:: Isomorphisms of Categories
:: by Andrzej Trybulec
::
:: Received November 22, 1991
:: Copyright (c) 1991-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies CAT_1, FUNCT_1, ZFMISC_1, RELAT_1, MCART_1, STRUCT_0, ALGSTR_0,
GRAPH_1, PZFMISC1, NATTRA_1, XBOOLE_0, WELLORD1, TARSKI, PARTFUN1,
VALUED_1, REWRITE1, ISOCAT_1, MONOID_0;
notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1,
FUNCT_2, BINOP_1, FUNCT_3, STRUCT_0, GRAPH_1, CAT_1, CAT_2, CAT_3,
NATTRA_1;
constructors NATTRA_1, FUNCOP_1, RELSET_1, CAT_3, FUNCT_4;
registrations RELSET_1, FUNCT_2, STRUCT_0, CAT_1, FUNCT_3;
requirements SUBSET, BOOLE;
definitions TARSKI, CAT_1, FUNCT_1, NATTRA_1, XBOOLE_0, FUNCT_2;
equalities CAT_1, XBOOLE_0, CAT_2, BINOP_1;
expansions CAT_1, FUNCT_1, NATTRA_1;
theorems FUNCT_2, CAT_1, ZFMISC_1, FUNCT_1, FUNCT_3, NATTRA_1, RELAT_1,
TARSKI, CAT_3, XTUPLE_0;
schemes FUNCT_2;
begin
:: Auxiliary theorems
reserve A,B,C,D for Category,
F for Functor of A,B,
G for Functor of B,C;
::$CT 2
theorem Th1:
for a,b being Object of A
for f being Morphism of a,b st f is invertible holds F/.f is invertible
proof let a,b be Object of A;
let f be Morphism of a,b such that
A1: Hom(a,b) <> {} & Hom(b,a) <> {};
given g being Morphism of b,a such that
A2: f*g = id b and
A3: g*f = id a;
A4: dom g = b by A1,CAT_1:5 .= cod f by A1,CAT_1:5;
A5: cod g = a by A1,CAT_1:5 .= dom f by A1,CAT_1:5;
A6: cod f = b by A1,CAT_1:5;
A7: dom f = a by A1,CAT_1:5;
A8: f(*)g = id cod f by A2,A1,A6,CAT_1:def 13;
A9: g(*)f = id dom f by A3,A1,A7,CAT_1:def 13;
thus
A10: Hom(F.a,F.b) <> {} & Hom(F.b,F.a) <> {} by A1,CAT_1:84;
take F/.g;
A11: F/.f = F.f by A1,CAT_3:def 10;
A12: F/.g = F.g by A1,CAT_3:def 10;
thus (F/.f)*(F/.g) = (F.f)(*)(F.g) by A10,A11,A12,CAT_1:def 13
.= F.(f(*)g) by A5,CAT_1:64
.= id cod(F/.f) by A8,A11,CAT_1:63
.= id(F.b) by A10,CAT_1:5;
thus (F/.g)*(F/.f) = (F.g)(*)(F.f) by A10,A11,A12,CAT_1:def 13
.= F.(g(*)f) by A4,CAT_1:64
.= id dom(F/.f) by A9,A11,CAT_1:63
.= id(F.a) by A10,CAT_1:5;
end;
theorem Th2:
for F1,F2 being Functor of A,B st F1 is_transformable_to F2 for t
being transformation of F1,F2, a being Object of A holds t.a in Hom(F1.a,F2.a)
proof
let F1,F2 be Functor of A,B such that
A1: F1 is_transformable_to F2;
let t be transformation of F1,F2, a be Object of A;
Hom(F1.a,F2.a)<>{} by A1;
hence thesis by CAT_1:def 5;
end;
theorem Th3:
for F1,F2 being Functor of A,B, G1,G2 being Functor of B,C st F1
is_transformable_to F2 & G1 is_transformable_to G2 holds G1*F1
is_transformable_to G2*F2
proof
let F1,F2 be Functor of A,B, G1,G2 be Functor of B,C such that
A1: F1 is_transformable_to F2 and
A2: G1 is_transformable_to G2;
let a be Object of A;
Hom(F1.a,F2.a) <> {} by A1;
then
A3: Hom(G1.(F1.a),G1.(F2.a)) <> {} by CAT_1:84;
A4: G1.(F1.a) = (G1*F1).a & G2.(F2.a) = (G2*F2).a by CAT_1:76;
Hom(G1.(F2.a),G2.(F2.a)) <> {} by A2;
hence thesis by A4,A3,CAT_1:24;
end;
theorem Th4:
for F1,F2 being Functor of A,B st F1 is_transformable_to F2 for t
being transformation of F1,F2 st t is invertible for a being Object of A holds
F1.a, F2.a are_isomorphic
proof
let F1,F2 be Functor of A,B such that
F1 is_transformable_to F2;
let t be transformation of F1,F2 such that
A1: t is invertible;
let a be Object of A;
take t.a;
thus thesis by A1;
end;
definition
let C,D;
redefine mode Functor of C,D means
(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 dom(it.f) & it.id cod f = id cod(it.f)) & for f,g being Morphism of C st dom
g = cod f holds it.(g(*)f) = (it.g)(*)(it.f);
compatibility by CAT_1:61,62,63,64;
end;
reserve o,m for set;
theorem Th5:
F is isomorphic implies for g being Morphism of B ex f being
Morphism of A st F.f = g
proof
assume
A1: F is isomorphic;
let g be Morphism of B;
rng F = the carrier' of B by A1;
then consider f being object such that
A2: f in dom F and
A3: F.f = g by FUNCT_1:def 3;
reconsider f as Morphism of A by A2;
take f;
thus thesis by A3;
end;
theorem Th6:
F is isomorphic implies for b being Object of B ex a being
Object of A st F.a = b
proof
assume
A1: F is isomorphic;
let b be Object of B;
rng Obj F = the carrier of B by A1;
then consider a being object such that
A2: a in dom Obj F and
A3: (Obj F).a = b by FUNCT_1:def 3;
reconsider a as Object of A by A2;
take a;
thus thesis by A3;
end;
theorem Th7:
F is one-to-one implies Obj F is one-to-one
proof
assume
A1: F is one-to-one;
let x1,x2 be object;
assume x1 in dom Obj F & x2 in dom Obj F;
then reconsider a1 = x1, a2 = x2 as Object of A;
assume (Obj F).x1 = (Obj F).x2;
then F.a1 = F.a2;
then
A2: F.(id a1 qua Morphism of A) = id(F.a2) by CAT_1:71
.= F.(id a2 qua Morphism of A)by CAT_1:71;
dom F = the carrier' of A by FUNCT_2:def 1;
then id a1 = id a2 by A1,A2;
hence thesis by CAT_1:59;
end;
definition
let A,B,F;
assume
A1: F is isomorphic;
func F" -> Functor of B,A equals
:Def2:
F";
coherence
proof
A2: F is one-to-one by A1;
rng F = the carrier' of B by A1;
then reconsider
G = F" as Function of the carrier' of B, the carrier' of A by A2,FUNCT_2:25
;
A3: dom F = the carrier' of A by FUNCT_2:def 1;
A4: Obj F is one-to-one by A2,Th7;
G is Functor of B,A
proof
thus for c being Object of B ex d being Object of A st G.(id c) = id d
proof
let b be Object of B;
consider a being Object of A such that
A5: F.a = b by A1,Th6;
take a;
thus G.(id b) = G.(F.(id a qua Morphism of A)) by A5,CAT_1:71
.=id a by A2,A3,FUNCT_1:34;
end;
thus for f being Morphism of B holds G.(id dom f) = id dom(G.f) & G.(id
cod f) = id cod(G.f)
proof
let f be Morphism of B;
consider g being Morphism of A such that
A6: f = F.g by A1,Th5;
thus G.(id dom f) = G.(id(F.dom g)) by A6,CAT_1:72
.= G.(F.(id dom g qua Morphism of A)) by CAT_1:71
