"Sheaves in geometry and logic" for dummies

“Sheaves in geometry and logic” for dummies Fernando Tohmé and Ignacio Viglizzo August 14, 2024 This is a rough draft of notes taken while reading the book “Sheaves in Geometry and Logic”[MLM94]. Here we record some examples and additional material or changes of notation that helped us understand the material, and we think someone else may find them useful as well. Frankly, we are surprised by the number of times this has been read in a few days. If you find any errors, or have questions or suggestions, please write to [email protected]. 1 1.1 Categories of Functors The Categories at Issue Examples of topoi: (iv) and (v) BG or G-Sets for a group G and BM or M -Sets for a monoid M , are categories of representations of G or M . Each representation is a pair (X, µ) where X is a set and µ is a right actions on sets. BG can also be seen as a category of functors SetsG . A functor R from G to Sets chooses a set and then each group element g is represented by a function R(g) : X → X. G(g)(x) = x · g. Notice how this is generalized in example (viii). (vi) Sets2 is also presented in [Gol84] as Sets→ . (viii) SetsC op = Ĉ. If f : D → C, P a functor from C op to Sets, P f : P C → P D, and x ∈ P C, the restriction of x along f is: x · f = x|f = P f (x) This operation satisfies x · (f ◦ g) = (x · f ) · g In this way, the morphisms in C act on the right on the sets P X. See [Ros22], section 2.4.1. for more on this approach. Yoneda embedding: C → Ĉ. For objects: yC = HomC (−, C) for a morphism α : D′ → D, u : D → C, yC(α) : yC(D) → yC(D′ ), that is yC(α) : HomC (D, C) → HomC (D′ , C) yC(α)(u) = u ◦ α 1 For Yoneda’s Lemma, we consulted [Awo10]. [MLM94] cites instead [ML98], where it makes use of universal arrows: Definition 1.1. If S : D → C is a functor and c an object of C, a universal arrow from c to S is a pair (r, u) consisting of an object r of D and an arrow u : c → Sr of C, such that to every pair (d, f ) with d an object of D and f : c → Sd an arrow of C, there is a unique arrow f ′ : r → d of D with Sf ′ ◦ u = f . In other words, every arrow f to S factors uniquely through the universal arrow u, as in the commutative diagram u c / Sr r Sf ′ f f′  Sd  d The arrow u is initial among those from c to any Sd. (xii) We studied [Fri12] and [Rie08] to get an idea of how this works. 1.2 Pullbacks In the category of Sets, the pullbackof a diagram: B g A f  /C can be constructed as the set P = {⟨a, b⟩|f (a) = g(b)} = a f −1 (c) × g −1 (c) = A ×C B c∈C Consider the pullback P of f with itself, where f : X → B. In any category, if the pullback exists, is a parallel pair of arrows P ⇒ X called the kernel pair of f . f is monic precisely when, up to isomorphism, both arrows in its kernel pair are the identity 1X : X → X. To prove this first we solved this exercise from [Awo10]: Section 5.7, page 114, 2 Let C be a category with pullbacks. (a) Show that an arrow m : M → X in C is monic if and only if the diagram below is a pullback. M 1M 1M  M /M m m 2  /X Proof. Assume we have Y, x and y such that mx = my: Y y M x 1M /( M 1M m  M  /X m then, since m is monic x = y, so the diagram Y x=y y M x 1M /( M 1M m  M  /X m commutes and yields a pullback. For the other implication: if the diagram is a pullback and mx = my, then there is a unique arrow u : Y → M such that 1M u = x = y. Now assume that f is monic and that p1 , p2 is a kernel pair. That is, the following diagram is a pullback: p2 /X P p1 f  X f  /B Since f is monic, by the preceding exercise, X 1X /X 1X  X f f  /B is a pullback as well, so we have that P and X are isomorphic. Let φ : P → X be the isomorphism. Then p1 = p2 = φ. For the converse, assume that p2 /X P p1  X f f 3  /B is a pullback and both p1 and p2 are isomorphic to 1X . From the hypothesis, we get that the following diagram commutes, X 1X P 1X p2 '/ X p1 f  X  /B f and therefore there is a unique morphism u : X → P such that p1 u = 1X = p2 u. It also follows that p1 = u−1 = p2 ,and that 1X /X X 1X  X f f  /B is a pullback, so by the previous exercise, f is monic. All the categories presented in 1.1 have pullbacks and terminal objects. Thus, they have finite products (X × Y is the pullback of X → 1 ← Y and 1 is the product of no factors) and equalizers (the equalizer e of f, g : X ⇒ Y obtains as a pullback of (f, g) : X → Y × Y and ∆ : Y → Y × Y ). Proposition 5.2.1 in [Awo10] indicates that a category has all finite limits iff it has products and equalizers. Given any diagram D : J → C, the equalizer of two maps Y Y ϕ, ψ : Di ⇒ Dj i∈Ob(J) α∈M orph(J),α:i→j is a limit of D. These arrows are defined by πα ϕ = πcod(α) and πα ψ = Dα πdom(α) Accordingly, all categories (i) − (xiv) have finite limits. 