OPPCAT_1: Opposite Categories and Contravariant Functors

:: Opposite Categories and Contravariant Functors
:: by Czes\l aw Byli\'nski
::
:: Received February 13, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users

:: Opposite Category

definition

let C be Category;

func C opp -> non empty non void strict CatStr equals :: OPPCAT_1:def 1
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #);

coherence ;

end;

:: deftheorem defines opp OPPCAT_1:def 1 :

definition

let C be Category;
let c be Object of C;

func c opp -> Object of (C opp) equals :: OPPCAT_1:def 2
c;

coherence ;

end;

:: deftheorem defines opp OPPCAT_1:def 2 :

registration

let C be Category;

cluster C opp -> non empty non void strict Category-like transitive associative reflexive with_identities ;

coherence

proof end;

end;

definition

let C be Category;
let c be Object of (C opp);

func opp c -> Object of C equals :: OPPCAT_1:def 3
c opp ;

coherence ;

end;

:: deftheorem defines opp OPPCAT_1:def 3 :

theorem :: OPPCAT_1:1

canceled;

::$CT

theorem :: OPPCAT_1:2

for C being Category
for c being Object of C holds (c opp) opp = c ;

theorem :: OPPCAT_1:3

for C being Category
for c being Object of C holds opp (c opp) = c ;

theorem :: OPPCAT_1:4

for C being Category
for c being Object of (C opp) holds (opp c) opp = c ;

theorem Th4: :: OPPCAT_1:5

for C being Category
for a, b being Object of C holds Hom (a,b) = Hom ((b opp),(a opp))

proof end;

theorem Th5: :: OPPCAT_1:6

for C being Category
for a, b being Object of (C opp) holds Hom (a,b) = Hom ((opp b),(opp a))

proof end;

definition

let C be Category;
let f be Morphism of C;

func f opp -> Morphism of (C opp) equals :: OPPCAT_1:def 4
f;

coherence ;

end;

:: deftheorem defines opp OPPCAT_1:def 4 :

definition

let C be Category;
let f be Morphism of (C opp);

func opp f -> Morphism of C equals :: OPPCAT_1:def 5
f opp ;

coherence ;

end;

:: deftheorem defines opp OPPCAT_1:def 5 :

definition

let C be Category;
let a, b be Object of C;
assume A1: Hom (a,b) <> {} ;
let f be Morphism of a,b;

func f opp -> Morphism of b opp ,a opp equals :Def6: :: OPPCAT_1:def 6
f;

coherence

proof end;

end;

:: deftheorem Def6 defines opp OPPCAT_1:def 6 :

definition

let C be Category;
let a, b be Object of C;
assume A1: Hom ((a opp),(b opp)) <> {} ;
let f be Morphism of a opp ,b opp ;

func opp f -> Morphism of b,a equals :Def7: :: OPPCAT_1:def 7
f;

coherence

proof end;

end;

:: deftheorem Def7 defines opp OPPCAT_1:def 7 :

theorem :: OPPCAT_1:7

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

proof end;

theorem :: OPPCAT_1:8

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

proof end;

theorem :: OPPCAT_1:9

for C being Category
for a, b being Object of (C opp)
for f being Morphism of a,b holds (opp f) opp = f ;

theorem Th9: :: OPPCAT_1:10

for C being Category
for a, b being Object of C st Hom (a,b) <> {} holds
for f being Morphism of a,b holds
( dom (f opp) = cod f & cod (f opp) = dom f )

proof end;

theorem :: OPPCAT_1:11

for C being Category
for a, b being Object of (C opp)
for f being Morphism of a,b holds
( dom (opp f) = cod f & cod (opp f) = dom f ) ;

theorem :: OPPCAT_1:12

for C being Category
for a, b being Object of C st Hom (a,b) <> {} holds
for f being Morphism of a,b holds
( (dom f) opp = cod (f opp) & (cod f) opp = dom (f opp) ) by Th9;

theorem :: OPPCAT_1:13

for C being Category
for a, b being Object of (C opp) st Hom (a,b) <> {} holds
for f being Morphism of a,b holds
( opp (dom f) = cod (opp f) & opp (cod f) = dom (opp f) ) ;

theorem :: OPPCAT_1:14

canceled;

