Some notes on sheaves (Edrx, 2007)

2007oct28: You are not expected to understand this!
I am only using this file to store some draft notes about sheaves, in DNC notation (see my math page)...

• 1. Sheafification
• 2. Terminology and notation
• 3. Amazing fact
• 4. Finer covers

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: