:: Natural Transformations. Discrete Categories
:: by Andrzej Trybulec
::
:: Received May 15, 1991
:: Copyright (c) 1991-2011 Association of Mizar Users

begin

definition

let A1 be set ;
let B1 be non empty set ;
let f be Function of A1,B1;
let Y1 be Subset of A1;
:: original: |
redefine func f | Y1 -> Function of Y1,B1;
coherence
f | Y1 is Function of Y1,B1 by FUNCT_2:38;

end;

theorem Th1: :: NATTRA_1:1

for B1, B2, A1, A2 being non empty set
for f being Function of A1,B1
for g being Function of A2,B2
for Y1 being non empty Subset of A1
for Y2 being non empty Subset of A2 holds [:f,g:] | [:Y1,Y2:] = [:(f | Y1),(g | Y2):]

proof end;

definition

let A, B be non empty set ;
let A1 be non empty Subset of A;
let B1 be non empty Subset of B;
let f be PartFunc of [:A1,A1:],A1;
let g be PartFunc of [:B1,B1:],B1;
:: original: |:
redefine func |:f,g:| -> PartFunc of [:[:A1,B1:],[:A1,B1:]:],[:A1,B1:];
coherence
|:f,g:| is PartFunc of [:[:A1,B1:],[:A1,B1:]:],[:A1,B1:] by FUNCT_4:62;

end;

theorem Th2: :: NATTRA_1:2

for A1, A2 being non empty set
for Y1 being non empty Subset of A1
for Y2 being non empty Subset of A2
for f being PartFunc of [:A1,A1:],A1
for g being PartFunc of [:A2,A2:],A2
for F being PartFunc of [:Y1,Y1:],Y1 st F = f || Y1 holds
for G being PartFunc of [:Y2,Y2:],Y2 st G = g || Y2 holds
|:F,G:| = |:f,g:| || [:Y1,Y2:]

proof end;

theorem Th3: :: NATTRA_1:3

for A being Category
for a being Object of A
for m being Morphism of a,a holds m in Hom (a,a)

proof end;

theorem Th4: :: NATTRA_1:4

for o, m being set
for f, g being Morphism of (1Cat (o,m)) holds f = g

proof end;

theorem Th5: :: NATTRA_1:5

for A being Category
for a being Object of A holds [[(id a),(id a)],(id a)] in the Comp of A

proof end;

theorem Th6: :: NATTRA_1:6

for o, m being set holds the Comp of (1Cat (o,m)) = {[[m,m],m]}

proof end;

theorem Th7: :: NATTRA_1:7

for A being Category
for a being Object of A holds 1Cat (a,(id a)) is Subcategory of A

proof end;

theorem Th8: :: NATTRA_1:8

for A being Category
for C being Subcategory of A holds
( the Source of C = the Source of A | the carrier' of C & the Target of C = the Target of A | the carrier' of C & the Comp of C = the Comp of A || the carrier' of C & the Id of C = the Id of A | the carrier of C )

proof end;

theorem Th9: :: NATTRA_1:9

for A being Category
for O being non empty Subset of the carrier of A
for M being non empty Subset of the carrier' of A
for DOM, COD being Function of M,O st DOM = the Source of A | M & COD = the Target of A | M holds
for COMP being PartFunc of [:M,M:],M st COMP = the Comp of A || M holds
for ID being Function of O,M st ID = the Id of A | O holds
CatStr(# O,M,DOM,COD,COMP,ID #) is Subcategory of A

proof end;

theorem Th10: :: NATTRA_1:10

for C being strict Category
for A being strict Subcategory of C st the carrier of A = the carrier of C & the carrier' of A = the carrier' of C holds
A = C

proof end;

begin

definition

let A, B be Category;
let F be Functor of A,B;
let a, b be Object of A;
assume A1: Hom (a,b) <> {} ;
let f be Morphism of a,b;
func F . f -> Morphism of F . a,F . b equals :Def1: :: NATTRA_1:def 1
F . f;
coherence
F . f is Morphism of F . a,F . b by A1, CAT_1:125;

end;