.= id dom g by A2,A3,FUNCT_1:34
.= id dom(G.f) by A2,A3,A6,FUNCT_1:34;
thus G.(id cod f) = G.(id(F.cod g)) by A6,CAT_1:72
.= G.(F.(id cod g qua Morphism of A)) by CAT_1:71
.= id cod g by A2,A3,FUNCT_1:34
.= id cod(G.f) by A2,A3,A6,FUNCT_1:34;
end;
let f,g be Morphism of B;
A7: dom Obj F = the carrier of A by FUNCT_2:def 1;
assume
A8: dom g = cod f;
consider f9 being Morphism of A such that
A9: f = F.f9 by A1,Th5;
consider g9 being Morphism of A such that
A10: g = F.g9 by A1,Th5;
(Obj F).dom g9 = F.dom g9 .= cod f by A10,A8,CAT_1:72
.= F.cod f9 by A9,CAT_1:72
.= (Obj F).cod f9;
then dom g9 = cod f9 by A4,A7;
hence G.(g(*)f) = G.(F.(g9(*)f9)) by A9,A10,CAT_1:64
.= g9(*)f9 by A2,A3,FUNCT_1:34
.= g9(*)(G.f) by A2,A3,A9,FUNCT_1:34
.= (G.g)(*)(G.f) by A2,A3,A10,FUNCT_1:34;
end;
hence thesis;
end;
end;
definition
let A,B,F;
redefine attr F is isomorphic means
F is one-to-one & rng F = the carrier' of B;
compatibility
proof
thus F is isomorphic implies F is one-to-one & rng F = the carrier' of B;
assume that
A1: F is one-to-one and
A2: rng F = the carrier' of B;
thus F is one-to-one & rng F = the carrier' of B by A1,A2;
thus rng Obj F c= the carrier of B;
let x be object;
assume x in the carrier of B;
then reconsider d = x as Object of B;
consider f being object such that
A3: f in dom F and
A4: id d = F.f by A2,FUNCT_1:def 3;
reconsider f as Morphism of A by A3;
reconsider c = id dom f as Morphism of A;
F.c = id dom(id d) by A4,CAT_1:63
.= id d;
then dom Obj F = the carrier of A & (Obj F).(dom f) = d by CAT_1:67
,FUNCT_2:def 1;
hence x in rng Obj F by FUNCT_1:def 3;
end;
end;
theorem Th8:
F is isomorphic implies F" is isomorphic
proof
assume F is isomorphic;
then
A1: F is one-to-one & F" = (F qua Function of the carrier' of A, the
carrier' of B)" by Def2;
hence F" is one-to-one by FUNCT_1:40;
thus rng(F") = dom F by A1,FUNCT_1:33
.= the carrier' of A by FUNCT_2:def 1;
end;
theorem
F is isomorphic implies (Obj F)" = Obj F"
proof
assume
A1: F is isomorphic;
then
A2: F is one-to-one;
A3: rng Obj F = the carrier of B by A1;
then reconsider
G = (Obj F)" as Function of the carrier of B, the carrier of A by A2,Th7,
FUNCT_2:25;
A4: Obj F is one-to-one by A2,Th7;
now
let b be Object of B;
F.(id(G.b) qua Morphism of A) = id((Obj F).(G.b)) by CAT_1:68
.= id b by A3,A4,FUNCT_1:35;
then
id(G.b) = (F qua Function of the carrier' of A, the carrier' of B)".id
b by A2,FUNCT_2:26
.= F".(id b qua Morphism of B) by A1,Def2;
hence G.b = (Obj F").b by CAT_1:67;
end;
hence thesis by FUNCT_2:63;
end;
theorem
F is isomorphic implies F"" = F
proof
reconsider f = F as Function of the carrier' of A, the carrier' of B;
reconsider g = F" as Function of the carrier' of B, the carrier' of A;
assume
A1: F is isomorphic;
then
A2: F is one-to-one;
thus F"" = g" by A1,Def2,Th8
.= f"" by A1,Def2
.= F by A2,FUNCT_1:43;
end;
theorem Th11:
F is isomorphic implies F*F" = id B & F"*F = id A
proof
reconsider f = F as Function of the carrier' of A, the carrier' of B;
A1: dom f = the carrier' of A by FUNCT_2:def 1;
assume
A2: F is isomorphic;
then
A3: F is one-to-one;
A4: rng f = the carrier' of B by A2;
thus F*F" = f*f" by A2,Def2
.= id B by A3,A4,FUNCT_1:39;
thus F"*F = f"*f by A2,Def2
.= id A by A3,A1,FUNCT_1:39;
end;
theorem Th12:
F is isomorphic & G is isomorphic implies G*F is isomorphic
proof
assume that
A1: F is one-to-one and
A2: rng F = the carrier' of B and
A3: G is one-to-one and
A4: rng G = the carrier' of C;
thus G*F is one-to-one by A1,A3,FUNCT_1:24;
dom G = the carrier' of B by FUNCT_2:def 1;
hence thesis by A2,A4,RELAT_1:28;
end;
:: Isomorphism of categories
definition
let A,B;
pred A,B are_isomorphic means
ex F being Functor of A,B st F is isomorphic;
reflexivity
proof
let A;
id A is isomorphic by CAT_1:80;
hence thesis;
end;
symmetry
proof
let A,B;
given F being Functor of A,B such that
A1: F is isomorphic;
take F";
thus thesis by A1,Th8;
end;
end;
notation
let A,B;
synonym A ~= B for A,B are_isomorphic;
end;
theorem
A ~= B & B ~= C implies A ~= C
proof
given F1 being Functor of A,B such that
A1: F1 is isomorphic;
given F2 being Functor of B,C such that
A2: F2 is isomorphic;
take F2*F1;
thus thesis by A1,A2,Th12;
end;
theorem
[:1Cat(o,m),A:] ~= A
proof
take F = pr2(1Cat(o,m),A);
set X = [:the carrier' of 1Cat(o,m), the carrier' of A:];
now
let x1,x2 be object;
assume x1 in X;
then consider x11,x12 being object such that
A1: x11 in the carrier' of 1Cat(o,m) and
A2: x12 in the carrier' of A and
A3: x1 = [x11,x12] by ZFMISC_1:def 2;
assume x2 in X;
then consider x21,x22 being object such that
A4: x21 in the carrier' of 1Cat(o,m) and
A5: x22 in the carrier' of A and
A6: x2 = [x21,x22] by ZFMISC_1:def 2;
reconsider f11 = x11, f21 = x21 as Morphism of 1Cat(o,m) by A1,A4;
assume
A7: F.x1 = F.x2;
reconsider f12 = x12, f22 = x22 as Morphism of A by A2,A5;
A8: f11 = m by TARSKI:def 1
.= f21 by TARSKI:def 1;
f12 = F.(f11,f12) by FUNCT_3:def 5
.= F.(f21,f22) by A3,A6,A7
.= f22 by FUNCT_3:def 5;
hence x1 = x2 by A3,A6,A8;
end;
hence F is one-to-one by FUNCT_2:19;
thus thesis by FUNCT_3:46;
end;
theorem
[:A,B:] ~= [:B,A:]
proof
take F = <:pr2(A,B),pr1(A,B):>;
A1: dom pr1(A,B) = the carrier' of [:A,B:] & dom pr2(A,B) = the carrier' of
[:A, B:] by FUNCT_2:def 1;
now
let x1,x2 be object;
assume x1 in [:the carrier' of A,the carrier' of B:];
then consider x11,x12 being object such that
A2: x11 in the carrier' of A and
A3: x12 in the carrier' of B and
A4: x1 = [x11,x12] by ZFMISC_1:def 2;
assume x2 in [:the carrier' of A,the carrier' of B:];
then consider x21,x22 being object such that
A5: x21 in the carrier' of A and
A6: x22 in the carrier' of B and
