. The Chu category

We begin this section with a brief description of the Chu categories and chu categories that concern us in this article. They are instances of a much more general construction. For more details, see [Barr (1998), Barr (1999)].

4.1. Definition. We denote by D the category of (discrete) R-modules. We define a category Chu(D, R) as follows. An object of Chu(D, R) is a pair (A, X) in which A and X are R-modules together with a pairing ⟨−, −⟩ : A ⊗ X [figure] R. A morphism (f, g) : (A, X) [figure] (B, Y) consists of R-linear homomorphisms f : A [figure] B and g : Y [figure] X such that ⟨fa, y⟩ = ⟨a, gy⟩ for a ∈ A and y ∈ Y. The definition of morphism is equivalent to the commutativity of either of the squares

[figure: B; hom(Y, R); A; B; f; A; hom(X, R); hom(X, R); hom(Y, R); hom(g,R); X; hom(A, R); Y; X; g; Y; hom(B, R); hom(B, R); hom(A, R); hom(f,R)]  (∗)

in which the horizontal arrows are the adjoint transposes of the two ⟨−, −⟩.

One way of getting an object of Chu(D, R) is to begin with a topological R-module C and forming the pair (|C|, hom(C, R)) with the pairing given by evaluation. Then, if C and C′ are topological modules, it follows immediately that every continuous homomorphism from C to C′ induces a morphism (|C|, hom(C, R)) [figure] (|C′|, hom(C′, R)).

If U = (A, X) is an object of Chu(D, R), we denote by U^{⊥} the object (X, A) with the evident pairing. If U = (A, X) and V = (B, Y) are objects of Chu(D, R), the set of morphisms U [figure] V is obviously an R-module that we denote [U, V]. Then Chu(D, R) becomes a ∗-autonomous category when we define

U −∘ V = ([U, V], A ⊗ Y)

with pairing ⟨(f, g), (a, y)⟩ = ⟨fa, y⟩ = ⟨a, gy⟩. When R = (R, R) with the R-module structure as pairing, one easily sees that U −∘ R = U^{⊥}. We call R the dualizing object. There is also a tensor product given by

U ⊗ V = (U −∘ V^{⊥})^{⊥}

We say that the object (A, X) is separated if the induced map A [figure] hom(X, R) is monic and that it is extensional if X [figure] hom(A, R) is monic. (Incidentally, extensionality is the property of functions that two are equal if they are equal for all possible arguments. Thus extensionality here means that, in effect, X is a module of homomorphisms A [figure] R). A pair is called non-singular if it is both separated and extensional. This means that for all 0 ≠ a ∈ A there is an x ∈ X with ⟨a, x⟩ ≠ 0 and, symmetrically, that for all 0 ≠ x ∈ X, there is an a ∈ A with ⟨a, x⟩ ≠ 0. The results in the theorem that follows are proved in detail in [Barr (1998)]. Since D is abelian the factorization referred to in that citation can only be the standard one into epics and monics. Let Chu_{s}(D, R), Chu_{e}(D, R), and chu(D, R) denote, respectively, the full subcategories of Chu(D, R) consisting of the separated, the extensional, and the separated extensional objects.

4.2. Theorem. [Barr (1998)]

1. The inclusion Chu_{s}(D, R) [figure]Chu(D, R) has a left adjoint S;

2. the inclusion Chu_{e}(D, R) [figure]Chu(D, R) has a right adjoint E;

3. SE ≅ ES;

4. when (A, X) is extensional and (B, Y) is separated, then (A, X) −∘(B, Y) is separated; equivalently, the tensor product of extensional objects is extensional;

5. chu(D, R) becomes a ∗-autonomous category when we define

(A, X) −∘(B, Y) = E((A, X) −∘(B, Y))
(A, X) ⊗ (B, Y) = S((A, X) ⊗ (B, Y))

where the right hand sides of these formulas refer to the operations in Chu(D, R) and the left hand sides define the operations in chu(D, R).

4.3. Conventions. Here and later, when S is a set and A is an object of some category, we denote by S · A an S-fold coproduct of copies of A. When A = R, this is simply the free R-module generated by S.

From now on, we will be restricting our attention to objects of chu(D, R), unless explicitly stated otherwise.

4.4. Self-injective rings. In a number of cases the category of chu objects is equivalent to one or more concrete categories of topological objects. See [Barr & Kleisli (2001)] or the unpublished note [Barr, (unpublished)] for an example that generalizes Pontrjagin duality.

