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Appendix A. Preliminaries UvA-DARE (Digital Academic Repository) Logic, algebra and topology: investigations into canonical extensions, duality theory and point-free topology Vosmaer, J. Publication date 2010 Link to publication Citation for published version (APA): Vosmaer, J. (2010). Logic, algebra and topology: investigations into canonical extensions, duality theory and point-free topology. Institute for Logic, Language and Computation. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:28 Sep 2021 Appendix A Preliminaries In this appendix, we will briefly discuss some of the mathematical background knowledge that we rely on elsewhere in this dissertation. The presentation of this appendix is not linear: when explaining one subject, we will sometimes refer to another one which may lie further ahead in the text. A.1 Set theory Throughout this dissertation, we assume that the reader is familiar with elementary set theoretical notions such as membership x 2 X, intersection X \ Y , union X [ Y and set theoretic difference X n Y . An exception is Chapter 5, where we use some extra notation which does not occur in the rest of the dissertation. The additional preliminaries for Chapter 5 are discussed in x5.2. The only thing we would like to briefly mention at this point is the power- set construction. If f : X ! Y is a function between sets X and Y , then by f −1 : P(Y ) !P(X) we denote the inverse image function, which maps any set U ⊆ Y to f −1(U) := fx 2 X j f(x) 2 Ug: At the same time, if U ⊆ X, then we define f[U] := ff(x) j x 2 Ug; this yields a function f[·]: P(X) !P(Y ). Using these two mappings on subsets based on a function f : X ! Y , we can define two functors on Set, the category of sets, with the same action on objects, viz. sending X to P(X). The covariant powerset functor P : Set ! Set maps f : X ! Y to Pf : P(X) !P(Y ), where Pf : U 7! f[U]. The contravariant powerset functor P˘ : Set ! Setop maps f : X ! Y to P˘ f : P(Y ) !P(X), where P˘ f : U 7! f −1(U). 199 200 Appendix A. Preliminaries A.2 Category theory Our main reference for category theory is Mac Lane [69]; alternatively, one can consult Adamek, Herrlich & Strecker [3]. A.2.1 Categories and functors A category is a structure C consisting of a class of objects X; Y; Z; : : : and a binary function HomC which assigns to any two objects X; Y a class of morphisms, or 1 arrows, denoted HomC(X; Y ). If f 2 HomC(X; Y ) then we write f : X ! Y or f X −! Y: 0 0 0 0 We require that if (X; Y ) 6= (X ;Y ), then HomC(X; Y ) and HomC(X ;Y ) are disjoint. In other words, given an arrow f in C, f has a unique domain X and codomain Y such that f 2 HomC(X; Y ). Every category comes equipped with an associative composition operation ◦ for morphisms, and identity morphisms idX , one for each object X. If f : X ! Y and g : Y ! Z then g ◦ f : X ! Z is a morphism from X to Z. Saying that composition is associative means that for all f : X ! Y , g : Y ! Z and h: Z ! W , we have h ◦ (g ◦ f) = (h ◦ g) ◦ f: The identity arrows idX are characterized by the following property: for all arrows f : X ! Y and g : Y ! X, f ◦ idX = f and idX ◦g = g. We also use the arrows to define the notion of isomorphism. An arrow f : X ! Y is an isomorphism if there exists an arrow g : Y ! X such that gf = idX and fg = idY : A.2.1. Example. As a prototypical example of a category, consider Set, which has as its objects the class of all sets, and as its arrows all functions between sets. Composition of arrows is then simply composition of functions; the identity arrows are the identity functions and isomorphisms are precisely the bijective functions. Note that this is not the prototypical example of a category, see [69, xI.2] for a list of further basic examples. Let C and C0 be categories. We say C0 is a subcategory of C if every object 0 0 of C is also an object of C, and if for all objects X; Y of C , HomC0 (X; Y ) ⊆ 0 0 HomC(X; Y ). We say C is a full subcategory of C if for all objects X; Y of C , HomC0 (X; Y ) = HomC(X; Y ). 1 In all categories we consider in this dissertation, HomC(X; Y ) is a set rather than a class. A.2. Category theory 201 A.2.2. Example. The category Setf , which has as its objects the class of all finite sets, and as its morphisms all functions between finite sets, forms a full subcategory of Set. Given a category C, we can construct its dual category Cop. The objects of Cop are simply those of C. The arrows of Cop are in 1-1 correspondence f 7! f op with those of C. The only difference between C and Cop is that if f : X ! Y is an arrow of C, then f op : Y ! X in Cop goes in the other direction. We can now define the composite of two arrows f op : Y ! X and gop : Z ! Y in Cop to be f op ◦ gop := (gf)op: op op op It is easy to see that (idX ) is the identity arrow of X in C , and that C indeed forms a category. Because of the way Cop is defined in terms of C, we can unravel statements about Cop into statements about C. Using this fact, we can automatically define the dual version of any categorical concept. Usually, we will name such a dual concept by prefixing `co-' to the name of the original concept. This process can be made much more precise, see [69, xII.1, II.2]. A.2.3. Example. Let C be a category and let F : C ! C be an endofunctor, i.e. a functor with the same domain and codomain. An F -algebra consists of an object X and a morphism h: F (X) ! X. The dual notion, that of an F -coalgebra, consists simply of an object X and a morphism r : X ! F (X). A functor is a structure-preserving map between categories. Concretely, if C; D are categories then a functor F : C ! D consists of an assignment of an object F (X) to every object X of C, and of an assignment of an arrow F (f) 2 HomD(F X; F Y ) to every arrow f 2 HomC(X; Y ) such that F (g◦f) = F (g)◦F (f) f g for all X −! Y −! Z in C and F (idX ) = idF (X) for all X in C. If it is not visually confusing, we will sometimes omit parentheses, writing e.g. F f instead of F (f). A.2.4. Example. As a trivial example of a functor, observe that given any category C we can define the identity functor IdC : C ! C which leaves all objects and arrows unchanged. A functor F : C ! Dop is called a contravariant functor from C to D; one can alternatively view F as an `arrow-reversing' functor from C to D. The notion of functor we introduced before is sometimes also called a covariant functor. A natural transformation ν between functors F; G: C ! D is a family of D-morphisms νX : F (X) ! G(X), one for each object X of C, such that for all f : X ! Y in C, the following diagram commutes: νX X F (X) / G(X) f F (f) G(f) F (Y ) / G(Y ) Y νY 202 Appendix A. Preliminaries If ν is a natural transformation such that for each X, νX is an isomorphism, then we call ν a natural isomorphism. If there exists a natural isomorphism between two functors F; G: C ! D, we say F and G are naturally isomorphic. A.2.2 Adjunctions of categories One of the fundamental notions of category theory is that of adjunctions between categories. A pair of functors F : C ! D and G: D ! C is called an adjunction, abbreviated F a G, if there exist natural transformations η : IdC ! GF and 0 : FG ! IdD such that for all f : X ! GY , there exists a unique f : F (X) ! Y 0 0 such that f = Gf ◦ ηX , and for all g : FX ! Y , there exists a unique g : X ! 0 G(Y ) such that g = Y ◦ F g : GF (X) F (X) FG(Y ) G(Y ) F η x; O F O X xx 0 0 0 FF Y 0 xx G(f ) f F (g ) FF g xx FF xx F" / G(Y ) F (X) / X f Y g YX A dual adjunction between categories C and D is simply an adjunction between C and Dop.
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