[INCLUDE TH/speedbar.blogme]
[SETFAVICON dednat4/dednat4-icon.png]
[lua: LR = R ]
[#
(defun c () (interactive) (find-blogme3-sh0-if "2007dnc-etc"))
;;    http://angg.twu.net/2007dnc-etc.html
;; file:///home/edrx/TH/L/2007dnc-etc.html
;; (eev-math-glyphs-edrx)
;; (find-eevfile "eev-math-glyphs.el")
;; (find-glyphashtml-links "&&")
;; (find-glyphashtml-links "ku")
;; (find-glyphashtml-links "^^")
;; (find-glyphashtml-links "GG")
;; (add-to-alist 'eev-math-glyphs-name-to-char '("sqcup" . 343252))
;; (add-to-alist 'eev-math-glyphs-name-to-char '("Gamma" . 332659))
;; (eev-math-glyphs-set 'eev-glyph-face-linear "ud&"   "&&" "Ñ")
;; (eev-math-glyphs-set 'eev-glyph-face-linear "sqcup" "ku" "Ñ")
;; (eev-math-glyphs-set 'eev-glyph-face-graphic "neblock" "^^" "£")
;; (eev-math-glyphs-set 'eev-glyph-face-Greek   "Gamma"   "GG" "£")
#]
[lua:
  -- (eev-math-glyphs-edrx)
  eev_math_glyphs_edrx()
  sgmlify_table["\04"]  = "δ"
  sgmlify_table["\05"]  = "ε"
  sgmlify_table["\209"] = "⊔"
  sgmlify_table["\209"] = "⊔"
  sgmlify_table["\163"] = "Γ"
  sgmlify_re = "([\1-\8\12\14-\31\128-\254])"
]

[htmlize [J Notes on other downcasings]

[P These are notes on downcasings that I have not completed yet. They
are for personal use; they contain errors.]

[WITHINDEX
[#
 # «.right-kan-extensions»	(to "right-kan-extensions")
 # «.enriched-cats»		(to "enriched-cats")
 # «.polynomials»		(to "polynomials")
 # «.geometric-morphisms»	(to "geometric-morphisms")
 # «.geometric-morphisms-exs»	(to "geometric-morphisms-exs")
 # «.germs-and-sections»	(to "germs-and-sections")
 # «.geometric-morphisms-shvs»	(to "geometric-morphisms-shvs")
 # «.filterpowers»		(to "filterpowers")
 #]
[RULE ----------------------------------------]



[tsec «right-kan-extensions»  (to ".right-kan-extensions")
[++N]. (Right) Kan extensions
=============================
A downcasing for the notation in CWM (sec X.3, p.232):

          C
 	  ^ \\		         m^K ==========> m^KS
 	  |  \\      		  /\ \\            -
 	  |   \\ S   S		  ||  \\           | 
 	  |  R \\    v		  ||   \\          v
 	K |     \\   R		  ||    \======> m^KR
 	  |     :\\		  ||               -
 	  |    : \\		  ||               | 
 	  |     v  vv		  ||               v
 	  M -------> A 		  m ============> m^T



               A^K
        A^M <------ A^C

         SK <-------| S          (m=>m^KS) <====== (m^K=>m^KS)
       | |            |   	     -                  -
       | | K  <--|   |  	     |       <--|       |
       | v            v   	     v                  v
  ·K | RK <-------| R   	 (m=>m^KR) <====== (m^K=>m^KR)
       | |		      	     -
       | | 	      		     | (univ)
       v v		      	     v
         T                	 (m=>m^T)


   Nat(SK,T) <--- Nat(S,Ran_K T)
        ·K <--| 
]



[tsec «enriched-cats»  (to ".enriched-cats")
[++N]. Enriched categories
==========================
Definition (from SLNM 752):
a category \catC is enriched over \catV
(motivations: FinVec is enriched over itself;
  any small category is enriched over Set)
when its hom-sets have more structure:
for any objects A,B,C of \catC,

     Hom(A,B) × Hom(B,C)    -> Hom(A,C)
   \catV(A,B) ¤ \catV(B,C)  -> \catV(A,C)
  FinVec(A,B) ¤ FinVec(B,C) -> FinVec(A,C)


]

[tsec «polynomials»  (to ".polynomials")
[++N]. Polynomials
==================
]