In order to motivate what follows, we begin with the special case in which R is selfinjective. This case is treated in more detail in Section 6 below. If U = (A, X) is an object of chu(D, R), the separability condition implies that A [figure] R^{||X||}. Let σU denote the topological module whose underlying module is A and whose topology is such that σU ⊆ R^{||X||} is a topological embedding, with R topologized discretely. We will see in 6.1 below that R continues to be injective in a category B that contains C as a full subcategory; a fortiori R is injective in C with respect to topological embeddings.

We claim that σU determines U. Observe that we can view hom(σU, R) as a submodule of hom(A, R) consisting of the maps continuous in the topology induced by R^{||X||}. Moreover, each element of X is continuous in that topology, so that we have X ⊆ hom(σU, R) ⊆ hom(A, R). It follows from Corollary 3.3 that the canonical map ||X|| · R [figure] hom(R^{||X||}, R) is an isomorphism. Injectivity of R applied to the embedding σU [figure] R^{||X||} implies that ||X|| · R [figure] hom(σU, R) is surjective. The image of this map is the submodule of hom(σU, R) generated by the elements of X, which is X itself, so that X = Hom(σU, A). Thus we conclude that X ≅ hom(σU, R) and so we recover U from σU as (|σU|, hom(σU, R)).

This argument evidently depends on the injectivity of R. In order to deal with the case that R is not injective (but its complete ring of quotients is) we will have to introduce new subcategories of chu(D, R).

4.5. The functors σ and ρ. We introduce functors σ : chu(D, R) [figure] C and ρ : C [figure]chu(D, R) as follows. If U = (A, X) is an object of chu(D, R), then σU is the module A topologized as a subobject of R^{||X||}. If C is an object of C, let ρC = (|C|, |C^{⋆}|) with the obvious pairing.

4.6. Proposition. ρ is left adjoint to σ.

Proof. As seen in Diagram (∗) on Page 382, a map ρC [figure] (A, X) is given by R-linear homomorphisms |C| [figure] A and X [figure] |C^{⋆}| for which the left-hand square of

[figure: A; Hom(X, R); |C|; A; |C|; Hom(|C^{⋆}|, R); Hom(|C^{⋆}|, R); Hom(X, R); Hom(X, R); R^{||X||}; Hom(|C^{⋆}|, R); Hom(X, R); Hom(|C^{⋆}|, R); R^{||C^{⋆}||}; R^{||C^{⋆}||}; R^{||X||}]

commutes, while the right-hand one obviously does. But the commutation of the outer square is the condition required for continuity of C [figure] σ(A, X) when A is topologized by the embedding into R^{||X||}. Thus we get a map C [figure] σ(A, X).

In the other direction, a map C [figure] σ(A, X) consists of a map |C| [figure] A for which the composite C [figure] A [figure] R^{||X||} is continuous. Dualizing gives a map X · R [figure] |C^{⋆}| which when composed with the inclusion X [figure] X · R gives a map X [figure] |C^{⋆}|. This function is actually a homomorphism as it factors X [figure] X · R [figure] Hom(A, R) [figure] |C^{⋆}| and the composite of the first two as well as the third are homomorphisms. The remaining details are left to the reader.

4.7. Proposition. For any C ∈ C, the inner adjunction C [figure] σρC is an isomorphism.

Proof. This follows from the facts that ρC = (|C|, |C^{⋆}|) and that σρC is just |C| topologized as a subspace of R^{||C^{⋆}||}, which is just the original topology on C by 3.1.

Next we identify the objects of chu(D, R) on which ρσ is the identity. If U = (A, X) is an object of chu(D, R), then σU = A, topologized as a subobject of R^{||X||}. Then ρσU = (A, hom(σU, R)). Since the elements of X induce continuous maps on σU, we have X [figure] hom(σU, R) [figure] hom(A, R). Since U is extensional the composite is monic and hence so is the first map. In other words, we can assume X ⊆ hom(σU, R). There is, however, no reason to suppose that X = hom(σU, R) so that ρσ is not generally the identity. To deal with this situation, we introduce new conditions on objects of the chu category.

4.8. Definition. We say that U is high when ρσU = U. We also say that U is wide when U^{⊥} is high.

We let chu(D, R)_{h}, chu(D, R)_{w} and chu(D, R)_{hw} denote the full subcategories of chu(D, R) consisting, respectively, of the high objects, the wide objects and the high, wide objects.

