Theory and Applications of Categories, Vol. 4, No. 6, 1998, pp. 137–147.

THE SEPARATED EXTENSIONAL CHU CATEGORY

MICHAEL BARR

Transmitted by Robert Rosebrugh

ABSTRACT. This paper shows that, given a factorization system, E/M on a closed symmetric monoidal category, the full subcategory of separated extensional objects of the Chu category is also ∗-autonomous under weaker conditions than had been given previously ([Barr, 1991)]. In the process we find conditions under which the intersection of a full reflective subcategory and its coreflective dual in a Chu category is ∗-autonomous.

1. Introduction

1.1. Chu categories. An appendix to [Barr, 1979] was an extract from the master’s thesis of P.-H. Chu that described what seemed at the time a too-simple-to-be-interesting construction of ∗-autonomous categories [Chu, 1979]. In fact, this construction, now called the Chu construction has turned out to be surprisingly interesting, both as a way of providing models of Girard’s linear logic [Seely, 1988], in theoretical computer science [Pratt, 1993a, 1993b, 1995] and as a general approach to duality [Barr and Kleisli, to apear] and [Schläpfer, 1998]..

Given an autonomous (symmetric, closed monoidal) category A and an object ⊥ of A, the category we denote Chu(A, ⊥) has as objects pairs (A_{1}, A_{2}) of objects equipped with a pairing A_{1} ⊗ A_{2} ⟶ ⊥. An arrow (f_{1}, f_{2}) : (A_{1}, A_{2}) ⟶ (B_{1}, B_{2}) consists of arrows of A, f_{1} : A_{1} ⟶ B_{1} and f_{2} : B_{2} ⟶ A_{2} such that the square

[figure: A_{1} ⊗ A_{2}; ⊥; A_{1} ⊗ B_{2}; B_{1} ⊗ B_{2}; f_{1} ⊗ B_{2}; A_{1} ⊗ f_{2}]

commutes. It is evident that the endofunctor that interchanges the two components is a contravariant equivalence, so that Chu(A, ⊥) is a self-dual category. Less obvious, but true is that, provided A has pullbacks, Chu(A, ⊥) is also an autonomous category so that it is, in fact, a ∗-autonomous category. The object ⊥ will be called the dualizing object of the Chu construction. Details can all be found in [Chu, 1979].

I would like to thank the Canadian NSERC, the FCAR of Québec, and the Swiss National Science Foundation (Project 2000-050579.97).

Received by the editors 21 April 1998 and, in revised form, 8 October 1998.

Published on 31 October 1998.

1991 Mathematics Subject Classification: 18D15, 46A20.

Key words and phrases: ∗-autonomous categories, Chu construction, separated, extensional.

\begin{matrix}c \\ ◯ \end{matrix} Michael Barr 1998. Permission to copy for private use granted.

1.2. Separated and extensional objects. Now suppose A is an autonomous category with pullbacks and E/M is a factorization system in A. See Section 5 for definitions and elementary properties of factorization systems. Let ⊥ be an arbitrary object of A, which we call our dualizing object. We will write Chu for Chu(A, ⊥) and write A^{⊥} = A −∘ ⊥ for an object A of A. We say that an object (A_{1}, A_{2}) is M-separated, or simply separated if the transpose A_{1} ⟶ A_{2}^{⊥} of the structure arrow A_{1} ⊗ A_{2} ⟶ ⊥ belongs to M and M-extensional, or simply extensional, if the other transpose A_{2} ⟶ A_{1}^{⊥} does.

We denote the full subcategories of separated objects by Chu_{s} = Chu_{s}(A, ⊥) and of extensional objects by Chu_{e} = Chu_{e}(A, ⊥). We follow Pratt in denoting the full subcategory of objects that are both separated and extensional by chu = chu(A, ⊥). Since Chu_{s} is evidently the dual of Chu_{e}, it is immediate that chu is self-dual. It is useful to ask if chu is also ∗-autonomous. In [Barr, 1991], I showed the following result (more or less). Suppose A is an autonomous category with pullbacks, D is an object of A and E/M is a factorization system on A that satisfies the following conditions

FS–1. Every arrow in E is an epimorphism;

FS–2. if m ∈ M, then for any object A of A, the induced A −∘ m is in M;

FS–3. if e ∈ E then for any object A of A, the induced e −∘ A is in M.