A7: x2 = [x21,x22] by ZFMISC_1:def 2;
reconsider f12 = x12, f22 = x22 as Morphism of B by A3,A6;
reconsider f11 = x11, f21 = x21 as Morphism of A by A2,A5;
A8: [f21,f22] in the carrier' of [:A,B:];
assume
A9: F.x1 = F.x2;
A10: [f11,f12] in the carrier' of [:A,B:];
A11: [f12,f11] = [f12,pr1(A,B).(f11,f12)] by FUNCT_3:def 4
.= [pr2(A,B).(f11,f12),pr1(A,B).(f11,f12)] by FUNCT_3:def 5
.= F.(f21,f22) by A1,A4,A7,A10,A9,FUNCT_3:49
.= [pr2(A,B).(f21,f22),pr1(A,B).(f21,f22)]by A1,A8,FUNCT_3:49
.= [f22,pr1(A,B).(f21,f22)]by FUNCT_3:def 5
.= [f22,f21] by FUNCT_3:def 4;
then x12 = x22 by XTUPLE_0:1;
hence x1 = x2 by A4,A7,A11,XTUPLE_0:1;
end;
hence F is one-to-one by FUNCT_2:19;
thus rng F c= the carrier' of [:B,A:];
let x be object;
assume x in the carrier' of [:B,A:];
then consider x1,x2 being object such that
A12: x1 in the carrier' of B and
A13: x2 in the carrier' of A and
A14: x = [x1,x2] by ZFMISC_1:def 2;
reconsider x2 as Morphism of A by A13;
reconsider x1 as Morphism of B by A12;
F.[x2,x1] = [pr2(A,B).(x2,x1),pr1(A,B).(x2,x1)] by A1,FUNCT_3:49
.= [x1,pr1(A,B).(x2,x1)] by FUNCT_3:def 5
.= [x1,x2] by FUNCT_3:def 4;
hence thesis by A14,FUNCT_2:4;
end;
theorem
[:[:A,B:],C:] ~= [:A,[:B,C:]:]
proof
set X = pr1(A,B)*pr1([:A,B:],C);
set Y = <:pr2(A,B)*pr1([:A,B:],C),pr2([:A,B:],C):>;
take F = <:X, Y:>;
A1: dom pr2([:A,B:],C) = the carrier' of [:[:A,B:],C:] & dom(pr2(A,B)*pr1([:
A,B :],C)) = the carrier' of [:[:A,B:],C:] by FUNCT_2:def 1;
A2: dom X = the carrier' of [:[:A,B:],C:] & dom Y = the carrier' of [:[:A,B
:],C :] by FUNCT_2:def 1;
now
let x,y be object;
assume x in [:[:the carrier' of A,the carrier' of B:],the carrier' of C:];
then consider x1,x2 being object such that
A3: x1 in [:the carrier' of A,the carrier' of B:] and
A4: x2 in the carrier' of C and
A5: x = [x1,x2] by ZFMISC_1:def 2;
assume y in [:[:the carrier' of A,the carrier' of B:],the carrier' of C :];
then consider y1,y2 being object such that
A6: y1 in [:the carrier' of A,the carrier' of B:] and
A7: y2 in the carrier' of C and
A8: y = [y1,y2] by ZFMISC_1:def 2;
reconsider f2 = x2, g2 = y2 as Morphism of C by A4,A7;
assume
A9: F.x = F.y;
consider x11,x12 being object such that
A10: x11 in the carrier' of A and
A11: x12 in the carrier' of B and
A12: x1 = [x11,x12] by A3,ZFMISC_1:def 2;
consider y11,y12 being object such that
A13: y11 in the carrier' of A and
A14: y12 in the carrier' of B and
A15: y1 = [y11,y12] by A6,ZFMISC_1:def 2;
reconsider f12 = x12, g12 = y12 as Morphism of B by A11,A14;
reconsider f11 = x11, g11 = y11 as Morphism of A by A10,A13;
A16: [f11,[f12,f2]] = [pr1(A,B).(f11,f12),[f12,f2]] by FUNCT_3:def 4
.= [pr1(A,B).(pr1([:A,B:],C).([f11,f12],f2)),[f12,f2]] by FUNCT_3:def 4
.= [pr1(A,B).(pr1([:A,B:],C).[[f11,f12],f2]),[f12,f2]]
.= [X.([f11,f12],f2),[f12,f2]] by FUNCT_2:15
.= [X.[[f11,f12],f2],[pr2(A,B).(f11,f12),f2]] by FUNCT_3:def 5
.= [X.[[f11,f12],f2],[pr2(A,B).(pr1([:A,B:],C).([f11,f12],f2)),f2]] by
FUNCT_3:def 4
.= [X.[[f11,f12],f2],[(pr2(A,B)*pr1([:A,B:],C)).[[f11,f12],f2],f2]] by
FUNCT_2:15
.= [X.[[f11,f12],f2],[(pr2(A,B)*pr1([:A,B:],C)).[[f11,f12],f2], pr2([:
A,B:],C).([f11,f12],f2)]] by FUNCT_3:def 5
.= [X.[[f11,f12],f2],Y.[[f11,f12],f2]] by A1,FUNCT_3:49
.= F.[[g11,g12],g2] by A2,A5,A12,A8,A15,A9,FUNCT_3:49
.= [X.[[g11,g12],g2],Y.[[g11,g12],g2]] by A2,FUNCT_3:49
.= [X.[[g11,g12],g2],[(pr2(A,B)*pr1([:A,B:],C)).[[g11,g12],g2], pr2([:
A,B:],C).([g11,g12],g2)]] by A1,FUNCT_3:49
.= [X.[[g11,g12],g2],[(pr2(A,B)*pr1([:A,B:],C)).[[g11,g12],g2],g2]] by
FUNCT_3:def 5
.= [X.[[g11,g12],g2],[pr2(A,B).(pr1([:A,B:],C).([g11,g12],g2)),g2]] by
FUNCT_2:15
.= [X.[[g11,g12],g2],[pr2(A,B).(g11,g12),g2]]by FUNCT_3:def 4
.= [X.[[g11,g12],g2],[g12,g2]]by FUNCT_3:def 5
.= [pr1(A,B).(pr1([:A,B:],C).([g11,g12],g2)),[g12,g2]]by FUNCT_2:15
.= [pr1(A,B).(g11,g12),[g12,g2]]by FUNCT_3:def 4
.= [g11,[g12,g2]] by FUNCT_3:def 4;
then [x12,x2] = [y12,y2] by XTUPLE_0:1;
then x12 = y12 & x2 = y2 by XTUPLE_0:1;
hence x = y by A5,A12,A8,A15,A16,XTUPLE_0:1;
end;
hence F is one-to-one by FUNCT_2:19;
thus rng F c= the carrier' of [:A,[:B,C:]:];
let x be object;
assume x in the carrier' of [:A,[:B,C:]:];
then consider x1,x2 being object such that
A17: x1 in the carrier' of A and
A18: x2 in the carrier' of [:B,C:] and
A19: x = [x1,x2] by ZFMISC_1:def 2;
reconsider x1 as Morphism of A by A17;
consider x21,x22 being object such that
A20: x21 in the carrier' of B and
A21: x22 in the carrier' of C and
A22: x2 = [x21,x22] by A18,ZFMISC_1:def 2;
reconsider x22 as Morphism of C by A21;
reconsider x21 as Morphism of B by A20;
F.[[x1,x21],x22] = [X.[[x1,x21],x22],Y.[[x1,x21],x22]] by A2,FUNCT_3:49
.= [pr1(A,B).(pr1([:A,B:],C).([x1,x21],x22)),Y.[[x1,x21],x22]] by
FUNCT_2:15
.= [pr1(A,B).(x1,x21),Y.[[x1,x21],x22]] by FUNCT_3:def 4
.= [x1,Y.[[x1,x21],x22]] by FUNCT_3:def 4
.= [x1,[(pr2(A,B)*pr1([:A,B:],C)).[[x1,x21],x22], pr2([:A,B:],C).[[x1,
x21],x22]]] by A1,FUNCT_3:49
.= [x1,[pr2(A,B).(pr1([:A,B:],C).([x1,x21],x22)), pr2([:A,B:],C).[[x1,
x21],x22]]] by FUNCT_2:15
.= [x1,[pr2(A,B).(x1,x21),pr2([:A,B:],C).([x1,x21],x22)]] by FUNCT_3:def 4
.= [x1,[x21,pr2([:A,B:],C).([x1,x21],x22)]] by FUNCT_3:def 5
.= [x1,[x21,x22]] by FUNCT_3:def 5;
hence thesis by A19,A22,FUNCT_2:4;
end;
theorem
A ~= B & C ~= D implies [:A,C:] ~= [:B,D:]
proof
given F being Functor of A,B such that
A1: F is isomorphic;
A2: F is one-to-one by A1;
given G being Functor of C,D such that
A3: G is isomorphic;
take [:F,G:];
G is one-to-one by A3;
hence [:F,G:] is one-to-one by A2;
thus rng [:F,G:] = [:rng F, rng G:] by FUNCT_3:67
.= [: the carrier' of B, rng G:] by A1
.= the carrier' of [:B,D:] by A3;
