:: Object-Free Definition of Categories
:: by Marco Riccardi
::
:: Received October 7, 2013
:: Copyright (c) 2013 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 FUNCT_1, TARSKI, ZFMISC_1, XBOOLE_0, RELAT_1, SUBSET_1, WELLORD1,
CAT_1, ALTCAT_1, GRAPH_1, BINOP_1, STRUCT_0, PARTFUN1, FUNCOP_1,
MONOID_0, NATTRA_1, MSSUBFAM, ALTCAT_2, ARYTM_0, NUMBERS, ORDINAL1,
ARYTM_3, CARD_1, FUNCTOR0, MOD_4, CAT_6;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, FUNCT_3, FUNCT_4, RELAT_1,
FUNCT_1, FUNCT_2, ORDINAL1, PARTFUN1, FINSEQ_1, STRUCT_0, FUNCOP_1,
RELSET_1, BINOP_1, GRAPH_1, CAT_1, CAT_2, NUMBERS, NAT_1, XXREAL_0,
XTUPLE_0, ALG_1, XCMPLX_0, WELLORD2, CARD_1, INT_1, VALUED_0, ISOCAT_1;
constructors NUMBERS, FUNCT_2, CLASSES2, FINSEQ_1, STRUCT_0, FUNCOP_1,
RELAT_2, PARTFUN1, RELSET_1, BINOP_1, FUNCT_4, CAT_1, CAT_2, NAT_1,
XXREAL_0, WELLORD2, XREAL_0, CARD_1, XCMPLX_0, INT_1, VALUED_0, ISOCAT_1;
registrations XBOOLE_0, SUBSET_1, ZFMISC_1, RELAT_1, FUNCT_1, ORDINAL1,
XTUPLE_0, CARD_1, CLASSES2, STRUCT_0, RELSET_1, FUNCT_2, PARTFUN1,
GRFUNC_1, BINOP_1, CAT_1, CARD_2, NAT_LAT, FINSET_1, XXREAL_0, XREAL_0,
NAT_1, NUMBERS, VALUED_0, FINSEQ_1;
requirements SUBSET, BOOLE, REAL, NUMERALS, ARITHM;
theorems TARSKI, XBOOLE_0, XTUPLE_0, ZFMISC_1, RELAT_1, FUNCT_1, STRUCT_0,
FUNCT_2, SUBSET_1, BINOP_1, GRAPH_1, CAT_1, FUNCOP_1, FUNCT_4, NAT_1,
ORDINAL1, ISOCAT_1;
schemes FUNCT_2;
begin :: Yet another definition of category
definition
struct (1-sorted) CategoryStr
(# carrier -> set,
composition -> PartFunc of [:the carrier, the carrier:], the carrier #);
end;
reserve C for CategoryStr;
definition
let C;
func Mor(C) -> set equals the carrier of C;
correctness;
end;
definition
let C;
mode morphism of C is Element of Mor(C);
end;
reserve f,f1,f2,f3 for morphism of C;
definition
let C,f1,f2;
pred f1,f2 are_composable means :Def2:
[f1,f2] in dom the composition of C;
end;
notation
let C,f1,f2;
synonym f1 |> f2 for f1,f2 are_composable;
end;
definition
let C,f1,f2;
assume
A1: f1 |> f2;
func f1 (*) f2 -> morphism of C equals :Def3:
(the composition of C).(f1,f2);
correctness
proof
per cases by A1;
suppose [f1,f2] in dom the composition of C;
then (the composition of C).[f1,f2] in rng(the composition of C)
by FUNCT_1:3;
hence thesis by BINOP_1:def 1;
end;
suppose
A2: Mor(C) is empty;
(the composition of C).[f1,f2] = (the composition of C).(f1,f2)
by BINOP_1:def 1;
hence thesis by A2,SUBSET_1:def 1;
end;
end;
end;
definition
let C,f;
attr f is left_identity means :Def4:
for f1 being morphism of C st f |> f1 holds f (*) f1 = f1;
attr f is right_identity means :Def5:
for f1 being morphism of C st f1 |> f holds f1 (*) f = f1;
end;
definition
let C;
attr C is with_left_identities means :Def6:
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f |> f1 & f is left_identity;
attr C is with_right_identities means :Def7:
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f1 |> f & f is right_identity;
attr C is left_composable means
for f,f1,f2 being morphism of C st f1 |> f2 holds f1(*)f2 |> f iff f2 |> f;
attr C is right_composable means :Def9:
for f,f1,f2 being morphism of C st f1 |> f2 holds f |> f1(*)f2 iff f |> f1;
attr C is associative means :Def10:
for f1,f2,f3 being morphism of C
st f1 |> f2 & f2 |> f3 & (f1(*)f2) |> f3 & f1 |> (f2(*)f3)
holds f1(*)(f2(*)f3) = (f1(*)f2)(*)f3;
end;
definition
let C;
attr C is composable means :Def11:
C is left_composable right_composable;
attr C is with_identities means :Def12:
C is with_left_identities with_right_identities;
end;
definition
let X be set;
let f be PartFunc of [:X,X:],X;
redefine func ~f -> PartFunc of [:X,X:],X;
coherence by FUNCT_4:48;
end;
definition
let C;
func C opp -> strict CategoryStr equals
CategoryStr (#the carrier of C, ~the composition of C#);
coherence;
end;
theorem
C is empty implies not f1 |> f2;
reserve g1,g2 for morphism of C opp;
theorem
f1 = g1 & f2 = g2 implies (f1 |> f2 iff g2 |> g1) by FUNCT_4:42;
theorem Th3:
f1 = g1 & f2 = g2 & f1 |> f2 implies f1 (*) f2 = g2 (*) g1
proof
assume
A1: f1 = g1 & f2 = g2;
assume
A2: f1 |> f2;
A3: g2 |> g1 by A1,A2,FUNCT_4:42;
thus f1 (*) f2 = (the composition of C).(f1,f2) by A2,Def3
.= (~the composition of C).(f2,f1) by A2,FUNCT_4:def 2
.= g2 (*) g1 by A1,A3,Def3;
end;
theorem Th4:
C is left_composable iff C opp is right_composable
proof
hereby
assume
A1: C is left_composable;
for g,g1,g2 being morphism of C opp st g1 |> g2
holds (g |> g1(*)g2 iff g |> g1)
proof
let g,g1,g2 be morphism of C opp;
reconsider f=g,f2=g1,f1=g2 as morphism of C;
assume g1 |> g2;
then
A2: f1 |> f2 by FUNCT_4:42;
then
A3: f1(*)f2 = g1(*)g2 by Th3;