Then the category chu(A, D) of separated, extensional objects (with respect to M) in Chu(A, D) is a ∗-autonomous category.

The main purpose of this note is to show that FS–3 is unnecessary. We also show two conditions that are equivalent to FS–3 and draw a consequence. However, FS–1 and FS–2 still appear to be needed. It should be observed that although FS–2 and FS–3 seem in some sense to be dual, they are actually of a quite different character. For example, both extremal monics and all monics are automatically preserved by right adjoint functors, so that FS–2 is automatic in those cases. But, for example, when M is the extremal monics, E consists of all epics. While it might be reasonable that the duality take epics to monics, it seems unlikely that they take them to extremal monics.

2. Reflections in ∗-autonomous categories

2.1. We begin by treating a somewhat common situation in a ∗-autonomous category. Suppose A is a such a category and A_{l} is a reflective full subcategory with reflector l. Assume that A_{l} is isomorphism closed, although that is no fundamental importance, but simplifies the treatment. The full subcategory A_{r} = A_{l}^{⊥} is quite evidently a coreflective subcategory with rA = (lA^{⊥})^{⊥} as coreflector. Let A_{rl} = A_{r} ∩ A_{l}. Assume that when A is an object of A_{r}, then so lA, which is then an object of A_{rl}. This is readily seen to imply that when A is an object of A_{l}, so is rA. It is also immediate that under these conditions, l and r induce left and right adjoints, resp., to the inclusions of A_{rl} ⟶ A_{r} and A_{rl} ⟶ A_{l}. 2.2. Proposition. Under the above assumptions A_{r}⊗A_{r} ⊆ A_{r} if and only if A_{r} −∘ A_{l} ⊆ A_{l}.

Proof. These follow immediately from the identities, valid in any ∗-autonomous category

A −∘ B ≅ (A ⊗ B^{⊥})^{⊥}
A ⊗ B = (A −∘ B^{⊥})^{⊥}  □

2.3. Theorem. Suppose A is a ∗-autonomous category and A_{l} is a reflective subcategory whose reflector l leaves A_{l}^{⊥} invariant as in 2.1. Suppose that the unit object belongs to A_{r} and that the equivalent conditions of 2.2 are satisfied. Then A_{rl} is a ∗-autonomous category with unit object l⊤, dualizing object r⊥, tensor A □ × B = l(A ⊗ B) and A − B = r(A −∘ B).

Proof. We have, for any objects A and B of A_{rl}, using the facts that B and A −∘ B belong to A_{l},

Hom(l⊤ □ × A, B) ≅ Hom(l(l⊤ ⊗ A), B) ≅ Hom(l⊤ ⊗ A, B)
≅ Hom(l⊤, A −∘ B) ≅ Hom(⊤, A −∘ B) ≅ Hom(A, B)

and so, by Yoneda, we conclude that l⊤ □ × A ≅ A.

We have, for any objects A, B, C, and D of A_{rl},

Hom(A, (B □ × C) − D) ≅ Hom(A, r((B □ × C) −∘ D)) ≅ Hom(A, (B □ × C) −∘ D)
≅ Hom(A ⊗ (B □ × C), D) ≅ Hom((B □ × C), A −∘ D)
≅ Hom(l(B ⊗ C), A −∘ D) ≅ Hom(B ⊗ C, A −∘ D)
≅ Hom(A ⊗ B ⊗ C, D)

and

Hom(A, B − (C − D)) ≅ Hom(A, r(B −∘(C − D))) ≅ Hom(A, B −∘(C − D))
≅ Hom(A ⊗ B, C − D) ≅ Hom(A ⊗ B, r(C −∘ D))
≅ Hom(A ⊗ B, C −∘ D) ≅ Hom(A ⊗ B ⊗ C, D)

By letting A = l⊤, we see that Hom(B □ × C, D) ≅ Hom(B, C − D) and by Yoneda we get (B □ × C) − D ≅ B − (C − D), which is the internal version and implies the associativity of the tensor. Thus we have a ∗-autonomous category.

