NATTRA_1: Natural Transformations. Discrete Categories

:: Natural Transformations. Discrete Categories
:: by Andrzej Trybulec
::
:: Received May 15, 1991
:: Copyright (c) 1991-2012 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 by FUNCT_2:32;

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 by FUNCT_4:59;

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 :: NATTRA_1:3

canceled;

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 )

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 st ( for o being Element of A st o in O holds
id o in M ) holds
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
CatStr(# O,M,DOM,COD,COMP #) 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

canceled;

end;

:: deftheorem NATTRA_1:def 1 :

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;

:: The following theorems are analogues of theorems from CAT_1, with
:: the new concept of the application of a functor to a morphism

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)) <> {} ;

end;

:: deftheorem Def2 defines is_transformable_to NATTRA_1:def 2 :

theorem Th18: :: NATTRA_1:18

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;

proof end;

end;

:: deftheorem Def3 defines transformation NATTRA_1:def 3 :

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);

proof end;

proof end;

end;

:: deftheorem Def4 defines id NATTRA_1:def 4 :

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 by A1, Def3;

end;

:: deftheorem Def5 defines . NATTRA_1:def 5 :

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);

proof end;

proof end;

end;

:: deftheorem Def6 defines `*` NATTRA_1:def 6 :

theorem Th19: :: NATTRA_1:19

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 Th20: :: NATTRA_1:20

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 Th21: :: NATTRA_1:21

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 Th22: :: NATTRA_1:22

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) );

proof end;

end;

:: deftheorem Def7 defines is_naturally_transformable_to NATTRA_1:def 7 :

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 Th23: :: NATTRA_1:23

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 by A1, Def7;

end;

:: deftheorem Def8 defines natural_transformation NATTRA_1:def 8 :

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

proof end;

end;

definition

let A, B be Category;
let F, F1, F2 be Functor of A,B;
assume that
B1: F is_naturally_transformable_to F1 and
B2: 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;

proof end;

end;

:: deftheorem Def9 defines `*` NATTRA_1:def 9 :

theorem Th24: :: NATTRA_1:24

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 Th25: :: NATTRA_1:25

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 Th26: :: NATTRA_1:26

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;

:: Natural equivalences

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 :

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 );

proof end;

end;

:: deftheorem Def11 defines are_naturally_equivalent NATTRA_1:def 11 :

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 B1: F1 is_transformable_to F2 ;
let t1 be transformation of F1,F2;
assume B2: 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) " ;

proof end;

proof end;

end;

:: deftheorem Def12 defines " NATTRA_1:def 12 :

Lm4: now :: thesis:

let A, B be Category; :: thesis:
let F1, F2 be Functor of A,B; :: thesis:
let t1 be natural_transformation of F1,F2; :: thesis:
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: t1 is invertible ; :: thesis:
let a, b be Object of A; :: thesis:
assume A3: Hom (a,b) <> {} ; :: thesis:
A4: Hom ((F1 . a),(F1 . b)) <> {} by A3, CAT_1:84;
let f be Morphism of a,b; :: thesis:
A5: Hom ((F2 . a),(F2 . b)) <> {} by A3, CAT_1:84;
A6: F1 is_transformable_to F2 by A1, Def7;
A7: Hom ((F1 . b),(F2 . b)) <> {} by Def2, A1, Def7;
A8: t1 . b is invertible by A2, Def10;
then A9: Hom ((F2 . b),(F1 . b)) <> {} by A7, CAT_1:41;
A10: Hom ((F1 . a),(F2 . a)) <> {} by A1, Def7, 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_1:25
.= (((t1 . b) ") * (t1 . b)) * (F1 /. f) by A4, A7, A9, CAT_1:25
.= (id (F1 . b)) * (F1 /. f) by A8, CAT_1:def 17
.= F1 /. f by A4, CAT_1:28 ;
A12: t1 . a is invertible by A2, Def10;
then A13: Hom ((F2 . a),(F1 . a)) <> {} by A10, CAT_1:41;
then A14: Hom ((F2 . a),(F1 . b)) <> {} by A4, CAT_1:24;
then ((t1 . b) ") * (F2 /. f) = (((t1 . b) ") * (F2 /. f)) * (id (F2 . a)) by CAT_1:29
.= (((t1 . b) ") * (F2 /. f)) * ((t1 . a) * ((t1 . a) ")) by A12, CAT_1:def 17
.= (F1 /. f) * ((t1 . a) ") by A10, A13, A14, A11, CAT_1:25 ;
hence ((t1 ") . b) * (F2 /. f) = (F1 /. f) * ((t1 . a) ") by A2, A6, Def12
.= (F1 /. f) * ((t1 ") . a) by A2, A6, Def12 ;
:: thesis:

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
B1: F1 is_naturally_transformable_to F2 and
B2: t1 is invertible ;

func t1 " -> natural_transformation of F2,F1 equals :Def13: :: NATTRA_1:def 13
t1 " ;

proof end;

end;

:: deftheorem Def13 defines " NATTRA_1:def 13 :

theorem Th27: :: NATTRA_1:27

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_1:28

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

proof end;

theorem Th29: :: NATTRA_1:29

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 by A1, Def11;

end;

:: deftheorem Def14 defines natural_equivalence NATTRA_1:def 14 :

theorem :: NATTRA_1:30

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_1:31

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 );

proof end;

end;

:: deftheorem Def15 defines NatTrans-DOMAIN NATTRA_1:def 15 :

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 ) );

proof end;

proof end;

end;

:: deftheorem Def16 defines NatTrans NATTRA_1:def 16 :

theorem Th32: :: NATTRA_1:32

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

let A, B be Category;

func Functors (A,B) -> strict Category means :Def17: :: NATTRA_1:def 17
( 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)] ) );

proof end;

proof end;

end;

:: deftheorem Def17 defines Functors NATTRA_1:def 17 :

:: As immediate consequences of the definition we get

theorem Th33: :: NATTRA_1:33

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_1:34

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 Th35: :: NATTRA_1:35

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_1:36

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 :Def18: :: NATTRA_1:def 18
for f being Morphism of C ex a being Object of C st f = id a;

end;

:: deftheorem Def18 defines discrete NATTRA_1:def 18 :

registration

cluster non empty non void V55() Category-like transitive associative reflexive with_identities discrete for CatStr ;

proof end;

end;

theorem Th37: :: NATTRA_1:37

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

proof end;

theorem Th38: :: NATTRA_1:38

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 Th39: :: NATTRA_1:39

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

proof end;

theorem :: NATTRA_1:40

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

proof end;

theorem :: NATTRA_1:41

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

proof end;

theorem Th42: :: NATTRA_1:42

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 Th43: :: NATTRA_1:43

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_1:44

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 V55() strict Category-like transitive associative reflexive with_identities discrete for Subcategory of C;

proof end;

end;

definition

let C be Category;

func IdCat C -> strict discrete Subcategory of C means :Def19: :: NATTRA_1:def 19
( the carrier of it = the carrier of C & the carrier' of it = { (id a) where a is Object of C : verum } );

proof end;

proof end;

end;

:: deftheorem Def19 defines IdCat NATTRA_1:def 19 :

theorem Th45: :: NATTRA_1:45

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

proof end;

theorem :: NATTRA_1:46

for C being Category holds IdCat (IdCat C) = IdCat C by Th45;

theorem :: NATTRA_1:47

for o, m being set holds IdCat (1Cat (o,m)) = 1Cat (o,m)

proof end;

theorem :: NATTRA_1:48

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

proof end;