If (A, X) is an object of chu(D, R), the topology induced on A by its embedding into R^{X} has a subbase at 0 given by the kernels of the composites A [figure] R^{X} [figure: p_{x}] R. It follows that φ : A [figure] R is continuous in this topology if and only if there are finitely many elements x_{1}, …, x_{n} ∈ X such that ker φ ⊇ ⋂ ker⟨−, x_{i}⟩. Thus we conclude:

4.9. Proposition. The object (A, X) ∈ chu(D, R) is high if and only if, for all φ : A [figure] R and all x_{1}, …, x_{n} ∈ X such that ker φ ⊇ ⋂ ker⟨−, x_{i}⟩, there is an x ∈ X such that φ = ⟨−, x⟩.

4.10. Proposition. If U = (A, X) ∈ chu(D, R), then the cokernel of the map ||X|| · R [figure] σU that is induced by the inclusion U ⊆ R^{||X||} is weak torsion.

Proof. This is immediate from 3.3 and 3.4.

4.11. Proposition. The inclusion chu(D, R)_{h} [figure] chu(D, R) has a right adjoint H and the inclusion chu(D, R)_{w} [figure]chu(D, R) has a left adjoint W.

Proof. We claim that H = ρσ. In fact, suppose U is high and V is arbitrary. Then, since σ is full and faithful and ρ is its left adjoint, we have

Hom(U, ρσV) ≅ Hom(σU, σρσV) ≅ Hom(σU, σV)
≅ Hom(ρσU, V) ≅ Hom(U, V)

which shows the first claim. For the second, let W V = (H(V^{⊥}))^{⊥}.

4.12. Notation. We will denote H(A, X) by (A, X̄) since the first coordinate is A and the second is the subset of hom(A, R) consisting of the continuous maps. That X ⊆ X̄ follows from extensionality. Similarly, we denote W (A, X) by (Ā, X). However, since, as we will see in Example 4.19, HW is not naturally equivalent to WH, we will never use the ambiguous (Ā, X̄) outside of this very sentence.

4.13. Remark. Since X [figure] X̄ is essentially induced from the topological inclusion A [figure] R^{X}, it follows from Theorem 3.4 that the cokernel of X [figure] X̄ is weak torsion.

Recall from 4.1 and Theorem 4.2.4 that when U = (A, X) and V = (B, Y) are objects of chu(D, R), the tensor product in Chu(D, R) is given by (A, X) ⊗ (B, Y) = (A⊗B, [U, V^{⊥}]) and is extensional (but not generally separated). Its separated reflection is gotten by factoring out of A ⊗ B the elements that are annihilated by every map U [figure] V^{⊥}. It nonetheless makes sense to talk of continuous maps A ⊗ B [figure] R.

In the study of Chu objects that are separated and extensional, a crucial point was that the separated reflection commuted with the extensional coreflection. One would similarly hope here that the wide reflection might commute with the high coreflection. That this fails will be shown in Example 4.19. However, the only real consequence of that hoped-for commutation that matters to us remains true:

4.14. Proposition. If U is high, so is W U; dually if U is wide, so is HU.

Proof. It suffices to prove the first claim. Let U = (A, X) be high, with W U = (Ā, X). As noted above in 4.13, the cokernel of A[figure] Ā is weak torsion. Since weak torsion modules have no non-zero homomorphisms into R, we see that two homomorphisms Ā [figure] R that agree on A are equal. Now suppose that φ : Ā [figure] R and x_{1}, …, x_{n} ∈ X such that ker φ ⊇ ⋂ ker⟨−, x_{i}⟩. Then ker(φ|A) ⊇ ⋂ ker(⟨−, x_{i}⟩|A). Since A is wide, there is an x ∈ X such that φ|A = ⟨−, x⟩|A. But then φ = ⟨−, x⟩ on all of Ā.