3. The separated extensional subcategory

We apply the above construction to show that under the conditions FS–1 and FS–2 on a factorization system E/M, the full subcategory of M-separated and M-extensional objects forms a ∗-autonomous category.

3.1. Theorem. Suppose the category A has pullbacks and E/M is a factorization system that satisfies FS–1 and FS–2. Then for any object D, the category chu(A, D) of Mseparated, and M-extensional objects of Chu is ∗-autonomous.

We give the proof as a series of propositions.

3.2. Proposition. The inclusion of Chu_{s} ⟶ Chu has a left adjoint.

Proof. Let A = (A_{1}, A_{2}) be an object of Chu. Factor A_{1} ⟶ A_{2}^{⊥} as A_{1} ⟶ Ã_{1} \underrightarrow{}A_{2}^{⊥}, where we adopt the usual notation of writing ⟶ for an arrow of E and \underrightarrow{}for an arrow of M. Then sA = (Ã_{1}, A_{2}) is a separated object and we have an obvious map A ⟶ sA. If (f_{1}, f_{2}) : (A_{1}, A_{2}) ⟶ (B_{1}, B_{2}) is a map in Chu(A, K) and (B_{1}, B_{2}) is separated, the unique diagonal fill-in in the square

[figure: B_{1}; B_{2}^{⊥}; A_{1}; Ã_{1}; A_{2}^{⊥}]

gives the required map sA ⟶ B.

It follows that the inclusion Chu_{e} ⟶ Chu has a right adjoint, which we denote e.

3.3. Proposition. FS–1 implies that when the object (A_{1}, A_{2}) of Chu(A, D) is separated, so is e(A_{1}, A_{2}) and similarly if (A_{1}, A_{2}) is extensional, so is s(A_{1}, A_{2}).

Proof. Suppose that A = (A_{1}, A_{2}) is a separated object. Then eA = (A_{1}, Ã_{2}) where A_{2} ⟶ Ã_{2} \underrightarrow{}A_{1}^{⊥}. By transposing, this gives A_{1} ⟶ (Ã_{2})^{⊥} ⟶ A_{2}^{⊥} whose composite belongs to M and so, by Proposition 5.6, does the first factor A_{1} ⟶ (Ã_{2})^{⊥} which means that eA is still separated. The proof that sA is extensional when A is is similar (also dual).

3.4. Proposition. FS–2 implies that when A and B are extensional, so is A ⊗ B.

Proof. Let A = (A_{1}, A_{2}) and B = (B_{1}, B_{2}) be extensional. The second component of A ⊗ B is the pullback

[figure: B_{1} −∘ A_{2}; B_{1} −∘ A_{1}^{⊥} ≅ A_{1} −∘ B_{1}^{⊥} ≅ (A_{1} ⊗ B_{1})^{⊥}; P; A_{1} −∘ B_{2}]

It follows from FS–2 that both B_{1} −∘ A_{2} ⟶ B_{1} −∘ A_{1}^{⊥} and A_{1} −∘ B_{2} ⟶ A_{1} −∘ B_{1}^{⊥} lie in M. But it is a general property of factorization systems that M is closed under pullback and composition, whence the arrow P ⟶ (A_{1} ⊗ B_{1})^{⊥} also lies in M.

The results of Section 2 now prove the theorem.

4. About FS–3

In this section, we take a brief look at FS–3. Although it is not necessary for our main theorem, it has the consequence that se = es. This obviously implies that the separated objects are invariant under the extensional coreflection and vice versa, but, as we have seen, that is true under the hypothesis FS–1. Thus our main results are true under the hypotheses FS–2 and either FS–1 or FS–3.