[tsec «geometric-morphisms»  (to ".geometric-morphisms")
[++N]. Geometric morphisms
==========================
A geometric morphism, f,
        f
  \tF ------> \tE

is an adjunction:

        f^*         <- "inverse image"
  \tF <------ \tE      (left exact, i.e.
        _|_	        preserves finite limits)
      ------>
        f_*         <- "direct image"

If f^* has a left adjoint -
which is a bit stronger than
preserving finite limits -

        f_!
      ------>
        _|_
  \tF <------ \tE
        f^*

then f is said to be _essential_.
]

[tsec «geometric-morphisms-exs»  (to ".geometric-morphisms-exs")
[++N]. Geometric morphisms: simple examples
====================================

Set -> Set^2:

   A |---> (A,0)       a ======> a;®    .
   |         |         -          -
   |  <-->   |	       |   <-->   |
   v         v	       v          v
   B <---| (B,B')      b <====== b;c
   |         |         -          -
   |  <-->   |         |   <-->   |
   v         v         v          v
   D |---> (D,1)       d ======> d;*

  {1} ---> {1,2}



Set^N -> Set:

  (A_i)_{iÝN} |--> Æi:N.A_i      i;a_i ====> i,a_i
       |              |            -           -
       |     <-->     |            |    <->    |
       v              v            v           v
   (B)_{iÝN} <------| B           i;b <======= b
       |              |            -           -
       |     <-->     |            |    <->    |
       v              v            v           v
  (C_i)_{iÝN} |--> åi:N.C_i      i;c_i ===> i|->c_i

       N -----------> 1
]


[tsec «germs-and-sections»  (to ".germs-and-sections")
[++N]. Germs and sections
=========================
(Johnstone, sec. 0.24):

  / L(P)\                             /x,a_{®_x}\
  |  |  |     L	  		      |    -    |
  | |  | <------| (U^op|->P(U))      |    |    | <====== (u^op=>a_U)
  |  v  |                |	      |    v    | 	   -
  \  X  /      		 |	      \    x    / 	   |
     |	       		 |	           -		   |
     |	      <--->    	 |	           |       <--->   |
     v	       	   	 |	           v	    	   |
  /  E  \      	   	 |	         / e \	    	   |
  |  |  |      	   	 v	         | - |	    	   v
  | p|  | |----> (U^op|->£(E,p)(U))      | | | ========> (u^op=>u|->e)
  |  v  |    £ 			         | v |
  \  X  /			         \ x /
               L
  esp/(X,T) <------ Set^{T^op}
              _|_
            ------>
               £                      .

(Johnstone, sec. 0.25):
                                           shv
  (U^op|->£LA(U)) <--| (U^op|->A(U))   (u^op=>u|->x,a_{®_x}) <=== (u^op=>a_U)
          |                   |                  -                     -
          |       <-->        |                  |         <-->        |
          v                   v                  v                     v
   (U^op|->B(U)) |---> (U^op|->B(U))        (u^op=>b_U) ========> (u^op=>b_U)
                                                shv
                  £L
             <----------
      Shv(X)     _|_     Set^{T^op}
             ---------->
                incl

(Johnstone, sec. 0.26):

  (T^op|->£(f^*(L(E)),f^*)(T)) <--> (U^op|->E(U))

]


[tsec «geometric-morphisms-shvs»  (to ".geometric-morphisms-shvs")
[++N]. Geometric morphisms: examples with sheaves
=================================================

  Shv(X) ----> Shv(Y)

       X ----> Y



             sheafification
  Shv(\C,J) <--------------- Set^{\C^op}
             --------------->
                inclusion
]

[tsec «filterpowers»  (to ".filterpowers")
[++N]. Filterpowers
===================
(As in Johnstone's "Topos Theory". pp.319-322)

  n;*_{nÝU}
     /
     |
     v
    n;* ============> * |-> (n|->Ï)|_big |-----------> §            .
       /                 /       /                     /
        \       |-->      \      |                     |
         v                 v     v                     v
         n;Ï[nÝU] =========> (n|->Ï) |-------> Ï[n|->Ï) is big]

    U
    v
    |
    v       L
  1_\tE |------> 1_\tF                    1_\tF
       \              \                     v
      u \     |-->     \ L(u)               | t
         v              v                   v
          Ø_\tE |-----> L(Ø_\tE) -------> Ø_\tF
                   L               \Phi




]

]
]
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