Representable presheaves

Transcription

1 Representable presheaves March 15, 2017 A presheaf on a category C is a contravariant functor F on C. In particular, for any object X Ob(C) we have the presheaf (of sets) represented by X, that is Hom C (, X). I d like to explain the formula F = (Hom C (,X) F) C F Hom C (, X), both very useful and tautological, which says that any presheaf is a it of representable presheaves. This comes from the Yoneda lemma. Presheaves Definition. A presheaf of sets on C is a contravariant functor F : C Set. A morphism of presheaves F G is a natural transformation of functors, i.e. a collection of morphisms θ X : F(X) G(X) for every object X Ob(C) such that for every morphism X f Y the diagram below is commutative: Y F(Y) θ Y G(Y) f X F( f ) F(X) θ X G( f ) G(X) Following SGA 4, I will denote by Ĉ the category of presheaves on C. The most basic example is the following. Let X be a topological space and let Open(X) be the category of open subsets of X partially ordered by inclusion: { {U V}, if U V Hom Open(X) (U, V) :=, otherwise. A presheaf on the category Open(X) is the same thing as a presheaf on X in the usual sense: for each V U we have the corresponding restriction map and by functoriality, res UU = id F(U) and res UV : F(U) F(V), res UW = res VW res UV for W V U. 1

2 Size issues There is a problem with the definition of Ĉ. Recall that one says that C is a small category if the objects Ob(C) form a set. For instance, the category Open(X) is small. The majority of interesting categories are not. Even the categories of finite sets, finite dimensional vector spaces, etc. are not small (even one-element sets { } do not form a set, since can be any set here). However, these categories are equivalent to small categories. It is very common to say small category when in fact the category in question is equivalent to some small category. In the definition of a category, one normally assumes that the morphisms Hom C (X, Y) between two fixed objects X, Y Ob(C) form a set. Some authors say in this case that the category is locally small. In general, natural transformations F G between two functors F, G : C D do not form a category, unless C is small. In particular, if C is small, then the presheaves on C form a category in the usual sense. To resolve the arising problems, Grothendieck in the appendix to exposé I of SGA 4 developed the theory of universes. Curiously, the text says Nous reproduisons ici, avec son accord, des papiers secrets de N. Bourbaki, but most likely it was written by Grothendieck himself, and it has never appeared in Bourbaki s books. You can check exposé I of SGA 4 to see the statements of some basic results (as the Yoneda lemma) with universes, but I won t go into details related to set theory. Here are a couple of links: Aise Johan de Jong (the maintainer of the Stacks Project) about set theory: (check the comments!) Zhen Lin Low, Universes for category theory : for those who are interested in logic. Limits and its Let s revise some definitions and basic properties, mainly to fix the notation and terminology. Let I be a (small) category and let F : I C be a functor. A limit of F is an object lim I F Ob(C) together with a collection of morphisms {lim I F F(i)} such that every morphism i j in I induces a commutative diagram lim I F F(i) F(j) Moreover, we ask for the following universal property: for each other object X Ob(C) with a collection of morphisms {X F(i)} that commute with the morphisms in I in the above sense (one says that X is a cone with respect to F) there exists a unique morphism X lim I F that gives commutative diagrams lim I F X! F(i) F(j) 2

3 A it of F is an object I F Ob(C) together with a collection of morphisms {F(i) I F} that give for each morphism i j in I a commutative diagram F(i) F(j) I F And one asks that I F satisfies the universal property of its F(i) F(j) I F When a limit or it exists, it is unique up to isomorphism. The terminology varies: X! limit inverse limit projective limit it direct limit inductive limite Actually, the terms inverse limit and direct limit, as well as the common notation lim and lim, should be reserved for the case where the indexing category enjoys some special properties (when it is a directed set, or in general a filtered category). That s why I prefer writing lim and. Instead of lim I F and I F one often writes lim F(i) and F(i), and I will use this notation, even though it s slightly misleading: limits and its are indexed by both objects and morphisms in I. Of course, the basic examples of limits and its are the following: A terminal object in C is a limit over the empty category I = and an initial object is a it over I =. Products X i, fibered products X Z Y (pullbacks) and equalizers Eq(X Y) are particular cases of limits. Coproducts X i, fibered coproducts X Z Y (pushouts) and coequalizers Coeq(X Y) are particular cases of its. Many important categories are complete and cocomplete, meaning that limits and its exist over any small category I. Among those is the category of sets Set, and many categories of sets with additional structure: topological spaces Top, groups Grp, rings Ring, modules R-Mod, and so on. Limits and its in these cases have an explicit description: lim I I F = {(x i ) F(i) F( f )(x i ) = x j for each f : i j}. F = F(i) /, 3

