Theory and Applications of Categories, Vol. 25, No. 20, 2011, pp. 533–536.

ON REFLECTIVE-COREFLECTIVE EQUIVALENCE AND ASSOCIATED PAIRS

ERIK BÉDOS, S. KALISZEWSKI, AND JOHN QUIGG

Abstract. We show that a reflective/coreflective pair of full subcategories satisfies a “maximal-normal”-type equivalence if and only if it is an associated pair in the sense of Kelly and Lawvere.

1. Introduction

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 crossedproduct 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 cross-fertilization 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.

2. Maximal-normal equivalences and associated pairs

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;

Received by the editors 2011-11-09 and, in revised form, 2011-12-07.

Transmitted by R. Street. Published on 2011-12-18.

2000 Mathematics Subject Classification: Primary 18A40; Secondary 46L55, 46L89.

Key words and phrases: reflective and coreflective subcategories, equivalent categories, associated pairs of subcategories.

\begin{matrix}c \\ ◯ \end{matrix} Erik Bédos, S. Kaliszewski, and John Quigg, 2011. Permission to copy for private use granted.

534  ERIK BÉDOS, S. KALISZEWSKI, AND JOHN QUIGG

• M: C → M is a coreflector and ψ: IM → 1_{C} denotes the counit of the adjunction

I ⊣ M. In [1, Corollary 4.4] we showed that the adjunction NI ⊣ MJ is an adjoint equivalence between M and N if and only if

(I) for each y ∈ Obj N, (y, ψ_{y}) is an initial object in the comma category My ↓ N; and (F) for each x ∈ Obj M, (x, θ_{x}) is a final object in the comma category M ↓ Nx.

In all our examples in [1], the adjoint equivalence NI ⊣ MJ between M and N was what we called the “maximal-normal” type (recall that this terminology was motivated by the particular example of maximal and normal coactions on C^{∗}-algebras; see [1, Corollary 6.16]): in addition to (I) and (F), such an adjunction satisfies (A) for each z ∈ Obj C, (Nz, θ_{z} ∘ ψ_{z}) is an initial object in Mz ↓ N. Equivalently, by [1, Theorem 3.4], (I) and (F) hold, and (B) for each z ∈ Obj C, (Mz, θ_{z} ∘ ψ_{z}) is a final object in M ↓ Nz. In fact, conditions (A) and (B) alone suffice: 2.1. Proposition. The adjunction NI ⊣ MJ between M and N is a “maximal-normal” adjoint equivalence if and only if (A) and (B) hold. Proof. By [1, Theorem 4.3], (I) is equivalent to (I′) for each y ∈ Obj N , Nψ_{y} : NMy → Ny is an isomorphism, while (F) is equivalent to (F′) for each x ∈ Obj M, Mθ_{x} : Mx → MNx is an isomorphism. On the other hand, by [1, Theorem 3.4], (A) is equivalent to (A′) for each z ∈ Obj C, Nψ_{z} is an isomorphism, while (B) is equivalent to (B′) for each z ∈ Obj C, Mθ_{z} is an isomorphism. Now clearly, (A′) implies (I′) and (B′) implies (F′), so (A) implies (I) and (B) implies (F).

ON REFLECTIVE-COREFLECTIVE EQUIVALENCE AND ASSOCIATED PAIRS 535

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 N^{⊥}.

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 θ_{y} ∘ f = Nf ∘ θ_{x}, the diagram

\xymatrix{N (Ny, z) \ar[r]^{Ψ_{f,z}} \ar[d]_{τ_{y,z}} & N (Nx, z) \ar[d]^{τ_{x,z}} \\ C(y, z) \ar[r]_{Φ_{f,z}} & C(x, z)}

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 co-orthogonal 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).

2.2. Theorem. 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_{θ_{x},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.

2.3. Remark. 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.

References

[1] E. Bédos, S. Kaliszewski, and J. Quigg, Reflective-coreflective equivalence, Th. Appl. Cat. 25 (2011), 142–179.

[2] C. Cassidy, M. Hébert, and G.M. Kelly, Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. 38 (Séries A) (1985), 287–329.

[3] F.W. Lawvere, and G.M. Kelly, On the Complete Lattice of Essential Localizations, Bull. Soc. Math. Belg. Sér A 41 (1989), 289–319.

[4] S. Mac Lane, Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998.

Institute of Mathematics University of Oslo P.B. 1053 Blindern, 0316 Oslo, Norway

School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287

School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287

Email: bedos@math.uio.no
      kaliszewski@asu.edu
      quigg@asu.edu

This article may be accessed at http://www.tac.mta.ca/tac/ or by anonymous ftp at

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/25/20/25-20.{dvi,ps,pdf}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor Gavin Seal, Ecole Polytechnique Fédérale de Lausanne:

gavin seal@fastmail.fm

Transmitting editors

Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne : kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Glasgow, Tom.Leinster@glasgow.ac.uk Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: street@math.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 Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca