Comorphism-based Grothendieck logics Till Mossakowski BISS

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cb^d ]` d``l`li db` bha`m a]n`h _o` lp`]`d^ a`b^ ^_^ ]` n`^
a`bc`l d lp`]`d^ fbq r bhnd^d e ^_` j`l fb jl n`bh`
^ bha`m d hdg b`q ]`o`]i a`bs`l dfjf` dl ^ e^`d _o`
^_`] ^]`df^_ d a]^bj] a`b^q tdf _`^`]f`d`j a`bcb^di ^_``
^]`df^_ bd n` bhnd`l ^_ bha]ng h `p]^q
ud ^_` ^`]^j]`i `o`] aa]b_` ^ _`^`]f`d`j a`bcb^d _o` n``d
l`o`a`l vwixi ywiz{|q _` h^ a]hd`d^ aa]b_ }e`~€ ^_ ^ bjn` e
`f_^ fb dl ^`o` a]`b^d ‚e]hs`l ƒ„…†ƒ†‡†ƒˆ„ ‰ˆŠ‹Œ ƒ…‰… hdf
^_`h vŽ|i _odf `hd^b n`l d ^_` d^d e ]^_`dl`b
d^^j^d
vx|q `o`]i d^ dg a]`b^d n`^``d fbi nj^ fb `dbldf ‚e]k
hs`l b`l ‘ˆ‰ˆŠ‹Œ ƒ…‰… ]` ]``od^ ^ _`^`]f`d`j a`bcb^d
vz{i yw|q ]`o`]i n`l` ^_`` hl` ^_`]`^b aa]b_`i ^_` d``l e
d^`f]^df lp`]`d^ a]e bbj o ’n]lf`“ _ n``d ^]``l vw|q _` f
e ^_ aa`] ^ `m^`dl ^_` }e`~€ ]`aq ]^_`dl`b d^^j^d aa]b_
^ bo`] ^_`` a`b^q
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›` d ]`b `o`] d^d h`d^d`l d ^_` d^]ljb^dq œ„…†ƒ†‡†ƒˆ„… vy|
ba^j]` ^_` hl` ^_`]g e fbi ž„†Ÿƒ ‰ž„† …¡…†ž‰… vy¢| ba^j]` a]e ^_`]gi
_` ˆ£ ƒ‘… vy¢| bhnd` n^_q œ„…†ƒ†‡†ƒˆ„ ‰ˆŠ‹Œ ƒ…‰… vy| ba^j]` ^_` d^j^d
^_^ d` fb ¤‡ƒ † ‡‹ ˆ„i ] ‹ Šˆ¥ž‘†ž¦ ˆ„†ˆ d^_`] d`i _` ƒ„…†ƒ†‡†ƒˆ„ ‘ˆ§
‰ˆŠ‹Œ ƒ…‰… vy{|i b`l ƒ„…†ƒ†‡†ƒˆ„ Šž‹ Šž…ž„†Ÿ†ƒˆ„… vzz| ] ‰Ÿ‹ … ˆ¨ ƒ„…†ƒ†‡†ƒˆ„…
vy¢| ba^j]` ^_` d^j^d ^_^ d` fb ž„‘ˆ¦ž¦ ƒ„†ˆ d^_`] d`q ^_ d^d
bd n` `m^`dl`l ^ ej fbq ›` n`fd ^_ ^_` bhad`d^ e d^^j^di
_b_ ]` b`l Šˆˆ‰…i dl ^gabg a]ol` ^eb^d g^`h b ^ h`
fd^j]`q
rd ƒ„…†ƒ†‡†ƒˆ„ Šˆˆ‰ ‚  bd^ e
`^ e e …ž„†ž„‘ž…i
b^`f]g e ‰ˆ¦ž …i dl
^eb^d ]`^d q
h ]` bdd`b^`l o b]]l] ‚_b_ hl` b_df` e d^^d ^_d
d` fbi ` ^]d^d n`^``d fbq
rd ƒ„…†ƒ†‡†ƒˆ„ ‘ˆŠŠƒ¦ˆŠ ‚  ‚  ‚  bd^ e
`d^`db` ^]d^d ejdb^d i dl
hl` ]`ljb^d ejdb^] i jb_ ^_^
‚  ‚  _l e] `b_ dl `b_ ‚…Ÿ†ƒ…¨Ÿ‘†ƒˆ„ ‘ˆ„¦ƒ†ƒˆ„q
