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A the Language of Categories 541 A The language of categories We assume that the reader is familiar with the concept of a category and of a functor. In this appendix we briefly fix some notation and recall some results which are used in Volume I. Everything in this appendix can be found in any of the standard text books to category theory, such as [McL], [Mit], or [Sch]. We avoid any discussion of set-theoretical difficulties and refer for this to the given references. (A.1) Categories. Recall that a category C consists of (1) a collection of objects, (2) for any two objects X and Y a set HomC(X, Y ) = Hom(X, Y ) of morphisms from X to Y ,(3)forevery object X an element idX ∈ Hom(X, X), (4) for any three objects X, Y , Z a composition map ◦: Hom(X, Y ) × Hom(Y,Z) → Hom(X, Z), such that composition of morphisms is associative, and the elements idX are neutral elements with respect to composition. We write f : X → Y if f is a morphism from X to Y . If f : X → Y is a morphism, then a morphism g : Y → X is called a section (resp. a retraction)iff ◦ g =idY (resp. if g ◦ f =idX ). A morphism f : X → Y is called an isomorphism if there exists a morphism g : Y → X ∼ such that f ◦ g =idY and g ◦ f =idX .Weoftenwritef : X → Y to indicate that f is an isomorphism. We also write X =∼ Y and say that X and Y are isomorphic if there exists ∼ an isomorphism X → Y . A subcategory of a category C is a category C such that every object of C is an object of C and such that HomC (X ,Y ) ⊆ HomC(X ,Y ) for any pair (X ,Y ) of objects of C , compatibly with composition of morphisms and identity elements. The subcategory C is called full if HomC (X ,Y ) = HomC(X ,Y ) for all objects X and Y of C . For every category C the opposite category, denoted by Copp, is the category with the opp same objects as C and where for two objects X and Y of C we set HomCopp (X, Y ):= HomC(Y,X) with the obvious composition law. A morphism f : X → Y in a category C is called monomorphism (resp. epimorphism)if for every object Z left composition with f is an injective map Hom(Z, X) → Hom(Z, Y ) (resp. right composition with f is an injective map Hom(Y,Z) → Hom(X, Z)). We often write f : X→ Y (resp. f : X Y ) to indicate that f is a monomorphism (resp. epimor- phism). Every isomorphism is a monomorphism and an epimorphism. The converse does not hold in general. An object Z in C is called final (resp. initial), if HomC(X, Z) (resp. HomC(Z, X)) has exactly one element for all objects X.Foranytwofinal(resp.initial)objectsZ and Z ∼ there is a unique isomorphism Z → Z. Notation A.1. Throughout this book we use the following notation for specific cate- gories: (Sets) the category of sets, C the category of all contravariant functors of C to the category of sets (see also (4.2)), 542 A The language of categories (Ring) the category of (commutative) rings, (A-Mod) the category of A-modules for a ring A, (R-Alg) the category of (commutative) R-algebras for a ring R, (Sch) the category of schemes (see (3.1)), (Sch/S) the category of S-schemes for a scheme S (if S =SpecR is affine we also write (Sch/R)) (see (3.1)). (Aff) the category of affine schemes (see (2.11)). (OX -Mod) the category of OX -modules for a sheaf of rings OX (see (7.1)). (A.2) Functors, equivalence of categories, adjoint functors. Given categories C and D, a (covariant) functor F : C→Dis given by attaching to each object C of C an object F (C)ofD, and to each morphism f : C → C in C a morphism F (f): F (C) → F (C), compatibly with composition of morphisms and identity elements. A contravariant functor from C to D is by definition a covariant functor F : Copp →D, where Copp is the opposite category of C. Sometimes we use the notation F : C→Dfor a contravariant functor, in which case we explicitly state that F is contravariant. If F is a functor from C to D and G a functor from D→E,wewriteG ◦ F for the composition. For two functors F, G: C→Dwe call a family of morphisms α(S): F (S) → G(S) for every object S of C functorial in S or a morphism of functors if for every