Ultracategories seminar Talk 1: Sheafs on topological spaces Julie Zangenberg Rasmusen February 2019 Contents 1 Introduction to sheaves 1 2 Sheaf of cross-sections 3 3 Bundles of germs 4 4 Étale spaces 6 5 Stalks 10 6 Direct and inverse image sheaves 12 These notes are based on chapter 2 of "Sheaves in Geometry and Logic" by Saunders Mac Lane and leke Moerdijk. 1 Introduction to sheaves Denition 1.1. Let X be a topological space. Then we dene the category O(X) to have all open subsets U of X as objects and dening the morphisms V ! U to be the inclusion V ⊂ U. Denition 1.2. A presheaf is a functor P : O(X)op ! Sets. This means that every inclusion of open sets V ⊂ U in X determines a functor P (V ⊂ U): P (U) ! P (V ) which we will often denote as t 7! tjV for each t 2 P (U) (as if they were restriction of ordinary maps) 1 1 INTRODUCTION TO SHEAVES Denition 1.3. A sheaf of sets on a topological space X is a presheaf F : O(X)op ! Sets, such that each open covering U = [iUi; i 2 I of an open set U of X yields an equalizer diagram p e Q / Q FU / FUi / F (Ui \ Uj) i q i;j where for t 2 FU e(t) = ftjUi ji 2 Ig and for a family ti 2 FUi, pftig = ftijUi\Uj g; qftig = ftjjUi\Uj g Denition 1.4. Let X be a topological space. We dene a new category Sh(X): Objects: All sheaves F of sets on X Morphisms F ! G: Natural transformations F ) G. Note that Sh(X) is a full subcategory of the functor category O\(X) := SetsO(X)op which is the category of presheaves on X. Example 1.5. Let X be a topological space and U ⊆ X an open subset. Dene CU := ffjf : U ! R is continuous g: If V ⊆ U then restricting each f 7! fjV gives us a function CU ! CV . This restriction is transitive in the sense that if W ⊆ V ⊆ U are three nested open subsets, then (fjV )jW = fjW . This gives us that the assignments U 7! CU; fC ⊆ Ug 7! fCU ! CV by f 7! fjV g denes a presheaf C : O(X)op ! Sets. We will prove that this is indeed a sheaf. Let U = Ui be an open covering with index set I. Then an I-indexed family of continuous functions fi : Ui ! R is an element of the product set ΠiCUi. The assignments and ffig 7! ffijUi\Uj g; ffig 7! ffjjUi\Uj g gives two maps p and q from I-indexed set to I × I-indexed set as in q Q / Q CUi / C(Ui \ Uj): i p i;j Since being continuous is a local property this gives us an equalizer diagram q e Q / Q CU / CUi / C(Ui \ Uj): i p i;j where maps to the -indexed set . This proves that is a sheaf on . e f I ffijUi g C X 2 2 SHEAF OF CROSS-SECTIONS Example 1.6. Let X be a locally connected space, then a sheaf F on X is locally constant if each point x 2 X has a basis of open neighbourhoods Nx such that whenever U; V 2 Nx with U ⊂ V , the restriction FV ! FU is a bijection. Example 1.7. Let X be a topological space, x 2 X a point and S 2 Set a set. We op dene the skyscraper sheaf concentrated at x, denoted by Skyx(A): O(X) ! Sets, by ( A ; x 2 U Skyx(A)(U) = 1 ; otherwise where 1 denotes a xed point and U an open subset of X. We can turn this into a functor Skyx : Sets ! Sh(X) by saying that for an inclusion V,! U of open sets, the restriction map Skyx(A)(U) ! Skyx(A)(V ) is the evident maps: A −!id A if x 2 V ⊆ U unique A −−−−! 1 if x2 = V ⊆ U: Proposition 1.8. Let X be a topological space, then the category Sh(X) has all small limits and the inclusion of sheaves into presheaves Sh(X) ! O\(X) preserve all small limits. 2 Sheaf of cross-sections Denition 2.1. Let X be a topological space. A continuous map p : Y ! X is called a bundle over X. Denition 2.2. Let p : Y ! X be a bundle and U an open subset of X, then pU : p−1U ! U is a bundle over U. Moreover the square diagram −1 incl: / p U < Y s pU p U / X incl: is a pullback diagram in Top. A cross-section s of the bundle pU is a continuous map s : U ! Y such that p ◦ s is the inclusion i : U ! X. Let ΓpU = fsjs : U ! Y and p ◦ s = incl:g denote the set of all such cross-sections over U. 3 3 BUNDLES OF GERMS If we have V ⊆ U in the above context, then we have a restriction ΓpU ! ΓpV so we get a functor op Γp(−): O(X) ! Sets This can be proven to be a sheaf since it's sucient to check locally if a given function is a cross-section. Hence Γp is a sheaf of sets X, called the sheaf of cross-sections of the bundle p. So we have a functor Γ: Bund(X) ! Sh(X): 3 Bundles of germs Denition 3.1. Let P : O(X)op ! Sets be a presheaf, x a point and U; V two open neighbourhoods of x. Further let u 2 PU and v 2 PV . We say that u and v has the same germ if there exists an open set W ⊂ U \ V such that ujW = vjW 2 PW . This is an equivalence. The equivalence class of any one such u is called the germ of u at x. This will be denoted germxu. Further let Px = fgermxu j u 2 P U; x 2 U with U open in Xg be the set of all germs at x. This is called the stalk of P at x. The denition of stalks allow us to consider our sheaf locally. 4 3 BUNDLES OF GERMS Example 3.2. Let X be a topological space. A constant sheaf F associated to some set A is a sheaf F 2 Sh(X) for which it is true that Fx = A for all x 2 X. A sheaf F 2 Sh(X) is locally constant if for each x 2 X there exists an open neighbourhood U ⊆ X of x such that the restriction F jU is a constant sheaf on U. Denition 3.3. ` ΛP := x Px = fgermxs j x 2 X; s 2 PUg Let p :Λp ! X, p(germxs) = x Let s 2 PU. This determines a function s_ by s_ : U ! ΛP ; sx_ = germxs which is a section of p. Topologize ΛP by taking as a base of open sets all the images s_(U) ⊂ ΛP , so an open set in ΛP is a union of images of s_. This makes every p and every s_ continuous - in particular is s_ : U ! s_(U) a homeomorphism. Using this above construction we get a functor Λ: O\(X) ! Bund(X) P 7! ΛP where we by ΛP mean the bundle p :ΛP ! X as dened above (We will often use this abuse of notation). We note that for a given presheaf P , ΓΛP is a sheaf of sections of bundles, since we have O\(X) Λ / Bund(X) Γ / Sh(X) : Denition 3.4. Let P be a presheaf over X. Then for each open subset U ⊆ X there is a function ηU : PU ! ΓΛP (U) s 7! s_ Note that η : P ! ΓΛP is a natural transformation of functors. Will only state the following theorem: op Theorem 3.5. If F : O(X) ! Sets is a sheaf, then η : F ! ΓΛF is an isomorphism. This means that every sheaf is a sheaf of cross-sections. Theorem 3.6. Let P be a presheaf. Then η is universal from P to sheaves. 5 4 ÉTALE SPACES This means that if F is a sheaf and θ : P ! F is any map of presheafes, then there exists a unique map σ : ΛΓP ! F of sheafs, such that the triangle commutes: η P / ΛΓP 9!σ θ ! F Proof of Theorem 3.6. By theorem 3.5 we know that η : F ! ΓΛF is an isomorphism, −1 so we may dene a map σ : ΛΓP ! F of sheaves as σ = η ΓΛθ such that the lower triangle in the following commutes: η P / ΓΛP σ θ ΓΛθ } / F η ΓΛF Since η is natural (as noted before it is a natural transformation) we know that the outer square commutes as well. But again, by theorem 3.5, we know that η : F ! ΓΛF is an isomorphism, so the upper triangle commutes. It remains to prove that this is indeed unique. Corollary 3.7. For any topological space X the inclusion functor Sh(X) ,! O\(X) admits a left adjoint. Proof. From theorem 3.6 we get that ΓΛ is a left adjoint to the inclusion with η as the unit. Denition 3.8. A left adjoint functor to the inclusion functor Sh(X) ,! O\(X) is called a sheacation functor. so we have that ΓΛ is a sheacation functor, and in particular a sheacation functor preserves all nite colimits. 4 Étale spaces Denition 4.1. A bundle P : E ! X is said to be étale when P is a local homeomor- phism in the following sense: To each e 2 E there is an open set V with e 2 V ⊂ E such that PV is open in X and P jV is a homeomorphism.