:: deftheorem Def1 defines . NATTRA_1:def 1 :
for A, B being Category
for F being Functor of A,B
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds F . f = F . f;

theorem :: NATTRA_1:11

for B, C, A being Category
for F being Functor of A,B
for G being Functor of B,C
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (G * F) . f = G . (F . f)

proof end;

theorem :: NATTRA_1:12

for A, B being Category
for F1, F2 being Functor of A,B st ( for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds F1 . f = F2 . f ) holds
F1 = F2

proof end;

theorem Th13: :: NATTRA_1:13

for B, A being Category
for F being Functor of A,B
for a, b, c being Object of A st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds F . (g * f) = (F . g) * (F . f)

proof end;

theorem :: NATTRA_1:14

for A, B being Category
for F being Functor of A,B
for c being Object of A
for d being Object of B st F . (id c) = id d holds
F . c = d

proof end;

theorem Th15: :: NATTRA_1:15

for B, A being Category
for F being Functor of A,B
for a being Object of A holds F . (id a) = id (F . a)

proof end;

theorem :: NATTRA_1:16

for A being Category
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (id A) . f = f

proof end;

theorem Th17: :: NATTRA_1:17

for A being Category
for a, b, c, d being Object of A st Hom (a,b) meets Hom (c,d) holds
( a = c & b = d )

proof end;

begin

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
pred F1 is_transformable_to F2 means :Def2: :: NATTRA_1:def 2
for a being Object of A holds Hom ((F1 . a),(F2 . a)) <> {} ;
reflexivity
for F1 being Functor of A,B
for a being Object of A holds Hom ((F1 . a),(F1 . a)) <> {} by CAT_1:56;

end;

:: deftheorem Def2 defines is_transformable_to NATTRA_1:def 2 :
for A, B being Category
for F1, F2 being Functor of A,B holds
( F1 is_transformable_to F2 iff for a being Object of A holds Hom ((F1 . a),(F2 . a)) <> {} );

theorem :: NATTRA_1:18

canceled;

theorem Th19: :: NATTRA_1:19

for A, B being Category
for F, F1, F2 being Functor of A,B st F is_transformable_to F1 & F1 is_transformable_to F2 holds
F is_transformable_to F2

proof end;

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
assume A1: F1 is_transformable_to F2 ;
mode transformation of F1,F2 -> Function of the carrier of A, the carrier' of B means :Def3: :: NATTRA_1:def 3
for a being Object of A holds it . a is Morphism of F1 . a,F2 . a;
existence
ex b₁ being Function of the carrier of A, the carrier' of B st
for a being Object of A holds b₁ . a is Morphism of F1 . a,F2 . a

proof end;

end;

:: deftheorem Def3 defines transformation NATTRA_1:def 3 :
for A, B being Category
for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds
for b₅ being Function of the carrier of A, the carrier' of B holds
( b₅ is transformation of F1,F2 iff for a being Object of A holds b₅ . a is Morphism of F1 . a,F2 . a );

definition

let A, B be Category;
let F be Functor of A,B;
func id F -> transformation of F,F means :Def4: :: NATTRA_1:def 4
for a being Object of A holds it . a = id (F . a);
existence
ex b₁ being transformation of F,F st
for a being Object of A holds b₁ . a = id (F . a)

proof end;

uniqueness
for b₁, b₂ being transformation of F,F st ( for a being Object of A holds b₁ . a = id (F . a) ) & ( for a being Object of A holds b₂ . a = id (F . a) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines id NATTRA_1:def 4 :
for A, B being Category
for F being Functor of A,B
for b₄ being transformation of F,F holds
( b₄ = id F iff for a being Object of A holds b₄ . a = id (F . a) );

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
assume A1: F1 is_transformable_to F2 ;
let t be transformation of F1,F2;
let a be Object of A;
func t . a -> Morphism of F1 . a,F2 . a equals :Def5: :: NATTRA_1:def 5
t . a;
coherence
t . a is Morphism of F1 . a,F2 . a by A1, Def3;

end;

:: deftheorem Def5 defines . NATTRA_1:def 5 :
for A, B being Category
for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds
for t being transformation of F1,F2
for a being Object of A holds t . a = t . a;

definition

let A, B be Category;
let F, F1, F2 be Functor of A,B;
assume that
A1: F is_transformable_to F1 and
A2: F1 is_transformable_to F2 ;
let t1 be transformation of F,F1;
let t2 be transformation of F1,F2;
func t2 `*` t1 -> transformation of F,F2 means :Def6: :: NATTRA_1:def 6
for a being Object of A holds it . a = (t2 . a) * (t1 . a);
existence
ex b₁ being transformation of F,F2 st
for a being Object of A holds b₁ . a = (t2 . a) * (t1 . a)