1.3 Characteristic Functions of Subobjects [The category C is said to be well-powered if there exists an object Ω such that for all objects X of C, SubC (X) is isomorphic to a small Hom-set HomC (X, Ω).] 1.4 Typical Subobject Classifiers [For an arbitrary small category C, a subfunctor of P : C op → Sets is defined to be another functor Q : C op → Sets with each QC a subset of P C and each Qf : QD → QC a restriction of P f , for all arrows f : C → D of C.] As a consequence, for each x ∈ QD, Qf (x) ∈ QC ⊆ P C. In other words: P f [QD] ⊆ QC. (1) [Given an object C in the category C, a sieve on C is a set S of arrows with codomain C such that: f ∈ S and the compositef h is defined implies f h ∈ S. 4 Now if Q ⊆ HomC (−, C) is a subfunctor, the set S = {f | for some object A, f : A → C and f ∈ Q(A)} is clearly a sieve on C.] Proof of the claim: Notice that S = ∪A∈ob(C) Q(A). Then, if f ∈ S, for some A, f : A → C and f ∈ Q(A). If h : B → A, then f h is defined. Furthermore, f h = HomC (h, C)(f ), where HomC (h, C) : HomC (A, C) → HomC (B, C). So f h ∈ Q(B) ⊆ S. S · g = Hom(−, g)−1 (S) = {h|g ◦ h ∈ S} 1.5 Colimits op The main result in this section is that every contravariant functor in the category SetsC is a colimit of representable functors. This is done in [MLM94] in an indirect way, proving first a stronger result, using an adjunction. In [Awo10], a direct proof is given, and the result with adjunction is given in a later chapter. Proposition 8.10. (from [Awo10]. This is Proposition 1= Corollary 3 in [MLM94])) For any op small category B, every object P in the functor category SetsC is a colimit of representable functors, lim yCj ∼ = P. −−→ j∈J More precisely, there is a canonical choice of an index category J and a functor π : J → C such ∼ that there is a natural isomorphism − lim →J y ◦ π(j) = P . op Proof. Given P : C → Sets, the index category we need is the so-called category of elements of P , written, Z P C and defined as follows. Objects: pairs (x, C) where C ∈ C0 and x ∈ P C. Arrows: an h : (x′ , C ′ ) → (x, C) is an arrow h : C ′ → C ∈ C such that P (h)(x) = x′ (2) actually, the arrows are triples of the form (h, (x′ , C ′ ), (x, C)) satisfying (2). The reader can easily work out the obvious identities and composites. R Note that P is a small category since C is small. There is a “projection” functor, π : C R ′ ′ P → C defined by π(x, C) = C and C R π(h, (x , C ), (x, C)) = h. To define the cocone of the form y ◦ π → P , take an object (x, C) ∈ C P observe that (by the Yoneda lemma) there is a natural, bijective correspondence between x ∈ P (C) x : yC → P which we simply identify notationally. Moreover, given any arrow h : (x′ , C ′ ) → (x, C) naturality in C implies that there is a commutative triangle 5 yh yC ′ x′ P / yC x ~ The naturality of Yoneda’s lemma in C can be read in the diagram: CO HomĈ (yC, P ) / PC  Hom(yC ′ , P )  / P C′ yh h C′ ηC,F yC O Ph yC ′ Remember that: • y : C → Ĉ is a covariant functor. • HomĈ (−, P ) is contravariant. • HomĈ (yh, P ) is calculated by composing with yh. • We ignore the functions ηC,F , identifying natural transformations and elements of P C. Now if we write P h(x) instead of x′ and use the commuting square above we get: ✤ x ❴ /x ❴  x ◦ yh ✤  / P h(x) That is, x ◦ yh = P h(x). We can therefore take the component of the desired cocone yπ → P at (x, C) to be simply x : yC → P . To see that this is a colimiting cocone, take any cocone yπ → Q with components θ(x,C) : yC → Q and we require a unique natural transformation θ : P → Q as indicated in the following diagram: yh yC ′ P h(x) P θ(P h(x),C ′ ) ~ / yC x   Q We can define θC : P C → QC by setting θC (x) = θ(x,C) 6 θ(x,C) (3) where we again identify, θ(x,C) ∈ Q(C) θ(x,C) : yC → Q This assignment is clearly natural in C by the commutativity of the diagram (3). In more detail: θC / QC CO PC h C′ Qh Ph  P C′ θC ′  / QC ′ commutes because for every x ∈ P C: Qh(θC (x)) = Qh(θ(x,C) ) = θ(x,C) ◦ yh = θ(P h(x),C ′ ) = θC ′ (P h(x)) Here the equation Qh(θ(x,C) ) = θ(x,C) ◦ yh follows from the same argument we used for showing that x ◦ yh = P h(x). For uniqueness, given any φ : P → Q such that φ ◦ x = θ(x,C) , again by Yoneda we must have φ ◦ x = θ(x,C) = θ ◦ x. ✷ Theorem 2 [MLM94] If A : C → E is a functor from a small category C to a cocomplete op category E, the functor RA : E → SetsC given by RA (E) : C 7→ HomE (A(C), E) has a left adjoint LA : SetsC op → E defined for each presheaf P as the colimit A ◦ π. LA (P ) = R−lim −− → C LA Ĉ o RA P / ?E A C Proof. To prove the adjunction is to prove the isomorphism: N at(P, RA (E)) ∼ = HomE (LA (P ), E). Corollary 3 [MLM94] Every presheaf is a colimit of representable presheaves. Proof. Take in the Theorem: LA / Ĉ o ? Ĉ RA y C 7 Then, from the definition of Ry , Ry and Ly must be both isomorphic to the identity functor in Ĉ, so y ◦ π. P ∼ = Ly (P ) = R−lim −− → C P Corollary 4 [MLM94] compare with Proposition 8.11 and 9.16 in [Awo10] LA ĈO o RA / F! ĈO o ?E y F / ∗ ?E y A F C C The diagram on the right corresponds to the notation in [Awo10]. Notice that: • Ĉ is in this sense a free cocompletion of C. • In these cocompletions, every functor has a left and right adjoint, see Corollary 9.17 in [Awo10]. They are the left and right Kan extensions. • In both proofs, the following Lemma is used. Lemma 1.2. (see [Kav16]) If an index category J has a final element 1J , then a diagram A : J → C has a colimit in C and it is A(1J ). Proof. For each i ∈ J, the arrow A(!i ) : Ai → A(1J ) and the functoriality of A prove that A(1J ) is a cocone for A. Given another cocone C, {τi }i∈J , there exists an arrow τ1J : A(1J ) → C, because C is a cocone. Af Ai A!i τi / Aj A!j " | A(1J ) τj τ 1J    C To see the uniqueness of τ1J consider the diagram: A(1J ) A!1J τ 1J $ A(1J ) ϕ   C Since !1J = 11J , A!1J = 1A(1J ) , so if ϕ makes the diagram commute, we have τ1J = ϕ ◦ 1A(1J ) = ϕ. On fibrations: On page 44, we read: 8 the projection π has the property that for each such p ∈ P (C) and each u : C ′ → C there is a unique pair p′ , u′ : (C ′ , p′ ) → (C, p) with πp′ = C ′ , π(u′ ) = u. Any functor π : E → C with this latter property is called a fibration of categories. In the Epilog, page 598 it says: Fibrations-or the essentially equivalent notion of indexed categories-occur frequently in topos theory. Pare and Schumacher (1978) describe indexed categories; Gray (1966) [Gra66] has an extensive description of fibrations, while Benabou’s article (1985) provides some controversy as well as a good list of references on fibrations. In turn, in [Gra66], the introduction reads: Fibred categories were introduced by Gkothendieck in [SGA] and [BB190]. As far as I know these are the only easily available references to the subject. Through sheer luck, during the final preparation of this paper I obtained a copy of handwritten notes [BN] of a seminar given by Chevalley at Berkeley in 1962 which treated these questions from a slightly different point of view. Francis Borceux says in an e-mail from January 21, 2024: When I wrote the three volumes of my “Handbook of categorical algebra” [Bor94a], I wanted of course to include a chapter on fibred or indexed categories: I chose fibred categories (Chapter 8 of Volume 2) [Bor94b]. Before sending the chapter to the editor, I sent a copy of it to Jean, asking for his comments, but making clear that I was not at all asking permission to publish this chapter, and that I would be the only one (with the referees) to decide of the final form of the text and take the responsibility of it. I definitely wanted to avoid a new endless story, in the vein of what was happening to the Jean-Roger notes. I remember the upset answer that Jean gave me; he was focusing on three points. • I do not appreciate that you include a chapter containing many results of mine before I myself publish them. • At least, it is a relief to notice that your text reflects faithfully my own views on this topic. • Thank you for putting emphasize on the notion of decidability, whose importance does not seem to have been recognized by the categorical community. Thomas Streicher, on the categories list said on January 18, 2024: In this context I want to bring up the issue of retyping some old stuff of Benabou and alike which I have made accessible via my homepage some time ago with the help of my former student J. Weinberger. [...] as well as Chaper 1 of his never finished book on Fibered Categories (about 100 pages). [Str23] 9 1.6 Exponentials Y ×X →Z Y → ZX op X ⟨X, Z⟩ 7→ Z is a functor C × C → C. The evaluation map eX,Z : Z X × X → Z is the co-unit of the adjunction, and the transpose of the identity 1Z X , while the unit ηZ : Z → (Z × X)X is the transpose of the identity 1Z×X . eZ,X : Z X ×X → Z can be seen as a natural transformation in Z, and a dinatural transformation ′ in the variable X: for each fixed Z, the functor Z X × X is covariant in X but contravariant in X ′ . The definition of dinaturality ( see [ML98], page 218, and [Gav]) requires that the following diagram commute: S(X, X) 8 S(f,1X ) αX,X / T (X, X) T (1X ′ ,f ) & T (X, X ′ ) 8 S(X ′ , X) S(1X ′ ,f ) & S(X ′ , X ′ ) α T (f,1X ′ ) X ′ ,X ′ / T (X ′ , X ′ ) Here we have: • S an T are functors from Cop × C to some category A, • f : X → X ′ in C, and f : X ′ → X in Cop • α is the dinatural transformation from S to T . ′ Taking S(X ′ , X) to be Z X × X and T the constant functor Z, the diagram reduces to: t Z ×1X Z8 X × X eZ,X # ′ ZX × X Z 1X ′ ×t & ′ ZX × X′ ;Z eZ,X ′ In Sets, we have for a function g : X ′ → Z, t : X → X ′ , and x ∈ X: (g ◦ t, x) : ☛ ✹ (g, x) ✡ % g(t(x)) 9 ✸ $ (g, t(x)) 10 We can use the Yoneda principle to prove some of the isomorphism: 1X ≡ 1 y1X = Hom(−, 1X ). Then, for every Y , Hom(Y, 1X ) ≡ Hom(X × Y, 1) ≡ {∗} ≡ Hom(Y, 1). X1 ≡ X For every Y , Hom(Y, X 1 ) ≡ Hom(1 × Y, X) ≡≡ Hom(Y, X). X Y ×Z ≡ (X Y )Z For every T , Hom(T, X Y ×Z ) ≡ Hom((Y × Z) × T, X) ≡ Hom(Y × (Z × T ), X) ≡ Hom(Z × T, X Y ) ≡ Hom(T, (X Y )Z ). There is an adjunction: ∆S → P S → ΓP Where ∆ : Sets → Ĉ sends a set S to the constant functor with S as the value, and Γ : Ĉ → Sets takes each functor P to the set of all its global sections: {γC ∈ P C} such that if f : C → D is a morphism in C, then P f (γD ) = γC , written as γD · f = γC . If we put S = 1, we have two forms of seeing global sections. For a general S, we have an S-indexed family of global sections. A left exact functor is a functor that preserves finite limits. A right exact functor is a functor that preserves finite colimits. 1.7 Propositional Calculus Ina topological space, if U and V are open sets, U ⇒ V = (−U ∪ V )◦ , and ¬U = U ⇒ ∅ = (−U )◦ . Here −X denotes the boolean, set theoretic complement of X, and ¬ is the intuitionistic negation. On page 50, the boolean complement in a lattice is also denoted with ¬. 1.8 Heyting Algebras [− ⇒ y is a contravariant functor in the argument y]: if a ≤ b, then b ⇒ y ≤ a ⇒ y. Also: z ≤ x ⇒ y iff x ≤ z ⇒ y This can be seen as an adjunction: we can think of − ⇒ y as a functor L : H → H op and also as a functor R : H op → H, so the equivalence above can be read as: x≤z⇒y z≤x⇒y 11 L(z) ≥ x z ≤ R(x) Then L preserves coproducts, so it carries coproducts to products (coproducts in H op ): L(x ∨ z) = L(x) ∧ L(z) (x ∨ z) ⇒ y = ((x ⇒ y) ∧ (z ⇒ y)) [... ¬¬U is the interior of the closure of U , which may be larger than U , as for example when U ] is (0, 1) ∪ (1, 2). [Proposition 5 The set SubĈ (P ) of all the subfunctors of P is a complete lattice satisfying the infinite distributive law.] If S and T are subfunctors of P , then we have for each object C, (S∨T )C = SC∪T C. To see that S ∨T is indeed a subfunctor of P , we need to check the condition (1). Consider f : C ′ → C. Since S and T are subfunctors, Sf : SC → SC ′ is P f |SC , and similarly for T . Thus (S ∨ T )f = P f |SC∪T C . Furthermore, we need to check that P f [SC ∪ T C] ⊆ (S ∨ T )C ′ . P f [SC ∪ T C] = P f SC ∪ P f T C ⊆ SC ′ ∪ T C ′ A similar reasoning S the intersection (using that P f [SC ∩ T C] ⊆ P f SC ∩ P f T C), W works for completeness, with ( i Si )C = i (Si C), and for the infinite distributivity, since it holds for all the subsets of each P C. Why this reasoning doesn’t work for complements? Consider C to be the category 2, with objects 0, 1 and f : 0 → 1. If P is a presheaf, we have sets P 0, P 1 and P f : P 1 → P 0. A subfunctor S of P is a pair of subsets, S0 ⊆ P 0, S1 ⊆ P 1 such that P f [S1] ⊆ S0. If we took (¬S)C = P C \ SC for all objects C, we do not get a functor: suppose that there exists x ∈ P 1 such that x ∈ / S1 but (P f )x = x · f ∈ S0. Then (¬S)f [(¬S)1] ⊆ (¬S)0 fails. So the right definition of the intuitionistic negation is (¬S)C = {x ∈ P C|for all f : D → C, x · f = P f (x) ∈ / SD} Similarly: (S ⇒ T )C = {x ∈ P C|for all f : D → C, if x · f ∈ SD then x · f ∈ T D} (S ⇒ T )C = {x ∈ P C|for all f : D → C, x · f ∈ / SD or x · f ∈ T D} 1.9 Quantifiers as adjoints Given the projection function p : X × Y → Y , consider the inverse image as a functor between the poset categories p∗ : P(Y ) → P(X × Y ), with p∗ (T ) = p−1 (T ) = X × T for every T ⊆ Y . Theorem 1. The adjunctions ∃p ⊣ p∗ ⊣ ∀p can be seen in the correspondences: ∃p S ⊆ T S ⊆ p∗ T where ∃p S = {y ∈ Y | there exists x ∈ X such that ⟨x, y⟩ ∈ S}, given by the equivalences: ∃p S ⊆ T iff (there exists x ∈ X such that ⟨x, y⟩ ∈ S implies y ∈ T ) iff ( ⟨x, y⟩ ∈ S implies ⟨x, y⟩ ∈ X × T ) iff S ⊆ p∗ T . 