Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Natural transformations. Discrete categories

**by****Andrzej Trybulec**- Received May 15, 1991
- MML identifier: NATTRA_1

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, RELAT_1, PARTFUN1, FUNCT_4, BOOLE, CAT_1, CAT_2, MCART_1, FUNCT_2, FINSET_1, CARD_1, NATTRA_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MCART_1, FINSET_1, PARTFUN1, RELAT_1, FUNCT_1, FUNCT_2, FUNCT_4, CARD_1, CAT_1, CAT_2; constructors MCART_1, CARD_1, CAT_2, PARTFUN1, MEMBERED, XBOOLE_0; clusters FUNCT_1, FINSET_1, CAT_1, CAT_2, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin :: Preliminaries reserve A1,A2,B1,B2 for non empty set, f for Function of A1,B1, g for Function of A2,B2, Y1 for non empty Subset of A1, Y2 for non empty Subset of A2; definition let A1 be set, B1 be non empty set, f be Function of A1, B1, Y1 be Subset of A1; redefine func f|Y1 -> Function of Y1,B1; end; theorem :: NATTRA_1:1 [:f,g:]|[:Y1,Y2:] = [:f|Y1,g|Y2:]; definition let A,B be non empty set; let A1 be non empty Subset of A, B1 be non empty Subset of B; let f be PartFunc of [:A1,A1:],A1; let g be PartFunc of [:B1,B1:],B1; redefine func |:f,g:| -> PartFunc of [:[:A1,B1:],[:A1,B1:]:],[:A1,B1:]; end; theorem :: NATTRA_1:2 for f being PartFunc of [:A1,A1:],A1, g being PartFunc of [:A2,A2:],A2 for F being PartFunc of [:Y1,Y1:],Y1 st F = f|([:Y1,Y1:] qua set) for G being PartFunc of [:Y2,Y2:],Y2 st G = g|([:Y2,Y2:] qua set) holds |:F,G:| = |:f,g:| |([:[:Y1,Y2:],[:Y1,Y2:]:] qua set); reserve A,B,C for Category, F,F1,F2,F3 for Functor of A,B, G for Functor of B,C; scheme M_Choice{A()-> non empty set, B()-> non empty set, F(set) -> set}: ex t being Function of A(),B() st for a being Element of A() holds t.a in F(a) provided for a being Element of A() holds B() meets F(a); theorem :: NATTRA_1:3 for a being Object of A, m being Morphism of a,a holds m in Hom(a,a); reserve m,o for set; theorem :: NATTRA_1:4 for f,g being Morphism of 1Cat(o,m) holds f = g; theorem :: NATTRA_1:5 for a being Object of A holds [[id a,id a],id a] in the Comp of A; theorem :: NATTRA_1:6 the Comp of 1Cat(o,m) = {[[m,m],m]}; theorem :: NATTRA_1:7 for a being Object of A holds 1Cat(a,id a) is Subcategory of A; theorem :: NATTRA_1:8 for C being Subcategory of A holds the Dom of C = (the Dom of A)|the Morphisms of C & the Cod of C = (the Cod of A)|the Morphisms of C & the Comp of C = (the Comp of A)|[:the Morphisms of C, the Morphisms of C:] & the Id of C = (the Id of A)|the Objects of C; theorem :: NATTRA_1:9 for O being non empty Subset of the Objects of A, M being non empty Subset of the Morphisms of A for DOM,COD being Function of M,O st DOM = (the Dom of A) |M & COD = (the Cod of A)|M for COMP being PartFunc of [:M,M qua non empty set:], M st COMP = (the Comp of A)|([:M,M:] qua set) 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; theorem :: NATTRA_1:10 for C being strict Category, A being strict Subcategory of C st the Objects of A = the Objects of C & the Morphisms of A = the Morphisms of C holds A = C; begin :: Application of a functor to a morphism definition let A,B,F; let a,b be Object of A such that Hom(a,b) <> {}; let f be Morphism of a,b; func F.f -> Morphism of F.a, F.b equals :: NATTRA_1:def 1 F.f; end; theorem :: NATTRA_1:11 for a,b being Object of A st Hom(a,b) <> {} for f being Morphism of a,b holds (G*F).f = G.(F.f); :: The following theorems are analogues of theorems from CAT_1.MIZ, with :: the new concept of the application of a functor to a morphism theorem :: NATTRA_1:12 :: CAT_1:95 for F1,F2 being Functor of A,B st for a,b being Object of A st Hom(a,b) <> {} for f being Morphism of a,b holds F1.f = F2.f holds F1 = F2; theorem :: NATTRA_1:13 :: CAT_1:99 for a,b,c being Object of A st Hom(a,b) <> {} & Hom(b,c) <> {} for f being Morphism of a,b, g being Morphism of b,c holds F.(g*f) = F.g*F.f; theorem :: NATTRA_1:14 :: CAT_1:107 for c being Object of A, d being Object of B st F.