Some notes on sheaves (Edrx, 2007)2007oct28: You are not expected to understand this!
1. Sheafification ================= 2. Terminology and notation =========================== Ω is a frame (we will only use this frame, Ω). an /open set/ is an element of Ω. U, V, U∩V, ...: open sets. a /cover/ is a subset of Ω. \cU^-, \cV^-, ...: covers. a /dense cover/ is a cover that is ``downward-closed'': ∀ U ∈ \cU^-. ∀ V ∈ Ω. V \subseteq U => V ∈ \cU^- \cU, \cV, ...: dense covers. \cU^- /covers/ U when \bigcup \cU^- = U; \cU^- /covers at least/ U when \bigcup \cU^- \supseteq U. \cU^- is a /cover of/ U when \cU^- covers U. \cU^ is a /dense cover of/ U when \cU^ is a dense cover with \bigcup \cU^- = U. We are primarily interested on the fibration of dense covers over open sets, DCov(Ω) \to Ω. The projection functor is the union. Notation: $\cU_U$ is a cover ``over $U$'', i.e., whose union is $U$. The subscript will be ommited often. Morphisms: in the base, Ω: V |-> U when V \subseteq U in a fiber DCov(U): \cV_U |-> \cU_U when \cV \subseteq \cU in DCov(Ω): \cV_V |-> \cU_U when \cV \subseteq \cU We are only secondarily interested on the fibration of covers over open sets, Cov(Ω) \to Ω. Covers can be completed to dense covers: \downarrow \cU^- := { V∈Ω | ∃U ∈ \cU^-.V \subseteq U } The morphisms on Cov(Ω) are inherited from DCov(Ω), but let's skip the details - only ``\downarrow'' matters. 3. Amazing fact =============== Unions and intersections of dense covers are very well-behaved. (We will use a ``logical notation'' for them - `∨' and `∧' instead of `∪' and `∩'). If \cU is a dense cover of U and \cV is a dense cover of V, then \cU∨\cV := \cU∪\cV is a dense cover of U∪V, \cU∧\cV := \cU∩\cV is a dense cover of U∩V. (\cU∧\cV)_{U∩V} |------> \cV_V |--> |--> \cU_U |------> (\cU∨\cV)_{U∪V} U∩V |--------------> V |--------> |-------> U |-------------> U∪V Note: on non-dense covers we would have to define \cU^-∧\cV^- := { U∩V | U∈\cU^-, V∈\cV^- }, as \cU^-∩\cV^- may cover too little... 4. Finer covers =============== If \cU and \cV are dense covers for U then we say that \cV is /finer/ than \cU when \cV \subseteq \cU. (The corresponding definition for non-dense covers is harder). In each fiber DCov(U) the morphisms go from finer covers to coarser ones - \cV_U |-> \cU_U. A corollary from one of the ``amazing facts'': if \cU and \cV are two dense covers for \cU then \cU∧\cV is a dense cover for \cU that is finer than both \cU and \cV. ---| \cW |--- / - \ / | \ v v v \cU <--| \cU∧\cV |--> \cV Each fiber DCov(U) also has a top element, ⊤_U := \downarrow {U}. We will use the notation ⊥_U to refer to the imaginary bottom element of DCov(U) - an imaginary dense cover of U that is finer than all dense covers of U. Stacks ====== Stacks are equivalent to presheaves, but easier to define. We will define presheaves later. Some of our archetypical stacks: Ω := { U ∈ \bboldC | U open } C^∞(Ω) := { f_U: U -> \bboldC | f_U is C^∞ } C^∞_bounded(Ω) := { f_U: U -> \bboldC | f_U is C^∞ and bounded } A /stack over Ω/, C^∞(Ω) -> Ω, is a set C^∞(Ω) plus an ``extent function'' [[·]]: C^∞(Ω) -> ( [[ f_U ]] = U ) plus an ``action'' ·: C^∞(Ω)×Ω -> that acts as the restriction: f_U·V = f_U|(U∩V). The extent and the action must obey: |