Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Subcategories and Products of Categories

**by****Czeslaw Bylinski**- Received May 31, 1990
- MML identifier: CAT_2

- [ Mizar article, MML identifier index ]

environ vocabulary FRAENKEL, FUNCT_1, FUNCT_5, FUNCT_2, RELAT_1, BOOLE, CAT_1, FUNCOP_1, FUNCT_3, TARSKI, PARTFUN1, FUNCT_4, MCART_1, CAT_2; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, DOMAIN_1, RELAT_1, FUNCT_1, FUNCT_2, PARTFUN1, FUNCT_3, FUNCT_4, FUNCT_5, FRAENKEL, CAT_1; constructors DOMAIN_1, FUNCT_3, FUNCT_4, FUNCT_5, FRAENKEL, CAT_1, MEMBERED, PARTFUN1, XBOOLE_0; clusters FUNCT_1, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin reserve X for set; reserve C,D,E for non empty set; reserve c for Element of C, d for Element of D; :: Auxiliary theorems definition let D,X,E; let F be FUNCTION_DOMAIN of X,E; let f be Function of D,F; let d be Element of D; redefine func f.d -> Element of F; end; reserve f for Function of [:C,D:],E; theorem :: CAT_2:1 curry f is Function of C,Funcs(D,E); theorem :: CAT_2:2 curry' f is Function of D,Funcs(C,E); definition let C,D,E,f; redefine func curry f -> Function of C,Funcs(D,E); func curry' f -> Function of D,Funcs(C,E); end; theorem :: CAT_2:3 f.[c,d] = ((curry f).c).d; theorem :: CAT_2:4 f.[c,d] = ((curry' f).d).c; reserve B,C,D,C',D' for Category; :: Auxiliary theorems on Functors definition let B,C; let c be Object of C; func B --> c -> Functor of B,C equals :: CAT_2:def 1 (the Morphisms of B) --> (id c); end; canceled; theorem :: CAT_2:6 for c being Object of C, f being Morphism of B holds (B --> c).f = id c; theorem :: CAT_2:7 for c being Object of C, b being Object of B holds (Obj (B --> c)).b = c; definition let C,D; func Funct(C,D) -> set means :: CAT_2:def 2 for x being set holds x in it iff x is Functor of C,D; end; definition let C,D; cluster Funct(C,D) -> non empty; end; definition let C,D; mode FUNCTOR-DOMAIN of C,D -> non empty set means :: CAT_2:def 3 for x being Element of it holds x is Functor of C,D; end; definition let C,D; let F be FUNCTOR-DOMAIN of C,D; redefine mode Element of F -> Functor of C,D; end; definition let A be non empty set; let C,D; let F be FUNCTOR-DOMAIN of C,D, T be Function of A,F, x be Element of A; redefine func T.x -> Element of F; end; definition let C,D; redefine func Funct(C,D) -> FUNCTOR-DOMAIN of C,D; end; :: Subcategory definition let C; mode Subcategory of C -> Category means :: CAT_2:def 4 the Objects of it c= the Objects of C & (for a,b being Object of it, a',b' being Object of C st a = a' & b = b' holds Hom(a,b) c= Hom(a',b')) & the Comp of it <= the Comp of C & (for a being Object of it, a' being Object of C st a = a' holds id a = id a'); end; definition let C; cluster strict Subcategory of C; end; reserve E for Subcategory of C; canceled 4; theorem :: CAT_2:12 for e being Object of E holds e is Object of C; theorem :: CAT_2:13 the Morphisms of E c= the Morphisms of C; theorem :: CAT_2:14 for f being Morphism of E holds f is Morphism of C; theorem :: CAT_2:15 for f being (Morphism of E), f' being Morphism of C st f = f' holds dom f = dom f' & cod f = cod f'; theorem :: CAT_2:16 for a,b being Object of E, a',b' being Object of C,f being Morphism of a,b st a = a' & b = b' & Hom(a,b)<>{} holds f is Morphism of a',b'; theorem :: CAT_2:17 for f,g being (Morphism of E), f',g' being Morphism of C st f = f' & g = g' & dom g = cod f holds g*f = g'*f'; theorem :: CAT_2:18 C is Subcategory of C; theorem :: CAT_2:19 id E is Functor of E,C; definition let C,E; func incl(E) -> Functor of E,C equals :: CAT_2:def 5 id E; end; canceled; theorem :: CAT_2:21 for f being Morphism of E holds (incl E).f = f; theorem :: CAT_2:22 for a being Object of E holds (Obj incl E).a = a; theorem :: CAT_2:23 for a being Object of E holds (incl E).a = a; theorem :: CAT_2:24 incl E is faithful; theorem :: CAT_2:25 incl E is full iff for a,b being Object of E, a',b' being Object of C st a = a' & b = b' holds Hom(a,b) = Hom(a',b'); definition