hereby
assume g |> g1(*)g2;
then f1(*)f2 |> f by A3,FUNCT_4:42;
then f2 |> f by A1,A2;
hence g |> g1 by FUNCT_4:42;
end;
assume g |> g1;
then f2 |> f by FUNCT_4:42;
then f1(*)f2 |> f by A1,A2;
hence g |> g1(*)g2 by A3,FUNCT_4:42;
end;
hence C opp is right_composable;
end;
assume
A4: C opp is right_composable;
for f,f1,f2 being morphism of C st f1 |> f2
holds (f1(*)f2 |> f iff f2 |> f)
proof
let f,f1,f2 be morphism of C;
reconsider g=f,g2=f1,g1=f2 as morphism of C opp;
assume
A5: f1 |> f2;
then
A6: g1 |> g2 by FUNCT_4:42;
A7: f1(*)f2 = g1(*)g2 by A5,Th3;
hereby
assume f1(*)f2 |> f;
then g |> g1(*)g2 by A7,FUNCT_4:42;
then g |> g1 by A4,A6;
hence f2 |> f by FUNCT_4:42;
end;
assume f2 |> f;
then g |> g1 by FUNCT_4:42;
then g |> g1(*)g2 by A4,A6;
hence f1(*)f2 |> f by A7,FUNCT_4:42;
end;
hence C is left_composable;
end;
theorem Th5:
C is right_composable iff C opp is left_composable
proof
hereby
assume
A1: C is right_composable;
for g,g1,g2 being morphism of C opp st g1 |> g2
holds (g1(*)g2 |> g iff g2 |> g)
proof
let g,g1,g2 be morphism of C opp;
reconsider f=g,f2=g1,f1=g2 as morphism of C;
assume g1 |> g2;
then
A2: f1 |> f2 by FUNCT_4:42;
then
A3: f1(*)f2 = g1(*)g2 by Th3;
hereby
assume g1(*)g2 |> g;
then f |> f1(*)f2 by A3,FUNCT_4:42;
then f |> f1 by A1,A2;
hence g2 |> g by FUNCT_4:42;
end;
assume g2 |> g;
then f |> f1 by FUNCT_4:42;
then f |> f1(*)f2 by A1,A2;
hence g1(*)g2 |> g by A3,FUNCT_4:42;
end;
hence C opp is left_composable;
end;
assume
A4: C opp is left_composable;
for f,f1,f2 being morphism of C st f1 |> f2
holds (f |> f1(*)f2 iff f |> f1)
proof
let f,f1,f2 be morphism of C;
reconsider g=f,g2=f1,g1=f2 as morphism of C opp;
assume
A5: f1 |> f2;
then
A6: g1 |> g2 by FUNCT_4:42;
A7: f1(*)f2 = g1(*)g2 by A5,Th3;
hereby
assume f |> f1(*)f2;
then g1(*)g2 |> g by A7,FUNCT_4:42;
then g2 |> g by A4,A6;
hence f |> f1 by FUNCT_4:42;
end;
assume f |> f1;
then g2 |> g by FUNCT_4:42;
then g1(*)g2 |> g by A4,A6;
hence f |> f1(*)f2 by A7,FUNCT_4:42;
end;
hence C is right_composable;
end;
theorem Th6:
C is with_left_identities iff C opp is with_right_identities
proof
hereby
assume
A1: C is with_left_identities;
for g1 being morphism of C opp st g1 in the carrier of C opp holds
ex g being morphism of C opp st g1 |> g & g is right_identity
proof
let g1 be morphism of C opp;
assume
A2: g1 in the carrier of C opp;
reconsider f1 = g1 as morphism of C;
consider f be morphism of C such that
A3: f |> f1 & f is left_identity by A1,A2;
reconsider g = f as morphism of C opp;
take g;
thus g1 |> g by A3,FUNCT_4:42;
for g2 being morphism of C opp st g2 |> g holds g2 (*) g = g2
proof
let g2 be morphism of C opp;
reconsider f2 = g2 as morphism of C;
assume g2 |> g;
then
A4: f |> f2 by FUNCT_4:42;
then f (*) f2 = f2 by A3;
hence g2 (*) g = g2 by A4,Th3;
end;
hence g is right_identity;
end;
hence C opp is with_right_identities;
end;
assume
A5: C opp is with_right_identities;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f |> f1 & f is left_identity
proof
let f1 be morphism of C;
assume
A6: f1 in the carrier of C;
reconsider g1 = f1 as morphism of C opp;
consider g be morphism of C opp such that
A7: g1 |> g & g is right_identity by A5,A6;
reconsider f = g as morphism of C;
take f;
thus f |> f1 by A7,FUNCT_4:42;
for f2 being morphism of C st f |> f2 holds f (*) f2 = f2
proof
let f2 be morphism of C;
reconsider g2 = f2 as morphism of C opp;
assume
A8: f |> f2;
then g2 |> g by FUNCT_4:42;
then g2 (*) g = g2 by A7;
hence f (*) f2 = f2 by A8,Th3;
end;
hence f is left_identity;
end;
hence C is with_left_identities;
end;
theorem Th7:
C is with_right_identities iff C opp is with_left_identities
proof
hereby
assume
A1: C is with_right_identities;
for g1 being morphism of C opp st g1 in the carrier of C opp holds
ex g being morphism of C opp st g |> g1 & g is left_identity
proof
let g1 be morphism of C opp;
assume
A2: g1 in the carrier of C opp;
reconsider f1 = g1 as morphism of C;
consider f be morphism of C such that
A3: f1 |> f & f is right_identity by A1,A2;
reconsider g = f as morphism of C opp;
take g;
thus g |> g1 by A3,FUNCT_4:42;
for g2 being morphism of C opp st g |> g2 holds g (*) g2 = g2
proof
let g2 be morphism of C opp;
reconsider f2 = g2 as morphism of C;
assume g |> g2;
then
A4: f2 |> f by FUNCT_4:42;
then f2 (*) f = f2 by A3;
hence g (*) g2 = g2 by A4,Th3;
end;
hence g is left_identity;
end;
hence C opp is with_left_identities;
end;
assume
A5: C opp is with_left_identities;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f1 |> f & f is right_identity
proof
let f1 be morphism of C;
assume
A6: f1 in the carrier of C;