12 p∗ T ⊆ S T ⊆ ∀p S where ∀p S = {y ∈ Y |for every x ∈ X, ⟨x, y⟩ ∈ S}, and the equivalence goes like this: p∗ T ⊆ S iff (if a ∈ p∗ T , then a ∈ S) iff (if a = ⟨x, y⟩ ∈ X × T then a ∈ S) iff (if y ∈ T , then for any x ∈ X, ⟨x, y⟩ ∈ S) iff T ⊆ ∀p S. Exercise 10 [Generalize Theorem 2 of Section 9 to presheaf categories. More precisely, prove op that for a morphism (i.e., a natural transformation) f : Z → Y in Ĉ = SetsC , the pullback functor f ∗ : SubC b (Y ) → SubC b (Z) has both a left adjoint :∃f and a right adjoint ∀f [Hint: the left adjoint can be constructed by taking the pointwise image. Define the right adjoint ∀f on a subfunctor S of Z by (∀f S)C = {y ∈ Y (C)| for all u : D → C in C and z ∈ ZD, z ∈ SD whenever fD (z) = yu}.]] In first place, we check that if P ⊆ Y , f ∗ P , which can be computed as (f ∗ P )C = fC (P C) is a subfunctor of Z. Consider an arrow u : D → C. We need to check that Zu((f ∗ P )C) ⊆ (f ∗ P )D. Since f is a natural transformation, we have: ZC fC Zu  ZD / YC Yu fD  / YD Now consider x ∈ Zu(fC−1 (P C)), so for some y ∈ fC−1 (P C), x = Zu(y). This is, for some y ∈ ZC, x = Zu(y) and fC (y) ∈ P C. Since P is a subfunctor of Y , (by (1)), (Y u)P C ⊆ P D, so Y ufC (y) ∈ P D, and by the commuting diagram above we have that fD Zu(y) ∈ P D, so x = −1 (P D) = (f ∗ P )D. Zu(y) ∈ fD Following the hint, we define for a subfunctor S of Z, (∃f S)C = fC [SC] ⊆ Y C. Now we prove that ∃f S is a subfunctor of Y , that is, that for every C and u, Y u(∃f S)C ⊆ (∃f S)D. For this, let x ∈ Y u(∃f S)C, so there exists y ∈ (∃f S)C such that Y u(y) = x. This means that there exists s ∈ SC such that fC (s) = y and (Y u)(y) = x, so Y ufC (s) = x. By the naturality of f , fD Zu(s) = x, but since S is a subfunctor of Z, Zu(s) ∈ SD, so x is the image by fD of some element in SD, that is, x ∈ fD [SD] = (∃f S)D. Now we check the adjunction: ∃f S ⊆ T S ⊆ f ∗T For any C (∃f S)C ⊆ T C is a condition on sets and we can omit the C. ∃f S = f [S] ⊆ T iff s ∈ S implies f (s) ∈ T iff s ∈ S implies s ∈ f −1 (T ) iff S ⊆ f −1 T = f ∗ T . Next we define (∀f S)C = {y ∈ Y C|for every u : D → C, z ∈ ZD, if fD (z) = Y u(y), then z ∈ SD}. First we check that ∀f S is a subfunctor of Y . Let v : D′ → C, so Y v : Y C → Y D′ . We want to prove that Y v(∀f S)C ⊆ (∀f S)D′ ). Let x ∈ Y v(∀f S)C. Then there exists y ∈ Y C such that (Y v)y = x and for any u : D → C and z ∈ ZD, fD (z) = Y u(y) implies z ∈ ZD. 13 To prove that x ∈ (∀f S)D′ , we have to show that for any α : A → D′ , z ∈ ZA, if fA (z) = Y α(x), then z ∈ SA. We have that fA (z) = Y α(x) = Y αY v(y) = Y (v ◦ α)y, so by the hypothesis z ∈ SA. For checking the adjunction: f ∗S ⊆ T S ⊆ ∀f T again we consider S and T to be sets (omitting the parameter C). In this instance, we have that ∀f T = {y ∈ Y |for all z, f (z) = y implies z ∈ T }. Now assume that f ∗ (S) = f −1 S ⊆ T and s ∈ S. If f z = s, then z ∈ f −1 (S) ⊆ T , so s ∈ ∀f T . In the other direction, assume that S ⊆ ∀f T . If z ∈ f ∗ S = f −1 S then f z ∈ S ⊆ ∀f T . To prove that z ∈ T , observe that for all z ′ , if f z = f z ′ then z ′ ∈ T . Since f z = f z, we conclude that z ∈ T . Reindexing. Some notation: ` a family of sets, and x ∈ Ai , we denote with ⟨x, i⟩ ` If {Ai }i∈I is the corresponding element in I Ai . Let pi : I Ai → I be the mapping ⟨x, i⟩ 7→ i. If we consider in Sets the pullback of the diagram: ` J I Ai p=[pi ]  /I α ` it can be constructed as J`×I I Ai = {⟨j, ⟨x, i⟩⟩|α(j) = p(⟨x, i⟩)} = {⟨j, ⟨x, i⟩⟩|α(j) = i}. This is equivalent to considering J Aα(j) . There is an equivalence between Sets/Y and SetsY . Then, if f : Z → Y , the pullback functor f ∗ : Sets/Y → Sets/Z can be thought of as the re-indexing f ∗ : SetsY → SetsZ given by f ∗ ({Ay |y ∈ Y }) = {Af (z) |z ∈ Z} Theorem 3 f ∗ has both a left and right adjoint. (therefore, it preserves both limits and colimits). Let us see these adjunctions through a simple example. Let Y = {1, 2, 3}, Z = {a, b, c} and f : Z → Y be given by f (a) = f (b) = 1, f (c) = 2. X Bz → Ay f (z)=y Bz → f ∗ Ay Then we have on top the mappings Ba + Bb → A1 and Bc → A2 , and below we have Ba → A1 , Bb → A1 , and Bc → A2 . The other adjunction is: f ∗ Ay → C z Y Ay → C f (z)=y On top we have maps A1 → Ca , A1 → Cb , and A2 → Cc and these correspond to maps A1 → Ca ×Cb and A2 → Cc below. 