4.15. Proposition. Let U = (A, X) and V = (B, Y) be high and extensional. Then U ⊗ V is high (and extensional).

Proof. By definition U ⊗ V = (A ⊗ B, [U, V^{⊥}]). This means that the topology on A ⊗ B is induced by its map to R^{Hom(U,V^{⊥})}. Note that even when U and V are separated this map is not necessarily injective. If φ : A ⊗ B [figure] R is continuous in this topology, then, as in Proposition 4.9, there are maps (f_{1}, g_{1}), … (f_{n}, g_{n}) : U [figure] V^{⊥} such that ker φ ⊇ ⋂ ker(f_{i}, g_{i}). Here (f_{i}, g_{i}) acts on A ⊗ B by (f_{i}, g_{i})(a ⊗ b) = ⟨b, f_{i}a⟩ = ⟨a, g_{i}b⟩. Now fix a ∈ A. If b ∈ ⋂ ker⟨−, f_{i}a⟩ then b ∈ ker φ(a, −). Since for any y ∈ Y, the map ⟨−, y⟩ is continuous on B, it follows that φ(a, −) : B [figure] R is continuous. Since B is high, there is a unique y = fa ∈ Y such that φ(a ⊗ b) = ⟨b, fa⟩ for all b ∈ B. The usual arguments involving uniqueness show that f is an R-linear homomorphism A [figure] Y. There is similarly an R-linear map g : B [figure] X such that φ(a ⊗ b) = ⟨a, gx⟩. It follows that φ is in the image of [U, V^{⊥}] [figure] hom(A ⊗ B, R). Extensionality is proved in [Barr(1998)].

4.16. Corollary. If U is high and V is wide, then U −∘ V is wide.

Proof. This is immediate from the fact that U −∘ V = (U ⊗ V^{⊥})^{⊥}.

Now we can define the ∗-autonomous structure on chu(D, R)_{hw}. Assume that U and V are high and wide. Then U ⊗ V is high and U −∘ V is wide. So we define U −_{h}−∘ V = H(U −∘ V) and U ⊗_{w} V = W (U ⊗ V).

4.17. Theorem. For any U, V, W ∈ chu(D, R)_{hw}, we have

Hom(U ⊗_{w} V, W) ≅ (U, V −_{h}−∘ W)

Proof.

Hom(U ⊗_{w} V, W) = Hom(W (U ⊗ V), W) ≅ Hom(U ⊗ V, W)
≅ Hom(U, V −∘ W) ≅ Hom(U, H(V −∘ W))
= Hom(U, V −_{h}−∘ W)

Since it is evident that U −_{h}−∘ V ≅ V^{⊥} −_{h}−∘ U^{⊥}, we conclude that chu_{hw} is a ∗- autonomous category.

4.18. Theorem. chu(D, R)_{hw} is complete and cocomplete.

Proof. chu(D, R)_{h} is a coreflective subcategory of chu(D, R) and is therefore complete and cocomplete and chu(D, R)_{hw} is a reflective subcategory of chu(D, R)_{h} and the same is true of it.

4.19. Example. In general, HW is not naturally isomorphic to WH.

Assume that R contains an element r that is neither invertible nor a zero divisor. Let (1/r)R denote the R-submodule of the classical ring of quotients (gotten by inverting all the non zero-divisors of R) generated by 1/r. Then as modules, we have proper inclusions rR ⊆ R ⊆ (1/r)R.

Let U = (R, R) and V = (R, rR) both using multiplication as pairing. It is clear that U ∈ chu(D, R)_{hw}. As for V, it is evident that σV = R. The topology is discrete since under the inclusion R ⊆ R^{||rR||}, the only element of R that goes to 0 under the projection on the coordinate r is 0, since r is not a zero-divisor. Then HV = ρσV = (R, R). Moreover if (f, g) : U [figure] V is the identity on the first coordinate and inclusion on the second, it is clear that H(f, g) is just the identity. Since HV is obviously also wide, we see that WH(f, g) is also the identity.

Now let us calculate HW V. Since V^{⊥} = (rR, R), it is clear that σ(V^{⊥}) = rR. Moreover σ(g, f) : σ(V^{⊥}) [figure] σ(U^{⊥}) is the proper inclusion rR ⊆ R and is not the identity. We can identify ρσ(V^{⊥}) as (rR, (1/r)R) so that W V = ((1/r)R, rR), which is isomorphic to U and is therefore high and wide. Thus HW V = W V (assuming, as we may, that H is the identity on chu_{h} and W is the identity on chu_{w}). So WH(f, g) is an isomorphism but HW (f, g) is not, which readily implies that WH cannot be naturally isomorphic to HW.

5. The ∗-autonomous structure in C

Recall that for C ∈ C, C^{⋆} is the object hom(C, R) topologized as a subset of R^{||C||}.