4.1. Proposition. the following three conditions are equivalent:

FS–3. if e ∈ E then for any object A of A, the induced e −∘ A is in M;

FS–4. if e ∈ E, then for any object B, the induced e ⊗ B ∈ E;

FS–5. If m : A ⟶ A′ ∈ M and e : C′ ⟶ C ∈ E, then

[figure: C −∘ A′; C′ −∘ A′; e −∘ A′; C −∘ A; C′ −∘ A; e −∘ A; C −∘ m; C′ −∘ m]

is a pullback;

and imply that es = se.

Proof. The filling in of the commutative diagram

[figure: C −∘ A′; C′ −∘ A′; C −∘ A; C′ −∘ A; B]

by a map B ⟶ C −∘ A is equivalent, using transpose, to a diagonal fill-in in either of the squares:

[figure: A; A′; C′ ⊗ B; C ⊗ B; B −∘ A; B −∘ A′; C′; C]

Thus from FS–5 we can infer both FS–3 and FS–4, while either of FS–3 or FS–4 similarly allows us to conclude FS–5.

Suppose that A = (A_{1}, A_{2}) is an object of Chu. Then sA = (Ã_{1}, A_{2}), where

A_{1} ⟶ Ã_{1} \underrightarrow{}A_{2}^{⊥}

is the E/M factorization. We compute eA = (A_{1}, Ã_{2}) using the factorization

A_{2} ⟶ Ã_{2} \underrightarrow{}A_{1}^{⊥}

Assuming FS–3, we have that Ã_{1}^{⊥} \underrightarrow{}A_{1}^{⊥} which gives us the square

[figure: Ã_{1}^{⊥}; A_{1}^{⊥}; A_{2}; Ã_{2}]

whose diagonal fill-in belongs to M by 5.6. But then it follows that A_{2} ⟶ Ã_{2} \underrightarrow{}Ã_{1}^{⊥} is also an E/M factorization, which implies that esA = (Ã_{1}, Ã_{2}). Dualizing, we conclude that seA is exactly the same.

4.2. An example. Here is an example using Banach spaces (see [Kleisli, et al, 1996] for the ∗-autonomous category of Banach spaces) in which FS–3 fails and es ≠ se, while FS–1 and FS–2 hold so the results of Section 3 are valid. Take the category of Banach spaces and linear contractions. It is a closed monoidal category with bounded linear maps and sup norm on the unit ball as the closed structure. The tensor product is the one with greatest cross-norm. Take as factorization system the epics and regular monics. The epics are actually maps with dense image and the regular monics are the closed isometric inclusions. Let ℓ^{p} denote the space of absolutely pth power summable sequences. Then there is a Chu object (ℓ^{1}, ℓ^{1}) with the inner product ⟨a_{i}, b_{i}⟩ = ∑ a_{i}b_{i}. Then it is not hard to see that s(ℓ^{1}, ℓ^{1}) = (ℓ^{∞}, ℓ^{1}) and that is separated and extensional, so that es(ℓ^{1}, ℓ^{1}) = (ℓ^{∞}, ℓ^{1}), while for exactly the same reason, se(ℓ^{1}, ℓ^{1}) = (ℓ^{1}, ℓ^{∞}).

5. Appendix: A factorization primer

In this section, we collect in one place well known, mainly folkloric, results on factorization systems mainly to have one place to refer to it in the future. We take the minimal hypotheses necessary to derive the standard results.

5.1. Definition. A factorization system in a category C consists of two subclasses E and M of the arrows of C subject to the conditions

FS–1. If I is the class of isomorphisms, then M^{∘} I ⊆ M and I^{∘} E ⊆ E.

FS–2. Every arrow f in C factors as f = m^{∘} e with m ∈ M and e ∈ E.

FS–3. In any commutative square

[figure: C; D; m; A; B; e; f; g]

with e ∈ E and m ∈ M, there is a unique h : B ⟶ C such that h^{∘} e = f and m^{∘} h = g.

The last condition is referred to as the “diagonal fill-in”. Note that we are following the usual convention of denoting an element of M with a tailed arrow and an element of E with a tailed arrow.