14 The functors Σf and Πf are sometimes suggestively called dependent sum and dependent product, see [Hua22]. In [Awo10], the pullback functor for f , f ∗ : SetsY → SetsZ gets the name f # when considered going from Sets/Y → Sets/Z. The right adjoint Πf is described for π : A → Z (that is, π is an object in Sets/Z) as (f# (A))y = {s : f −1 (y) → A|“s is a partial section of π”} where the condition “s is a partial section of π” means that the following triangle commutes with the canonical inclusion f −1 (y) ⊆ Z at the base. <A s f −1 (y)  π  /Z Since π ◦ s = 1f −1 (y) , then s is injective, and we are looking at the injective functions from f −1 (y) to A. In other words, at the product Πf (y)=z Az . Theorems 1 through 4 are a sequence of ever more general statements. Here is a comparison: Theorem 1 ∀p p X ×Y Sets p P(X × Y ) o /Y ∗  P(Y ) B ∃p Theorem 2 ∀f Sets f Z /Y P(Z) o f∗  P(Y ) D ∃f Exercise 10 ∀f SetsC op Z f /Y o SubC b (Z) f∗ ∃f 15  SubC (Y ) @ b Theorem 3 Πf Sets Z f Sets/Z o /Y f∗  Sets/Y B Σf Theorem 4 Πf C Z f /Y C/Z o f∗  C/Y E Σf Theorem 4 Let C be a category with pullbacks, and let B be an object of C. For each f : Z → Y , the change of base (pullback) functor f ∗ : C/Y → C/Z has a left adjoint; moreover, if C /Y is cartesian closed, each such f ∗ also has a right adjoint. Proof. The left adjoint Σf (called f! in [Hua22]) is given by composition with f : if a : A → Z is an object in C/Z, then f ◦ a : A → Y is in C/Y . If we take Y = 1, C/1 is isomorphic to C, and the pullback along f =!Z : Z → 1 is the functor − × Z : C → C/Z sending each object X to the object p : X × Z → Z. In this case, Σ!Z : C → C/Z is the forgetful functor that sends each object in the slice category to its domain. Now recall the general adjunction of exponentials: X ×Z →H X → HZ In particular, for X = 1 and H = Z, we have an isomorphism between Hom(1 × Z, Z) and Hom(1, Z Z ). We call j the morphism corresponding to 1Z . An arrow from π : X × Z → Z to h : H → Z is just an arrow t : X → H in C such that ht is the projection π: t /H X ×Z π h  Z  Z 16 These arrows correspond to the arrows t′ : X → H Z such that hZ t′ = j◦!X . ZO Z ZZ × O Z eval hZ HO Z t′ X eval ZZ × O Z ZO Z 1Z h H Z O× Z eval ZO o /Z O /H ; j eval Z co 1 ×O Z !X π t 1O X ×Z X ×Z X These arrows t′ in turn correspond by pullback exactly to the arrows t′′ : X → Γh, where Γh is the pullback in the square: X t′′ t′ (/ Γh !X HZ hZ   1 j / ZZ Therefore, Γh, the pullback of hZ along j, is the desired right adjoint to − × Z. Γ Z !Z  −×Z C/Z o C G /1 Σ! Z Note that, if C = Sets, this pullback Γh is just the set of those functions s from Z to H whose composite with h : H → Z is the identity of Z; that is, the set of cross sections of the map h. Hence, in general, we might call Γh the object of ”cross sections” of the arrow h. Γh = {s : Z → H|h ◦ s = 1Z } These cross sections are in particular, injective. This set can also be regarded as the product Q −1 h (z): each element (function) in the product is a selection of an element of H for each z∈Z z ∈ Z. Now return to the general case of any f : Z → Y . This arrow f is also an object (f ) in the slice category C/Y ; moreover, an object over (f ) is just a commutative square /Y X  Z f 17 /Y and this square is determined by the arrow X → Z; that is, by an object in C/Z. This correspondence is an isomorphism of slice categories (C/Y )/(f ) ≡ C/Z, and pullback along f ∗ : C/Y → C/Z = (C/Y )/(f ) is reduced to the previous case: the terminal object in C/Y is 1Y and we can find as before the right adjoint Γ to !∗(f ) where !(f ) : (f ) → 1Y . Another approach to similar results can be found in [Awo10]. Corollary 9.17, [Awo10] is a corollary to the Proposition 9.16 from this book that we saw in section 1.5. Let f : C → D be a functor between small categories. The precomposition functor f ∗ : SetsD op → SetsC op given by f ∗ (Q)(C) = Q(f C) has both left and right adjoints f! ⊣ f ∗ ⊣ f∗ Moreover, there is a natural isomorphism f! ◦ yC ∼ = yD ◦ f as indicated in the following diagram: f∗ SetsO Cop o f∗ * D Sets 8 O f! yC C op yD /D f The proof uses the construction of adjoints from Proposition 9.16 of [Awo10]. For !Z : Z → 1, we have the adjunctions ΣZ ⊣ Z ∗ ⊣ ΠZ with ΣZ (h : H → Z) = H Z ∗ (H) = (p : Z × H → Z) ΠZ (h : H → Z) = {s : Z → H|h ◦ s = 1Z } Moreover one therefore has ΣZ Z ∗ (H) = Z × H and ΠZ Z ∗ (H) = H Z Also, the following factorization of the product ⊣ exponential adjunction: Z×− Setsd o Z∗ ΠZ −Z / ΣZ $ z Sets/Z 18 Z∗ :Sets 2 Sheaves of Sets 2.1 Sheaves Some observations on the definition of sheaf : • It is interesting to compare the definition of sheaves given with the one in [Rie16]: ... let us remark that limit-preservation can be an important hypothesis: Definition 3.3.4. Let X be a topological space and write O(X) for the poset of open subsets, ordered by inclusion. An I-indexed family of open subsets Ui ⊆ U is said to cover U if the full diagram comprised of the sets Ui and the inclusions of their pairwise intersections Ui ∩ Uj has colimit U . A presheaf F : O(X)op → Sets is a sheaf if it preserves these colimits, sending them to limits in Sets. Applying Theorem 3.2.13 [ Any small limit in Sets may be expressed