reconsider g1 = f1 as morphism of C opp;
consider g be morphism of C opp such that
A7: g |> g1 & g is left_identity by A5,A6;
reconsider f = g as morphism of C;
take f;
thus f1 |> f by A7,FUNCT_4:42;
for f2 being morphism of C st f2 |> f holds f2 (*) f = f2
proof
let f2 be morphism of C;
reconsider g2 = f2 as morphism of C opp;
assume
A8: f2 |> f;
then g |> g2 by FUNCT_4:42;
then g (*) g2 = g2 by A7;
hence f2 (*) f = f2 by A8,Th3;
end;
hence f is right_identity;
end;
hence C is with_right_identities;
end;
theorem Th8:
C is associative iff C opp is associative
proof
hereby
assume
A1: C is associative;
for g1,g2,g3 being morphism of C opp st g1 |> g2 & g2 |> g3 &
g1(*)g2 |> g3 & g1 |> g2(*)g3 holds g1(*)(g2(*)g3) = (g1(*)g2)(*)g3
proof
let g1,g2,g3 be morphism of C opp;
reconsider f3=g1,f2=g2,f1=g3 as morphism of C;
assume g1 |> g2;
then
A2: f2 |> f3 by FUNCT_4:42;
assume g2 |> g3;
then
A3: f1 |> f2 by FUNCT_4:42;
assume
A4: g1(*)g2 |> g3;
assume
A5: g1 |> g2(*)g3;
A6: g1(*)g2 = f2(*)f3 by A2,Th3;
then
A7: f1 |> f2(*)f3 by A4,FUNCT_4:42;
A8: f1(*)f2 = g2(*)g3 by A3,Th3;
then
A9: f1(*)f2 |> f3 by A5,FUNCT_4:42;
thus g1(*)(g2(*)g3) = (f1(*)f2)(*)f3 by A8,A9,Th3
.= f1(*)(f2(*)f3) by A1,A2,A3,A7,A9
.= (g1(*)g2)(*)g3 by A6,A7,Th3;
end;
hence C opp is associative;
end;
assume
A10: C opp is associative;
for f1,f2,f3 being morphism of C st f1 |> f2 & f2 |> f3 &
f1(*)f2 |> f3 & f1 |> f2(*)f3 holds f1(*)(f2(*)f3) = (f1(*)f2)(*)f3
proof
let f1,f2,f3 be morphism of C;
reconsider g3=f1,g2=f2,g1=f3 as morphism of C opp;
assume
A11: f1 |> f2;
then
A12: g2 |> g3 by FUNCT_4:42;
assume
A13: f2 |> f3;
then
A14: g1 |> g2 by FUNCT_4:42;
assume
A15: f1(*)f2 |> f3;
assume
A16: f1 |> f2(*)f3;
A17: f1(*)f2 = g2(*)g3 by A11,Th3;
then
A18: g1 |> g2(*)g3 by A15,FUNCT_4:42;
A19: g1(*)g2 = f2(*)f3 by A13,Th3;
then
A20: g1(*)g2 |> g3 by A16,FUNCT_4:42;
thus f1(*)(f2(*)f3) = (g1(*)g2)(*)g3 by A19,A16,Th3
.= g1(*)(g2(*)g3) by A10,A12,A14,A18,A20
.= (f1(*)f2)(*)f3 by A17,A15,Th3;
end;
hence C is associative;
end;
registration
cluster with_left_identities non with_right_identities
composable associative for CategoryStr;
correctness
proof
reconsider X = {0,1} as set;
set comp = {[[0,0],0],[[0,1],1]};
A1: for x,y1,y2 being element st [x,y1] in comp & [x,y2] in comp holds y1 = y2
proof
let x,y1,y2 be element;
assume
A2: [x,y1] in comp;
assume
A3: [x,y2] in comp;
per cases by A2,TARSKI:def 2;
suppose
A4: [x,y1] = [[0,0],0];
then
A5: x = [0,0] & y1 = 0 by XTUPLE_0:1;
per cases by A3,TARSKI:def 2;
suppose [x,y2] = [[0,0],0];
hence thesis by A4,XTUPLE_0:1;
end;
suppose [x,y2] = [[0,1],1];
then x = [0,1] & y2 = 1 by XTUPLE_0:1;
hence thesis by A5,XTUPLE_0:1;
end;
end;
suppose
A6: [x,y1] = [[0,1],1];
then
A7: x = [0,1] & y1 = 1 by XTUPLE_0:1;
per cases by A3,TARSKI:def 2;
suppose [x,y2] = [[0,0],0];
then x = [0,0] & y2 = 0 by XTUPLE_0:1;
hence thesis by A7,XTUPLE_0:1;
end;
suppose [x,y2] = [[0,1],1];
hence thesis by A6,XTUPLE_0:1;
end;
end;
end;
for x being element st x in comp holds x in [:[:X,X:],X:]
proof
let x be element;
assume
A8: x in comp;
per cases by A8,TARSKI:def 2;
suppose
A9: x = [[0,0],0];
A10: 0 in X by TARSKI:def 2;
then [0,0] in [:X,X:] by ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A9,A10,ZFMISC_1:def 2;
end;
suppose
A11: x = [[0,1],1];
A12: 0 in X & 1 in X by TARSKI:def 2;
then [0,1] in [:X,X:] by ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A11,A12,ZFMISC_1:def 2;
end;
end;
then reconsider comp as PartFunc of [:X,X:],X
by A1,TARSKI:def 3,FUNCT_1:def 1;
set C = CategoryStr(# X, comp #);
A13: for f1,f2 being morphism of C st f1 |> f2 holds
(f1 = 0 & f2 = 0 & f1(*)f2 = 0) or
(f1 = 0 & f2 = 1 & f1(*)f2 = 1)
proof
let f1,f2 be morphism of C;
assume
A14: f1 |> f2;
assume
A15: not (f1 = 0 & f2 = 0 & f1(*)f2 = 0);
consider y be element such that
A16: [[f1,f2],y] in the composition of C by A14,XTUPLE_0:def 12;
A17: f1(*)f2 = (the composition of C).(f1,f2) by Def3,A14
.= (the composition of C).[f1,f2] by BINOP_1:def 1
.= y by A16,FUNCT_1:1;
per cases by A16,TARSKI:def 2;
suppose [[f1,f2],y] = [[0,0],0];
then [f1,f2] = [0,0] & y = 0 by XTUPLE_0:1;
hence thesis by A17,A15,XTUPLE_0:1;
end;
suppose [[f1,f2],y] = [[0,1],1];
then [f1,f2] = [0,1] & y = 1 by XTUPLE_0:1;
hence thesis by A17,XTUPLE_0:1;
end;
end;
A18: for f1,f2 being morphism of C st f1 |> f2 holds f1 = 0 &
(f2 = 0 or f2 = 1)
proof
let f1,f2 be morphism of C;
assume f1 |> f2;
then consider y be element such that
A19: [[f1,f2],y] in the composition of C by XTUPLE_0:def 12;
per cases by A19,TARSKI:def 2;
suppose [[f1,f2],y] = [[0,0],0];
then [f1,f2] = [0,0] & y = 0 by XTUPLE_0:1;