5.1. Proposition. The map C [figure] C^{⋆}^{⋆} that takes an element of C to evaluation at that element is a topological embedding.

Proof. By hypothesis, there is an embedding C [figure] R^{S} for some set S. This transposes to a function S [figure] Hom(C, R) = ||C^{⋆}|| and then R^{||C^{⋆}||} [figure] R^{S}. The diagram

[figure: C^{⋆}^{⋆}; R^{||C^{⋆}||}; C; C^{⋆}^{⋆}; C; R^{S}; R^{S}; R^{||C^{⋆}||}]

commutes and the result is that the left-hand arrow, being an initial factor of an embedding, is one itself.

We say that the object C is reflexive if the canonical map C [figure] C^{⋆}^{⋆} is an isomorphism. Let C_{r} denote the full subcategory of reflexive objects.

5.2. Proposition. For any C ∈ C, the object C^{⋆} is reflexive.

Proof. We have C [figure] C^{⋆}^{⋆} which gives C^{⋆}^{⋆}^{⋆} [figure] C^{⋆} and composes with the canonical C^{⋆} [figure] C^{⋆}^{⋆}^{⋆} to give the identity. Thus C^{⋆}^{⋆}^{⋆} ≅ C^{⋆} ⊕ C′ for a submodule C′ ⊆ C^{⋆}^{⋆}^{⋆}. In particular, C′ is weak torsion free. On the other hand, the inclusion C^{⋆}^{⋆} [figure] R^{||C^{⋆}||} gives a map ||C^{⋆}|| · R [figure] C^{⋆}^{⋆}^{⋆} whose cokernel T is, by 3.4, weak torsion. But since the canonical map ||C^{⋆}|| · R [figure] C^{⋆} is obviously surjective, we conclude that T ≅ C′, which implies that both are zero.

5.3. Proposition. Let U ∈ chu(D, R) be high. Then σU is reflexive if and only if U is also wide.

Proof. Let U = (A, X). In general (σU)^{⋆} = (X̄, A), which is the same as (X, A) = σ(U^{⊥}) since U is assumed high. The same argument implies that (σU)^{⋆}^{⋆} = σ(Ā, X) and this is σU if and only if Ā = A, that if and only if is U is also wide.

5.4. Corollary. The functors σ and ρ induce inverse equivalences between the categories chu(D, R)_{hw} and C_{r}.

5.5. The internal hom in C_{r}. The obvious candidate for an internal hom in C_{r} is to let C −∘ C′ be hom(C, C′) embedded as a topological subobject of R^{||C||×||C′||}. But it is not obvious that this defines a reflexive object. With the help of the ∗-autonomous structure on chu(D, R)_{hw}, we can prove this.

5.6. Theorem. For C, C′ ∈ C_{r}, the object C −∘ C′, as defined in the preceding paragraph, is reflexive. In fact, C −∘ C′ ≅ σ(ρC −_{h}−∘ ρC′).

Proof. For this argument, when C, C′ ∈ C_{r}, we let C −_{r}−∘ C′ = σ(ρC −_{h}−∘ ρC′). We wish to show that C −_{r}−∘ C′ = C −∘ C′. We have that

C −_{r}−∘ C′ = σ(ρC −_{h}−∘ ρC′) = σH(ρC −∘ ρC′)
= σρσ(ρC −∘ ρC′) (because H = ρσ, 4.11)
= σ(ρC −∘ ρC′) (from 4.7)

Now, we have

ρC −∘ ρC′ = ([ρC, ρC′], |C| ⊗ |C′^{⋆}|) ≅ (hom(C, C′), |C| ⊗ |C′^{⋆}|)

since ρ is full and faithful on C_{r}. Thus σ(ρC −∘ ρC′) is just hom(C, C′) equipped with the topology it inherits from R^{|||C|⊗|C′^{⋆}|||}. But the definition of C −∘ C′ is just hom(C, C′) equipped with the topology it inherits from R^{||C||×||C′^{⋆}||} so it suffices to show that those topologies coincide. There is a natural map ||C|| × ||C′^{⋆}|| [figure] |||C| ⊗ |C′^{⋆}||| which gives rise to a map R^{||C⊗C′^{⋆}||}. From the commutative triangle

[figure: hom(C, C′); R^{||C||×||C′||}; hom(C, C′); R^{|||C|⊗|C′|||}; R^{||C||×||C′||}; R^{|||C|⊗|C′|||}; ‘]

we see that hom(C, C′) [figure] R^{|||C|⊗|C′|||}, as a first factor in a topological embedding, is also a topological embedding. Thus C −∘ C′ is just σ(ρC −_{h}−∘ ρC′).