4 where the equivalence relation is generated by F(i) x i F( f )(x i ) F(j) for each f : i j. One can make the (obvious) adjustments for the case of sets with additional structure. For instance, in the case of topological spaces, F(i) / should be the space with the quotient topology. Here is an important fact that may be verified from the definitions: Observation. The contravariant Hom C (, X) converts its to limits: Hom C ( F(i), X) = lim Hom C (F(i), X). Limits and its are used in some canonical constructions, so the following is also important: Another observation. (1) If for a fixed small category I and a category C all limits and its exist, then one can choose their particular representatives (in general, limits and its are defined only up to an isomorphism), and one has functors lim I, I : Fun(I, C) C. Indeed, for any natural transformation η : F G, the limit lim I F gives a cone for G, and hence a canonical morphism lim I F lim I G. Similarly, the it I G gives a cocone for F and a canonical morphism I F I G. F(i) X F(j) G(i) η i F(i) G(j) η j F(j) η i G(i) η j G(j) X (2) Every functor between indexing categories φ : J I induces canonical morphisms φ : lim I F lim J F φ, φ : F φ F, J I and these constructions are functorial in the sense that (φ ψ) = φ ψ and (φ ψ) = ψ φ. Limits and its of presheaves For every small category C the category of presheaves Ĉ is complete and cocomplete. That is, for every functor I Ĉ, i F i 4

5 on a small category I, there exist its limit and it lim F i and F i. That s because we can see our functor as I C Set, i.e. calculate the limit and it pointwise : (lim F i )(X) = lim F i (X) and ( and the category Set is complete and cocomplete. In particular, F i )(X) = F i (X), For products of presheaves For coproducts of presheaves (F G)(X) = F(X) G(X). (F G)(X) = F(X) G(X). The category of presheaves has a terminal object, which is the presheaf F such that F(X) = { } is a one-element set a terminal object in the category of sets. The category of presheaves has an initial object, which is the presheaf F such that F(X) = for each X, where is an initial object in the category of sets. Very often (for instance, to develop sheaf cohomology) one considers not presheaves of sets but presheaves of, say, abelian groups. But we can use the fact that the products and coproducts are calculated pointwise and say that a presheaf of abelian groups F : C Ab is an abelian group object in the category of presheaves of sets Ĉ. The words abelian group object mean that for a presheaf F we specify morphisms + : F F F (addition), : F F (subtraction), and 0: { } F (zero) that fit into commutative diagrams that express that + is associative and commutative (i.e. x + (y + z) = x + (y + z) and x + y = y + x), that 0 is the neutral element with respect to + (i.e. x + 0 = x), and that is inverse to + (i.e. x + ( x) = 0). For instance, the associativity is expressed by F F F + id F F and the commutativity is expressed by id + F F + F + F F + F where τ is the canonical morphism defined by τ F F + p 1 τ = p 2 and p 2 τ = p 1, and p 1, p 2 : A A A are the canonical projections. 5

6 In general, we have that A presheaf of groups abelian groups rings modules..... is a group abelian group ring module..... object in the category Ĉ. So all the constructions for Ĉ may be generalized to presheaves with any algebraic structure defined by commutative diagrams involving terminal and initial objects, products and coproducts, and so on. One should be careful though: this does not work, for instance, for presheaves of topological spaces F : C Top, because one can t describe topologies via such diagrams. Yoneda lemma: the importance of representable presheaves The most basic and important example of presheaves is the presheaf represented by a fixed object X Ob(C). Namely, it is the functor Hom C (, X) : C Set. A morphism Y f Z induces a natural morphism Hom C (Z, X) Hom C (Y, X) which maps Z g X to Y g f X. Now the famous Yoneda lemma says that For every X Ob(C) and every presheaf F : C Set there is a natural bijection HomĈ(Hom C (, X), F) = F(X). The construction goes as follows. For a morphism of presheaves θ Y : Hom C (Y, X) F(Y) the only obvious element of F(X) that one can produce is θ X (id X ). In the other direction, starting from an element x F(X), one can define a morphism of presheaves θy x : Hom C(Y, X) F(Y) by sending Y f X to F( f )(x) F(Y), which is again the only obvious possibility. It only remains to check that this gives us a natural bijection. In particular, for two representable presheaves Hom C (, X) and Hom C (, Y) the above bijection is given by HomĈ(Hom C (, X), Hom C (, Y)) = Hom C (X, Y). This means that we can think of C as of a full subcategory of Ĉ: C Ĉ, X Hom C (, X). This is known as the Yoneda embedding. By abuse of notation, in certain diagrams in SGA 4, X in fact denotes the corresponding presheaf Hom C (, X). 6