::$CT

theorem Th13: :: OPPCAT_1:15

for C being Category
for a, b being Object of (C opp)
for f being Morphism of a,b st Hom (a,b) <> {} holds
opp f is Morphism of opp b, opp a

proof end;

theorem Th14: :: OPPCAT_1:16

for C being Category
for a, b, c being Object of C st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds (g (*) f) opp = (f opp) (*) (g opp)

proof end;

theorem :: OPPCAT_1:17

for C being Category
for a, b, c being Object of C st Hom ((b opp),(a opp)) <> {} & Hom ((c opp),(b opp)) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds (g (*) f) opp = (f opp) (*) (g opp)

proof end;

theorem Th16: :: OPPCAT_1:18

for C being Category
for f, g being Morphism of (C opp) st dom g = cod f holds
opp (g (*) f) = (opp f) (*) (opp g)

proof end;

theorem :: OPPCAT_1:19

for C being Category
for a, b, c being Object of C
for f being Morphism of a,b
for g being Morphism of b,c st Hom (a,b) <> {} & Hom (b,c) <> {} holds
(g * f) opp = (f opp) (*) (g opp)

proof end;

Lm1: for C being Category
for a being Object of C
for b being Object of (C opp) holds
( ( Hom ((a opp),b) <> {} implies for f being Morphism of a opp ,b holds f (*) ((id a) opp) = f ) & ( Hom (b,(a opp)) <> {} implies for f being Morphism of b,a opp holds ((id a) opp) (*) f = f ) )

proof end;

theorem Th18: :: OPPCAT_1:20

for C being Category
for a being Object of C holds (id a) opp = id (a opp)

proof end;

Lm2: for C being Category
for a being Object of C holds id a = id (a opp)

proof end;

theorem Th19: :: OPPCAT_1:21

for C being Category
for a being Object of (C opp) holds opp (id a) = id (opp a)

proof end;

Lm3: for C being Category
for a, b, c being Object of C st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds g * f = (f opp) * (g opp)

proof end;

theorem :: OPPCAT_1:22

for C being Category
for a, b being Object of C
for f being Morphism of a,b holds
( f opp is monic iff f is epi )

proof end;

theorem :: OPPCAT_1:23

for C being Category
for b, c being Object of C st Hom (b,c) <> {} holds
for f being Morphism of b,c holds
( f opp is epi iff f is monic )

proof end;

theorem :: OPPCAT_1:24

for C being Category
for a, b being Object of C
for f being Morphism of a,b holds
( f opp is invertible iff f is invertible )

proof end;

theorem :: OPPCAT_1:25

for C being Category
for c being Object of C holds
( c is initial iff c opp is terminal )

proof end;

theorem :: OPPCAT_1:26

for C being Category
for c being Object of C holds
( c opp is initial iff c is terminal )

proof end;

:: Contravariant Functors

definition

let C, B be Category;
let S be Function of the carrier' of (C opp), the carrier' of B;

func /* S -> Function of the carrier' of C, the carrier' of B means :Def8: :: OPPCAT_1:def 8
for f being Morphism of C holds it . f = S . (f opp);

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines /* OPPCAT_1:def 8 :

theorem :: OPPCAT_1:27

for B, C being Category
for S being Function of the carrier' of (C opp), the carrier' of B
for f being Morphism of (C opp) holds (/* S) . (opp f) = S . f

proof end;

Lm4: for B, C being Category
for S being Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj S) . (c opp))

proof end;

theorem Th26: :: OPPCAT_1:28

for B, C being Category
for S being Functor of C opp ,B
for c being Object of C holds (Obj (/* S)) . c = (Obj S) . (c opp)

proof end;

theorem :: OPPCAT_1:29

for B, C being Category
for S being Functor of C opp ,B
for c being Object of (C opp) holds (Obj (/* S)) . (opp c) = (Obj S) . c

proof end;

Lm5: for B, C being Category
for S being Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj (/* S)) . c)

proof end;

Lm6: now :: thesis:

let C, B be Category; :: thesis:
let S be Functor of C opp ,B; :: thesis:
let c be Object of C; :: thesis:
(/* S) . (id c) = id ((Obj (/* S)) . c) by Lm5;
hence ex d being Object of B st (/* S) . (id c) = id d ; :: thesis:

end;

Lm7: for B, C being Category
for S being Functor of C opp ,B
for f being Morphism of C holds
( (Obj (/* S)) . (dom f) = cod ((/* S) . f) & (Obj (/* S)) . (cod f) = dom ((/* S) . f) )

proof end;