hence thesis by XTUPLE_0:1;
end;
suppose [[f1,f2],y] = [[0,1],1];
then [f1,f2] = [0,1] & y = 1 by XTUPLE_0:1;
hence thesis by XTUPLE_0:1;
end;
end;
A20: for f1,f2 being morphism of C st f1 = 0 holds f1 |> f2
proof
let f1,f2 be morphism of C;
assume
A21: f1 = 0;
f2 = 0 or f2 = 1 by TARSKI:def 2;
then [[f1,f2],f2] in the composition of C by A21,TARSKI:def 2;
hence thesis by FUNCT_1:1;
end;
for f,f1,f2 being morphism of C st f1 |> f2 holds
f1(*)f2 |> f iff f2 |> f
proof
let f,f1,f2 be morphism of C;
assume
A22: f1 |> f2;
per cases by A22,A13;
suppose f1 = 0 & f2 = 0 & f1(*)f2 = 0; hence thesis; end;
suppose f1 = 0 & f2 = 1 & f1(*)f2 = 1; hence thesis; end;
end;
then
A23: C is left_composable;
for f,f1,f2 being morphism of C st f1 |> f2 holds
f |> f1(*)f2 iff f |> f1
proof
let f,f1,f2 be morphism of C;
assume
A24: f1 |> f2;
per cases by A24,A13;
suppose f1 = 0 & f2 = 0 & f1(*)f2 = 0; hence thesis; end;
suppose f1 = 0 & f2 = 1 & f1(*)f2 = 1;
hereby
assume f |> f1(*)f2;
then f = 0 by A18;
hence f |> f1 by A20;
end;
assume f |> f1;
then f = 0 by A18;
hence f |> f1(*)f2 by A20;
end;
end;
then
A25: C is composable by A23,Def9;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f |> f1 & f is left_identity
proof
let f1 be morphism of C;
assume f1 in the carrier of C;
reconsider f = 0 as morphism of C by TARSKI:def 2;
take f;
thus f |> f1 by A20;
for f2 being morphism of C st f |> f2 holds f (*) f2 = f2
proof
let f2 be morphism of C;
assume
A26: f |> f2;
f2 = 0 or f2 = 1 by TARSKI:def 2;
hence f (*) f2 = f2 by A26,A13;
end;
hence f is left_identity;
end;
then
A27: C is with_left_identities;
ex f1 being morphism of C st f1 in the carrier of C &
for f being morphism of C st f1 |> f holds not f is right_identity
proof
reconsider f1 = 1 as morphism of C by TARSKI:def 2;
take f1;
thus f1 in the carrier of C;
let f be morphism of C;
assume f1 |> f;
hence thesis by A18;
end;
then
A28: C is non with_right_identities;
for f1,f2,f3 being morphism of C st f1 |> f2 & f2 |> f3 &
f1 (*) f2 |> f3 & f1 |> f2 (*) f3 holds
f1 (*) (f2 (*) f3) = (f1 (*) f2) (*) f3
proof
let f1,f2,f3 be morphism of C;
assume
A29: f1 |> f2 & f2 |> f3;
assume
A30: f1 (*) f2 |> f3;
assume
A31: f1 |> f2 (*) f3;
per cases by TARSKI:def 2;
suppose
A32: f3 = 0;
then f2 (*) f3 = 0 by A29,A13;
hence f1 (*) (f2 (*) f3) = 0 by A31,A13
.= (f1 (*) f2) (*) f3 by A30,A32,A13;
end;
suppose
A33: f3 = 1;
then f2 (*) f3 = 1 by A29,A13;
hence f1 (*) (f2 (*) f3) = 1 by A31,A13
.= (f1 (*) f2) (*) f3 by A30,A33,A13;
end;
end;
hence thesis by A25,A27,A28,Def10;
end;
end;
registration
cluster non with_left_identities with_right_identities
composable associative for CategoryStr;
correctness
proof
set C1 = the with_left_identities non with_right_identities
composable associative CategoryStr;
set C = C1 opp;
take C;
thus C is non with_left_identities by Th7;
thus C is with_right_identities by Th6;
thus C is composable by Th4,Th5,Def11;
thus thesis by Th8;
end;
end;
registration
cluster non left_composable right_composable
with_identities associative for CategoryStr;
correctness
proof
set X1 = NAT \ {0};
set X2 = {[n1,n2] where n1,n2 is Element of NAT: n2 = n1 + 1};
reconsider X = X1 \/ X2 as set;
set c0 = {[[[n1,n2],[n1,n2]],[n1,n2]] where n1,n2 is Element of NAT:
n2 = n1 + 1};
set c1 = {[[[n1,n2],n2],n2] where n1,n2 is Element of NAT: n2 = n1 + 1};
set c2 = {[[n1,[n1,n2]],n1] where n1,n2 is Element of NAT: n2 = n1 + 1 &
n1 <> 0};
set c3 = {[[n1,n2],n1] where n1,n2 is Element of NAT: n2 = n1 + 1 &
n1 <> 0};
set comp = c0 \/ c1 \/ c2 \/ c3;
1 in NAT & not 1 in {0} by TARSKI:def 1;
then
A1: 1 in X1 by XBOOLE_0:def 5;
A2: for x,y1,y2 being element st [x,y1] in comp & [x,y2] in comp holds y1 = y2
proof
let x,y1,y2 be element;
assume [x,y1] in comp;
then [x,y1] in c0 \/ c1 \/ c2 or [x,y1] in c3 by XBOOLE_0:def 3;
then
A3: [x,y1] in c0 \/ c1 or [x,y1] in c2 or [x,y1] in c3
by XBOOLE_0:def 3;
assume [x,y2] in comp;
then [x,y2] in c0 \/ c1 \/ c2 or [x,y2] in c3 by XBOOLE_0:def 3;
then
A4: [x,y2] in c0 \/ c1 or [x,y2] in c2 or [x,y2] in c3
by XBOOLE_0:def 3;
per cases by A3,XBOOLE_0:def 3;
suppose [x,y1] in c0;
then consider n11,n12 be Element of NAT such that
A5: [x,y1] = [[[n11,n12],[n11,n12]],[n11,n12]] & n12 = n11 + 1;
A6: x = [[n11,n12],[n11,n12]] & y1 = [n11,n12] by A5,XTUPLE_0:1;
per cases by A4,XBOOLE_0:def 3;
suppose [x,y2] in c0;
then consider n21,n22 be Element of NAT such that
A7: [x,y2] = [[[n21,n22],[n21,n22]],[n21,n22]] & n22 = n21 + 1;
x = [[n21,n22],[n21,n22]] & y2 = [n21,n22] by A7,XTUPLE_0:1;
hence thesis by A6,XTUPLE_0:1;
end;
suppose [x,y2] in c1;