proof end;

uniqueness
for b₁, b₂ being transformation of F,F2 st ( for a being Object of A holds b₁ . a = (t2 . a) * (t1 . a) ) & ( for a being Object of A holds b₂ . a = (t2 . a) * (t1 . a) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines `*` NATTRA_1:def 6 :
for A, B being Category
for F, F1, F2 being Functor of A,B st F is_transformable_to F1 & F1 is_transformable_to F2 holds
for t1 being transformation of F,F1
for t2 being transformation of F1,F2
for b₈ being transformation of F,F2 holds
( b₈ = t2 `*` t1 iff for a being Object of A holds b₈ . a = (t2 . a) * (t1 . a) );

theorem Th20: :: NATTRA_1:20

for B, A being Category
for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds
for t1, t2 being transformation of F1,F2 st ( for a being Object of A holds t1 . a = t2 . a ) holds
t1 = t2

proof end;

theorem Th21: :: NATTRA_1:21

for B, A being Category
for F being Functor of A,B
for a being Object of A holds (id F) . a = id (F . a)

proof end;

theorem Th22: :: NATTRA_1:22

for A, B being Category
for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds
for t being transformation of F1,F2 holds
( (id F2) `*` t = t & t `*` (id F1) = t )

proof end;

theorem Th23: :: NATTRA_1:23

for A, B being Category
for F, F1, F2, F3 being Functor of A,B st F is_transformable_to F1 & F1 is_transformable_to F2 & F2 is_transformable_to F3 holds
for t1 being transformation of F,F1
for t2 being transformation of F1,F2
for t3 being transformation of F2,F3 holds (t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1)

proof end;

begin

Lm1: for B, A being Category
for F being Functor of A,B
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds ((id F) . b) * (F . f) = (F . f) * ((id F) . a)

proof end;

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
pred F1 is_naturally_transformable_to F2 means :Def7: :: NATTRA_1:def 7
( F1 is_transformable_to F2 & ex t being transformation of F1,F2 st
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t . b) * (F1 . f) = (F2 . f) * (t . a) );
reflexivity
for F1 being Functor of A,B holds
( F1 is_transformable_to F1 & ex t being transformation of F1,F1 st
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t . b) * (F1 . f) = (F1 . f) * (t . a) )

proof end;

end;

:: deftheorem Def7 defines is_naturally_transformable_to NATTRA_1:def 7 :
for A, B being Category
for F1, F2 being Functor of A,B holds
( F1 is_naturally_transformable_to F2 iff ( F1 is_transformable_to F2 & ex t being transformation of F1,F2 st
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t . b) * (F1 . f) = (F2 . f) * (t . a) ) );

Lm2: for B, A being Category
for F, F1, F2 being Functor of A,B st F is_transformable_to F1 & F1 is_transformable_to F2 holds
for t1 being transformation of F,F1 st ( for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t1 . b) * (F . f) = (F1 . f) * (t1 . a) ) holds
for t2 being transformation of F1,F2 st ( for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t2 . b) * (F1 . f) = (F2 . f) * (t2 . a) ) holds
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds ((t2 `*` t1) . b) * (F . f) = (F2 . f) * ((t2 `*` t1) . a)

proof end;

theorem :: NATTRA_1:24

canceled;

theorem Th25: :: NATTRA_1:25

for A, B being Category
for F, F1, F2 being Functor of A,B st F is_naturally_transformable_to F1 & F1 is_naturally_transformable_to F2 holds
F is_naturally_transformable_to F2

proof end;

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
assume A1: F1 is_naturally_transformable_to F2 ;
mode natural_transformation of F1,F2 -> transformation of F1,F2 means :Def8: :: NATTRA_1:def 8
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (it . b) * (F1 . f) = (F2 . f) * (it . a);
existence
ex b₁ being transformation of F1,F2 st
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (b₁ . b) * (F1 . f) = (F2 . f) * (b₁ . a) by A1, Def7;

end;