Lm8: now :: thesis:

let C, B be Category; :: thesis:
let S be Functor of C opp ,B; :: thesis:
let f be Morphism of C; :: thesis:
thus (/* S) . (id (dom f)) = id ((Obj (/* S)) . (dom f)) by Lm5
.= id (cod ((/* S) . f)) by Lm7 ; :: thesis:
thus (/* S) . (id (cod f)) = id ((Obj (/* S)) . (cod f)) by Lm5
.= id (dom ((/* S) . f)) by Lm7 ; :: thesis:

end;

Lm9: for B, C being Category
for S being Functor of C opp ,B
for a, b, c being Object of C st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds (/* S) . (g (*) f) = ((/* S) . f) (*) ((/* S) . g)

proof end;

definition

let C, D be Category;

mode Contravariant_Functor of C,D -> Function of the carrier' of C, the carrier' of D means :Def9: :: OPPCAT_1:def 9
( ( for c being Object of C ex d being Object of D st it . (id c) = id d ) & ( for f being Morphism of C holds
( it . (id (dom f)) = id (cod (it . f)) & it . (id (cod f)) = id (dom (it . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds
it . (g (*) f) = (it . f) (*) (it . g) ) );

existence

proof end;

end;

:: deftheorem Def9 defines Contravariant_Functor OPPCAT_1:def 9 :

theorem Th28: :: OPPCAT_1:30

for C, D being Category
for S being Contravariant_Functor of C,D
for c being Object of C
for d being Object of D st S . (id c) = id d holds
(Obj S) . c = d

proof end;

theorem Th29: :: OPPCAT_1:31

for C, D being Category
for S being Contravariant_Functor of C,D
for c being Object of C holds S . (id c) = id ((Obj S) . c)

proof end;

theorem Th30: :: OPPCAT_1:32

for C, D being Category
for S being Contravariant_Functor of C,D
for f being Morphism of C holds
( (Obj S) . (dom f) = cod (S . f) & (Obj S) . (cod f) = dom (S . f) )

proof end;

theorem Th31: :: OPPCAT_1:33

for C, D being Category
for S being Contravariant_Functor of C,D
for f, g being Morphism of C st dom g = cod f holds
dom (S . f) = cod (S . g)

proof end;

theorem Th32: :: OPPCAT_1:34

for B, C being Category
for S being Functor of C opp ,B holds /* S is Contravariant_Functor of C,B

proof end;

theorem Th33: :: OPPCAT_1:35

for B, C, D being Category
for S1 being Contravariant_Functor of C,B
for S2 being Contravariant_Functor of B,D holds S2 * S1 is Functor of C,D

proof end;

Lm10: for B, C being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj S) . (c opp))

proof end;

theorem Th34: :: OPPCAT_1:36

for B, C being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of C holds (Obj (/* S)) . c = (Obj S) . (c opp)

proof end;

theorem :: OPPCAT_1:37

for B, C being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of (C opp) holds (Obj (/* S)) . (opp c) = (Obj S) . c

proof end;

Lm11: for B, C being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj (/* S)) . c)

proof end;

Lm12: for B, C being Category
for S being Contravariant_Functor of C opp ,B
for f being Morphism of C holds
( (Obj (/* S)) . (dom f) = dom ((/* S) . f) & (Obj (/* S)) . (cod f) = cod ((/* S) . f) )

proof end;

theorem :: OPPCAT_1:38

for B, C being Category
for S being Contravariant_Functor of C opp ,B holds /* S is Functor of C,B

proof end;

:: Dualization functors

definition

let C, B be Category;
let S be Function of the carrier' of C, the carrier' of B;

func *' S -> Function of the carrier' of (C opp), the carrier' of B means :Def10: :: OPPCAT_1:def 10
for f being Morphism of (C opp) holds it . f = S . (opp f);

existence

proof end;

uniqueness

proof end;

func S *' -> Function of the carrier' of C, the carrier' of (B opp) means :Def11: :: OPPCAT_1:def 11
for f being Morphism of C holds it . f = (S . f) opp ;

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def10 defines *' OPPCAT_1:def 10 :

:: deftheorem Def11 defines *' OPPCAT_1:def 11 :

theorem :: OPPCAT_1:39

for B, C being Category
for S being Function of the carrier' of C, the carrier' of B
for f being Morphism of C holds (*' S) . (f opp) = S . f

proof end;