morphism f : T → S in C the diagram α(T ) F (T ) /G/(T ) F (f) G(f) α(S) F (S) /G/(S) commutes. With this notion of morphism we obtain the category of all functors from C to D.WedenotebyC the category of all contravariant functors of C into the category of sets. A functor F : C→Dis called faithful (resp. fully faithful) if for all objects X and Y of C the map HomC(X, Y ) → HomD(F (X),F(Y )), f → F (f) is injective (resp. bijective). The functor F is called essentially surjective if for every object Y of D there exists an object X of C and an isomorphism F (X) =∼ Y . A functor F : C→Dis called an equivalence of categories if it is fully faithful and essentially surjective. This is equivalent to the existence of a quasi-inverse functor G, ∼ ∼ i.e., of a functor G: D→Csuch that G◦F = idC and F ◦G = idD. Similarly, considering contravariant functors, we obtain the notion of an anti-equivalence of categories. Let F : C→Dand G: D→Cbe functors. Then G is said to be right adjoint to F and F is said to be left adjoint to G if for all objects X in C and Y in D there exists a bijection ∼ HomC(X, G(Y )) = HomD(F (X),Y) which is functorial in X and in Y . If a functor F (resp. a functor G) has a right adjoint functor (resp. a left adjoint functor), this adjoint functor is unique up to isomorphism. If F is an equivalence of categories, then a quasi-inverse functor is right adjoint and left adjoint to F . 543 (A.3) Inductive and projective limits. Ordered sets. Let I be a set. (1) A relation ≤ on I is called partial preorder or simply preorder,ifi ≤ i for all i ∈ I and i ≤ j, j ≤ k imply i ≤ k for all i, j, k ∈ I. (2) A preorder ≤ is called partial order or simply order if i ≤ j and j ≤ i imply i = j for all i, j ∈ I. (3) A preorder ≤ is called filtered if for all i, j ∈ I there exists a k ∈ I with i ≤ k and j ≤ k. (4) A partial order ≤ is called total order if for all i, j ∈ I one has i ≤ j or j ≤ i. Every preordered set I can be made into a category, again denoted by I, whose objects are the elements of I and for two elements i, j ∈ I the set of morphisms HomI (i, j) consists of one element if i ≤ j and is empty otherwise. There is a unique way to define a composition law in this category. Projective Limits and Products. Let I be a preordered set and let C be a category. A projective system in C indexed by opp I is a functor I →C. In other words, it is a tuple (Xi)i∈I of objects in C together → ∈ ≤ with morphisms ϕij : Xj Xi for all i, j I with i j such that ϕii =idXi and ϕij ◦ ϕjk = ϕik for all i ≤ j ≤ k. A projective limit (or simply limit) of a projective system ((Xi)i∈I , (ϕij)i≤j)inC is an object X in C together with a family of morphisms (ϕi : X → Xi)i∈I such that ϕi = ϕij ◦ ϕj for all i ≤ j which is final with this property, i.e., if (Y,(ψi)i∈I )isatupleof an object Y and morphisms ψi : Y → Xi such that ψi = ϕij ◦ ψj for all i ≤ j, then there exists a unique morphism ψ : Y → X such that ϕi ◦ ψ = ψi for all i ∈ I. A projective limit (X, (ϕi)i) need not exist, but if it exists, then it is uniquely determined up to unique isomorphism, and is denoted by lim X . ←− i i∈I Example A.2. (1) If I is endowed with the discrete order, i.e., i ≤ j if and only if i = j,then X := lim X i ←− i i∈I i∈I is called the product of the family (Xi)i∈I . (2) If I consists of three elements j1,j2,k whose nontrivial order relations are k ≤ j1 and k ≤ j2,then X × X := lim X j1 Xk j2 ←− i i∈{j1,j2,k} is called the fiber product; see also (4.4). (3) For I = ∅, the notion of projective limit is the same as that of final object of C. 544 A The language of categories Example A.3. C (1) Let be the category of sets. Then i∈I Xi is the usual cartesian product, and the final object is the singleton, i.e., the set consisting of one element. For an arbitrary preordered set I the projective limit of a projective system ((Xi)i∈I , (ϕij)i≤j) exists and is given by X = lim X = { (x ) ∈ ∈ X ; ϕ (x )=x for all i ≤ j }, ←− i i i I i ij j i i∈I i∈I → → where the maps ϕi : X Xi are the restrictions of the projections i Xi Xi.
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