`hd^b `d^h`d^ d d d^^j^d ]h l`cd`l jj e] i ` ]^` i e hl` ^egdf ^eg q
r ˆ£ ƒ‘ Šˆˆ‰ ‚  d d^^j^d ]h ‚  ` jaa`l ^_ d
ž„†Ÿƒ ‰ž„†
Šž Ÿ†ƒˆ„ ‚  i jb_ ^_^ ^_` edf bdl^d ]` ^c`l yq Šž ž ƒ ƒ†¡ e] dg i i
zq ‰ˆ„ˆ†ˆ„ƒ‘ƒ†¡ e dl ^_`d i
{q †ŠŸ„…ƒ†ƒ ƒ†¡ e ! i e] " # i dl $ ! " # % i ^_`d % i
q …ˆ‡„¦„ž…… e] dg dl i
ha` &
r fb ]h n` b`l ‘ˆ‰‹ ž†ž ei d ll^di ^_` bdo`]` e ^_` no`
hab^d _lq
r ˆ ƒ‘ ‘ˆŠŠƒ¦ˆŠ ‚  ‚  ‚  d d^^j^d
b]]l]£ ‚  ‚  ‚  jb_ ^_^ e i ^_`d ‚  ‚  ‚ k†ŠŸ„… Ÿ†ƒˆ„q f`^_`] ^_ noj d^d e bha^d dl l`d^^gi
^_ fo` j b^`f]` '()* + +, dl - +.* + +, q
`d`]gi `d^`db` dl hl` l`a`dl d fo`d obnj]g e ddkfb
ghn a]ol`l ng …ƒ£ „Ÿ†‡Šžq _`]`e]`i d ƒ„…†ƒ†‡†ƒˆ„ ejdb^] # /0.( '()* + +, ‚_`]` /0.( b`l ^_` b^`f]g e …ƒ „Ÿ†‡Šž…i dl ˆ£ ƒ‘ ejdb^] 1 /0.( - +.* + +, q ›` j` ^_`£ h]` ^dl]l d^^d
‚/2( 3 ‚4  5 + 63 ‚4  7  e] ^_` d^^j^d ]h # ‚4 q 5 + 63 dl /2( 3 bd
`g n` ``d ^ n` ejdb^]i
dl ` ]]o` ^ ^_` ^dl]l l`cd^d e dk
^^j^d jl]ja` ‚/0.( /2( 5 + 6 q ›` e]``g j` ^_ d^^d
_`d`o`] d``l`li dl ]^` 8 e] 5 + 6 ‚9 ‚ q
\] ^_` h]a_h n`^``d d^^j^d dl fbi ^_`]` ]` ^ noj
b_b` o`d d^^j^d # /0.( '( )* + +, dl # /0.( '( )* + +, i
d ƒ„…†ƒ†‡†ƒˆ„ ‰ˆŠ‹Œ ƒ…‰ ‚: ;  # # bd^ e ejdb^] : /0.( /0.( dl d^j] ^]de]h^d ; # < : # q ud bd^]^i d ƒ„…†ƒ†‡†ƒˆ„ ‘ˆ‰ˆŠ§
‹Œ ƒ…‰ ‚= >  # # bd^ e ejdb^] = /0.( /0.( dl d^j]
z
^]de]h^d > # # < : q f`^_`] ^_ noj bha^d dl l`d^k
^`i ^_ fo` j b^`f]` '( ) dl +'( ) q fb h]a_h dl bh]a_h
]` l`cd`l dfjgi `ldf ^ b^`f]` - +. dl +- +. q
o`d d^^j^d h]a_h ‚: ;  # # dl ‚: ;  # # i d ƒ„§
…†ƒ†‡†ƒˆ„ ‰ˆŠ‹Œ ƒ…‰ †ŠŸ„…¨ˆŠ‰Ÿ†ƒˆ„ ‚: ;  ‚: ;  j^ d^j] ^]dk
e]h^d : : jb_ ^_^ ; ; < ‚# q h]gi fo`d d^^j^d
bh]a_h ‚= >  # # dl ‚= >  # # i d ƒ„…†ƒ†‡†ƒˆ„ ‘ˆ‰ˆŠ‹Œ ƒ…‰
†ŠŸ„…¨ˆŠ‰Ÿ†ƒˆ„ ‚= >  ‚= >  d^j] ^]de]h^d = = jb_
^_^ ‚#  < > > q
_` b]]`adldf d^d e] fb ]` `d^]`g dfjq _ _ ^_^