:: deftheorem Def8 defines natural_transformation NATTRA_1:def 8 :
for A, B being Category
for F1, F2 being Functor of A,B st F1 is_naturally_transformable_to F2 holds
for b₅ being transformation of F1,F2 holds
( b₅ is natural_transformation of F1,F2 iff for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (b₅ . b) * (F1 . f) = (F2 . f) * (b₅ . a) );

definition

let A, B be Category;
let F be Functor of A,B;
:: original: id
redefine func id F -> natural_transformation of F,F;
coherence
id F is natural_transformation of F,F

proof end;

end;

definition

let A, B be Category;
let F, F1, F2 be Functor of A,B;
assume that
A1: F is_naturally_transformable_to F1 and
A2: F1 is_naturally_transformable_to F2 ;
let t1 be natural_transformation of F,F1;
let t2 be natural_transformation of F1,F2;
func t2 `*` t1 -> natural_transformation of F,F2 equals :Def9: :: NATTRA_1:def 9
t2 `*` t1;
coherence
t2 `*` t1 is natural_transformation of F,F2

proof end;

end;

:: deftheorem Def9 defines `*` NATTRA_1:def 9 :
for A, B being Category
for F, F1, F2 being Functor of A,B st F is_naturally_transformable_to F1 & F1 is_naturally_transformable_to F2 holds
for t1 being natural_transformation of F,F1
for t2 being natural_transformation of F1,F2 holds t2 `*` t1 = t2 `*` t1;

theorem Th26: :: NATTRA_1:26

for A, B being Category
for F1, F2 being Functor of A,B st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds
( (id F2) `*` t = t & t `*` (id F1) = t )

proof end;

theorem Th27: :: NATTRA_1:27

for B, A being Category
for F, F1, F2 being Functor of A,B st F is_naturally_transformable_to F1 & F1 is_naturally_transformable_to F2 holds
for t1 being natural_transformation of F,F1
for t2 being natural_transformation of F1,F2
for a being Object of A holds (t2 `*` t1) . a = (t2 . a) * (t1 . a)

proof end;

theorem Th28: :: NATTRA_1:28

for A, B being Category
for F, F1, F2, F3 being Functor of A,B
for t being natural_transformation of F,F1
for t1 being natural_transformation of F1,F2 st F is_naturally_transformable_to F1 & F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to F3 holds
for t3 being natural_transformation of F2,F3 holds (t3 `*` t1) `*` t = t3 `*` (t1 `*` t)

proof end;

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
let IT be transformation of F1,F2;
attr IT is invertible means :Def10: :: NATTRA_1:def 10
for a being Object of A holds IT . a is invertible ;

end;

:: deftheorem Def10 defines invertible NATTRA_1:def 10 :
for A, B being Category
for F1, F2 being Functor of A,B
for IT being transformation of F1,F2 holds
( IT is invertible iff for a being Object of A holds IT . a is invertible );

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
pred F1,F2 are_naturally_equivalent means :Def11: :: NATTRA_1:def 11
( F1 is_naturally_transformable_to F2 & ex t being natural_transformation of F1,F2 st t is invertible );
reflexivity
for F1 being Functor of A,B holds
( F1 is_naturally_transformable_to F1 & ex t being natural_transformation of F1,F1 st t is invertible )

proof end;

end;

:: deftheorem Def11 defines are_naturally_equivalent NATTRA_1:def 11 :
for A, B being Category
for F1, F2 being Functor of A,B holds
( F1,F2 are_naturally_equivalent iff ( F1 is_naturally_transformable_to F2 & ex t being natural_transformation of F1,F2 st t is invertible ) );

notation

let A, B be Category;
let F1, F2 be Functor of A,B;
synonym F1 ~= F2 for F1,F2 are_naturally_equivalent ;

end;

Lm3: for A, B being Category
for F1, F2 being Functor of A,B
for t being transformation of F1,F2 st F1 is_transformable_to F2 & t is invertible holds
F2 is_transformable_to F1

proof end;

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
assume A1: F1 is_transformable_to F2 ;
let t1 be transformation of F1,F2;
assume A2: t1 is invertible ;
func t1 " -> transformation of F2,F1 means :Def12: :: NATTRA_1:def 12
for a being Object of A holds it . a = (t1 . a) " ;
existence
ex b₁ being transformation of F2,F1 st
for a being Object of A holds b₁ . a = (t1 . a) "

proof end;

uniqueness
for b₁, b₂ being transformation of F2,F1 st ( for a being Object of A holds b₁ . a = (t1 . a) " ) & ( for a being Object of A holds b₂ . a = (t1 . a) " ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def12 defines " NATTRA_1:def 12 :
for A, B being Category
for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds
for t1 being transformation of F1,F2 st t1 is invertible holds
for b₆ being transformation of F2,F1 holds
( b₆ = t1 " iff for a being Object of A holds b₆ . a = (t1 . a) " );