6. The case of a self-injective ring

Things get simpler and a new possibility opens when R is a commutative self-injective ring. This was done in [Barr (1999)] when R is a field and what was done there goes through unchanged when R is separable (that is, a product of finitely many fields), but there are many more rings that are self-injective, for example, an arbitrary product of fields, and any complete boolean ring.

Throughout this section we will suppose that R is a commutative self-injective ring. Let S and T be sets. We denote by Q(S, T) the module R^{S} ×|R^{T} |. The module R itself

is always topologized discretely. We let B denote the category of topological R-modules that can be embedded into a module of the form Q(S, T) for some sets S and T. The principal use we make of self-injectivity is contained in the following.

6.1. Theorem. R is injective in B with respect to topological inclusions.

Proof. (Compare the proof of Theorem 3.2) It is easy to reduce this to the case of an object embedded into an object of the form Q(S, T). So we begin with B [figure] Q(S, T) and a continuous homomorphism φ : B [figure] R. The topology on Q(S, T) has a basis at 0 of sets of the form Q(S − S_{0}, T) for a finite subset S_{0} ⊆ S. Then the kernel of φ contains a set of the form B ∩ Q(S − S_{0}, T). Let B_{0} = B/(B ∩ Q(S − S_{0}, T)). Then we get a commutative diagram

[figure: B_{0}; Q(S_{0}, T); B; B_{0}; B; Q(S, T); Q(S, T); Q(S_{0}, T); B; R; φ; B_{0}; R; φ_{0}; Q(S_{0}, T); R]

in which the diagonal map exists because B_{0} and Q(S_{0}, T) are discrete and R is injective.

6.2. Corollary. When R is injective, every object of chu(D, R) is high and wide.

Proof. The proof is left as a simple exercise (see the discussion in 4.4).

We will say that an object of B is weakly topologized if it is embedded in a power of R (with the product topology). Let B_{wk} be the full subcategory of B consisting of the weakly topologized objects. Evidently B_{wk} is the same as C, as used before this section. It is clear that every object of chu(D, R) is both high and wide and thus σ and ρ, as defined in 4.5 are actually equivalences between B_{wk} and chu(D, R). Since chu(D, R) is ∗-autonomous, so is B_{wk}. Thus we have,

6.3. Theorem. The category B_{wk} of R-cogenerated topological algebras is ∗-autonomous when B −∘ B′ is defined as hom(B, B′) topologized as subspace of B^{′||B||}.

6.4. The strongly topologized objects. Let B ∈ B. We say that B is strongly topologized if whenever B′ [figure]B is a bijection in B that induces an isomorphism Hom(B, R) [figure] Hom(B′, R), then B′ [figure] B is an isomorphism. We have already denoted the category of weakly topologized modules by B_{wk} and we denote by B_{st} the full subcategory of strongly topologized modules.

For B ∈ B, let S(B) index the set of all topological R-modules that have the same underlying R-module as B, a stronger topology and the same set of continuous homomorphisms into R.

6.5. Theorem. Let B be an object of B. Then among all objects B_{s} with s ∈ S(B), there is one whose topology is finer than all the others.

Proof. Let τB be defined so that

[figure: B; B^{S}; diag; τB; B; τB; ∏_{s∈S} B_{s}; ∏_{s∈S} B_{s}; B^{S}]

is a pullback. Since ∏ B_{s} [figure] B^{S} is a bijection, so is τB [figure] B so it suffices to show that Hom(B, R) [figure] Hom(τB, R) is an isomorphism. It is clearly monic, so it suffices to see that it is surjective. In this diagram, the bottom arrow is a topological embedding, hence so is the top arrow. If we apply the functor hom(−, R) to the above diagram and use Corollary 3.3, we get

[figure: hom(B, R); S · hom(B, R); hom(τB, R); hom(B, R); hom(τB, R); ∑ hom_{s∈S}(B_{s}, R); ∑ hom_{s∈S}(B_{s}, R); S · hom(B, R); ≅]

from which we see immediately that hom(B, R) [figure] hom(τB, R) is surjective.