5.2. Proposition. The classes E and M are closed under composition.

Proof. Suppose m_{1} : A \underrightarrow{}B and m_{2} : B \underrightarrow{}C with m_{1} and m_{2} in M. Factor m_{2} ^{∘} m_{1} = m^{∘} e with m ∈ M and e ∈ E. The diagonal fill-in in the square

[figure: D; C; m_{2}; A; B; e; m_{1}; m]

is an arrow f : B ⟶ D such that f ^{∘} e = m_{1} and m_{2} ^{∘} f = m. The diagonal fill-in in the square

[figure: A; D; m_{1}; A; B; e; id; f]

is a map g : B ⟶ A such that g^{∘} e = id and m_{1} ^{∘} g = f. Since m^{∘} e^{∘} g = m_{2} ^{∘} m_{1} ^{∘} g = m_{2} ^{∘} f = m = m^{∘} id and e^{∘} g^{∘} e = e = id^{∘} e both the identity and e^{∘} g supply a diagonal fill-in in the square

[figure: B; C; m; A; B; e; e; m]

and hence, by uniqueness, are equal. This shows that e is an isomorphism and hence that m_{2} ^{∘} m_{1} = m^{∘} e ∈ M. The argument for E is dual.

5.3. Proposition. If f : A ⟶ B factors as A \underrightarrow{e} C m ; \underrightarrow{}B and also as A \underrightarrow{e′} C′

m′ ; \underrightarrow{}B then there is a unique arrow g : C ⟶ C′ such that g^{∘} e = e′ and m′ ^{∘} g = m; moreover g is an isomorphism.

Proof. The arrow g is the diagonal fill-in in the square

[figure: C′; B; m′; A; C; e; e′; m]

To see that g is an isomorphism, we transpose the square to get a map g′ : C′ ⟶ C such that g′ ^{∘} e′ = e and m^{∘} g′ = m′. Then we note that these equations imply that both the identity and g′ ^{∘} g fill in the square

[figure: C; B; m; A; C; e; e; m]

and the uniqeness of the diagonal fill-in forces g′ ^{∘} g = id and similarly g^{∘} g′ = id.

5.4. Proposition. Suppose f : C ⟶ D satisfies the condition that for all e : A ⟶ B in E, any commutative square

[figure: C; D; f; A; B; e]

has a unique diagonal fill-in. Then f ∈ M.

Proof. Factor f as C \underrightarrow{e} A m ; \underrightarrow{}D. From the diagonal fill-in in the square

[figure: C; D; f; C; A; e; id; m]

we get a map g : A ⟶ C such that g^{∘} e = id and f ^{∘} g = m. Then from e^{∘} g^{∘} e = e and m^{∘} e^{∘} g = f ^{∘} g = m we see that both the identity and e^{∘} g fill in the diagonal of

[figure: A; C; m; C; A; e; e; m]

and uniqueness of the diagonal fill-in implies that e is an isomorphism, whence f = m^{∘} e ∈ M.

Of course, the dual statement is true of E.

5.5. Corollary. Every isomorphism is in E ∩ M.

5.6. Proposition. Suppose every arrow in E is an epimorphism. Then g^{∘} f ∈ M implies that f ∈ M. Conversely, provided C has cokernel pairs, the left cancellability of M implies that every arrow in E is an epimorphism.

Proof. Suppose the composite C \underrightarrow{f} D \underrightarrow{g} E is in M. Suppose we have a commutative square

[figure: C; D; f; A; B; e; h; k]

The diagonal fill-in in the square

[figure: C; E; g^{∘} f; A; B; e; h; g^{∘} k]

provides a map l : B ⟶ C such that l^{∘} e = h. Then e can be cancelled on the right of f ^{∘} l^{∘} e = f ^{∘} h = k^{∘} e to conclude that f ^{∘} l = k. If l′ were another choice, e can be cancelled from l^{∘} e = l′ ^{∘} e.