then consider n21,n22 be Element of NAT such that
A8: [x,y2] = [[[n21,n22],n22],n22] & n22 = n21 + 1;
x = [[n21,n22],n22] & y2 = n22 by A8,XTUPLE_0:1;
hence thesis by A6,XTUPLE_0:1;
end;
suppose [x,y2] in c2;
then consider n21,n22 be Element of NAT such that
A9: [x,y2] = [[n21,[n21,n22]],n21] & n22 = n21 + 1 & n21 <> 0;
x = [n21,[n21,n22]] & y2 = n21 by A9,XTUPLE_0:1;
hence thesis by A6,XTUPLE_0:1;
end;
suppose [x,y2] in c3;
then consider n21,n22 be Element of NAT such that
A10: [x,y2] = [[n21,n22],n21] & n22 = n21 + 1 & n21 <> 0;
x = [n21,n22] & y2 = n21 by A10,XTUPLE_0:1;
hence thesis by A6,XTUPLE_0:1;
end;
end;
suppose [x,y1] in c1;
then consider n11,n12 be Element of NAT such that
A11: [x,y1] = [[[n11,n12],n12],n12] & n12 = n11 + 1;
A12: x = [[n11,n12],n12] & y1 = n12 by A11,XTUPLE_0:1;
per cases by A4,XBOOLE_0:def 3;
suppose [x,y2] in c0;
then consider n21,n22 be Element of NAT such that
A13: [x,y2] = [[[n21,n22],[n21,n22]],[n21,n22]] & n22 = n21 + 1;
x = [[n21,n22],[n21,n22]] & y2 = [n21,n22] by A13,XTUPLE_0:1;
hence thesis by A12,XTUPLE_0:1;
end;
suppose [x,y2] in c1;
then consider n21,n22 be Element of NAT such that
A14: [x,y2] = [[[n21,n22],n22],n22] & n22 = n21 + 1;
x = [[n21,n22],n22] & y2 = n22 by A14,XTUPLE_0:1;
hence thesis by A12,XTUPLE_0:1;
end;
suppose [x,y2] in c2;
then consider n21,n22 be Element of NAT such that
A15: [x,y2] = [[n21,[n21,n22]],n21] & n22 = n21 + 1 & n21 <> 0;
x = [n21,[n21,n22]] & y2 = n21 by A15,XTUPLE_0:1;
hence thesis by A12,XTUPLE_0:1;
end;
suppose [x,y2] in c3;
then consider n21,n22 be Element of NAT such that
A16: [x,y2] = [[n21,n22],n21] & n22 = n21 + 1 & n21 <> 0;
x = [n21,n22] & y2 = n21 by A16,XTUPLE_0:1;
hence thesis by A12,XTUPLE_0:1;
end;
end;
suppose [x,y1] in c2;
then consider n11,n12 be Element of NAT such that
A17: [x,y1] = [[n11,[n11,n12]],n11] & n12 = n11 + 1 & n11 <> 0;
A18: x = [n11,[n11,n12]] & y1 = n11 by A17,XTUPLE_0:1;
per cases by A4,XBOOLE_0:def 3;
suppose [x,y2] in c0;
then consider n21,n22 be Element of NAT such that
A19: [x,y2] = [[[n21,n22],[n21,n22]],[n21,n22]] & n22 = n21 + 1;
x = [[n21,n22],[n21,n22]] & y2 = [n21,n22] by A19,XTUPLE_0:1;
hence thesis by A18,XTUPLE_0:1;
end;
suppose [x,y2] in c1;
then consider n21,n22 be Element of NAT such that
A20: [x,y2] = [[[n21,n22],n22],n22] & n22 = n21 + 1;
x = [[n21,n22],n22] & y2 = n22 by A20,XTUPLE_0:1;
hence thesis by A18,XTUPLE_0:1;
end;
suppose [x,y2] in c2;
then consider n21,n22 be Element of NAT such that
A21: [x,y2] = [[n21,[n21,n22]],n21] & n22 = n21 + 1 & n21 <> 0;
x = [n21,[n21,n22]] & y2 = n21 by A21,XTUPLE_0:1;
hence thesis by A18,XTUPLE_0:1;
end;
suppose [x,y2] in c3;
then consider n21,n22 be Element of NAT such that
A22: [x,y2] = [[n21,n22],n21] & n22 = n21 + 1 & n21 <> 0;
x = [n21,n22] & y2 = n21 by A22,XTUPLE_0:1;
hence thesis by A18,XTUPLE_0:1;
end;
end;
suppose [x,y1] in c3;
then consider n11,n12 be Element of NAT such that
A23: [x,y1] = [[n11,n12],n11] & n12 = n11 + 1 & n11 <> 0;
A24: x = [n11,n12] & y1 = n11 by A23,XTUPLE_0:1;
per cases by A4,XBOOLE_0:def 3;
suppose [x,y2] in c0;
then consider n21,n22 be Element of NAT such that
A25: [x,y2] = [[[n21,n22],[n21,n22]],[n21,n22]] & n22 = n21 + 1;
x = [[n21,n22],[n21,n22]] & y2 = [n21,n22] by A25,XTUPLE_0:1;
hence thesis by A24,XTUPLE_0:1;
end;
suppose [x,y2] in c1;
then consider n21,n22 be Element of NAT such that
A26: [x,y2] = [[[n21,n22],n22],n22] & n22 = n21 + 1;
x = [[n21,n22],n22] & y2 = n22 by A26,XTUPLE_0:1;
hence thesis by A24,XTUPLE_0:1;
end;
suppose [x,y2] in c2;
then consider n21,n22 be Element of NAT such that
A27: [x,y2] = [[n21,[n21,n22]],n21] & n22 = n21 + 1 & n21 <> 0;
x = [n21,[n21,n22]] & y2 = n21 by A27,XTUPLE_0:1;
hence thesis by A24,XTUPLE_0:1;
end;
suppose [x,y2] in c3;
then consider n21,n22 be Element of NAT such that
A28: [x,y2] = [[n21,n22],n21] & n22 = n21 + 1 & n21 <> 0;
x = [n21,n22] & y2 = n21 by A28,XTUPLE_0:1;
hence thesis by A24,XTUPLE_0:1;
end;
end;
end;
for x being element st x in comp holds x in [:[:X,X:],X:]
proof
let x be element;
assume x in comp;
then x in c0 \/ c1 \/ c2 or x in c3 by XBOOLE_0:def 3;
then
A29: x in c0 \/ c1 or x in c2 or x in c3 by XBOOLE_0:def 3;
per cases by A29,XBOOLE_0:def 3;
suppose x in c0;
then consider n1,n2 be Element of NAT such that
A30: x = [[[n1,n2],[n1,n2]],[n1,n2]] & n2 = n1 + 1;
[n1,n2] in X2 by A30;
then
A31: [n1,n2] in X by XBOOLE_0:def 3;
then [[n1,n2],[n1,n2]] in [:X,X:] by ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A30,A31,ZFMISC_1:def 2;
end;
suppose x in c1;
then consider n1,n2 be Element of NAT such that
A32: x = [[[n1,n2],n2],n2] & n2 = n1 + 1;
[n1,n2] in X2 by A32;