This theorem and its proof are essentially identical to that of [Barr & Kleisli (2001), Theorem 4.1 (proof that 2 implies 3)]. Another approach is used for the similar [Barr & Kleisli (1999), Proposition 3.8]. The latter paper omitted, however to prove that τ was functorial. We give the proof here, essentially the same as that of the former citation.

6.6. Theorem. The object function τ determines a functor τ : B [figure] B_{st} which is right adjoint to the inclusion.

Proof. We begin by showing that whenever B′  [figure] B is an arrow in B and B_{s}  [figure] B for

s ∈ S(B), then there is an s′ ∈ S(B′) and a commutative square

[figure: B_{s}; B; B_{s′}; B_{s}; B_{s′}; B′; B′; B]

We let B′′ be a pullback B_{s} ×_{B} B′. Since B_{s} has the same underlying module as B, we can assume that B′′ has the same underlying module as B′. Thus it suffices to show it has the same set of continuous maps to R as B′ does. The map B′′ [figure] B_{s} × B′ is an equalizer of two maps B_{s} × B′ [figure] B so that B′′ is embedded in the product B_{s} × B. Then we have a commutative square

[figure: B′; B′ × B; B′′; B′; B′′; B′ × B_{s}; B′ × B_{s}; B′ × B]

If we apply the functor hom(−, R) we get the square

[figure: hom(B′, R); hom(B′, R) × hom(B, R); hom(B′′, R); hom(B′, R); hom(B′′, R); hom(B′, R) × hom(B_{s}, R); hom(B′, R) × hom(B_{s}, R); hom(B′, R) × hom(B, R); ≅]

from which it follows that hom(B′, R) [figure] hom(B′′, R) is surjective and it is obviously injective so that B′′ = B_{s′} for some s′ ∈ S(B′). This shows that τ is a functor and the adjunction is clear.

If we let ωB denote the module |B| retopologized by its embedding into R^{Hom(B,R)} then ωB is the weakest topology with the same set of homomorphisms into R as B and τB is the strongest. We leave it to the reader to prove the easy fact that ω is left adjoint to the inclusion of B_{wk} into B. Clearly τω = τ and ωτ = ω.

6.7. Theorem. τ and ω induce inverse equivalences between B_{wk} and B_{st}.

Proof. If B ∈ B_{wk}, then τB ∈ B_{st} and ωτB = ωB = B, while if B ∈ B_{st}, then ωB ∈ B_{wk} and τωB = τB = B.

Consequently, B_{st} is also ∗-autonomous. The internal hom is gotten by first forming B −∘ B′ and then applying τ.

7. Discussion

It is obvious that the ∗-autonomous structure on the category C_{r} depends crucially on 5.6. But we could not find a proof of that fact independent of the chu category, in particular, the high wide subcategory. While it is certainly true that you use the methods that work, an independent argument would still be desirable.

For example, suppose R is a ring that is not necessarily commutative. The category of two-sided R-modules has an obvious structure of a biclosed monoidal category. If A is a topological module, it has both a left dual ^{∗}A (consisting of the left R-linear homomorphisms into R) and a right dual A^{∗}. The two duals commute and there is a canonical map A [figure]^{∗}A^{∗}. It is natural to call an object reflexive if that map is an isomorphism. One can now ask if the internal homs (that is, the left and right homs) of two reflexive objects is reflexive. It might require something like the left and right complete rings of quotients being isomorphic and also R-injective. Even the case that R has no zero divisors would be interesting.

Although there a Chu construction for a biclosed monoidal category, with an infinite string of left and right duals (see [Barr, 1995]), there does not seem to be any obvious way of defining separated or extensional objects. An object might be separated, say, with respect to its right dual, but not its left. Factoring out elements that are annihilated by the left dual would usually lead to the right dual being undefined. And even if a notion of separated, extensional objects was definable, what possible functor to the topological category would exist that would transform to all the infinite string of duals? Thus we would require an independent argument for the analog of Theorem 5.6.

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Department of Mathematics and Statistics McGill University, Montreal, QC, H3A 2K6

Department of Mathematics and Computer Science Clark University, Worcester, MA 01610

Department of Mathematics and Statistics Concordia University, Montreal, QC, H4B 1R6

Email: barr@math.mcgill.ca, jkennison@clarku.edu, raphael@alcor.concordia.ca

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