end;
definition
let A,B,C;
let F1,F2 be Functor of A,B such that
A1: F1 is_transformable_to F2;
let t be transformation of F1,F2;
let G be Functor of B,C;
func G*t -> transformation of G*F1,G*F2 equals
:Def5:
G*t;
coherence
proof
reconsider Gt = G*t as Function of the carrier of A, the carrier' of C;
Gt is transformation of G*F1, G*F2
proof
thus G*F1 is_transformable_to G*F2 by A1,Th3;
let a be Object of A;
A2: G.(F1.a) = (G*F1).a & G.(F2.a) = (G*F2).a by CAT_1:76;
t.a in Hom(F1.a,F2.a) by A1,Th2;
then
A3: G.(t.a qua Morphism of B) in Hom((G*F1).a,(G*F2).a) by A2,CAT_1:81;
Gt.a = G.((t qua Function of the carrier of A, the carrier' of B).a)
by FUNCT_2:15
.= G.(t.a qua Morphism of B) by A1,NATTRA_1:def 5;
hence thesis by A3,CAT_1:def 5;
end;
hence thesis;
end;
correctness;
end;
definition
let A,B,C;
let G1,G2 be Functor of B,C such that
A1: G1 is_transformable_to G2;
let F be Functor of A,B;
let t be transformation of G1,G2;
func t*F -> transformation of G1*F,G2*F equals
:Def6:
t*Obj F;
coherence
proof
reconsider tF = t*Obj F as Function of the carrier of A, the carrier' of C;
tF is transformation of G1*F, G2*F
proof
thus G1*F is_transformable_to G2*F by A1,Th3;
let a be Object of A;
A2: G1.(F.a) = (G1*F).a by CAT_1:76;
tF.a = t.((Obj F).a qua set) by FUNCT_2:15
.= t.(F.a) by A1,NATTRA_1:def 5;
hence thesis by A2,CAT_1:76;
end;
hence thesis;
end;
correctness;
end;
theorem Th18:
for G1,G2 be Functor of B,C st G1 is_transformable_to G2 for F
be Functor of A,B, t be transformation of G1,G2, a be Object of A holds (t*F).a
= t.(F.a)
proof
let G1,G2 be Functor of B,C such that
A1: G1 is_transformable_to G2;
let F be Functor of A,B, t be transformation of G1,G2, a be Object of A;
thus (t*F).a = (t*F).(a qua set) by A1,Th3,NATTRA_1:def 5
.= ((t qua Function of the carrier of B, the carrier' of C)*Obj F).a by A1
,Def6
.= t.((Obj F).a qua set) by FUNCT_2:15
.= t.(F.a) by A1,NATTRA_1:def 5;
end;
theorem Th19:
for F1,F2 be Functor of A,B st F1 is_transformable_to F2 for t
be transformation of F1,F2, G be Functor of B,C, a being Object of A
holds (G*t).a = G/.(t.a)
proof
let F1,F2 be Functor of A,B such that
A1: F1 is_transformable_to F2;
let t be transformation of F1,F2, G be Functor of B,C, a be Object of A;
A2: Hom(F1.a,F2.a) <> {} by A1;
thus (G*t).a = (G*t).(a qua set) by A1,Th3,NATTRA_1:def 5
.= ((G qua Function of the carrier' of B, the carrier' of C)* (t qua
Function of the carrier of A, the carrier' of B)).a by A1,Def5
.= G.(t.(a qua set)) by FUNCT_2:15
.= G.(t.a qua Morphism of B) by A1,NATTRA_1:def 5
.= G/.(t.a) by A2,CAT_3:def 10;
end;
theorem Th20:
for F1,F2 being Functor of A,B, G1,G2 being Functor of B,C st F1
is_naturally_transformable_to F2 & G1 is_naturally_transformable_to G2 holds G1
*F1 is_naturally_transformable_to G2*F2
proof
let F1,F2 be Functor of A,B, G1,G2 be Functor of B,C such that
A1: F1 is_naturally_transformable_to F2 and
A2: G1 is_naturally_transformable_to G2;
set t1 = the natural_transformation of F1,F2,t2 = the natural_transformation of
G1,G2;
A3: G1 is_transformable_to G2 by A2;
A4: F1 is_transformable_to F2 by A1;
hence
A5: G1*F1 is_transformable_to G2*F2 by A3,Th3;
take t = (t2*F2)`*`(G1*t1);
let a,b be Object of A;
A6: Hom(G1.(F2.b),G2.(F2.b)) <> {} by A3;
A7: Hom((G1*F1).a,(G2*F2).a) <> {} by A5;
A8: G1*F1 is_transformable_to G1*F2 by A4,Th3;
then
A9: Hom((G1*F1).b,(G1*F2).b) <> {};
A10: Hom((G1*F1).b,(G2*F2).b) <> {} by A5;
assume
A11: Hom(a,b) <> {};
then
A12: Hom((G2*F2).a,(G2*F2).b) <> {} by CAT_1:84;
A13: Hom(F1.b,F2.b) <> {} by A4;
then
A14: Hom(G1.(F1.b),G1.(F2.b)) <> {} by CAT_1:84;
then
A15: Hom(G1.(F1.b),G2.(F2.b)) <> {} by A6,CAT_1:24;
A16: G1*F2 is_transformable_to G2*F2 by A3,Th3;
then
A17: Hom((G1*F2).b,(G2*F2).b) <> {};
A18: Hom((G1*F1).a,(G1*F2).a) <> {} by A8;
A19: Hom((G1*F2).a,(G2*F2).a) <> {} by A16;
A20: Hom(F1.a,F2.a) <> {} by A4;
then
A21: Hom(G1.(F1.a),G1.(F2.a)) <> {} by CAT_1:84;
let f be Morphism of a,b;
A22: Hom(F2.a,F2.b) <> {} by A11,CAT_1:84;
then
A23: Hom(G1.(F2.a),G1.(F2.b)) <> {} by CAT_1:84;
A24: Hom(F1.a,F1.b) <> {} by A11,CAT_1:84;
then
A25: Hom(G1.(F1.a),G1.(F1.b)) <> {} by CAT_1:84;
A26: Hom(G1.(F2.a),G2.(F2.a)) <> {} by A3;
then
A27: Hom(G1.(F1.a),G2.(F2.a)) <> {} by A21,CAT_1:24;
A28: Hom(G2.(F2.a),G2.(F2.b)) <> {} by A22,CAT_1:84;
Hom((G1*F1).a,(G1*F1).b) <> {} by A11,CAT_1:84;
hence t.b*(G1*F1)/.f = (t.b)(*)((G1*F1)/.f qua Morphism of C)
by A10,CAT_1:def 13
.= (t.b)(*)(G1/.(F1/.f) qua Morphism of C) by A11,NATTRA_1:11
.= ((t2*F2).b*(G1*t1).b)(*)(G1/.(F1/.f)) by A8,A16,NATTRA_1:def 6
.= ((t2*F2).b qua Morphism of C)(*)((G1*t1).b)(*)(G1/.(F1/.f))
by A17,A9,CAT_1:def 13
.= ((t2*F2).b)(*)(G1/.(t1.b))(*)(G1/.(F1/.f)) by A4,Th19
.= (t2.(F2.b) qua Morphism of C)(*)(G1/.(t1.b))(*)(G1/.(F1/.f)) by A3,Th18
.= (t2.(F2.b)*G1/.(t1.b) qua Morphism of C)(*)(G1/.(F1/.f))
by A6,A14,CAT_1:def 13
.= t2.(F2.b)*G1/.(t1.b)*G1/.(F1/.f) by A25,A15,CAT_1:def 13
.= t2.(F2.b)*(G1/.(t1.b)*G1/.(F1/.f)) by A6,A14,A25,CAT_1:25
.= t2.(F2.b)*(G1/.(t1.b*F1/.f)) by A13,A24,NATTRA_1:13
.= t2.(F2.b)*(G1/.(F2/.f*t1.a)) by A1,A11,NATTRA_1:def 8
.= t2.(F2.b)*(G1/.(F2/.f)*G1/.(t1.a)) by A22,A20,NATTRA_1:13
.= t2.(F2.b)*G1/.(F2/.f)*G1/.(t1.a) by A6,A23,A21,CAT_1:25
.= G2/.(F2/.f)*t2.(F2.a)*G1/.(t1.a) by A2,A22,NATTRA_1:def 8
.= G2/.(F2/.f)*(t2.(F2.a)*G1/.(t1.a)) by A21,A28,A26,CAT_1:25
.= (G2/.(F2/.f))(*)(t2.(F2.a)*G1/.(t1.a) qua Morphism of C) by A28,A27,
CAT_1:def 13
.= (G2/.(F2/.f))(*)((t2.(F2.a) qua Morphism of C)(*)(G1/.(t1.a)))
by A21,A26,CAT_1:def 13
.= (G2/.(F2/.f))(*)(((t2*F2).a)(*)(G1/.(t1.a))) by A3,Th18
.= (G2/.(F2/.f))(*)(((t2*F2).a qua Morphism of C)(*)((G1*t1).a)) by A4,Th19