Lm13: for B, C being Category
for S being Functor of C,B
for c being Object of (C opp) holds (*' S) . (id c) = id ((Obj S) . (opp c))

proof end;

theorem Th38: :: OPPCAT_1:40

for B, C being Category
for S being Functor of C,B
for c being Object of (C opp) holds (Obj (*' S)) . c = (Obj S) . (opp c)

proof end;

theorem :: OPPCAT_1:41

for B, C being Category
for S being Functor of C,B
for c being Object of C holds (Obj (*' S)) . (c opp) = (Obj S) . c

proof end;

Lm14: for B, C being Category
for S being Functor of C,B
for c being Object of C holds (S *') . (id c) = id (((Obj S) . c) opp)

proof end;

theorem Th40: :: OPPCAT_1:42

for B, C being Category
for S being Functor of C,B
for c being Object of C holds (Obj (S *')) . c = ((Obj S) . c) opp

proof end;

Lm15: for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of (C opp) holds (*' S) . (id c) = id ((Obj S) . (opp c))

proof end;

theorem Th41: :: OPPCAT_1:43

for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of (C opp) holds (Obj (*' S)) . c = (Obj S) . (opp c)

proof end;

theorem :: OPPCAT_1:44

for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (Obj (*' S)) . (c opp) = (Obj S) . c

proof end;

Lm16: for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (S *') . (id c) = id (((Obj S) . c) opp)

proof end;

theorem Th43: :: OPPCAT_1:45

for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (Obj (S *')) . c = ((Obj S) . c) opp

proof end;

Lm17: for C, D being Category
for F being Function of the carrier' of C, the carrier' of D
for f being Morphism of (C opp) holds ((*' F) *') . f = (F . (opp f)) opp

proof end;

theorem Th44: :: OPPCAT_1:46

for C, D being Category
for F being Function of the carrier' of C, the carrier' of D
for f being Morphism of C holds ((*' F) *') . (f opp) = (F . f) opp

proof end;

theorem Th45: :: OPPCAT_1:47

for C, D being Category
for S being Function of the carrier' of C, the carrier' of D holds /* (*' S) = S

proof end;

theorem :: OPPCAT_1:48

for C, D being Category
for S being Function of the carrier' of (C opp), the carrier' of D holds *' (/* S) = S

proof end;

theorem :: OPPCAT_1:49

for C, D being Category
for S being Function of the carrier' of C, the carrier' of D holds (*' S) *' = *' (S *')

proof end;

theorem :: OPPCAT_1:50

for C being Category
for D being strict Category
for S being Function of the carrier' of C, the carrier' of D holds (S *') *' = S

proof end;

theorem :: OPPCAT_1:51

for D being Category
for C being strict Category
for S being Function of the carrier' of C, the carrier' of D holds *' (*' S) = S

proof end;

Lm18: for B, C, D being Category
for S being Function of the carrier' of (C opp), the carrier' of B
for T being Function of the carrier' of B, the carrier' of D holds /* (T * S) = T * (/* S)

proof end;

theorem :: OPPCAT_1:52

for B, C, D being Category
for S being Function of the carrier' of C, the carrier' of B
for T being Function of the carrier' of B, the carrier' of D holds *' (T * S) = T * (*' S)

proof end;

theorem :: OPPCAT_1:53

for B, C, D being Category
for S being Function of the carrier' of C, the carrier' of B
for T being Function of the carrier' of B, the carrier' of D holds (T * S) *' = (T *') * S

proof end;

theorem :: OPPCAT_1:54

for B, C, D being Category
for F1 being Function of the carrier' of C, the carrier' of B
for F2 being Function of the carrier' of B, the carrier' of D holds (*' (F2 * F1)) *' = ((*' F2) *') * ((*' F1) *')

proof end;

Lm19: for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of (C opp) holds (*' S) . (id c) = id ((Obj (*' S)) . c)

proof end;

Lm20: for B, C being Category
for S being Contravariant_Functor of C,B
for f being Morphism of (C opp) holds
( (Obj (*' S)) . (dom f) = dom ((*' S) . f) & (Obj (*' S)) . (cod f) = cod ((*' S) . f) )

proof end;

theorem Th53: :: OPPCAT_1:55

for C, D being Category
for S being Contravariant_Functor of C,D holds *' S is Functor of C opp ,D

proof end;

Lm21: for B, C being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (S *') . (id c) = id ((Obj (S *')) . c)

proof end;