Lm4: now

let A, B be Category; :: thesis: for F1, F2 being Functor of A,B
for t1 being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 & t1 is invertible holds
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds ((t1 ") . b) * (F2 . f) = (F1 . f) * ((t1 ") . a)
let F1, F2 be Functor of A,B; :: thesis: for t1 being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 & t1 is invertible holds
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds ((t1 ") . b) * (F2 . f) = (F1 . f) * ((t1 ") . a)
let t1 be natural_transformation of F1,F2; :: thesis: ( F1 is_naturally_transformable_to F2 & t1 is invertible implies for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds ((t1 ") . b) * (F2 . f) = (F1 . f) * ((t1 ") . a) )
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: t1 is invertible ; :: thesis: for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds ((t1 ") . b) * (F2 . f) = (F1 . f) * ((t1 ") . a)
let a, b be Object of A; :: thesis: ( Hom (a,b) <> {} implies for f being Morphism of a,b holds ((t1 ") . b) * (F2 . f) = (F1 . f) * ((t1 ") . a) )
assume A3: Hom (a,b) <> {} ; :: thesis: for f being Morphism of a,b holds ((t1 ") . b) * (F2 . f) = (F1 . f) * ((t1 ") . a)
A4: Hom ((F1 . a),(F1 . b)) <> {} by A3, CAT_1:126;
let f be Morphism of a,b; :: thesis: ((t1 ") . b) * (F2 . f) = (F1 . f) * ((t1 ") . a)
A5: Hom ((F2 . a),(F2 . b)) <> {} by A3, CAT_1:126;
A6: F1 is_transformable_to F2 by A1, Def7;
then A7: Hom ((F1 . b),(F2 . b)) <> {} by Def2;
A8: t1 . b is invertible by A2, Def10;
then A9: Hom ((F2 . b),(F1 . b)) <> {} by A7, CAT_1:70;
A10: Hom ((F1 . a),(F2 . a)) <> {} by A6, Def2;
(F2 . f) * (t1 . a) = (t1 . b) * (F1 . f) by A1, A3, Def8;
then A11: (((t1 . b) ") * (F2 . f)) * (t1 . a) = ((t1 . b) ") * ((t1 . b) * (F1 . f)) by A10, A9, A5, CAT_1:54
.= (((t1 . b) ") * (t1 . b)) * (F1 . f) by A4, A7, A9, CAT_1:54
.= (id (F1 . b)) * (F1 . f) by A8, A7, CAT_1:def 14
.= F1 . f by A4, CAT_1:57 ;
A12: t1 . a is invertible by A2, Def10;
then A13: Hom ((F2 . a),(F1 . a)) <> {} by A10, CAT_1:70;
then A14: Hom ((F2 . a),(F1 . b)) <> {} by A4, CAT_1:52;
then ((t1 . b) ") * (F2 . f) = (((t1 . b) ") * (F2 . f)) * (id (F2 . a)) by CAT_1:58
.= (((t1 . b) ") * (F2 . f)) * ((t1 . a) * ((t1 . a) ")) by A12, A10, CAT_1:def 14
.= (F1 . f) * ((t1 . a) ") by A10, A13, A14, A11, CAT_1:54 ;
hence ((t1 ") . b) * (F2 . f) = (F1 . f) * ((t1 . a) ") by A2, A6, Def12
.= (F1 . f) * ((t1 ") . a) by A2, A6, Def12 ;
:: thesis: verum

end;

Lm5: for A, B being Category
for F1, F2 being Functor of A,B
for t1 being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 & t1 is invertible holds
F2 is_naturally_transformable_to F1

proof end;

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
let t1 be natural_transformation of F1,F2;
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: t1 is invertible ;
func t1 " -> natural_transformation of F2,F1 equals :Def13: :: NATTRA_1:def 13
t1 " ;
coherence
t1 " is natural_transformation of F2,F1

proof end;

end;

