Sheaves over Topological Spaces

The category of sheaves over a topological space has a rich internal logic, but they also have a very rich geometric interpretation. In these notes I review the basic notions of sheaves over topological spaces, with an emphasis on the geometry of sheaves. I prove that sheaves are toposes, present the proof of the equivalence between the category of sheaves over X and the set of local homeomorphisms into X . Finally, I define geometric morphisms and show that there is a correspondence between continuous functions and geometric morphisms. This material can mostly be found in Chapter II of Sheaves in Geometry and Logic by Mac Lane and Moerdijk [MM12]. 1. PRELIMINARIES Definition 1.1. A topological space is a pair (X ,O(X )), where X is a set and O(X ) ⊆ P (X ) satisfies (i) ;, X ∈ O(X ), (ii) O(X ) is closed under arbitrary unions, (iii) O(X ) is closed under finite intersections. We call O(X ) the topology, and the elements of O(X ) are called the open sets. We will view O(X ) as a category where the objects are the open sets and the morphisms are determined by set inclusion. The main building block in sheaf theory is a presheaf. Definition 1.2. Let C be a category. The presheaf category is the usual functor category b C := [Cop,Set]. In the special case when we are working over topological spaces we say that presheaves over (X ,O(X )) are presheaves over the O(X ), viewed as a category. Definition 1.3. Let F be a presheaf (of sets) over a topological space (X ,O(X )). An element s ∈F (U) is called a section of F over U , and an element of F (X ) is called a global section. Notation 1.4. Consider a presheaf F over (X ,O(X )), and U , V ∈ O(X ) with V ⊂ U . Then we have a map F (V ⊂ U) : F (U) → F (V ). We call this map the restriction map and write the action of this map for every s ∈F (U) as s 7→ s|V , as if it is were actual restriction of functions. The notion of sheaves allows one to patch locally defined objects together to produce an object defined on a union of open sets. In some sense, we obtain a framework for dealing with local to global analysis. Definition 1.5. Let (X ,O(X )) be a topological space, and let F be a presheaf over X . We say that F is a sheaf if both of the following conditions hold: (i) if U ∈ O(X ) and {Uα}α∈Λ is an open cover of U , that is if U = ⋃ α∈Λ Uα and Uα ∈ O(X ), and if s, t ∈F (U) satisfy s|Uα = t|Uα for all α ∈ Λ then s = t. If just (i) holds then we call F separated. (ii) if U ∈ O(X ) and {Uα}α∈Λ is an open cover of U , and if for every α ∈ Λ there exists sα ∈F (Uα) with the family (sα)α∈Λ satisfying sα|Uα∩Uβ = sβ |Uα∩Uβ for all α,β ∈ Λ, then there exists some s ∈F (U) such that sα = s|Uα for all α ∈ Λ. The following categorical characterization of a sheaf is equivalent; it is just saying the same thing slightly differently. E-mail address: [email protected]. Date: April 10, 2017. 1 Categorical Logic Yousuf Soliman Definition 1.6. Let (X ,O(X )) be a topological space, and let F be a presheaf over X . We say that F is a sheaf if for every U ∈ O(X ) and any open cover {Uα}α∈Λ of U we have that F (U) e // ∏ α∈Λ F (Uα) πα // πβ // ∏