as an equalizer of a pair of maps between products.], the hypothesis is that for any open cover {Ui }i∈I of U , the following is an equalizer diagram: F (U ) F (Ui ֒→U ) / Y F (Ui ∩Uj ֒→Ui )◦πi F (Ui ) i∈I F (Ui ∩Uj ֒→Uj )◦πj // Y F (Ui ∩ Uj ) i,j∈I • In the category of Sets, the equalizer of any two arrows always exists. The definition of sheaf, however, says that F is a sheaf if F U is the equalizer of p and q. • It is remarked in the initial example of sheaf, the set of real-valued functions on a topological space, that the definition of sheaf amounts to saying that continuity can be tested for locally. This is split in two conditions: (i) If f : U → R is continuous and V ⊂ U is open, then the function f restricted to V is continuous, f |V : V → R. (ii) If U is covered by open sets Ui , and the functions fi : Ui → R are continuous for all i ∈ I, then there is at most one continuous f : U → R with restrictions f |Ui = fi for all i; moreover, such an f exists if and only if the various given fi ”match” on all the overlaps Ui ∩ Uj , in the sense that fi x = fj x for all x ∈ Ui ∩ Uj and all i, j in I. If we write C(X) for the set of continuous functions on X, the first condition is just “C” is a functor. The inclusions in O(X) get transformed into the restrictions of the continuous functions to open subsets. Condition (ii), as noted above is that the functor C preserves the limit U = ∪Ui in the poset category O(X). Proposition 1. If F is a sheaf on X, then a subfunctor S ⊂ F is a subsheaf S if and only if, for every open set U and every element f ∈ F U , and every open covering U = Ui , one has f ∈ SU if and only if f |Ui ∈ SUi for all i. 19 Proof. [The last condition of the proposition states precisely that the left-hand square is a pullback.] The pullback of the diagram: Q SUi  / Q F Ui FU can be constructed as the set of all pairs ⟨f, (yi )⟩ ∈ F U × hypothesis, this is SU . [diagram chasing] Q SUi such that yi = f |Ui , but the X f v SU u e′ (Q / q′ SUi // Q S(Ui ∩ Uj ) i′ i  FU p′ e Q  / F Ui i′′  Q // F (Ui ∩ Uj ) p q Here the arrows i, i′ , and i′′ are monic, andQe is an equalizer of p and q. To prove that e′ is an equalizer of p′ and q ′ , we take f : X → SUi such that p′ f = q ′ f . Then i′′ p′ f = i′′ q ′ f so ′ ′ pi f = qi f . Then there exists u : X → F U such that eu = i′ f . Then the pullback condition gives a unique v : X → SU such that e′ v = f , as needed. Theorem 2 Is this the diagram of the proof? FU 2.2 / Q // Q Fk (U ∩ Wk ) Q  F Ui  / Q Q Fk (Ui ∩ Wk )   F (Ui ∩ Uj )   / Q Q Fk (Ui ∩ Uj ∩ Wk ) Fkl (U ∩ Wk ∩ Wl ) // Q Q // Q Q  Fkl (Ui ∩ Wk ∩ Wl )   Fkl (Ui ∩ Uj ∩ Wk ∩ Wl ) Sieves and Sheaves When a poset is seen as a category, what is a sieve? We can identify a sieve on an object U of the poset with the set of elements of the poset such that the only arrow from this element to U is in the sieve. That is S = {V |V → U ∈ S}. By the definition of sieve, this is a decreasing set. Sieves on U are also the subfunctors of Hom(−, U ). A solution to Exercise II.1, stating that a sieve S on U is principal iff the subfunctor S of y(U ) is a subsheaf can be found at [jh]. Since Hom(V, U ) is always 1 or ∅, it us useful for the determination of subfunctors, to recall that there is an unique arrow in the cases ∅ → ∅, ∅ → 1 and 1 → 1 but no arrow from 1 to ∅ (this is just the order relation on the chain with two elements!). 20 Proposition 2. For any space X the category Sh(X) has all small limits, and the inclusion of sheaves in presheaves preserves all these limits. In the proof, we use the characterization of sheaves from Proposition 1, so to prove that Sh(X) has equalizers, we need to prove that i in the diagram below is an isomorphism, assuming that the top and bottom rows are equalizers in Sets, while i and j are isomorphisms. e′ Hom(yU, E) q′ / Hom(yU, F ) p′ // Hom(yU, G) j i h  Hom(S, E) e  / Hom(S, F ) q p //  Hom(S, G) We notice first that i−1 e equalizes p′ and q ′ : p′ i−1 e = j −1 pe = j −1 qe = q ′ i−1 e. Then, there exists u : Hom(S, E) → Hom(yU, E) such that e′ u = i−1 e. Then the usual argument proves that u is the inverse of h: Hom(yU, E) h  Hom(S, E) u  Hom(yU, E) e′ i−1 e e′ ( / Hom(yU, E) In Proposition 4 “1” is used for denoting the constant sheaf 1 = Hom(−, X), which is also the terminal object in O(X). 