.= (G2/.(F2/.f))(*)((t2*F2).a*(G1*t1).a) by A19,A18,CAT_1:def 13
.= (G2/.(F2/.f))(*)(t.a) by A8,A16,NATTRA_1:def 6
.= ((G2*F2)/.f qua Morphism of C)(*)(t.a) by A11,NATTRA_1:11
.= (G2*F2)/.f*t.a by A7,A12,CAT_1:def 13;
end;
definition
let A,B,C;
let F1,F2 be Functor of A,B such that
A1: F1 is_naturally_transformable_to F2;
let t be natural_transformation of F1,F2;
let G be Functor of B,C;
func G*t -> natural_transformation of G*F1,G*F2 equals
:Def7:
G*t;
coherence
proof
A2: F1 is_transformable_to F2 by A1;
G*t is natural_transformation of G*F1, G*F2
proof
thus G*F1 is_naturally_transformable_to G*F2 by A1,Th20;
then
A3: G*F1 is_transformable_to G*F2;
let a,b be Object of A such that
A4: Hom(a,b) <> {};
A5: Hom((G*F1).a,(G*F1).b) <> {} by A4,CAT_1:84;
A6: Hom((G*F2).a,(G*F2).b) <> {} by A4,CAT_1:84;
A7: Hom((G*F1).a,(G*F2).a) <> {} by A3;
let f be Morphism of a,b;
A8: Hom(F1.b,F2.b) <> {} by A2;
then
A9: Hom(G.(F1.b),G.(F2.b)) <> {} by CAT_1:84;
A10: Hom(F1.a,F2.a) <> {} by A2;
then
A11: Hom(G.(F1.a),G.(F2.a)) <> {} by CAT_1:84;
A12: Hom(F2.a,F2.b) <> {} by A4,CAT_1:84;
then
A13: Hom(G.(F2.a),G.(F2.b)) <> {} by CAT_1:84;
A14: Hom(F1.a,F1.b) <> {} by A4,CAT_1:84;
then
A15: Hom(G.(F1.a),G.(F1.b)) <> {} by CAT_1:84;
Hom((G*F1).b,(G*F2).b) <> {} by A3;
hence (G*t).b*(G*F1)/.f = ((G*t).b)(*)((G*F1)/.f qua Morphism of C)
by A5,CAT_1:def 13
.= ((G*t).b)(*)(G/.(F1/.f)) by A4,NATTRA_1:11
.= (G/.(t.b) qua Morphism of C)(*)(G/.(F1/.f)) by A2,Th19
.= G/.(t.b)*G/.(F1/.f) by A9,A15,CAT_1:def 13
.= G/.(t.b*F1/.f) by A8,A14,NATTRA_1:13
.= G/.(F2/.f*t.a) by A1,A4,NATTRA_1:def 8
.= G/.(F2/.f)*G/.(t.a)by A10,A12,NATTRA_1:13
.= (G/.(F2/.f))(*)(G/.(t.a) qua Morphism of C)by A13,A11,CAT_1:def 13
.= (G/.(F2/.f))(*)((G*t).a) by A2,Th19
.= ((G*F2)/.f qua Morphism of C)(*)((G*t).a) by A4,NATTRA_1:11
.= (G*F2)/.f*(G*t).a by A7,A6,CAT_1:def 13;
end;
hence thesis;
end;
correctness;
end;
theorem Th21:
for F1,F2 be Functor of A,B st F1 is_naturally_transformable_to
F2 for t be natural_transformation of F1,F2, G be Functor of B,C, a being
Object of A holds (G*t).a = G/.(t.a)
proof
let F1,F2 be Functor of A,B;
assume
A1: F1 is_naturally_transformable_to F2;
then
A2: F1 is_transformable_to F2;
let t be natural_transformation of F1,F2, G be Functor of B,C, a be Object
of A;
thus (G*t).a = (G*(t qua transformation of F1,F2)).a by A1,Def7
.= G/.(t.a) by A2,Th19;
end;
definition
let A,B,C;
let G1,G2 be Functor of B,C such that
A1: G1 is_naturally_transformable_to G2;
let F be Functor of A,B;
let t be natural_transformation of G1,G2;
func t*F -> natural_transformation of G1*F,G2*F equals
:Def8:
t*F;
coherence
proof
A2: G1 is_transformable_to G2 by A1;
t*F is natural_transformation of G1*F, G2*F
proof
thus G1*F is_naturally_transformable_to G2*F by A1,Th20;
then
A3: G1*F is_transformable_to G2*F;
let a,b be Object of A;
A4: Hom((G1*F).b,(G2*F).b) <> {} by A3;
A5: Hom((G1*F).a,(G2*F).a) <> {} by A3;
assume
A6: Hom(a,b) <> {};
then
A7: Hom((G2*F).a,(G2*F).b) <> {} by CAT_1:84;
let f be Morphism of a,b;
A8: Hom(F.a,F.b) <> {} by A6,CAT_1:84;
then
A9: Hom(G1.(F.a),G1.(F.b)) <> {} by CAT_1:84;
A10: Hom(G1.(F.a),G2.(F.a)) <> {} by A2;
A11: Hom(G1.(F.b),G2.(F.b)) <> {} by A2;
A12: Hom(G2.(F.a),G2.(F.b)) <> {} by A8,CAT_1:84;
Hom((G1*F).a,(G1*F).b) <> {} by A6,CAT_1:84;
hence (t*F).b*(G1*F)/.f
= ((t*F).b)(*)((G1*F)/.f qua Morphism of C) by A4,CAT_1:def 13
.= ((t*F).b)(*)(G1/.(F/.f)) by A6,NATTRA_1:11
.= (t.(F.b) qua Morphism of C)(*)(G1/.(F/.f)) by A2,Th18
.= t.(F.b)*G1/.(F/.f) by A11,A9,CAT_1:def 13
.= G2/.(F/.f)*t.(F.a) by A1,A8,NATTRA_1:def 8
.= (G2/.(F/.f))(*)(t.(F.a) qua Morphism of C) by A12,A10,CAT_1:def 13
.= (G2/.(F/.f))(*)((t*F).a) by A2,Th18
.= ((G2*F)/.f qua Morphism of C)(*)((t*F).a) by A6,NATTRA_1:11
.= (G2*F)/.f*(t*F).a by A7,A5,CAT_1:def 13;
end;
hence thesis;
end;
correctness;
end;
theorem Th22:
for G1,G2 be Functor of B,C st G1 is_naturally_transformable_to
G2 for F be Functor of A,B, t be natural_transformation of G1,G2, a be Object
of A holds (t*F).a = t.(F.a)
proof
let G1,G2 be Functor of B,C;
assume
A1: G1 is_naturally_transformable_to G2;
then
A2: G1 is_transformable_to G2;
let F be Functor of A,B, t be natural_transformation of G1,G2, a be Object
of A;
thus (t*F).a = ((t qua transformation of G1,G2)*F).a by A1,Def8
.= t.(F.a) by A2,Th18;
end;
reserve F,F1,F2,F3 for Functor of A,B,
G,G1,G2,G3 for Functor of B,C,
H,H1,H2 for Functor of C,D,
s for natural_transformation of F1,F2,
s9 for natural_transformation of F2,F3,
t for natural_transformation of G1,G2,
t9 for natural_transformation of G2,G3,
u for natural_transformation of H1,H2;
theorem Th23:
F1 is_naturally_transformable_to F2 implies for a being Object
of A holds Hom(F1.a,F2.a) <> {}
by NATTRA_1:def 2;
theorem Th24:
F1 is_naturally_transformable_to F2 implies for t1,t2 being
natural_transformation of F1,F2 st for a being Object of A holds t1.a = t2.a
holds t1 = t2
by NATTRA_1:19;
theorem Th25:
F1 is_naturally_transformable_to F2 & F2
is_naturally_transformable_to F3 implies G*(s9`*`s) = (G*s9)`*`(G*s)
proof
assume
A1: F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to F3;
then
A2: G*F1 is_naturally_transformable_to G*F2 & G*F2
is_naturally_transformable_to G*F3 by Th20;
now
let a be Object of A;
A3: G.(F1.a) = (G*F1).a & G.(F2.a) = (G*F2).a by CAT_1:76;
A4: G.(F3.a) = (G*F3).a by CAT_1:76;
A5: Hom(F1.a,F2.a) <> {} & Hom(F2.a,F3.a) <> {} by A1,Th23;
A6: G/.(s9.a) = (G*s9).a & G/.(s.a) = (G*s).a by A1,Th21;
thus (G*(s9`*`s)).a = G/.((s9`*`s).a) by A1,Th21,NATTRA_1:23
.= G/.((s9.a)*(s.a)) by A1,NATTRA_1:25
.= G/.(s9.a)*G/.(s.a) by A5,NATTRA_1:13