For the converse, suppose e : A ⟶ B in E is not an epimorphism. Then the cokernel

pair B f ; \overrightarrow{⟶} ; g C have a common left inverse h : C ⟶ B and h^{∘} f = id ∈ M. If f were in M, the diagonal fill-in in

[figure: B; C; f; A; B; e; e; g]

would provide a map k : B ⟶ B such that f ^{∘} k = l. But then k = h^{∘} f ^{∘} k = h^{∘} g = id so that f = g which contradicts the assumption that e is not epic.

For the following result, we define a category to be wide complete if it is complete and if any class of subobjects of an object has an intersection. Of course, a well-powered category is wide complete if and only if it is complete, but it costs no more to state the theorem in this way. We also define an epimorphism f : A ⟶ B to be extremal if it factors through no subobject of B.

5.7. Theorem. Suppose C is a wide complete category. The there is a factorization system E/M in which E consists of the class of extremal epimorphisms and M consists of the class of monomorphisms.

Proof. Given f : A ⟶ B, let C ⊆ B denote the intersection of the class of all subobjects of B′ ⊆ B for which there is a factorization of f as A ⟶ B′ ⟶ B. The universal mapping property of intersection implies that there is a factorization of f as A \underrightarrow{e} C \underrightarrow{m} B. If the arrow e : A ⟶ C were not epic, there would be two arrows, say g ≠ h : C ⟶ D with g^{∘} e = h^{∘} e. But then the equalizer of g and h would be a subobject of B strictly smaller than C that factors e and hence f, a contradiction. The fact that e is extremal is obvious. This gives the factorization. Now suppose

[figure: C; D; m; A; B; e; f; g]

is a commutative square with e ∈ E and m ∈ M. Let

[figure: C; D; m; A′; B; m′; f′; g]

be a pullback. Then m′ is monic since a pullback of a monic is always monic. There is a unique arrow h : A ⟶ A′ such that m′ ^{∘} h = e and f′ ^{∘} h = f. The fact that e is an extremal epic implies that m′ is an isomorphism and then f′ ^{∘} m′^{−1} is the required diagonal fill-in since f′ ^{∘} m′^{−1} ^{∘} e = f′ ^{∘} m′^{−1} ^{∘} m′ ^{∘} h = f′ ^{∘} h = f and m^{∘} f′ ^{∘} m^{−1} = g^{∘} m′ ^{∘} m′^{−1} = g. If k were another diagonal fill-in, then from m^{∘} k = g and m^{∘} f′ ^{∘} m^{−1} = g we can cancel the monic m to conclude that k = f′ ^{∘} m^{−1}.

Of course, there is a dual theorem with epics and extremal monics in a wide cocomplete category.

References

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M. Barr (1991), ∗-Autonomous categories and linear logic. Mathematical Structures Computer Science, 1, 159–178.

P.-H. Chu (1979), Constructing ∗-autonomous categories. Appendix to [Barr, 1979].

H. Kleisli, H. P. Künzi, J. Rosičký (1996), A topological Banach space model of linear logic. Proceedings of the Workshop on Categorical Topology, L’Aquila, 155–162.

V.R. Pratt (1993a), Linear Logic for Generalized Quantum Mechanics. In Proceedings Workshop on Physics and Computation, Dallas, IEEE Computer Society.

V.R. Pratt (1993b), The Second Calculus of Binary Relations. In Proceedings of MFCS’93, Gdańsk, Poland, 142–155.

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E. Schläpfer (1998), On group algebras, Chu-spaces and induced representations of topological Hausdorff groups. Ph.D. Thesis, Université de Fribourg.

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Michael Barr Department of Mathematics and Statistics McGill University, Montreal, Quebec Canada H3A 2K6

Email: barr@math.mcgill.ca

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John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@math.mcgill.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università della Calabria: carboni@unical.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly m@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.ruu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mscs.dal.ca Andrew Pitts, University of Cambridge: ap@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@charlie.math.unc.edu Ross Street, Macquarie University: street@macadam.mpce.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters b@maths.usyd.edu.au R. J. Wood, Dalhousie University: rjwood@mscs.dal.ca