'() i +'() i - +. dl +- +. ]` dl``l zkb^`f]`q ud ^_` ` j`i h^ e ^_`
l`cd^d dl ]`j^ _l e] n^_ d^^j^d dl fbi ^_jf_ ` d^ ]`a`^ ^_ `o`]g ^h` ‚dl``li h^ bd^]jb^d ] e] d ]n^]]g
b^`f]g e ]hq
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}dl`]i e] `mha`i ^_` d^^j^d d \fq yq a]a^d
fbi 1 c]^k]l`] fbi 1 _f_`]k]l`] fbi n^_ ^_ ` j^gq
vyŽ| `m^`dl 1 ^_ a]^^gi jn]^df dl f`d`]^d bd^]d^
‚h` e]h e dljb^dq k vz ! | d `m^`dd e ^_ } k
` n``l ^]d^d fbq ›` _o` dbjl`l " # 1 ^`ei ^`ha] fb
e`^j]df n^_ ^`ha] hl^` dl a^_ jd^c`]i ^_ hl`kb_`b`]
on`q k v| d `m^`dd e ^_^ e ^_` n^]b^ ^^`
hb_d` a]lfh i _`]` ^^` b]]`adl ^ f`n]q _`]` ]` noj a]k
`b^d h]a_h ‚l`d^`l ng l^^`l ]]i ^_^ g bh` d a] ^_
dbjd bh]a_h ‚l`d^`l l ]]q `d`]gi d _`^`]f`d`j
a`bcb^di ^_`` h]a_h bd n` hj^`l ng ^_` b]]`adldf bh]k
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+&&)1 <*$ A" ;& .&$ $>-.F $>=$ $>-1 "B$+= ?"."+=I-$2 -1 &' )*,> #+=,$-,=I *1"%
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a_h q h` e ^_`` fb ]` ]`a]``d^`l o h]` bha`m bh]a_h
d 1 i `` vyw|q \dgi bh]a_h ^]de]h^d ]` l`d^`l ng ljn`
]]q _`g ]` j^ dbjd _`]`q qfq ` bd f e]h 1 ^ 1 `^_`] l]`b^g ] o q ^_` `] ]f_^ ljn` ]] ^` j ^_^ ^_`
lp`]`db` n`^``d ^_`` ^ g bd n` ba^j]`l ng ehg e fd^j]`
h]a_h d 1 q ^`] di ^_ `dn` j ^ l`d^eg h` fd^j]`
h]a_h d ^_` ]^_`dl`b bd^]jb^di `ldf ^ n`^^`] aabn^g
e a]e ]j` d ^_` bbjj e] _`^`]f`d`j l`o`ah`d^ f]a_ d^]ljb`l
n`q
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h]a_h ^]de]h^dq ud b` _`]` ^_` zkb^`f]b ^]jb^j]` d^
d``l`li ` h^ ^_` a]`cm ’zk“q
h]gi §ƒ„¦ž ž¦ ‘ˆƒ„…†ƒ†‡†ƒˆ„ zkejdb^] #
+'() q
_` ]^_`dl`b bd^]jb^d e] dl`m`l d^^j^d _ n``d l`b]n`l d
vx|! ` l`o`a ^ lj _`]`q ud d dl`m`l bd^^j^d i ` j` ^_` d^^d
e] ‚" i ‚= >  e] ^_` bh]a_h ‚  d dq
2 (0 0+(
o`d §ƒ„¦ž ž¦ ‘ˆƒ„…†ƒ†‡†ƒˆ„ # +'( ) i l`cd` ^_`