.= ((G*s9)`*`(G*s)).a by A2,A6,A3,A4,NATTRA_1:25;
end;
hence thesis by A2,Th24,NATTRA_1:23;
end;
theorem Th26:
G1 is_naturally_transformable_to G2 & G2
is_naturally_transformable_to G3 implies (t9`*`t)*F = (t9*F)`*`(t*F)
proof
assume
A1: G1 is_naturally_transformable_to G2 & G2 is_naturally_transformable_to G3;
then
A2: G1*F is_naturally_transformable_to G2*F & G2*F
is_naturally_transformable_to G3*F by Th20;
now
let a be Object of A;
A3: G1.(F.a) = (G1*F).a & G2.(F.a) = (G2*F).a by CAT_1:76;
A4: G3.(F.a) = (G3*F).a by CAT_1:76;
A5: t9.(F.a) = (t9*F).a & t.(F.a) = (t*F).a by A1,Th22;
thus ((t9`*`t)*F).a = (t9`*`t).(F.a) by A1,Th22,NATTRA_1:23
.= (t9.(F.a))*(t.(F.a)) by A1,NATTRA_1:25
.= ((t9*F)`*`(t*F)).a by A2,A5,A3,A4,NATTRA_1:25;
end;
hence thesis by A2,Th24,NATTRA_1:23;
end;
theorem Th27:
H1 is_naturally_transformable_to H2 implies u*G*F = u*(G*F)
proof
assume
A1: H1 is_naturally_transformable_to H2;
A2: H1*(G*F) = H1*G*F by RELAT_1:36;
then reconsider v = u*(G*F) as natural_transformation of H1*G*F, H2*G*F by
RELAT_1:36;
A3: H2*(G*F) = H2*G*F by RELAT_1:36;
A4: now
let a be Object of A;
thus (u*G*F).a = (u*G).(F.a) by A1,Th20,Th22
.= u.(G.(F.a)) by A1,Th22
.= u.((G*F).a) by CAT_1:76
.= v.a by A1,A2,A3,Th22;
end;
H1*G is_naturally_transformable_to H2*G by A1,Th20;
hence thesis by A4,Th20,Th24;
end;
theorem Th28:
G1 is_naturally_transformable_to G2 implies H*t*F = H*(t*F)
proof
assume
A1: G1 is_naturally_transformable_to G2;
A2: H*(G1*F) = H*G1*F by RELAT_1:36;
then reconsider v = H*(t*F) as natural_transformation of H*G1*F, H*G2*F by
RELAT_1:36;
A3: H*(G2*F) = H*G2*F by RELAT_1:36;
A4: now
let a be Object of A;
A5: G1.(F.a) = (G1*F).a & G2.(F.a) = (G2*F).a by CAT_1:76;
thus (H*t*F).a = (H*t).(F.a) by A1,Th20,Th22
.= H/.(t.(F.a)) by A1,Th21
.= H/.((t*F).a) by A1,A5,Th22
.= v.a by A1,A2,A3,Th20,Th21;
end;
H*G1 is_naturally_transformable_to H*G2 by A1,Th20;
hence thesis by A4,Th20,Th24;
end;
theorem Th29:
F1 is_naturally_transformable_to F2 implies H*G*s = H*(G*s)
proof
assume
A1: F1 is_naturally_transformable_to F2;
A2: H*(G*F1) = H*G*F1 by RELAT_1:36;
then reconsider v = H*(G*s) as natural_transformation of H*G*F1, H*G*F2 by
RELAT_1:36;
A3: H*(G*F2) = H*G*F2 by RELAT_1:36;
now
let a be Object of A;
A4: G.(F1.a) = (G*F1).a & G.(F2.a) = (G*F2).a by CAT_1:76;
A5: Hom(F1.a,F2.a) <> {} by A1,Th23;
thus (H*G*s).a = (H*G)/.(s.a) by A1,Th21
.= H/.(G/.(s.a)) by A5,NATTRA_1:11
.= H/.((G*s).a) by A1,A4,Th21
.= v.a by A1,A2,A3,Th20,Th21;
end;
hence thesis by A1,Th20,Th24;
end;
theorem Th30:
(id G)*F = id(G*F)
proof
now
let a be Object of A;
thus ((id G)*F).a = id G.(F.a) by Th22
.= id(G.(F.a)) by NATTRA_1:20
.= id((G*F).a) by CAT_1:76
.= id(G*F).a by NATTRA_1:20;
end;
hence thesis by Th24;
end;
theorem Th31:
G*id F = id(G*F)
proof
now
let a be Object of A;
thus (G*id F).a = G/.(id F.a) by Th21
.= G/.id(F.a) by NATTRA_1:20
.= id(G.(F.a)) by NATTRA_1:15
.= id((G*F).a) by CAT_1:76
.= (id(G*F)).a by NATTRA_1:20;
end;
hence thesis by Th24;
end;
theorem Th32:
G1 is_naturally_transformable_to G2 implies t*id B = t
proof
assume
A1: G1 is_naturally_transformable_to G2;
A2: G1*id B = G1 by FUNCT_2:17;
then reconsider s = t*id B as natural_transformation of G1,G2 by FUNCT_2:17;
A3: G2*id B = G2 by FUNCT_2:17;
now
let b be Object of B;
thus s.b = t.((id B).b) by A1,A2,A3,Th22
.= t.b by CAT_1:79;
end;
hence thesis by A1,Th24;
end;
theorem Th33:
F1 is_naturally_transformable_to F2 implies (id B)*s = s
proof
assume
A1: F1 is_naturally_transformable_to F2;
A2: (id B)*F1 = F1 by FUNCT_2:17;
then reconsider t = (id B)*s as natural_transformation of F1,F2 by FUNCT_2:17
;
A3: (id B)*F2 = F2 by FUNCT_2:17;
now
let a be Object of A;
A4: Hom(F1.a,F2.a) <> {} by A1,Th23;
thus t.a = (id B)/.(s.a) by A1,A2,A3,Th21
.= (id B).(s.a qua Morphism of B) by A4,CAT_3:def 10
.= s.a by FUNCT_1:18;
end;
hence thesis by A1,Th24;
end;
definition
let A,B,C,F1,F2,G1,G2;
let s,t;
func t(#)s -> natural_transformation of G1*F1,G2*F2 equals
(t*F2)`*`(G1*s);
correctness;
end;
theorem Th34:
F1 is_naturally_transformable_to F2 & G1
is_naturally_transformable_to G2 implies t(#)s = (G2*s)`*`(t*F1)
proof
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: G1 is_naturally_transformable_to G2;
A3: G1*F1 is_naturally_transformable_to G1*F2 & G1*F2
is_naturally_transformable_to G2*F2 by A1,A2,Th20;
A4: G2*F1 is_naturally_transformable_to G2*F2 & G1*F1
is_naturally_transformable_to G2*F1 by A1,A2,Th20;
now
let a be Object of A;
A5: (G1*F1).a = G1.(F1.a) & (G1*F2).a = G1.(F2.a) by CAT_1:76;
A6: (G2*F2).a = G2.(F2.a) by CAT_1:76;
A7: (G2*s).a = G2/.(s.a) & (G1*F1).a = G1.(F1.a) by A1,Th21,CAT_1:76;
A8: Hom(F1.a,F2.a) <> {} & (t*F1).a = t.(F1.a) by A1,A2,Th22,Th23;
A9: (G2*F1).a = G2.(F1.a) & (G2*F2).a = G2.(F2.a) by CAT_1:76;
(t*F2).a = t.(F2.a) & (G1*s).a = G1/.(s.a) by A1,A2,Th21,Th22;
hence ((t*F2)`*`(G1*s)).a = t.(F2.a)*G1/.(s.a) by A3,A5,A6,NATTRA_1:25
.= (G2*s).a*(t*F1).a by A2,A8,A7,A9,NATTRA_1:def 8
.= ((G2*s)`*`(t*F1)).a by A4,NATTRA_1:25;
end;
hence thesis by A3,Th24,NATTRA_1:23;
end;
theorem
F1 is_naturally_transformable_to F2 implies (id id B)(#)s = s
proof
assume
A1: F1 is_naturally_transformable_to F2;
then
A2: (id B)*F1 is_naturally_transformable_to (id B)*F2 by Th20;
thus (id id B)(#)s = (id((id B)*F2))`*`((id B)*s) by Th30
.= (id B)*s by A2,NATTRA_1:24
.= s by A1,Th33;
end;
theorem
G1 is_naturally_transformable_to G2 implies t(#)(id id B) = t
proof
assume
A1: G1 is_naturally_transformable_to G2;
then
A2: G1*id B is_naturally_transformable_to G2*id B by Th20;
thus t(#)(id id B) = (t*id B)`*`id(G1*id B) by Th31
.= t*id B by A2,NATTRA_1:24
.= t by A1,Th32;
end;
theorem