:: deftheorem Def13 defines " NATTRA_1:def 13 :
for A, B being Category
for F1, F2 being Functor of A,B
for t1 being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 & t1 is invertible holds
t1 " = t1 " ;

theorem :: NATTRA_1:29

canceled;

theorem Th30: :: NATTRA_1:30

for A, B being Category
for F1, F2 being Functor of A,B
for t1 being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 & t1 is invertible holds
for a being Object of A holds (t1 ") . a = (t1 . a) "

proof end;

theorem :: NATTRA_1:31

for A, B being Category
for F1, F2 being Functor of A,B st F1 ~= F2 holds
F2 ~= F1

proof end;

theorem Th32: :: NATTRA_1:32

for A, B being Category
for F1, F2, F3 being Functor of A,B st F1 ~= F2 & F2 ~= F3 holds
F1 ~= F3

proof end;

definition

let A, B be Category;
let F1, F2 be Functor of A,B;
assume A1: F1,F2 are_naturally_equivalent ;
mode natural_equivalence of F1,F2 -> natural_transformation of F1,F2 means :Def14: :: NATTRA_1:def 14
it is invertible ;
existence
ex b₁ being natural_transformation of F1,F2 st b₁ is invertible by A1, Def11;

end;

:: deftheorem Def14 defines natural_equivalence NATTRA_1:def 14 :
for A, B being Category
for F1, F2 being Functor of A,B st F1,F2 are_naturally_equivalent holds
for b₅ being natural_transformation of F1,F2 holds
( b₅ is natural_equivalence of F1,F2 iff b₅ is invertible );

theorem :: NATTRA_1:33

for A, B being Category
for F being Functor of A,B holds id F is natural_equivalence of F,F

proof end;

theorem :: NATTRA_1:34

for A, B being Category
for F1, F2, F3 being Functor of A,B st F1 ~= F2 & F2 ~= F3 holds
for t being natural_equivalence of F1,F2
for t9 being natural_equivalence of F2,F3 holds t9 `*` t is natural_equivalence of F1,F3

proof end;

begin

definition

let A, B be Category;
mode NatTrans-DOMAIN of A,B -> non empty set means :Def15: :: NATTRA_1:def 15
for x being set st x in it holds
ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 );
existence
ex b₁ being non empty set st
for x being set st x in b₁ holds
ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 )

proof end;

end;

:: deftheorem Def15 defines NatTrans-DOMAIN NATTRA_1:def 15 :
for A, B being Category
for b₃ being non empty set holds
( b₃ is NatTrans-DOMAIN of A,B iff for x being set st x in b₃ holds
ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 ) );

definition

let A, B be Category;
func NatTrans (A,B) -> NatTrans-DOMAIN of A,B means :Def16: :: NATTRA_1:def 16
for x being set holds
( x in it iff ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 ) );
existence
ex b₁ being NatTrans-DOMAIN of A,B st
for x being set holds
( x in b₁ iff ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 ) )

proof end;

uniqueness
for b₁, b₂ being NatTrans-DOMAIN of A,B st ( for x being set holds
( x in b₁ iff ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 ) ) ) & ( for x being set holds
( x in b₂ iff ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def16 defines NatTrans NATTRA_1:def 16 :
for A, B being Category
for b₃ being NatTrans-DOMAIN of A,B holds
( b₃ = NatTrans (A,B) iff for x being set holds
( x in b₃ iff ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 ) ) );

theorem Th35: :: NATTRA_1:35

for A, B being Category
for F1, F2 being Functor of A,B
for t1 being natural_transformation of F1,F2 holds
( F1 is_naturally_transformable_to F2 iff [[F1,F2],t1] in NatTrans (A,B) )

proof end;

definition

canceled;
let A, B be Category;
func Functors (A,B) -> strict Category means :Def18: :: NATTRA_1:def 18
( the carrier of it = Funct (A,B) & the carrier' of it = NatTrans (A,B) & ( for f being Morphism of it holds
( dom f = (f `1) `1 & cod f = (f `1) `2 ) ) & ( for f, g being Morphism of it st dom g = cod f holds
[g,f] in dom the Comp of it ) & ( for f, g being Morphism of it st [g,f] in dom the Comp of it holds
ex F, F1, F2 being Functor of A,B ex t being natural_transformation of F,F1 ex t1 being natural_transformation of F1,F2 st
( f = [[F,F1],t] & g = [[F1,F2],t1] & the Comp of it . [g,f] = [[F,F2],(t1 `*` t)] ) ) & ( for a being Object of it
for F being Functor of A,B st F = a holds
id a = [[F,F],(id F)] ) );
existence
ex b₁ being strict Category st
( the carrier of b₁ = Funct (A,B) & the carrier' of b₁ = NatTrans (A,B) & ( for f being Morphism of b₁ holds
( dom f = (f `1) `1 & cod f = (f `1) `2 ) ) & ( for f, g being Morphism of b₁ st dom g = cod f holds
[g,f] in dom the Comp of b₁ ) & ( for f, g being Morphism of b₁ st [g,f] in dom the Comp of b₁ holds
ex F, F1, F2 being Functor of A,B ex t being natural_transformation of F,F1 ex t1 being natural_transformation of F1,F2 st
( f = [[F,F1],t] & g = [[F1,F2],t1] & the Comp of b₁ . [g,f] = [[F,F2],(t1 `*` t)] ) ) & ( for a being Object of b₁
for F being Functor of A,B st F = a holds
id a = [[F,F],(id F)] ) )