Šˆ†Œ ž„¦ƒž‘ ƒ„…†ƒ†‡†ƒˆ„ e ]` a] ‚4 " i _`]` " # dl 4 fd^j]` d ! i
fd^j]` d
fd^j]` h]a_h ‚9  ‚4 "  ‚4  bd^ e h]a_h "#
dl fd^j]` h]a_h 9 = ‚4  4 i
bha^d
fo`d ng ‚9  < ‚9  ‚9 < = ‚9  < i
‚4 "  ! ‚4 i dl ‚9  ! ‚4 
‚= ‚4  8 ‚4  q
\dgi d^^j^d bh]a_h ^]de]h^d `l ^ bdf]j`db` d
]^_`dl`b fd^j]` h]a_h ^_` bdf]j`db` f`d`]^`l ng
‚ 7 = ‚4  = ‚4  "  ‚" = ‚4  = ‚4  " 
e] # i " # i 4 /0.( ! q
+ +)0 0+(
bd^d`l d ^_` `]d` e ‚bdl`]`l ejdb^]q
+ +
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l`o`a`l ng bd`bj vx|q rb^jgi ^_` b]]`adldf ^_`]g e] dl`m`l
bd^^j^d ^j]d j^ ^ n` hjb_ ha`]q `]`i ` ebj d bbha`^`d`
dl hfh^d ]`j^i db` ^_`` ]` d``l`l e] ldf ^]jb^j]`l a]e vyx|q
+ +)0 0+( `^ # +'() n` d dl`m`l bd^^j^d jb_ ^_^
#
kbha`^` e] h` h b^`f]g i = bbd^djj e] `b_ " # i dl ^_` dl`m`l b^`f]g e fd^j]` e bg kbbha`^`
‚^_` ^^`] h`ddf ^_^ /0.( ! kbbha`^` e] `b_ " # q _`d ^_`
fd^j]` b^`f]g e ^_` ]^_`dl`b d^^j^d _ kbh^q
o`d d d^^j^d # dl lf]h /0.( 3 i ehg e hl`
‚  b`l §‘ˆ„…ƒ…†ž„† e e] `b_ q r bbd`
‚4 ‚;   o`] ^_` lf]h d /0.( 3 b`l žŸ ¡ Ÿ‰Ÿ Ÿ‰Ÿ¤ ž
e e] `b_ kbd^`d^ ehg e hl` ‚  i ^_`]` 4 khl` £ ^_
‚ q ue ^_ hl` jd j`i ^_` bbd` b`l Ÿ‰Ÿ £ Ÿ‰Ÿ¤ žq
_`` d^d `m^`dl ^ lf]h # dl bd` o`] ^_`` ^_` ] ` e ^_` d^]kd^^j^d hl` ]`ljb^d no` d ag`l ng ^_`
d^`]kd^^j^d hl` ^]d^dq
rd d^^j^d # b`l žŸ ¡ …ž‰ƒ§ž Ÿ‘†i e dg aj_j^ e fd^j]`
‚`g hfhn`q rd d^^j^d bh]a_h ‚= >  # # b`l
‚`g `mb^ ei e] `b_ fd^j]` h]a_h 9 4 4 d # i ^_` d^j]^g
lf]h
5 + 63 ‚4 
5 + 6 ‚= ‚4 
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5 + 6 ‚= ‚4 
‚`g hfhn`i q`q ‚` ajnbq
rd dl`m`l bd^^j^d # +'() b`l žŸ ¡ ˆ‘Ÿ ¡ …ž‰ƒ§
ž Ÿ‘†i e `b_ d^^j^d # ! ‚`g `hk`mb^ ‚" # q u^ b`l žŸ ¡
…ž‰ƒ§ž Ÿ‘† e e] `b_ ajnb d #
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5 + 6 ‚= ‚4 
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5 + 6 ‚= ‚= ‚4 
!