7 A little bit of history The Yoneda lemma and Yoneda embedding are named after the Japanese mathematician Nobuo Yoneda ( ). The Yoneda lemma was stated and popularized by Saunders Mac Lane after he met Yoneda. In texts like SGA, the Yoneda lemma is used extensively without ever mentioning Yoneda. The origin of the lemma is an article on homological algebra Nobuo Yoneda. On the Homology Theory of Modules. Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics Vol. 7 No. 2, p MR In this article Yoneda introduced what is known now as the Yoneda Ext, which is a construction of Ext n (M, N) that does not use derived functors and coincides with the usual definition Ext n (M, N) := R n Hom(M, )(N) when the base category has enough injective objects. Then Yoneda mentions something which corresponds to the following curious result. Let F : A Ab be an additive functor defined on an abelian category A (Yoneda uses the term A-module ). Then for each X Ob(A) one has the functor Hom A (X, ) : A Ab represented by X and a natural isomorphism of abelian groups Similarly, for representable contravariant functors Nat(Hom A (X, ), F) = F(X). (L 0 ) Nat(Hom A (, X), F) = F(X). (R 0 ) This is an abelian version of what we know nowadays as Yoneda lemma. Under the assumption that A has enough injective/projective objects, this generalizes to isomorphisms and Nat(Ext n A (X, ), F) = L n F(X) (L n ) Nat(Ext n A (, X), F) = R n F(X). (R n ) 7

8 This is an interesting interpretation of derived functors! One possible modern reference for this result is A Course in Homological Algebra by Hilton and Stammbach. I took some time to track Yoneda s paper in the library. On pages he mentions the formulas (L 0 ) and (R 0 ), and then uses (L n ) and (R n ) as the definitions for L n F and R n X. If I understand correctly, then Yoneda shows that these definitions coincide with the usual ones. Here s the relevant part of his text (Yoneda uses the term satelite, and for a right exact functor F its satellite S n F is naturally isomorphic to the left derived functor L n F, while for a left exact functor F its satellite S n F is naturally isomorphic to R n F). Every presheaf is a it of representable presheaves Observation. 1) Every presheaf may be seen in a canonical way as a it of the representable presheaves. Namely, we have F = (Hom C (,X) F) C F Hom C (, X), where C F is the category where the objects are presheaf morphisms Hom C(, X) F, and morphisms in C F are morphisms f : X Y in C that induce commutative diagrams of 8

9 presheaves Hom C (, X) f Hom C (, Y). F 2) In particular, for every pair of presheaves F and G there is a natural bijection HomĈ(F, G) = lim (Hom C (,X) F) G(X). C F First of all, we see that part 2) is a consequence of 1), the fact that HomĈ(, G) converts its to limits, and the Yoneda lemma: HomĈ(F, G) = HomĈ( (Hom C (,X) F) Hom C (, X), G) C F = lim (Hom C (,X) F) C F HomĈ(Hom C (, X), G) = lim (Hom C (,X) F) C F G(X). Part 1) seems to be more interesting, and in fact it s a very important principle, but it turns out to be something tautological as well, and it may be easily seen from the construction of the category C F. By the Yoneda lemma, each morphism of presheaves Hom C (, X) F corresponds to an element x F(X), and the category C F has the following equivalent description: the objects are pairs (X Ob(C), x F(X)), the morphisms are f : X Y in C such that F( f )(y) = x. Because of that the category C F is also known as the category of elements of F. Then, by the definition of C F, the presheaf F gives a cocone with respect to (Hom C (,X) F) C F Hom C(, X), i.e. there are commutative diagrams Hom C (, X) f Hom C (, Y). F Then, for any other presheaf G that gives a cocone Hom C (, X) f Hom C (, Y) F G 9