proof end;

uniqueness
for b₁, b₂ being strict Category st the carrier of b₁ = Funct (A,B) & the carrier' of b₁ = NatTrans (A,B) & ( for f being Morphism of b₁ holds
( dom f = (f `1) `1 & cod f = (f `1) `2 ) ) & ( for f, g being Morphism of b₁ st dom g = cod f holds
[g,f] in dom the Comp of b₁ ) & ( for f, g being Morphism of b₁ st [g,f] in dom the Comp of b₁ holds
ex F, F1, F2 being Functor of A,B ex t being natural_transformation of F,F1 ex t1 being natural_transformation of F1,F2 st
( f = [[F,F1],t] & g = [[F1,F2],t1] & the Comp of b₁ . [g,f] = [[F,F2],(t1 `*` t)] ) ) & ( for a being Object of b₁
for F being Functor of A,B st F = a holds
id a = [[F,F],(id F)] ) & the carrier of b₂ = Funct (A,B) & the carrier' of b₂ = NatTrans (A,B) & ( for f being Morphism of b₂ holds
( dom f = (f `1) `1 & cod f = (f `1) `2 ) ) & ( for f, g being Morphism of b₂ st dom g = cod f holds
[g,f] in dom the Comp of b₂ ) & ( for f, g being Morphism of b₂ st [g,f] in dom the Comp of b₂ holds
ex F, F1, F2 being Functor of A,B ex t being natural_transformation of F,F1 ex t1 being natural_transformation of F1,F2 st
( f = [[F,F1],t] & g = [[F1,F2],t1] & the Comp of b₂ . [g,f] = [[F,F2],(t1 `*` t)] ) ) & ( for a being Object of b₂
for F being Functor of A,B st F = a holds
id a = [[F,F],(id F)] ) holds
b₁ = b₂

proof end;

end;

:: deftheorem NATTRA_1:def 17 :
canceled;

:: deftheorem Def18 defines Functors NATTRA_1:def 18 :
for A, B being Category
for b₃ being strict Category holds
( b₃ = Functors (A,B) iff ( the carrier of b₃ = Funct (A,B) & the carrier' of b₃ = NatTrans (A,B) & ( for f being Morphism of b₃ holds
( dom f = (f `1) `1 & cod f = (f `1) `2 ) ) & ( for f, g being Morphism of b₃ st dom g = cod f holds
[g,f] in dom the Comp of b₃ ) & ( for f, g being Morphism of b₃ st [g,f] in dom the Comp of b₃ holds
ex F, F1, F2 being Functor of A,B ex t being natural_transformation of F,F1 ex t1 being natural_transformation of F1,F2 st
( f = [[F,F1],t] & g = [[F1,F2],t1] & the Comp of b₃ . [g,f] = [[F,F2],(t1 `*` t)] ) ) & ( for a being Object of b₃
for F being Functor of A,B st F = a holds
id a = [[F,F],(id F)] ) ) );

theorem :: NATTRA_1:36

canceled;

theorem :: NATTRA_1:37

canceled;

theorem :: NATTRA_1:38

canceled;

theorem Th39: :: NATTRA_1:39

for A, B being Category
for F, F1 being Functor of A,B
for t being natural_transformation of F,F1
for f being Morphism of (Functors (A,B)) st f = [[F,F1],t] holds
( dom f = F & cod f = F1 )

proof end;

theorem :: NATTRA_1:40

for A, B being Category
for a, b being Object of (Functors (A,B))
for f being Morphism of a,b st Hom (a,b) <> {} holds
ex F, F1 being Functor of A,B ex t being natural_transformation of F,F1 st
( a = F & b = F1 & f = [[F,F1],t] )

proof end;