let C be CatStr, D; pred C is_full_subcategory_of D means :: CAT_2:def 6 C is Subcategory of D & for a,b being Object of C, a',b' being Object of D st a = a' & b = b' holds Hom(a,b) = Hom(a',b'); end; canceled; theorem :: CAT_2:27 E is_full_subcategory_of C iff incl(E) is full; theorem :: CAT_2:28 for O being non empty Subset of the Objects of C holds union{Hom(a,b) where a is Object of C,b is Object of C: a in O & b in O} is non empty Subset of the Morphisms of C; theorem :: CAT_2:29 for O being non empty Subset of the Objects of C, M being non empty set st M = union{Hom(a,b) where a is Object of C,b is Object of C: a in O & b in O} holds (the Dom of C)|M is Function of M,O & (the Cod of C)|M is Function of M,O & (the Comp of C)|[:M,M:] is PartFunc of [:M,M:],M & (the Id of C)|O is Function of O,M; theorem :: CAT_2:30 for O being non empty Subset of the Objects of C, M being non empty set, d,c being Function of M,O, p being PartFunc of [:M,M:],M, i being Function of O,M st M = union{Hom(a,b) where a is Object of C,b is Object of C: a in O & b in O} & d = (the Dom of C)|M & c = (the Cod of C)|M & p = (the Comp of C)|[:M,M:] & i = (the Id of C)|O holds CatStr(#O,M,d,c,p,i#) is_full_subcategory_of C; theorem :: CAT_2:31 for O being non empty Subset of the Objects of C, M being non empty set, d,c being Function of M,O, p being PartFunc of [:M,M:],M, i being Function of O,M st CatStr(#O,M,d,c,p,i#) is_full_subcategory_of C holds M = union{Hom(a,b) where a is Object of C,b is Object of C: a in O & b in O} & d = (the Dom of C)|M & c = (the Cod of C)|M & p = (the Comp of C)|[:M,M:] & i = (the Id of C)|O; :: Product of Categories definition let X1,X2,Y1,Y2 be non empty set; let f1 be Function of X1,Y1; let f2 be Function of X2,Y2; redefine func [:f1,f2:] -> Function of [:X1,X2:],[:Y1,Y2:]; end; definition let A,B be non empty set; let f be PartFunc of [:A,A:],A; let g be PartFunc of [:B,B:],B; redefine func |:f,g:| -> PartFunc of [:[:A,B:],[:A,B:]:],[:A,B:]; end; definition let C,D; func [:C,D:] -> Category equals :: CAT_2:def 7 CatStr (# [:the Objects of C,the Objects of D:], [:the Morphisms of C,the Morphisms of D:], [:the Dom of C,the Dom of D:], [:the Cod of C,the Cod of D:], |:the Comp of C, the Comp of D:|, [:the Id of C,the Id of D:] #); end; definition let C,D; cluster [:C,D:] -> strict; end; canceled; theorem :: CAT_2:33 the Objects of [:C,D:] = [:the Objects of C,the Objects of D:] & the Morphisms of [:C,D:] = [:the Morphisms of C,the Morphisms of D:] & the Dom of [:C,D:] = [:the Dom of C,the Dom of D:] & the Cod of [:C,D:] = [:the Cod of C,the Cod of D:] & the Comp of [:C,D:] = |:the Comp of C, the Comp of D:| & the Id of [:C,D:] = [:the Id of C,the Id of D:]; theorem :: CAT_2:34 for c being Object of C, d being Object of D holds [c,d] is Object of [:C,D:]; definition let C,D; let c be Object of C; let d be Object of D; redefine func [c,d] -> Object of [:C,D:]; end; theorem :: CAT_2:35 for cd being Object of [:C,D:] ex c being Object of C, d being Object of D st cd = [c,d]; theorem :: CAT_2:36 for f being Morphism of C for g being Morphism of D holds [f,g] is Morphism of [:C,D:]; definition let C,D; let f be Morphism of C; let g be Morphism of D; redefine func [f,g] -> Morphism of [:C,D:]; end; theorem :: CAT_2:37 for fg being Morphism of [:C,D:] ex f being (Morphism of C), g being Morphism of D st fg = [f,g]; theorem :: CAT_2:38 for f being Morphism of C for g being Morphism of D holds dom [f,g] = [dom f,dom g] & cod [f,g] = [cod f,cod g]; theorem :: CAT_2:39 for f,f' being Morphism of C for g,g' being Morphism of D st dom f' = cod f & dom g' = cod g holds [f',g']*[f,g] = [f'*f,g'*g]; theorem :: CAT_2:40 for f,f' being Morphism of C for g,g' being Morphism of D st dom [f',g'] = cod [f,g] holds [f',g']*[f,g] = [f'*f,g'*g]; theorem :: CAT_2:41 for c being Object of C, d being Object of D holds id [c,d] = [id c,id d]; theorem :: CAT_2:42 for c,c' being Object of C, d,d' being Object of D holds Hom([c,d],[c',d']) = [:Hom(c,c'),Hom(d,d'):]; theorem :: CAT_2:43 for c,c' being Object of C, f being Morphism of c,c', d,d' being Object of D, g being Morphism of d,d' st Hom(c,c') <> {} & Hom(d,d') <> {} holds [f,g] is Morphism of [c,d],[c',d']; :: Bifunctors theorem :: CAT_2:44 for S being Functor of [:C,C':],D, c being Object of C holds (curry S).