F1 is_naturally_transformable_to F2 & G1 is_naturally_transformable_to
G2 & H1 is_naturally_transformable_to H2 implies u(#)(t(#)s) = (u(#)t)(#)s
proof
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: G1 is_naturally_transformable_to G2 and
A3: H1 is_naturally_transformable_to H2;
A4: u*(G2*F2) = u*G2*F2 & H1*(t*F2) = H1*t*F2 by A2,A3,Th27,Th28;
A5: H1*G1 is_naturally_transformable_to H1*G2 by A2,Th20;
then
A6: H1*G1*F2 is_naturally_transformable_to H1*G2*F2 by Th20;
A7: H1*(G1*s) = H1*G1*s & H1*G1*F1 is_naturally_transformable_to H1*G1*F2 by A1
,Th20,Th29;
A8: H1*G1*F1 = H1*(G1*F1) & H1*G1*F2 = H1*(G1*F2) by RELAT_1:36;
A9: H1*G2 is_naturally_transformable_to H2*G2 by A3,Th20;
then
A10: H1*G2*F2 is_naturally_transformable_to H2*G2*F2 by Th20;
A11: H1*G2*F2 = H1*(G2*F2) & H2*G2*F2 = H2*(G2*F2) by RELAT_1:36;
G1*F2 is_naturally_transformable_to G2*F2 & G1*F1
is_naturally_transformable_to G1*F2 by A1,A2,Th20;
hence u(#)(t(#)s) = (u*(G2*F2))`*`((H1*(t*F2))`*`(H1*(G1*s))) by Th25
.= (u*G2*F2)`*`(H1*t*F2)`*`((H1*G1)*s) by A8,A11,A4,A7,A6,A10,NATTRA_1:26
.= u(#)t(#)s by A5,A9,Th26;
end;
theorem
G1 is_naturally_transformable_to G2 implies t*F = t(#)id F
proof
assume G1 is_naturally_transformable_to G2;
then G1*F is_naturally_transformable_to G2*F by Th20;
hence t*F = (t*F)`*`id(G1*F) by NATTRA_1:24
.= t(#)id F by Th31;
end;
theorem
F1 is_naturally_transformable_to F2 implies G*s = (id G)(#)s
proof
assume F1 is_naturally_transformable_to F2;
then G*F1 is_naturally_transformable_to G*F2 by Th20;
hence G*s = id(G*F2)`*`(G*s) by NATTRA_1:24
.= (id G)(#)s by Th30;
end;
theorem
F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
F3 & G1 is_naturally_transformable_to G2 & G2 is_naturally_transformable_to G3
implies (t9`*`t)(#)(s9`*`s) = (t9(#)s9)`*`(t(#)s)
proof
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_naturally_transformable_to F3 and
A3: G1 is_naturally_transformable_to G2 and
A4: G2 is_naturally_transformable_to G3;
A5: t9(#)s9 = (G3*s9)`*`(t9*F2) & t(#)s = (G2*s)`*`(t*F1) by A1,A2,A3,A4,Th34;
A6: G1*F1 is_naturally_transformable_to G2*F1 by A3,Th20;
A7: G2*F1 is_naturally_transformable_to G2*F2 by A1,Th20;
then
A8: G1*F1 is_naturally_transformable_to G2*F2 by A6,NATTRA_1:23;
A9: G2*F2 is_naturally_transformable_to G3*F2 by A4,Th20;
A10: G1 is_naturally_transformable_to G3 by A3,A4,NATTRA_1:23;
then
A11: G1*F1 is_naturally_transformable_to G3*F1 by Th20;
A12: G3*F1 is_naturally_transformable_to G3*F2 by A1,Th20;
A13: G2*F1 is_naturally_transformable_to G3*F1 by A4,Th20;
A14: G3*F2 is_naturally_transformable_to G3*F3 by A2,Th20;
F1 is_naturally_transformable_to F3 by A1,A2,NATTRA_1:23;
hence (t9`*`t)(#)(s9`*`s) = (G3*(s9`*`s))`*`((t9`*`t)*F1) by A10,Th34
.= (G3*s9)`*`(G3*s)`*`((t9`*`t)*F1) by A1,A2,Th25
.= (G3*s9)`*`(G3*s)`*`((t9*F1)`*`(t*F1)) by A3,A4,Th26
.= (G3*s9)`*`((G3*s)`*`((t9*F1)`*`(t*F1))) by A14,A12,A11,NATTRA_1:26
.= (G3*s9)`*`((G3*s)`*`(t9*F1)`*`(t*F1)) by A12,A6,A13,NATTRA_1:26
.= (G3*s9)`*`((t9(#)s)`*`(t*F1)) by A1,A4,Th34
.= (G3*s9)`*`((t9*F2)`*`((G2*s)`*`(t*F1))) by A6,A9,A7,NATTRA_1:26
.= (t9(#)s9)`*`(t(#)s) by A14,A9,A8,A5,NATTRA_1:26;
end;
theorem Th41:
for F being Functor of A,B, G being Functor of C,D for I,J being
Functor of B,C st I ~= J holds G*I ~= G*J & I*F ~= J*F
proof
let F be Functor of A,B, G be Functor of C,D;
let I,J be Functor of B,C;
assume
A1: I is_naturally_transformable_to J;
given t being natural_transformation of I,J such that
A2: t is invertible;
thus G*I ~= G*J
proof
thus G*I is_naturally_transformable_to G*J by A1,Th20;
take G*t;
let b be Object of B;
A3: t.b is invertible by A2;
A4: G.(I.b) = (G*I).b & G.(J.b) = (G*J).b by CAT_1:76;
(G*t).b = G/.(t.b) by A1,Th21;
hence (G*t).b is invertible by A3,Th1,A4;
end;
thus I*F is_naturally_transformable_to J*F by A1,Th20;
take t*F;
let a be Object of A;
A5: I.(F.a) = (I*F).a & J.(F.a) = (J*F).a by CAT_1:76;
(t*F).a = t.(F.a) by A1,Th22;
hence (t*F).a is invertible by A2,A5;
end;
theorem Th42:
for F being Functor of A,B, G being Functor of B,A for I being
Functor of A,A st I ~= id A holds F*I ~= F & I*G ~= G
proof
let F be Functor of A,B, G be Functor of B,A;
let I be Functor of A,A;
F*id A = F & id A*G = G by FUNCT_2:17;
hence thesis by Th41;
end;
definition
let A,B be Category;
pred A is_equivalent_with B means
ex F being Functor of A,B, G being
Functor of B,A st G*F ~= id A & F*G ~= id B;
reflexivity
proof
let A;
take id A, id A;
thus thesis by FUNCT_2:17;
end;
symmetry;
end;
notation
let A,B be Category;
synonym A,B are_equivalent for A is_equivalent_with B;
end;
theorem
A ~= B implies A is_equivalent_with B
proof
given F being Functor of A,B such that
A1: F is isomorphic;
take F,F";
thus thesis by A1,Th11;
end;
theorem Th44:
A,B are_equivalent & B,C are_equivalent implies A,C are_equivalent
proof
given F1 being Functor of A,B, G1 being Functor of B,A such that
A1: G1*F1 ~= id A and
A2: F1*G1 ~= id B;
given F2 being Functor of B,C, G2 being Functor of C,B such that
A3: G2*F2 ~= id B and
A4: F2*G2 ~= id C;
take F2*F1,G1*G2;
(G1*G2)*F2 = G1*(G2*F2) by RELAT_1:36;
then
A5: (G1*G2)*F2 ~= G1 by A3,Th42;
(G1*G2)*(F2*F1) = ((G1*G2)*F2)*F1 by RELAT_1:36;
then (G1*G2)*(F2*F1) ~= G1*F1 by A5,Th41;
hence (G1*G2)*(F2*F1) ~= id A by A1,NATTRA_1:29;
(F2*F1)*G1 = F2*(F1*G1) by RELAT_1:36;
then
A6: (F2*F1)*G1 ~= F2 by A2,Th42;
(F2*F1)*(G1*G2) = ((F2*F1)*G1)*G2 by RELAT_1:36;
then (F2*F1)*(G1*G2) ~= F2*G2 by A6,Th41;
hence thesis by A4,NATTRA_1:29;
end;
definition
let A,B;
assume
A1: A,B are_equivalent;
mode Equivalence of A,B -> Functor of A,B means
:Def11:
ex G being Functor of B,A st G*it ~= id A & it*G ~= id B;
existence by A1;