2.3 Sheaves and Manifolds 2.4 Bundles If the space X is discrete, a sheaf on X can be recovered from Q the values it takes on singletons. ` If F is a sheaf on X, letting f x = F ({x}), we have that F U = x∈U f x. Letting Y = f x and p : Y → X the obvious projection function, we have that Γp U = F U for every subset U of X. 2.5 Sheaves and Cross-Sections Px = {germx s|s ∈ P U, x ∈ U ∈ O(X)}, the stalk of a presheaf P at a point x ∈ X is a colimit. Let P (x) be the restriction of P to neighborhoods of x. germx s : P U → Px is a cocone. Given another 21 cocone, {τU : P U → L}x∈U , we have: PU  PW τU } Lo germx τW "  Px t For this we define for each germx s ∈ Px , t(germx s) = τU s. t is well defined: if s, s′ ∈ P U are such that germx s = germx s′ , then there exists an open set W such that s|W = s′ |W , so τU s′ = τW s′ |W = τW s|W = τU s. a ΛP = Px x Each s ∈ P U determines a function ṡ : U → ΛP . ṡ is a section of p. pṡx = p(germx s) = x ṡp|U (germx s) = ṡx = germx s The topology on ΛP is the one generated by the sets ṡ(U ) = {ṡ(x)|x ∈ U } = {germx s|x ∈ U } (for all U ∈ O(X) and s ∈ P U ). With this topology: p is continuous: If U is open, p−1 (U ) = ∪x∈U Px = {germx s|s ∈ P U, x ∈ U } = ∪s∈P U ṡ(U ). [Every function ṡ is continuous]: for any basic open ṫ(V ), V ⊆ U , consider a point u ∈ ṡ−1 (ṫ(V )). This means that ṡ(u) ∈ ṫ(V ), so for some v ∈ V, ṡ(u) = ṫ(v). This is, germu s = germv t, and therefore u = v and there exists some open set W ⊆ V such that u ∈ W and s|W = t|W , and thus ṡ(w) = ṫ(w) for all w ∈ W , so ṡ(W ) = ṫ(W ) ⊆ ṫ(V ). We conclude that u ∈ W ⊆ ṡ−1 (ṫ(V )). [ṡ is trivially an open map and an injection]: for every open set U , ṡ(U ) is a (basic) open. If ṡ(u) = ṡ(v) then germu s = germv s, but this is only possible if u = v. [Finally, if h : P → Q is a natural transformation between presheaves, the disjoint union of the functions hx : Px → Qx of hU / QU PU germx germx  Px hx  / Qx ` is a map ΛP → ΛQ of bundles, readily shown continuous.] The map will be x hx and we want to justify calling it Λh. ` ` Consider germz t ∈ ΛP such that germz t ∈ ( x hx )−1 (ṡ(U )). Then ( x hx )(germz t) = germu s for some u ∈ U . Then hz germz t = germz hU t = germu s. This implies that z = u and that there exists W ⊆ U , open and such that hU t|W = s|W . ` Now we have an open set ṫ(W )`of ΛP such that germz t ∈ ṫ(W ) and ṫ(W ) ⊆ ( x hx )−1 (ṡ(U )): Indeed, if we take an element of ( x hx )ṫ(W ), it is of the form hx germx t for some x ∈ W ⊆ U , but hx germx t = germx hU t = germx s ∈ ṡ(U ). 22 In the proof of Theorem 2, σ = ηF−1 ΓΛθ . ηP P / ΓΛP σ θ  } F ΓΛθ  / ΓΛF ηF In Lemma 3, notice that ηP is not necessarily an isomorphism. \ This means that the inclusion functor has a Sh(X) is reflective in the presheaf category O(X). left adjoint: ΓΛ (the sheafification functor ), but also that for each presheaf P there exists a sheaf ΓΛP and a presheaf morphism ηP : P → ΓΛP such that for each presheaf morphism θ : P → F to a sheaf F there exists a unique sheaf morphism σ : ΓΛP → S with σηP = θ. ηP P / ΓΛP !σ θ !  S Another way of saying this is the statement of Theorem 2: For any presheaf P , the corresponding morphism ηP : P → ΓΛP of presheaves is universal from P to sheaves. Recall the definition of universal arrow in Definition 1.1, section 1.1 of these notes. 2.6 Sheaves as Étale Spaces In the proof of Theorem 2:[Similarly, one verifies the continuity of ϵY : ΛΓY → Y ] To prove this consider an open set V ⊆ Y and a point in ϵ−1 Y (V ). Since this point is in ΓΛY , it is of the form ṡx for some x ∈ X and some cross-section s : U → Y for some open set U ⊆ X. Since s is continuous, s−1 (V ) is an open subset of X. Take W = U ∩ s−1 (V ), which is open and x ∈ W . ṡ(W ) is an open set in ΓΛY (because ṡ is a homomorphism), ṡx ∈ ṡ(W ), and ϵY (ṡ(W )) = s(W ) = s(U ∩ s−1 (V ) ⊆ s(s−1 (V )) ⊆ V , so this proves that ϵ−1 Y (V ) is open. “triangular identities”: Γ ηΓ / ΓΛΓ Λ Γϵ 1Γ Λη ϵΛ 1Λ !  Γ !  Λ An auxiliary diagram for the proof of Lemma 4: Λ0 P0 o Γ0 / > B0 Λ′ j i Γ◦j  ~ P Λ 23 / ΛΓΛ  /B The “different, more conceptual, way to construct the adjunction of Theorem 2” uses Theorem 2 of I.5, and the cocompleteness of Top/X. ΛP → B P → ΓB 2.7 Sheaves with Algebraic Structure op Ab(SetsO(X) ) ∼ = AbO(X) 2.8 op Sheaves are Typical Sets V1 ⊆ U1 and V2 ⊆ U2 such that V1 ∪ V2 is not in Ω: In the following case V1 ∪ V2 is in the equalizer: V1 ∩ U2 = U1 ∩ V2 . 24 References [Awo10] Steve Awodey. Category theory, volume 52 of Oxford Logic Guides. Oxford University Press, Oxford, second edition, 2010. [Bor94a] Francis Borceux. Handbook of categorical algebra. 1, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994. Basic category theory. [Bor94b] Francis Borceux. Handbook of categorical algebra. 2, volume 51 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994. Categories and structures. 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