then
A33: [n1,n2] in X by XBOOLE_0:def 3;
not n2 in {0} by A32,TARSKI:def 1;
then n2 in X1 by XBOOLE_0:def 5;
then
A34: n2 in X by XBOOLE_0:def 3;
[[n1,n2],n2] in [:X,X:] by A33,A34,ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A32,A34,ZFMISC_1:def 2;
end;
suppose x in c2;
then consider n1,n2 be Element of NAT such that
A35: x = [[n1,[n1,n2]],n1] & n2 = n1 + 1 & n1 <> 0;
[n1,n2] in X2 by A35;
then
A36: [n1,n2] in X by XBOOLE_0:def 3;
not n1 in {0} by A35,TARSKI:def 1;
then n1 in X1 by XBOOLE_0:def 5;
then
A37: n1 in X by XBOOLE_0:def 3;
[n1,[n1,n2]] in [:X,X:] by A36,A37,ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A35,A37,ZFMISC_1:def 2;
end;
suppose x in c3;
then consider n1,n2 be Element of NAT such that
A38: x = [[n1,n2],n1] & n2 = n1 + 1 & n1 <> 0;
not n1 in {0} by A38,TARSKI:def 1;
then n1 in X1 by XBOOLE_0:def 5;
then
A39: n1 in X by XBOOLE_0:def 3;
not n2 in {0} by A38,TARSKI:def 1;
then n2 in X1 by XBOOLE_0:def 5;
then
A40: n2 in X by XBOOLE_0:def 3;
[n1,n2] in [:X,X:] by A39,A40,ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A38,A39,ZFMISC_1:def 2;
end;
end;
then reconsider comp as PartFunc of [:X,X:],X
by A2,TARSKI:def 3,FUNCT_1:def 1;
set C = CategoryStr(# X, comp #);
A41: for n1 being Element of NAT st n1 <> 0 holds n1 is morphism of C
proof
let n1 be Element of NAT;
assume n1 <> 0;
then not n1 in {0} by TARSKI:def 1;
then n1 in X1 by XBOOLE_0:def 5;
hence n1 is morphism of C by XBOOLE_0:def 3;
end;
A42: for f1,f2 being morphism of C, n1,n2 being Element of NAT st
f1 = n1 & f2 = n2 & n2 = n1 + 1 & n1 <> 0 holds f1 |> f2 & f1(*)f2 = f1
proof
let f1,f2 be morphism of C;
let n1,n2 be Element of NAT;
assume
A43: f1 = n1 & f2 = n2;
assume n2 = n1 + 1 & n1 <> 0;
then
A44: [[n1,n2],n1] in c3;
then
A45: [[n1,n2],n1] in c0 \/ c1 \/ c2 \/ c3 by XBOOLE_0:def 3;
A46: [[n1,n2],n1] in the composition of C by A44,XBOOLE_0:def 3;
A47: [f1,f2] in dom the composition of C by A43,A45,XTUPLE_0:def 12;
thus f1 |> f2 by A43,A45,XTUPLE_0:def 12;
thus f1(*)f2 = (the composition of C).(f1,f2) by A47,Def2,Def3
.= (the composition of C).[n1,n2] by A43,BINOP_1:def 1
.= f1 by A46,A43,FUNCT_1:1;
end;
A48: for f being morphism of C st f in X2 holds f is left_identity &
f is right_identity & f |> f
proof
let f be morphism of C;
assume f in X2;
then consider nf1,nf2 be Element of NAT such that
A49: f = [nf1,nf2] & nf2 = nf1 + 1;
for f1 being morphism of C st f |> f1 holds f (*) f1 = f1
proof
let f1 be morphism of C;
assume
A50: f |> f1;
then consider y be element such that
A51: [[f,f1],y] in the composition of C by XTUPLE_0:def 12;
[[f,f1],y] in c0 \/ c1 \/ c2 or [[f,f1],y] in c3
by A51,XBOOLE_0:def 3;
then
A52: [[f,f1],y] in c0 \/ c1 or [[f,f1],y] in c2
or [[f,f1],y] in c3 by XBOOLE_0:def 3;
per cases by A52,XBOOLE_0:def 3;
suppose [[f,f1],y] in c0;
then consider n1,n2 be Element of NAT such that
A53: [[f,f1],y] = [[[n1,n2],[n1,n2]],[n1,n2]] & n2 = n1 + 1;
[f,f1] = [[n1,n2],[n1,n2]] & y = [n1,n2] by A53,XTUPLE_0:1;
then
A54: f = [n1,n2] & f1 = [n1,n2] by XTUPLE_0:1;
thus f (*) f1 = (the composition of C).(f,f1) by A50,Def3
.= (the composition of C).[f,f1] by BINOP_1:def 1
.= f1 by A54,A51,FUNCT_1:1,A53;
end;
suppose [[f,f1],y] in c1;
then consider n1,n2 be Element of NAT such that
A55: [[f,f1],y] = [[[n1,n2],n2],n2] & n2 = n1 + 1;
[f,f1] = [[n1,n2],n2] & y = n2 by A55,XTUPLE_0:1;
then
A56: f = [n1,n2] & f1 = n2 by XTUPLE_0:1;
thus f (*) f1 = (the composition of C).(f,f1) by A50,Def3
.= (the composition of C).[f,f1] by BINOP_1:def 1
.= f1 by A56,A51,FUNCT_1:1,A55;
end;
suppose [[f,f1],y] in c2;
then consider n1,n2 be Element of NAT such that
A57: [[f,f1],y] = [[n1,[n1,n2]],n1] & n2 = n1 + 1 & n1 <> 0;
[f,f1] = [n1,[n1,n2]] by A57,XTUPLE_0:1;
hence thesis by A49,XTUPLE_0:1;
end;
suppose [[f,f1],y] in c3;
then consider n1,n2 be Element of NAT such that
A58: [[f,f1],y] = [[n1,n2],n1] & n2 = n1 + 1 & n1 <> 0;
[f,f1] = [n1,n2] by A58,XTUPLE_0:1;
hence thesis by A49,XTUPLE_0:1;
end;
end;
hence f is left_identity;
for f1 being morphism of C st f1 |> f holds f1 (*) f = f1
proof
let f1 be morphism of C;
assume
A59: f1 |> f;
then consider y be element such that
A60: [[f1,f],y] in the composition of C by XTUPLE_0:def 12;
[[f1,f],y] in c0 \/ c1 \/ c2 or [[f1,f],y] in c3
by A60,XBOOLE_0:def 3;
then
A61: [[f1,f],y] in c0 \/ c1 or [[f1,f],y] in c2
or [[f1,f],y] in c3 by XBOOLE_0:def 3;
per cases by A61,XBOOLE_0:def 3;
suppose [[f1,f],y] in c0;
then consider n1,n2 be Element of NAT such that
A62: [[f1,f],y] = [[[n1,n2],[n1,n2]],[n1,n2]] & n2 = n1 + 1;
[f1,f] = [[n1,n2],[n1,n2]] & y = [n1,n2] by A62,XTUPLE_0:1;
