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;;    http://angg.twu.net/2007sheaves.html
;; file:///home/edrx/TH/L/2007sheaves.html
   ;; (find-2005oct20seminarfile "")
   ;; (find-2005oct20seminar "")
   ;; (find-2005oct20seminar "" "original-motivation-2")
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   ;; (find-2005oct20seminarpage 31)
#]
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[htmlize [J Some notes on sheaves (Edrx, 2007)]

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

[WITHINDEX
[#
 # «.sheafification»		(to "sheafification")
 # «.sheaf-notation»		(to "sheaf-notation")
 # «.sheaf-amazing-fact»	(to "sheaf-amazing-fact")
 # «.sheaf-finer-covers»	(to "sheaf-finer-covers")
 #]
[RULE ----------------------------------------]

[tsec «sheafification»  (to ".sheafification")
[++N]. Sheafification
=====================
]

[tsec «sheaf-notation»  (to ".sheaf-notation")
[++N]. Terminology and notation
===============================
Ø is a frame
  (we will only use this frame, Ø).
an /open set/ is an element of Ø.
U, V, UÌV, ...: open sets.
a /cover/ is a subset of Ø.
\cU^-, \cV^-, ...: covers.

a /dense cover/ is a cover that is ``downward-closed'':
  ý U Ý \cU^-. ý V Ý Ø. V \subseteq U => V Ý \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(Ø) \to Ø.
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, Ø:     V |-> U         when V \subseteq U
  in a fiber DCov(U): \cV_U |-> \cU_U when \cV \subseteq \cU
  in DCov(Ø):         \cV_V |-> \cU_U when \cV \subseteq \cU

We are only secondarily interested on the fibration
of covers over open sets, Cov(Ø) \to Ø.
Covers can be completed to dense covers:
\downarrow \cU^- := { VÝØ | ÎU Ý \cU^-.V \subseteq U }
The morphisms on Cov(Ø) are inherited from DCov(Ø),
but let's skip the details - only ``\downarrow'' matters.
]

[tsec «sheaf-amazing-fact»  (to ".sheaf-amazing-fact")
[++N]. Amazing fact
===================
Unions and intersections of dense covers
are very well-behaved.
(We will use a ``logical notation'' for them -
`' and `´' instead of `þ' and `Ì').
If \cU is a dense cover of U
and \cV is a dense cover of V, then
\cU\cV := \cUþ\cV is a dense cover of UþV,
\cU´\cV := \cUÌ\cV is a dense cover of UÌV.

   (\cU´\cV)_{UÌV} |------> \cV_V
                  |-->           |-->
                      \cU_U |------> (\cU\cV)_{UþV}

          UÌV |--------------> V
              |-------->         |------->
                         U |-------------> UþV

Note: on non-dense covers we would have to define
\cU^-´\cV^- := { UÌV | UÝ\cU^-, VÝ\cV^- },
as \cU^-Ì\cV^- may cover too little...
]

[tsec «sheaf-finer-covers»  (to ".sheaf-finer-covers")
[++N]. 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:

               Ø := { U Ý \bboldC | U open }
          C^‚(Ø) := { f_U: U -> \bboldC | f_U is C^‚ }
  C^‚_bounded(Ø) := { f_U: U -> \bboldC | f_U is C^‚  and bounded }

A /stack over Ø/, C^‚(Ø) -> Ø,
is a set C^‚(Ø)
plus an ``extent function'' [[·]]: C^‚(Ø) ->
( [[ f_U ]] = U )
plus an ``action'' ·: C^‚(Ø)×Ø ->
that acts as the restriction: f_U·V = f_U|(UÌV).

The extent and the action must obey:
]

]]

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