On reflective-coreflective equivalence and associated pairs

In a recent paper [1] we explored a special type of category equivalence between reflective/coreflective pairs of subcategories that we first encountered in the context of crossed-product duality for C * -algebras. Because our main example of this phenomenon involved categories of maximal and normal C * -coactions of locally compact groups, we called it a "maximal-normal"-type equivalence.

Since then, F. W. Lawvere has drawn our attention to [3], where G. M. Kelly and he introduced the concept of associated pairs of subcategories. The purpose of this short note is to show that these two notions of equivalence are the same: a reflective/coreflective pair of full subcategories satisfies the “maximal-normal”-type equivalence considered in [1] if and only if it is an associated pair in the sense of [3].

As operator algebraists, we had hoped with [1] to initiate a crossfertilization between operator algebras and category theory, and we are grateful to Ross Street for the role he has played in helping this happen. Our understanding of the operator-algebraic examples has certainly been deepened by this connection; ideally, the techniques and examples of “maximal-normal”-type equivalence will in turn provide a way of looking at associated pairs that will also be useful to category theorists.

Our conventions regarding category theory follow [4]; see also [1]. Throughout this note, we let M and N denote full subcategories of a category C, with N reflective and M coreflective. The inclusion functors I : M → C and J : N → C are then both full and faithful. We also use the following notation:

• N : C → N is a reflector and θ : 1 C → JN denotes the unit of the adjunction N ⊣ J; We now recall from [2,3] that a morphism f in C(x, y) and an object z of C are said to be orthogonal when the map Φ f,z from C(y, z) into C(x, z) given by Φ f,z (g) = g • f is a bijection. The collection of all morphisms in C that are orthogonal to every object of N is denoted by

As shown in [3, Proposition 2.1], a morphism f : x → y in C belongs to N ⊥ if and only if f is inverted by N, that is, Nf is an isomorphism. (The standing assumption in [3] that N is replete is not necessary for this fact to be true. To see this, note that Nf is an isomorphism if and only if the map Ψ f,z from N (Ny, z) into N (Nx, z) given by Ψ f,z (h) = h • Nf is a bijection for each object z of N . For each such z, the universal properties of θ imply that the map τ w,z from N (Nw, z) into C(w, z) given by τ w,z (g) = g • θ w is a bijection for each object w of C. Now, as

is readily seen to commute. It follows that Ψ f,z is a bijection if and only if Φ f,z is a bijection. This shows that Nf is an isomorphism if and only if f is orthogonal to z for each object z of N , i.e., if and only if f belongs to N ⊥ .)

Similarly, a morphism f in C(x, y) and an object z in C are coorthogonal when the map g → f • g from C(z, x) into C(z, y) is a bijection. The collection of all morphisms in C that are co-orthogonal to every object in M is denoted by M ⊤ . Equivalently, a morphism f : x → y in C belongs to M ⊤ if and only if f is inverted by M, that is, if and only if Mf is an isomorphism.

The pair (N , M) is called an associated pair if N ⊥ = M ⊤ ; equivalently, if for every morphism f in C, N inverts f if and only if M does. We refer to [3, Section 2] for more information concerning this concept (in the case where both M and N are also assumed to be replete).

Theorem 2.2. The adjunction NI ⊣ MJ is a “maximal-normal” adjoint equivalence if and only if (N , M) is an associated pair.

Proof. First assume that (N , M) is an associated pair, and let x be an object in C. As pointed out above, the map τ x,z is a bijection from N (Nx, z) into C(x, z) for each object z of N . But Φ θx,z = τ x,z , so this means that θ x lies in N ⊥ , and therefore in M ⊤ . As M ⊤ consists of the morphisms in C that are inverted by M, we deduce that Mθ x is an isomorphism. This shows that (B ′ ) holds, and therefore that (B) holds. The argument that (A) holds is similar, so NI ⊣ MJ is a “maximal-normal” adjoint equivalence by Proposition 2.1. Now assume that the adjunction NI ⊣ MJ is a “maximal-normal” adjoint equivalence. Then N ∼ = NIM by [1, Proposition 5.3], and NI is an equivalence. So for any morphism f of C, we have Nf is an isomorphism ⇔ NIMf is an isomorphism ⇔ Mf is an isomorphism.

Thus (N , M) is an associated pair.

Remark 2.3. In the examples presented in [1, Section 6], the adjunctions NI ⊣ MJ are “maximal-normal” adjoint equivalences, so all the pairs (N , M) there are associated pairs. Moreover, all these pairs consist of subcategories that are easily seen to be replete. It follows from [3,Theorem 2.4] that M and N are uniquely determined as subcategories by each other, a fact that is not a priori obvious in any of the examples.

This content is AI-processed based on open access ArXiv data.