10 there is a unique morphism of presheaves F G making the diagram commute. Indeed, by the equivalent description of C F, the commutative diagrams as above mean that for each X Ob(C) and x F(X) one has x G(X), such that for every morphism f : X Y in C F( f )(y) = x and G( f )(y ) = x. (*) The required morphism of presheaves is given by τ X : F(X) G(X), x x, and its naturality in X is the condition (*). This shows the formula 1). Here are some observations: The Yoneda embedding C Ĉ preserves limits: for any functor F : I C we have Hom C (, lim F(i)) = lim Hom C (, F(i)) (in a sense, this is dual to the fact that Hom C ( F(i), ) = lim Hom C (F(i), )). So a limit of representable presheaves is also a representable presheaf. The above observation is about its. If F = Hom C (, X) is a representable presheaf, then C F has a terminal object, which is the identity morphism Hom C (, X) Hom C (, X): Hom C (, Y)! Hom C (, X) Hom C (, X) id and because of that C Hom C(,X) Hom C(, Y) is isomorphic to Hom C (, X). Let X be a topological space and let Open(X) be the corresponding category of open subsets. A representable presheaf in this case is something boring, as Hom Open(X) (U, V) is either a one element set or an empty set. The category Open(X) F is formed by pairs (V, x) where V X is an open subset and x F(V). A morphism (V, x) (W, y) is an inclusion V W such that res WV (y) = x. The it (V,x) Hom Open(X) (U, V) Open(X) F is given by a disjoint union / / V X x F(V) Hom Open(X) (U, V) = V X x F(V) U V { } = V X U V F(V) where in the las expression identifies x F(V) with y F(W) if res WV (y) = x. So the it formula gives in this case F(U) = F(V), V X U V which is something obvious. /, 10

11 Example: direct and inverse image of presheaves (SGA 4) Let s conclude with one particular example of the formula F = (X,x) C F Hom C(, X): the inverse image of a presheaf. A functor between two small categories u : C D induces the inverse image functor u : D Ĉ that sends any presheaf G on D to the presheaf u G := G u on C: (u G)(X) := G(u(X)). In fact, there is another, less obvious functor u! that goes in the opposite direction: u! : Ĉ D, that is left adjoint to u, meaning that there is a natural bijection Hom D (u!f, G) = HomĈ(F, u G) for every presheaf F on C and G on D. In order to discover this functor, we may use the following common trick: first we see what happens if F is a representable presheaf, and then generalize the construction to an arbitrary presheaf using the wonderful it formula. If F = Hom C (, X) for some X Ob(C), then by the Yoneda lemma, the natural bijection as above corresponds to Hom D (u! Hom C (, X), G) = G(u(X)). Again by the Yoneda lemma, we see that u! Hom C (, X) should be isomorphic to the representable presheaf Hom D (, u(x)). Now any presheaf F is naturally isomorphic to F = and the following construction comes to mind. (X,x) Hom C (, X), C F Definition. Let u : C D be a functor between two small categories. Then the direct image of a presheaf F on C is the presheaf on D given by u! F := (X,x) Hom D (, u(x)). C F We see that this defines a covariant functor u! : Ĉ D. With this definition, we get natural isomorphisms Hom D (u!f, G) = Hom D ( (X,x) Hom D (, u(x)), G) C F = lim (X,x) C F Hom D (Hom D(, u(x)), G) = lim (X,x) C F G(u(X)) = lim (X,x) C F HomĈ(Hom C (, X), u G) = HomĈ( (X,x) C F Hom C (, X), u G) = HomĈ(F, u G), 11

12 just as required. Here s a trivial example. Let be the category with one object and one arrow id and let C be another small category. A functor u : C is defined by one object X Ob(C) such that X, so specifying a presheaf on is equivalent to specifying a set S. Let s see what u! S is. We have to consider the category S with objects being the elements of S and no arrows (except for the identity for each object). Such a category without nontrivial arrows is called a discrete category, and a it over a discrete category is isomorphic to the coproduct indexed by its objects. Therefore, u! S = Hom C (, X). S To give a more interesting example, every continuous map between topological spaces induces a functor which goes in the opposite direction: f : X Y u := f 1 : Open(Y) Open(X). Now if F is a presheaf on X, then the presheaf u F is called the direct image of F and it is denoted by f F: ( f F)(U) := (u F)(X) = F( f 1 (U)). For a presheaf G on Y, the presheaf u! G on X is called the inverse image and it is denoted by f G. This presheaf is given by the it over the category Open(Y) G which boils down to ( f G)(U) = V Y open U f 1 (V) G(V) = V Y open V f (U) G(V). This is the well-known formula for the inverse image. This is simply G( f (U)) if f (U) is an open subset, but in general f (U) is not open, so one has to take the it. 12