(id c) is Functor of C',D; theorem :: CAT_2:45 for S being Functor of [:C,C':],D, c' being Object of C' holds (curry' S).(id c') is Functor of C,D; :: Partial Functors definition let C,C',D; let S be Functor of [:C,C':],D, c be Object of C; func S?-c -> Functor of C',D equals :: CAT_2:def 8 (curry S).(id c); end; canceled; theorem :: CAT_2:47 for S being Functor of [:C,C':],D, c being Object of C, f being Morphism of C' holds (S?-c).f = S.[id c,f]; theorem :: CAT_2:48 for S being Functor of [:C,C':],D, c being Object of C, c' being Object of C' holds (Obj S?-c).c' = (Obj S).[c,c']; definition let C,C',D; let S be Functor of [:C,C':],D, c' be Object of C'; func S-?c' -> Functor of C,D equals :: CAT_2:def 9 (curry' S).(id c'); end; canceled; theorem :: CAT_2:50 for S being Functor of [:C,C':],D, c' being Object of C', f being Morphism of C holds (S-?c').f = S.[f,id c']; theorem :: CAT_2:51 for S being Functor of [:C,C':],D, c being Object of C, c' being Object of C' holds (Obj S-?c').c = (Obj S).[c,c']; theorem :: CAT_2:52 for L being Function of the Objects of C,Funct(B,D), M being Function of the Objects of B,Funct(C,D) st ( for c being Object of C,b being Object of B holds (M.b).(id c) = (L.c).(id b) ) & ( for f being Morphism of B for g being Morphism of C holds ((M.(cod f)).g)*((L.(dom g)).f) = ((L.(cod g)).f)*((M.(dom f)).g) ) ex S being Functor of [:B,C:],D st for f being Morphism of B for g being Morphism of C holds S.[f,g] = ((L.(cod g)).f)*((M.(dom f)).g); theorem :: CAT_2:53 for L being Function of the Objects of C,Funct(B,D), M being Function of the Objects of B,Funct(C,D) st ex S being Functor of [:B,C:],D st for c being Object of C,b being Object of B holds S-?c = L.c & S?-b = M.b for f being Morphism of B for g being Morphism of C holds ((M.(cod f)).g)*((L.(dom g)).f) = ((L.(cod g)).f)*((M.(dom f)).g); theorem :: CAT_2:54 pr1(the Morphisms of C,the Morphisms of D) is Functor of [:C,D:],C; theorem :: CAT_2:55 pr2(the Morphisms of C,the Morphisms of D) is Functor of [:C,D:],D; definition let C,D; func pr1(C,D) -> Functor of [:C,D:],C equals :: CAT_2:def 10 pr1(the Morphisms of C,the Morphisms of D); func pr2(C,D) -> Functor of [:C,D:],D equals :: CAT_2:def 11 pr2(the Morphisms of C,the Morphisms of D); end; canceled 2; theorem :: CAT_2:58 for f being (Morphism of C),g being Morphism of D holds pr1(C,D).[f,g] = f; theorem :: CAT_2:59 for c being Object of C, d being Object of D holds (Obj pr1(C,D)).[c,d] = c; theorem :: CAT_2:60 for f being (Morphism of C),g being Morphism of D holds pr2(C,D).[f,g] = g; theorem :: CAT_2:61 for c being Object of C, d being Object of D holds (Obj pr2(C,D)).[c,d] = d; theorem :: CAT_2:62 for T being Functor of C,D, T' being Functor of C,D' holds <:T,T':> is Functor of C,[:D,D':]; definition let C,D,D'; let T be Functor of C,D, T' be Functor of C,D'; redefine func <:T,T':> -> Functor of C,[:D,D':]; end; theorem :: CAT_2:63 for T being Functor of C,D, T' being Functor of C,D', c being Object of C holds (Obj <:T,T':>).c = [(Obj T).c,(Obj T').c]; theorem :: CAT_2:64 for T being Functor of C,D, T' being Functor of C',D' holds [:T,T':] = <:T*pr1(C,C'),T'*pr2(C,C'):>; theorem :: CAT_2:65 for T being Functor of C,D, T' being Functor of C',D' holds [:T,T':] is Functor of [:C,C':],[:D,D':]; definition let C,C',D,D'; let T be Functor of C,D, T' be Functor of C',D'; redefine func [:T,T':] -> Functor of [:C,C':],[:D,D':]; end; theorem :: CAT_2:66 for T being Functor of C,D, T' being Functor of C',D', c being Object of C, c' being Object of C' holds (Obj [:T,T':]).[c,c'] = [(Obj T).c,(Obj T').c'];

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