The rudiments of the theoly of Galois connections (or residuation theory, as it is sometimes called) are provided, together with many examples and applications. Galois connections occur in profusion and are well known to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear, and thus, the whole situation typically becomes much easier to understand. Mathematics SubjectClassification. Primary 06A15, 06-01, 06A06; Secondary 54-01, 54B99, 54899,68F05. Key words and phrases. Galois connection, closure operation, interior operation, polarity, axiality. 'Galois' main results were published fourteen years after his early death (at the age of 21 in a duel) by Liouville in his Journal de mathimatiquespures et applqukes (1846). For a translation of Galois' original notes Memoire sur les conditions de rksolubiliti des iquationspar radicaux, see the text by Edwards [ 141. 103 104 ANNALS NEW YORK ACADEMY OF SCIENCES the world of intermediate fields of a field extension E : F and the world of subgroups of the group of automorphisms of E that fix the subfield F (cf. Examples 1 and 19).Today this area is known as Galois theory.Since the proofs of many results in this paper are either well known or easily obtained, we do not include them. Galois connections were originally expressed in a symmetric but contravariant form with transformations that reverse (rather than preserve) order. Early references to this form are [8], [22], [44], and [45]. We use the covariant form since it is more convenient, for example, compositions of Galois connections are handled more easily; it allows for more natural categorical explanations (e.g., by means of adjunctions); and it is more applicable to computer science situations (where relative information preservation is important). For references to the covariant form, see [7], [51], [9], [32], [25], [35], [15], [42], [31], and [19]. THE DEFINITION AND SOME OF ITS CONSEQUENCESWe formulate all our results in terms of partially ordered sets (orposets), that is, sets equipped with a reflexive, transitive, and antisymmetric relation. Everything can easily be generalized to preordered sets (i.e., one may drop the antisymmetry requirement) and even to preordered classes. Applications of these generalizations can be found in Examples 10, 22, and 23, and in [lo]. For a poset 9 = (P, s ) , the order-theoretical dual (P, 2 ) is denoted by 9 OP. =* DEFINITION 1: Consider posets 9 = (P, I) and d = (Q, E). If P -+ Q and Q 2 P are functions such that for allp E P and all q E Q p s a * ( q ) iff a * @ )~q (1)then the quadruple IT = (9, IT *, IT*, d ) is called a Galois connection. We also write 9 5 d (or sometimes just ( I T * , T*)) for the whole Gal...

Linear bicategories are a generalization of bicategories in which the one horizontal composition is replaced by two (linked) horizontal compositions. These compositions provide a semantic model for the tensor and par of linear logic: in particular, as composition is fundamentally non-commutative, they provide a suggestive source of models for non-commutative linear logic.In a linear bicategory, the logical notion of complementation becomes a natural linear notion of adjunction. Just as ordinary adjoints are related to (Kan) extensions, these linear adjoints are related to the appropriate notion of linear extension.There is also a stronger notion of complementation, which arises, for example, in cyclic linear logic. This sort of complementation is modelled by cyclic adjoints. This leads to the notion of a *ast;-linear bicategory and the coherence conditions that it must satisfy. Cyclic adjoints also give rise to linear monads: these are, essentially, the appropriate generalization (to the linear setting) of Frobenius algebras and the ambialgebras of Topological Quantum Field Theory.A number of examples of linear bicategories arising from different sources are described, and a number of constructions that result in linear bicategories are indicated.

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Basic results are obtained concerning Galois connections between collections of closure operators (of various types) and collections consisting of subclasses of (pairs of) morphisms in ..&' for an (E, ./)-category Z In effect, the "lattice" of closure operators on -# is shown to be equivalent to the futed-point lattice of the polarity induced by the orthogonality relation between cornposable pairs of morphisms in A.Galois "insertions" of idempotent (respectively, weakly hereditary) closure opera-Mathematics Subject Classification. Primary 18A32; Secondary 06A15,54B30. CASTELLINI et al.: CLOSURE OPERATORS 39tors into all closure operators. This twofold appearance helps to explain why the idempotent weakly hereditary closure operators have been dominant in the carly work in the field (cf.[3], [9], [ll] and [13]). The analogue between the results for. seems to make a strong case for the study of classes of composable pairs. Applications and examples can be found in the final section. PRELIMINARIESOur main tool will be a notion of orthogonality that generalizes the one introduced by Tholen (cf. [17]), and cncompasses part of the defining properties of factorization structures for sinks (cf. Definition 3) and for sources, as well as one of the essential features of closure operators (cf. Definition 2). Throughout we work in a category X DEFINITION 1 (cf. [13]): A pair (a, a ' ) consisting of a sink a = (A, 2 A ' ) , and a morphism A ' $ A " is called lefl orthogonal to a pair (b, b' ), consisting of a morphism B + B' and a source b' = (B' 4 B,"),, written as (a, a ' ) l. (h, b'), iff for any sink f = (A, --j B ) , and any source f " = $ 4" 4 B " ) with thc property that for each i E I and each j E J the outer square of the following diagram commutesthcre exists a unique ,X-morphismA' + B' such that all inner trapezoids commute. In this case the pair (h, b') is called right orthogonal to (a, a ' ) . We write a I b' rather than (a, id) I (id, b' ), that is, we suppress the morphism part of a pair in case it is an isomorphism.Notice that a sink a and an object X are separated in the sense of Pumpliin and Rohrl ( cf. [14]), if a is left-orthogonal to the 2-sourceX -X -X , and that a is an epi-sink iff a and every Z-object are separated in this sense. Now consider the following characterization of continuous functions between topological spaces (X, t ) and (Y, s), where "-" denotes the usual topological closure. PROPOSITION I: A function x 4 Yis coGinuous from (x, t ) to (Y, s) iff for every M C Xand every N C Y the direct image of M alongg is contained in m, provided that the direct image of M alongg is contained in N . id, id,y

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