theorem Th41: :: NATTRA_1:41

for A, B being Category
for F2, F3, F, F1 being Functor of A,B
for t being natural_transformation of F,F1
for t9 being natural_transformation of F2,F3
for f, g being Morphism of (Functors (A,B)) st f = [[F,F1],t] & g = [[F2,F3],t9] holds
( [g,f] in dom the Comp of (Functors (A,B)) iff F1 = F2 )

proof end;

theorem :: NATTRA_1:42

for A, B being Category
for F, F1, F2 being Functor of A,B
for t being natural_transformation of F,F1
for t1 being natural_transformation of F1,F2
for f, g being Morphism of (Functors (A,B)) st f = [[F,F1],t] & g = [[F1,F2],t1] holds
g * f = [[F,F2],(t1 `*` t)]

proof end;

begin

definition

let C be Category;
attr C is discrete means :Def19: :: NATTRA_1:def 19
for f being Morphism of C ex a being Object of C st f = id a;

end;

:: deftheorem Def19 defines discrete NATTRA_1:def 19 :
for C being Category holds
( C is discrete iff for f being Morphism of C ex a being Object of C st f = id a );

registration

cluster non empty non void Category-like discrete CatStr ;
existence
ex b₁ being Category st b₁ is discrete

proof end;

end;

theorem :: NATTRA_1:43

canceled;

theorem Th44: :: NATTRA_1:44

for A being discrete Category
for a being Object of A holds Hom (a,a) = {(id a)}

proof end;

theorem Th45: :: NATTRA_1:45

for A being Category holds
( A is discrete iff ( ( for a being Object of A ex B being finite set st
( B = Hom (a,a) & card B = 1 ) ) & ( for a, b being Object of A st a <> b holds
Hom (a,b) = {} ) ) )

proof end;

theorem Th46: :: NATTRA_1:46

for o, m being set holds 1Cat (o,m) is discrete

proof end;

theorem :: NATTRA_1:47

for A being discrete Category
for C being Subcategory of A holds C is discrete

proof end;

theorem :: NATTRA_1:48

for A, B being Category st A is discrete & B is discrete holds
[:A,B:] is discrete

proof end;

theorem Th49: :: NATTRA_1:49

for A being discrete Category
for B being Category
for F1, F2 being Functor of B,A st F1 is_transformable_to F2 holds
F1 = F2

proof end;

theorem Th50: :: NATTRA_1:50

for A being discrete Category
for B being Category
for F being Functor of B,A
for t being transformation of F,F holds t = id F

proof end;

theorem :: NATTRA_1:51

for A, B being Category st A is discrete holds
Functors (B,A) is discrete

proof end;

registration

let C be Category;
cluster non empty non void strict Category-like discrete Subcategory of C;
existence
ex b₁ being Subcategory of C st
( b₁ is strict & b₁ is discrete )

proof end;

end;

definition

let C be Category;
func IdCat C -> strict discrete Subcategory of C means :Def20: :: NATTRA_1:def 20
( the carrier of it = the carrier of C & the carrier' of it = { (id a) where a is Object of C : verum } );
existence
ex b₁ being strict discrete Subcategory of C st
( the carrier of b₁ = the carrier of C & the carrier' of b₁ = { (id a) where a is Object of C : verum } )

proof end;

uniqueness
for b₁, b₂ being strict discrete Subcategory of C st the carrier of b₁ = the carrier of C & the carrier' of b₁ = { (id a) where a is Object of C : verum } & the carrier of b₂ = the carrier of C & the carrier' of b₂ = { (id a) where a is Object of C : verum } holds
b₁ = b₂

proof end;

end;

:: deftheorem Def20 defines IdCat NATTRA_1:def 20 :
for C being Category
for b₂ being strict discrete Subcategory of C holds
( b₂ = IdCat C iff ( the carrier of b₂ = the carrier of C & the carrier' of b₂ = { (id a) where a is Object of C : verum } ) );

theorem Th52: :: NATTRA_1:52

for C being strict Category st C is discrete holds
IdCat C = C

proof end;

theorem :: NATTRA_1:53

for C being Category holds IdCat (IdCat C) = IdCat C by Th52;

theorem :: NATTRA_1:54

for o, m being set holds IdCat (1Cat (o,m)) = 1Cat (o,m) by Th46, Th52;

theorem :: NATTRA_1:55

for A, B being Category holds IdCat [:A,B:] = [:(IdCat A),(IdCat B):]

proof end;