end;
theorem
id A is Equivalence of A,A
proof
thus A is_equivalent_with A;
take id A;
thus thesis by FUNCT_2:17;
end;
theorem
A,B are_equivalent & B,C are_equivalent implies for F being
Equivalence of A,B, G being Equivalence of B,C holds G*F is Equivalence of A,C
proof
assume that
A1: A,B are_equivalent and
A2: B,C are_equivalent;
let F be Equivalence of A,B, G be Equivalence of B,C;
thus A,C are_equivalent by A1,A2,Th44;
consider F9 being Functor of B,A such that
A3: F9*F ~= id A and
A4: F*F9 ~= id B by A1,Def11;
(G*F)*F9 = G*(F*F9) by RELAT_1:36;
then
A5: (G*F)*F9 ~= G by A4,Th42;
consider G9 being Functor of C,B such that
A6: G9*G ~= id B and
A7: G*G9 ~= id C by A2,Def11;
take F9*G9;
(F9*G9)*G = F9*(G9*G) by RELAT_1:36;
then
A8: (F9*G9)*G ~= F9 by A6,Th42;
(F9*G9)*(G*F) = ((F9*G9)*G)*F by RELAT_1:36;
then (F9*G9)*(G*F) ~= F9*F by A8,Th41;
hence (F9*G9)*(G*F) ~= id A by A3,NATTRA_1:29;
(G*F)*(F9*G9) = ((G*F)*F9)*G9 by RELAT_1:36;
then (G*F)*(F9*G9) ~= G*G9 by A5,Th41;
hence thesis by A7,NATTRA_1:29;
end;
theorem Th47:
A,B are_equivalent implies for F being Equivalence of A,B ex G
being Equivalence of B,A st G*F ~= id A & F*G ~= id B
proof
assume
A1: A,B are_equivalent;
let F be Equivalence of A,B;
consider G be Functor of B,A such that
A2: G*F ~= id A & F*G ~= id B by A1,Def11;
G is Equivalence of B,A
proof
thus B,A are_equivalent by A1;
take F;
thus thesis by A2;
end;
hence thesis by A2;
end;
theorem Th48:
for F being Functor of A,B, G being Functor of B,A st G*F ~= id
A holds F is faithful
proof
let F be Functor of A,B, G be Functor of B,A;
assume G*F ~= id A;
then
A1: id A ~= G*F by NATTRA_1:28;
then
A2: id A is_naturally_transformable_to G*F;
consider s being natural_transformation of id A, G*F such that
A3: s is invertible by A1;
thus F is faithful
proof
let a,a9 be Object of A;
assume
A4: Hom(a,a9) <> {};
then
A5: Hom((id A).a,(id A).a9) <> {} by CAT_1:84;
let f1,f2 be Morphism of a,a9;
assume
A6: F.(f1 qua Morphism of A) = F.(f2 qua Morphism of A);
A7: (G*F)/.f1 = (G*F).(f1 qua Morphism of A) by A4,CAT_3:def 10
.= G.(F.(f2 qua Morphism of A)) by A6,FUNCT_2:15
.= (G*F).(f2 qua Morphism of A) by FUNCT_2:15
.= (G*F)/.f2 by A4,CAT_3:def 10;
A8: s.a9*(id A)/.f1 = (G*F)/.f1*s.a by A2,A4,NATTRA_1:def 8
.= s.a9*(id A)/.f2 by A2,A4,A7,NATTRA_1:def 8;
s.a9 is invertible by A3;
then
A9: s.a9 is monic by CAT_1:43;
thus f1 = (id A)/.f1 by A4,NATTRA_1:16
.= (id A)/.f2 by A5,A9,A8
.= f2 by A4,NATTRA_1:16;
end;
end;
theorem
A,B are_equivalent implies for F being Equivalence of A,B holds F is
full & F is faithful & for b being Object of B ex a being Object of A st b, F.a
are_isomorphic
proof
assume
A1: A,B are_equivalent;
let F be Equivalence of A,B;
consider G being Equivalence of B,A such that
A2: G*F ~= id A and
A3: F*G ~= id B by A1,Th47;
A4: id A ~= G*F by A2,NATTRA_1:28;
then
A5: id A is_naturally_transformable_to G*F;
consider s being natural_transformation of id A, G*F such that
A6: s is invertible by A4;
A7: G is faithful by A3,Th48;
thus F is full
proof
let a,a9 be Object of A such that
A8: Hom(F.a,F.a9) <> {};
reconsider f2 = s.a9 as Morphism of a9, (G*F).a9 by CAT_1:79;
reconsider f1 = s.a as Morphism of a, (G*F).a by CAT_1:79;
A9: s.a9 is invertible by A6;
A10: Hom((id A).a9,(G*F).a9) <> {} by A5,Th23;
let g be Morphism of F.a,F.a9;
A11: (G*F).a = G.(F.a) by CAT_1:76;
then reconsider h = G/.g as Morphism of (G*F).a, (G*F).a9 by CAT_1:76;
A12: Hom((id A).a,(G*F).a) <> {} by A5,Th23;
(G*F).a9 = G.(F.a9) by CAT_1:76;
then
A13: Hom((G*F).a,(G*F).a9) <> {} by A8,A11,CAT_1:84;
then
A14: Hom((id A).a,(G*F).a9) <> {} by A12,CAT_1:24;
s.a is invertible by A6;
then
A15: s.a is epi by CAT_1:43;
(G*F) is_naturally_transformable_to id A by A2;
then
A16: Hom((G*F).a9,(id A).a9) <> {} by Th23;
A17: (id A).a = a & (id A).a9 = a9 by CAT_1:79;
hence
A18: Hom(a,a9) <> {} by A16,A14,CAT_1:24;
take f = f2"*(h*f1);
A19: (id A)/.f = ((s.a9)"*(h*(s.a))) by A17,A18,NATTRA_1:16;
h*s.a = id((G*F).a9)*(h*(s.a)) by A14,CAT_1:28
.= s.a9*(s.a9)"*(h*(s.a)) by A9,CAT_1:def 17
.= s.a9*(id A)/.f by A16,A14,A10,A19,CAT_1:25
.= (G*F)/.f*s.a by A5,A18,NATTRA_1:def 8;
then
A20: h = (G*F)/.f by A13,A15;
G.(g qua Morphism of B) = G/.g by A8,CAT_3:def 10
.= (G*F).(f qua Morphism of A) by A18,A20,CAT_3:def 10
.= G.(F.(f qua Morphism of A)) by FUNCT_2:15
.= G.(F/.f qua Morphism of B) by A18,CAT_3:def 10;
hence g = F/.f by A7,A8
.= F.(f qua Morphism of A) by A18,CAT_3:def 10;
end;
thus F is faithful by A2,Th48;
let b be Object of B;
take G.b;
A21: F.(G.b) = (F*G).b & (id B).b = b by CAT_1:76,79;
A22: id B ~= F*G by A3,NATTRA_1:28;
then id B is_naturally_transformable_to F*G;
then
A23: id B is_transformable_to F*G;
ex t being natural_transformation of id B, F*G st t is invertible by A22;
hence thesis by A21,A23,Th4;
end;
:: The elimination of the Id selector caused the necessity to
:: introduce corresponding functor because 'the Id of C' is sometimes
:: separately used, not applied to an object (2012.01.25, A.T.)
definition let C be Category;
deffunc F(Object of C) = id $1;
func IdMap C -> Function of the carrier of C, the carrier' of C means
for o being Object of C holds it.o = id o;
existence
proof deffunc F(Object of C) = id $1;
consider f being Function of the carrier of C, the carrier' of C such that
A1: for o being Object of C holds f.o = F(o) from FUNCT_2:sch 4;
take f;
thus thesis by A1;
end;
uniqueness
proof
let F,G be Function of the carrier of C, the carrier' of C such that
A2: for o being Object of C holds F.o = id o and
A3: for o being Object of C holds G.o = id o;
let o be Object of C;
thus F.o = id o by A2 .= G.o by A3;
end;
end;