Lm22: for B, C being Category
for S being Contravariant_Functor of C,B
for f being Morphism of C holds
( (Obj (S *')) . (dom f) = dom ((S *') . f) & (Obj (S *')) . (cod f) = cod ((S *') . f) )

proof end;

theorem Th54: :: OPPCAT_1:56

for C, D being Category
for S being Contravariant_Functor of C,D holds S *' is Functor of C,D opp

proof end;

Lm23: for B, C being Category
for S being Functor of C,B
for c being Object of (C opp) holds (*' S) . (id c) = id ((Obj (*' S)) . c)

proof end;

Lm24: for B, C being Category
for S being Functor of C,B
for f being Morphism of (C opp) holds
( (Obj (*' S)) . (dom f) = cod ((*' S) . f) & (Obj (*' S)) . (cod f) = dom ((*' S) . f) )

proof end;

theorem Th55: :: OPPCAT_1:57

for C, D being Category
for S being Functor of C,D holds *' S is Contravariant_Functor of C opp ,D

proof end;

Lm25: for B, C being Category
for S being Functor of C,B
for c being Object of C holds (S *') . (id c) = id ((Obj (S *')) . c)

proof end;

Lm26: for B, C being Category
for S being Functor of C,B
for f being Morphism of C holds
( (Obj (S *')) . (dom f) = cod ((S *') . f) & (Obj (S *')) . (cod f) = dom ((S *') . f) )

proof end;

theorem Th56: :: OPPCAT_1:58

for C, D being Category
for S being Functor of C,D holds S *' is Contravariant_Functor of C,D opp

proof end;

theorem :: OPPCAT_1:59

for B, C, D being Category
for S1 being Contravariant_Functor of C,B
for S2 being Functor of B,D holds S2 * S1 is Contravariant_Functor of C,D

proof end;

theorem :: OPPCAT_1:60

for B, C, D being Category
for S1 being Functor of C,B
for S2 being Contravariant_Functor of B,D holds S2 * S1 is Contravariant_Functor of C,D

proof end;

theorem :: OPPCAT_1:61

for C, D being Category
for F being Functor of C,D
for c being Object of C holds (Obj ((*' F) *')) . (c opp) = ((Obj F) . c) opp

proof end;

theorem :: OPPCAT_1:62

for C, D being Category
for F being Contravariant_Functor of C,D
for c being Object of C holds (Obj ((*' F) *')) . (c opp) = ((Obj F) . c) opp

proof end;

theorem :: OPPCAT_1:63

for C, D being Category
for F being Functor of C,D holds (*' F) *' is Functor of C opp ,D opp

proof end;

theorem :: OPPCAT_1:64

for C, D being Category
for F being Contravariant_Functor of C,D holds (*' F) *' is Contravariant_Functor of C opp ,D opp

proof end;

:: Duality Functors

definition

let C be Category;

func id* C -> Contravariant_Functor of C,C opp equals :: OPPCAT_1:def 12
(id C) *' ;

coherence by Th56;

func *id C -> Contravariant_Functor of C opp ,C equals :: OPPCAT_1:def 13
*' (id C);

coherence by Th55;

end;

:: deftheorem defines id* OPPCAT_1:def 12 :

:: deftheorem defines *id OPPCAT_1:def 13 :

theorem Th63: :: OPPCAT_1:65

for C being Category
for f being Morphism of C holds (id* C) . f = f opp

proof end;

theorem :: OPPCAT_1:66

for C being Category
for c being Object of C holds (Obj (id* C)) . c = c opp

proof end;

theorem Th65: :: OPPCAT_1:67

for C being Category
for f being Morphism of (C opp) holds (*id C) . f = opp f

proof end;

theorem :: OPPCAT_1:68

for C being Category
for c being Object of (C opp) holds (Obj (*id C)) . c = opp c

proof end;

theorem :: OPPCAT_1:69

for C, D being Category
for S being Function of the carrier' of C, the carrier' of D holds
( *' S = S * (*id C) & S *' = (id* D) * S )

proof end;

theorem :: OPPCAT_1:70

for C being Category
for a, b, c being Object of C st Hom (a,b) <> {} & Hom (b,c) <> {} holds
for f being Morphism of a,b
for g being Morphism of b,c holds g * f = (f opp) * (g opp) by Lm3;

theorem Th69: :: OPPCAT_1:71

for C being Category
for a being Object of C holds id a = id (a opp) by Lm2;

theorem :: OPPCAT_1:72

for C being Category
for a being Object of (C opp) holds id a = id (opp a)

proof end;