(id c) = id d holds F.c = d; theorem :: NATTRA_1:15 :: CAT_1:108 for a being Object of A holds F.id a = id (F.a); theorem :: NATTRA_1:16 :: CAT_1:115 for a,b being Object of A st Hom(a,b) <> {} for f being Morphism of a,b holds (id A).f = f; theorem :: NATTRA_1:17 for a,b,c,d being Object of A st Hom(a,b) meets Hom(c,d) holds a = c & b = d; begin :: Transformations definition let A,B,F1,F2; pred F1 is_transformable_to F2 means :: NATTRA_1:def 2 for a being Object of A holds Hom(F1.a,F2.a) <> {}; reflexivity; end; canceled; theorem :: NATTRA_1:19 F is_transformable_to F1 & F1 is_transformable_to F2 implies F is_transformable_to F2; definition let A,B,F1,F2; assume F1 is_transformable_to F2; mode transformation of F1,F2 -> Function of the Objects of A, the Morphisms of B means :: NATTRA_1:def 3 for a being Object of A holds it.a is Morphism of F1.a,F2.a; end; definition let A,B; let F be Functor of A,B; func id F ->transformation of F,F means :: NATTRA_1:def 4 for a being Object of A holds it.a = id (F.a); end; definition let A,B,F1,F2; assume 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 :: NATTRA_1:def 5 t.a; end; definition let A,B,F,F1,F2; assume that F is_transformable_to F1 and 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 :: NATTRA_1:def 6 for a being Object of A holds it.a = (t2.a)*(t1.a); end; theorem :: NATTRA_1:20 F1 is_transformable_to F2 implies for t1,t2 being transformation of F1,F2 st for a being Object of A holds t1.a = t2.a holds t1 = t2; theorem :: NATTRA_1:21 for a being Object of A holds id F.a = id(F.a); theorem :: NATTRA_1:22 F1 is_transformable_to F2 implies for t being transformation of F1,F2 holds (id F2)`*`t = t & t`*`(id F1) = t; theorem :: NATTRA_1:23 F is_transformable_to F1 & F1 is_transformable_to F2 & F2 is_transformable_to F3 implies for t1 being transformation of F,F1, t2 being transformation of F1,F2, t3 being transformation of F2,F3 holds t3`*`t2`*`t1 = t3`*`(t2`*`t1); begin definition let A,B,F1,F2; pred F1 is_naturally_transformable_to F2 means :: 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) <> {} for f being Morphism of a,b holds t.b*F1.f = F2.f*t.a; reflexivity; end; canceled; theorem :: NATTRA_1:25 F is_naturally_transformable_to F1 & F1 is_naturally_transformable_to F2 implies F is_naturally_transformable_to F2; definition let A,B,F1,F2; assume F1 is_naturally_transformable_to F2; mode natural_transformation of F1,F2 -> transformation of F1,F2 means :: NATTRA_1:def 8 for a,b being Object of A st Hom(a,b) <> {} for f being Morphism of a,b holds it.b*F1.f = F2.f*it.a; end; definition let A,B,F; redefine func id F -> natural_transformation of F,F; end; definition let A,B,F,F1,F2 such that F is_naturally_transformable_to F1 and 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 :: NATTRA_1:def 9 t2`*`t1; end; theorem :: NATTRA_1:26 F1 is_naturally_transformable_to F2 implies for t being natural_transformation of F1,F2 holds (id F2)`*`t = t & t`*`(id F1) = t; reserve t for natural_transformation of F,F1, t1 for natural_transformation of F1,F2; theorem :: NATTRA_1:27 F is_naturally_transformable_to F1 & F1 is_naturally_transformable_to F2 implies 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); theorem :: NATTRA_1:28 F is_naturally_transformable_to F1 & F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to F3 implies for t3 being natural_transformation of F2,F3 holds t3`*`t1`*`t = t3`*`(t1`*`t) ; :: Natural equivalences definition let A,B,F1,F2; let IT be transformation of F1,F2; attr IT is invertible means :: NATTRA_1:def 10 for a being Object of A holds IT.a is invertible; end; definition let A,B,F1,F2; pred F1,F2 are_naturally_equivalent means :: NATTRA_1:def 11 F1 is_naturally_transformable_to F2 & ex t being natural_transformation of F1,F2 st t is invertible; reflexivity; synonym F1 ~= F2; end; definition let A,B,F1,F2 such that F1 is_transformable_to F2; let t1 be transformation of F1,F2 such that t1 is invertible; func t1" -> transformation of F2,F1 means :: NATTRA_1:def 12 for a being Object of A holds it.a = (t1.a)"; end; definition let A,B,F1,F2,t1 such that F1 is_naturally_transformable_to F2 and t1 is invertible; func t1" -> natural_transformation of F2,F1 