‚` ajnb e] `b_ fd^j]` 4 d /0.( q
+ +)0 0+( ! _` ]^_`dl`b d^^j^d
e d dl`m`l bd^^j^d
# +'( ) bd^df e bh]a_h ^_ bbd^djj fd^j]` ^]dk
^d ‚`g `hk`mb^ e dl dg e
‚`g bg `hk`mb^i
‚`g `hk`mb^i dl
d^^j^d bh]a_h d ]` ‚`g `mb^q
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bd`bj a]o` ^_` no` ]`j^ e] kb`l ž‰¤ž¦¦ƒ„ §ƒ„¦ž ž¦ ƒ„…†ƒ§
†‡†ƒˆ„…i _b_ h`d ^_^ `b_ fd^j]` ^]d^d : _ £`e^ ld^ = q
j^ ^_`` `e^ ld^ `l ^ b]]`adldf dl`m`l bd^^j^di dl d
eb^i ^]b^g a`df bd`bj j` ^_ dljb`l dl`m`l bd^^j^d d _
a]eq _ _ ^_^ dl`m`l bd^^j^d ]` ha`] dl h]` f`d`] ^_d
`hn`lldfkdl`m`l d^^j^d ‚dl dg e] ^_``i bd`bj _ ]`j^ nj^
`mb^d` dl hfhn^gq ud a]^bj]i ha`] a]e e bd`bj ]`k
j^ bd n` n^d`l ng ]`ljbdf ^_`h ^ ^_` no` ]`j^ o ^_` edf
f`d`]s^d e ]`j^ e]h vy| + +)0 0+(
o`d d `hn`lldfkdl`m`l d^^j^d # '( ) i
l`cd` ^_` dl`m`l bd^^j^d # +'( ) ng
‚"  ‚"  dl ‚ "  ‚= ‚; =  < ‚  _`]` ^_` jd^ e ^_` ljdb^d n`^``d = dl : q
_`d h]a_b ^ ‚  q
bd`bj ]`lg d^` ^_^ ^_` jha^d e ‚_ h]` a`b o`]d
e ]aq {q" ]` ^ ^]df ^ n` h`^ d a]b^b`q qfq ^_` d^^j^d
d^ `g bg `hk`mb^i dl ^ `dbldf d^ 1 d`^_`] `mb^i
d] l` ^ _o` bbd^djj fd^j]` ^]d^dq `i ` ^]g ^
``d ^_` jha^d ng ]df ^_ `g hfhn` bbd` ]^_`]
^_d ^_ hfhn` bh^ b d d^^j^d # ‡Ÿ…ƒ§ž Ÿ‘† e e] `b_ lk
f]h /0.(3 i ^_`]` h` `g hfhn` bbd` o`] q rd
dl`m`l bd^^j^d
# +'( ) b`l ˆ‘Ÿ ¡ ‡Ÿ…ƒ§ž Ÿ‘†i e `b_ dk
^^j^d ! jk`mb^ ‚" # q u^ b`l ‡Ÿ…ƒ§ž Ÿ‘†i e e] `b_ lf]h
# i ^_`]` h` bd` ‚ ‚   o`] ^_^ `g hfhk
n`q ‡Ÿ…ƒ§…ž‰ƒ§ž Ÿ‘†„ž…… ^_` ]`^]b^d e ^_`` d^d ^ lf]h e _a`
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bha`^`q
ud hdg b`i d^ `o`]g d^^j^d bh` ^_ bha`^` `d^h`d^
g^`h _`db`i ^ l bj^ ^ aag ^_` no` a]a^d d a]b^b`q ud vyw|i
` ^_`]`e]` _o` ]`a]``d^`l d h` `ma]`o` ’jdo`]“ d^^j^d ^_
fl a]e jaa]^q `o`]i e^`d ^ b]jb ^ j` a`bcb ^ l`fd`l e]
a`bcb fb ^ n^d fl ]`j^q `db`i ` d``l _`^`]f`d`j a]odf `q r c]^ ^^`ha^ d ^_ l]`b^d ]` Œ ž†žŠˆ£ ž„žˆ‡… ¤Šƒ¦£ ž… vw| _b_i _k
`o`]i _o` d b`] `hd^b nq ›` ^_`]`e]` h ^ b`] `hd^b n
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njdl“ e] h b^`f]`i _`d ` ]`g h`d ^_` a]`k]l`] n^d`l e]h
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jha^di bh]a_h ]` hl`k`mado`i ng ˆ¦ž § ‹ Ÿ„…ƒˆ„i ‚" 
bd`]o^o`q g ˆŠŠˆ ƒ„£ i 8 " q
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