then
A63: f1 = [n1,n2] & f = [n1,n2] by XTUPLE_0:1;
thus f1 (*) f = (the composition of C).(f1,f) by A59,Def3
.= (the composition of C).[f1,f] by BINOP_1:def 1
.= f1 by A63,A60,FUNCT_1:1,A62;
end;
suppose [[f1,f],y] in c1;
then consider n1,n2 be Element of NAT such that
A64: [[f1,f],y] = [[[n1,n2],n2],n2] & n2 = n1 + 1;
[f1,f] = [[n1,n2],n2] & y = n2 by A64,XTUPLE_0:1;
hence thesis by A49,XTUPLE_0:1;
end;
suppose [[f1,f],y] in c2;
then consider n1,n2 be Element of NAT such that
A65: [[f1,f],y] = [[n1,[n1,n2]],n1] & n2 = n1 + 1 & n1 <> 0;
[f1,f] = [n1,[n1,n2]] & y = n1 by A65,XTUPLE_0:1;
then
A66: f1 = n1 & f = [n1,n2] by XTUPLE_0:1;
thus f1 (*) f = (the composition of C).(f1,f) by A59,Def3
.= (the composition of C).[f1,f] by BINOP_1:def 1
.= f1 by A66,A60,FUNCT_1:1,A65;
end;
suppose [[f1,f],y] in c3;
then consider n1,n2 be Element of NAT such that
A67: [[f1,f],y] = [[n1,n2],n1] & n2 = n1 + 1 & n1 <> 0;
[f1,f] = [n1,n2] by A67,XTUPLE_0:1;
hence thesis by A49,XTUPLE_0:1;
end;
end;
hence f is right_identity;
[[f,f],f] in c0 by A49;
then [[f,f],f] in c0 \/ c1 by XBOOLE_0:def 3;
then [[f,f],f] in c0 \/ c1 \/ c2 by XBOOLE_0:def 3;
then [[f,f],f] in c0 \/ c1 \/ c2 \/ c3 by XBOOLE_0:def 3;
hence f |> f by FUNCT_1:1;
end;
A68: for f1,f2 being morphism of C st f1 in X1 & f1 |> f2 holds f1(*)f2 = f1
proof
let f1,f2 be morphism of C;
assume
A69: f1 in X1;
reconsider nf1 = f1 as Element of NAT by A69;
assume
A70: f1 |> f2;
per cases by A1,XBOOLE_0:def 3;
suppose
A71: f2 in X1;
reconsider nf2 = f2 as Element of NAT by A71;
consider y be element such that
A72: [[f1,f2],y] in the composition of C by A70,XTUPLE_0:def 12;
[[f1,f2],y] in c0 \/ c1 \/ c2 or [[f1,f2],y] in c3
by A72,XBOOLE_0:def 3;
then
A73: [[f1,f2],y] in c0 \/ c1 or [[f1,f2],y] in c2
or [[f1,f2],y] in c3 by XBOOLE_0:def 3;
per cases by A73,XBOOLE_0:def 3;
suppose [[f1,f2],y] in c0;
then consider n1,n2 be Element of NAT such that
A74: [[f1,f2],y] = [[[n1,n2],[n1,n2]],[n1,n2]] & n2 = n1 + 1;
[f1,f2] = [[n1,n2],[n1,n2]] by A74,XTUPLE_0:1;
hence thesis by A71,XTUPLE_0:1;
end;
suppose [[f1,f2],y] in c1;
then consider n1,n2 be Element of NAT such that
A75: [[f1,f2],y] = [[[n1,n2],n2],n2] & n2 = n1 + 1;
[f1,f2] = [[n1,n2],n2] & y = n2 by A75,XTUPLE_0:1;
hence thesis by A69,XTUPLE_0:1;
end;
suppose [[f1,f2],y] in c2;
then consider n1,n2 be Element of NAT such that
A76: [[f1,f2],y] = [[n1,[n1,n2]],n1] & n2 = n1 + 1 & n1 <> 0;
[f1,f2] = [n1,[n1,n2]] by A76,XTUPLE_0:1;
hence thesis by A71,XTUPLE_0:1;
end;
suppose [[f1,f2],y] in c3;
then consider n1,n2 be Element of NAT such that
A77: [[f1,f2],y] = [[n1,n2],n1] & n2 = n1 + 1 & n1 <> 0;
[f1,f2] = [n1,n2] & y = n1 by A77,XTUPLE_0:1;
then
A78: f1 = n1 & f2 = n2 by XTUPLE_0:1;
thus f1 (*) f2 = (the composition of C).(f1,f2) by A70,Def3
.= (the composition of C).[f1,f2] by BINOP_1:def 1
.= f1 by A78,A72,FUNCT_1:1,A77;
end;
end;
suppose f2 in X2;
then f2 is right_identity by A48;
hence thesis by A70;
end;
end;
ex f,f1,f2 being morphism of C st f1 |> f2 & not( f1(*)f2 |> f iff f2 |> f)
proof
reconsider f1=1,f2=2,f=3 as morphism of C by A41;
take f,f1,f2;
A79: 2 = 1 + 1;
hence f1 |> f2 by A42;
A80: not f1 |> f
proof
assume f1 |> f;
then consider y be element such that
A81: [[f1,f],y] in the composition of C by XTUPLE_0:def 12;
[[f1,f],y] in c0 \/ c1 \/ c2 or [[f1,f],y] in c3
by A81,XBOOLE_0:def 3;
then
A82: [[f1,f],y] in c0 \/ c1 or [[f1,f],y] in c2
or [[f1,f],y] in c3 by XBOOLE_0:def 3;
per cases by A82,XBOOLE_0:def 3;
suppose [[f1,f],y] in c0;
then consider n1,n2 be Element of NAT such that
A83: [[f1,f],y] = [[[n1,n2],[n1,n2]],[n1,n2]] & n2 = n1 + 1;
[f1,f] = [[n1,n2],[n1,n2]] by A83,XTUPLE_0:1;
hence contradiction by XTUPLE_0:1;
end;
suppose [[f1,f],y] in c1;
then consider n1,n2 be Element of NAT such that
A84: [[f1,f],y] = [[[n1,n2],n2],n2] & n2 = n1 + 1;
[f1,f] = [[n1,n2],n2] by A84,XTUPLE_0:1;
hence contradiction by XTUPLE_0:1;
end;
suppose [[f1,f],y] in c2;
then consider n1,n2 be Element of NAT such that
A85: [[f1,f],y] = [[n1,[n1,n2]],n1] & n2 = n1 + 1 & n1 <> 0;
[f1,f] = [n1,[n1,n2]] by A85,XTUPLE_0:1;
hence contradiction by XTUPLE_0:1;
end;
suppose [[f1,f],y] in c3;
then consider n1,n2 be Element of NAT such that
A86: [[f1,f],y] = [[n1,n2],n1] & n2 = n1 + 1 & n1 <> 0;
[f1,f] = [n1,n2] by A86,XTUPLE_0:1;
then f1 = n1 & f = n2 by XTUPLE_0:1;
hence contradiction by A86;
end;
end;
3 = 2 + 1;
hence thesis by A42,A80,A79;
end;
then
A87: C is non left_composable;
for f,f1,f2 being morphism of C st f1 |> f2 holds f |> f1(*)f2 iff f |> f1
proof
let f,f1,f2 be morphism of C;
assume
A88: f1 |> f2;
per cases by A1,XBOOLE_0:def 3;