equals :: NATTRA_1:def 13 (t1 qua transformation of F1,F2)"; end; canceled; theorem :: NATTRA_1:30 for A,B,F1,F2,t1 st F1 is_naturally_transformable_to F2 & t1 is invertible for a being Object of A holds t1".a = (t1.a)"; theorem :: NATTRA_1:31 F1 ~= F2 implies F2 ~= F1; theorem :: NATTRA_1:32 F1 ~= F2 & F2 ~= F3 implies F1 ~= F3; definition let A,B,F1,F2; assume F1,F2 are_naturally_equivalent; mode natural_equivalence of F1,F2 -> natural_transformation of F1,F2 means :: NATTRA_1:def 14 it is invertible; end; theorem :: NATTRA_1:33 id F is natural_equivalence of F,F; theorem :: NATTRA_1:34 F1 ~= F2 & F2 ~= F3 implies for t being natural_equivalence of F1,F2, t' being natural_equivalence of F2,F3 holds t'`*`t is natural_equivalence of F1,F3; begin :: Functor category definition let A,B; mode NatTrans-DOMAIN of A,B -> non empty set means :: NATTRA_1:def 15 for x being set holds x in it implies ex F1,F2 being Functor of A,B, t being natural_transformation of F1,F2 st x = [[F1,F2],t] & F1 is_naturally_transformable_to F2; end; definition let A,B; func NatTrans(A,B) -> NatTrans-DOMAIN of A,B means :: NATTRA_1:def 16 for x being set holds x in it iff ex F1,F2 being Functor of A,B, t being natural_transformation of F1,F2 st x = [[F1,F2],t] & F1 is_naturally_transformable_to F2; end; definition let A1,B1,A2,B2 be non empty set, f1 be Function of A1,B1, f2 be Function of A2,B2; redefine pred f1 = f2 means :: NATTRA_1:def 17 A1 = A2 & for a being Element of A1 holds f1.a = f2.a; end; theorem :: NATTRA_1:35 F1 is_naturally_transformable_to F2 iff [[F1,F2],t1] in NatTrans(A,B); definition let A,B; func Functors(A,B) -> strict Category means :: NATTRA_1:def 18 the Objects of it = Funct(A,B) & the Morphisms 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) ex F,F1,F2,t,t1 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, F st F = a holds id a = [[F,F],id F]; end; :: As immediate consequences of the definition we get canceled 3; theorem :: NATTRA_1:39 for f being Morphism of Functors(A,B) st f = [[F,F1],t] holds dom f = F & cod f = F1; theorem :: NATTRA_1:40 for a,b being Object of Functors(A,B), f being Morphism of a,b st Hom(a,b) <> {} ex F,F1,t st a = F & b = F1 & f = [[F,F1],t]; theorem :: NATTRA_1:41 for t' being natural_transformation of F2,F3 for f,g being Morphism of Functors(A,B) st f = [[F,F1],t] & g = [[F2,F3],t'] holds [g,f] in dom the Comp of Functors(A,B) iff F1 = F2; theorem :: NATTRA_1:42 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]; begin :: Discrete categories definition let C be Category; attr C is discrete means :: NATTRA_1:def 19 for f being Morphism of C ex a being Object of C st f = id a; end; definition cluster discrete Category; end; canceled; theorem :: NATTRA_1:44 for A being discrete Category, a being Object of A holds Hom(a,a) = { id a}; theorem :: NATTRA_1:45 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) = {}; theorem :: NATTRA_1:46 1Cat(o,m) is discrete; theorem :: NATTRA_1:47 for A being discrete Category, C being Subcategory of A holds C is discrete; theorem :: NATTRA_1:48 A is discrete & B is discrete implies [:A,B:] is discrete; theorem :: NATTRA_1:49 for A being discrete Category, B being Category, F1,F2 being Functor of B,A st F1 is_transformable_to F2 holds F1 = F2; theorem :: NATTRA_1:50 for A being discrete Category, B being Category, F being Functor of B,A, t being transformation of F,F holds t = id F; theorem :: NATTRA_1:51 A is discrete implies Functors(B,A) is discrete; definition let C be Category; cluster strict discrete Subcategory of C; end; definition let C; func IdCat(C) -> strict discrete Subcategory of C means :: NATTRA_1:def 20 the Objects of it = the Objects of C & the Morphisms of it = { id a where a is Object of C: not contradiction}; end; theorem :: NATTRA_1:52 for C being strict Category holds C is discrete implies IdCat(C) = C; theorem :: NATTRA_1:53 IdCat(IdCat(C)) = IdCat(C); theorem :: NATTRA_1:54 IdCat(1Cat(o,m)) = 1Cat(o,m); theorem :: NATTRA_1:55 IdCat([:A,B:]) = [:IdCat(A), IdCat(B):];

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