suppose
A89: f2 in X1;
per cases by A1,XBOOLE_0:def 3;
suppose f1 in X1;
hence thesis by A88,A68; end;
suppose
A90: f1 in X2;
then
A91: f1 is left_identity by A48;
then
A92: f1(*)f2 = f2 by A88;
consider n1,n2 be Element of NAT such that
A93: f1 = [n1,n2] & n2 = n1 + 1 by A90;
reconsider n3 = f2 as Element of NAT by A89;
A94: n2 = n3
proof
assume
A95: n2 <> n3;
consider y be element such that
A96: [[f1,f2],y] in the composition of C by A88,XTUPLE_0:def 12;
[[f1,f2],y] in c0 \/ c1 \/ c2 or [[f1,f2],y] in c3
by A96,XBOOLE_0:def 3;
then
A97: [[f1,f2],y] in c0 \/ c1 or [[f1,f2],y] in c2
or [[f1,f2],y] in c3 by XBOOLE_0:def 3;
per cases by A97,XBOOLE_0:def 3;
suppose [[f1,f2],y] in c0;
then consider n1,n2 be Element of NAT such that
A98: [[f1,f2],y] = [[[n1,n2],[n1,n2]],[n1,n2]] & n2 = n1 + 1;
[f1,f2] = [[n1,n2],[n1,n2]] by A98,XTUPLE_0:1;
hence contradiction by XTUPLE_0:1,A89;
end;
suppose [[f1,f2],y] in c1;
then consider n11,n22 be Element of NAT such that
A99: [[f1,f2],y] = [[[n11,n22],n22],n22] & n22 = n11 + 1;
[f1,f2] = [[n11,n22],n22] by A99,XTUPLE_0:1;
then f1 = [n11,n22] & f2 = n22 by XTUPLE_0:1;
hence contradiction by A95,A93,XTUPLE_0:1;
end;
suppose [[f1,f2],y] in c2;
then consider n1,n2 be Element of NAT such that
A100: [[f1,f2],y] = [[n1,[n1,n2]],n1] & n2 = n1 + 1 & n1 <> 0;
[f1,f2] = [n1,[n1,n2]] by A100,XTUPLE_0:1;
hence contradiction by XTUPLE_0:1,A89;
end;
suppose [[f1,f2],y] in c3;
then consider n1,n2 be Element of NAT such that
A101: [[f1,f2],y] = [[n1,n2],n1] & n2 = n1 + 1 & n1 <> 0;
[f1,f2] = [n1,n2] by A101,XTUPLE_0:1;
hence contradiction by A93,XTUPLE_0:1;
end;
end;
hereby
assume f |> f1(*)f2;
then consider y be element such that
A102: [[f,f2],y] in the composition of C by A92,XTUPLE_0:def 12;
[[f,f2],y] in c0 \/ c1 \/ c2 or [[f,f2],y] in c3
by A102,XBOOLE_0:def 3;
then
A103: [[f,f2],y] in c0 \/ c1 or [[f,f2],y] in c2
or [[f,f2],y] in c3 by XBOOLE_0:def 3;
per cases by A103,XBOOLE_0:def 3;
suppose [[f,f2],y] in c0;
then consider n,n2 be Element of NAT such that
A104: [[f,f2],y] = [[[n,n2],[n,n2]],[n,n2]] & n2 = n + 1;
[f,f2] = [[n,n2],[n,n2]] by A104,XTUPLE_0:1;
hence f |> f1 by XTUPLE_0:1,A89;
end;
suppose [[f,f2],y] in c1;
then consider n,n22 be Element of NAT such that
A105: [[f,f2],y] = [[[n,n22],n22],n22] & n22 = n + 1;
[f,f2] = [[n,n22],n22] by A105,XTUPLE_0:1;
then
A106: f = [n,n22] & f2 = n22 by XTUPLE_0:1;
[[f1,f1],f1] in c0 by A93;
then [[f1,f1],f1] in c0 \/ c1 by XBOOLE_0:def 3;
then [[f1,f1],f1] in c0 \/ c1 \/ c2 by XBOOLE_0:def 3;
then [[f1,f1],f1] in the composition of C by XBOOLE_0:def 3;
hence f |> f1 by A93,A106,A105,A94,XTUPLE_0:def 12;
end;
suppose [[f,f2],y] in c2;
then consider n,n2 be Element of NAT such that
A107: [[f,f2],y] = [[n,[n,n2]],n] & n2 = n + 1 & n <> 0;
[f,f2] = [n,[n,n2]] by A107,XTUPLE_0:1;
hence f |> f1 by A89,XTUPLE_0:1;
end;
suppose [[f,f2],y] in c3;
then consider n,n22 be Element of NAT such that
A108: [[f,f2],y] = [[n,n22],n] & n22 = n + 1 & n <> 0;
A109: [f,f2] = [n,n22] by A108,XTUPLE_0:1;
then
A110: f = n & f2 = n22 by XTUPLE_0:1;
A111: n + 1 = n1 + 1 by A94,A108,A93,A109,XTUPLE_0:1;
[[n1,[n1,n2]],n1] in c2 by A93,A108,A111;
then [[f,f1],f] in c0 \/ c1 \/ c2
by A93,A110,XBOOLE_0:def 3,A94,A108;
then [[f,f1],f] in the composition of C by XBOOLE_0:def 3;
hence f |> f1 by XTUPLE_0:def 12;
end;
end;
assume A112: f |> f1;
f |> f2
proof
consider y be element such that
A113: [[f,f1],y] in the composition of C by A112,XTUPLE_0:def 12;
[[f,f1],y] in c0 \/ c1 \/ c2 or [[f,f1],y] in c3
by A113,XBOOLE_0:def 3;
then
A114: [[f,f1],y] in c0 \/ c1 or [[f,f1],y] in c2
or [[f,f1],y] in c3 by XBOOLE_0:def 3;
per cases by A114,XBOOLE_0:def 3;
suppose [[f,f1],y] in c0;
then consider n,n1 be Element of NAT such that
A115: [[f,f1],y] = [[[n,n1],[n,n1]],[n,n1]] & n1 = n + 1;
[f,f1] = [[n,n1],[n,n1]] by A115,XTUPLE_0:1;
then f = [n,n1] & f1 = [n,n1] by XTUPLE_0:1;
hence f |> f2 by A88;
end;
suppose [[f,f1],y] in c1;
then consider n,n1 be Element of NAT such that
A116: [[f,f1],y] = [[[n,n1],n1],n1] & n1 = n + 1;
[f,f1] = [[n,n1],n1] by A116,XTUPLE_0:1;
hence f |> f2 by A93,XTUPLE_0:1;
end;
suppose [[f,f1],y] in c2;
then consider n,n11 be Element of NAT such that
A117: [[f,f1],y] = [[n,[n,n11]],n] & n11 = n + 1 & n <> 0;
[f,f1] = [n,[n,n11]] by A117,XTUPLE_0:1;
then
A118: f = n & f1 = [n,n11] by XTUPLE_0:1;
then
A119: n = n1 & n11 = n2 by A93,XTUPLE_0:1;
[[n1,n2],n1] in c3 by A117,A119;
then [[f,f2],f] in the composition of C
by A119,A118,A94,XBOOLE_0:def 3;
hence f |> f2 by XTUPLE_0:def 12;
end;
suppose [[f,f1],y] in c3;
then consider n,n1 be Element of NAT such that
A120: [[f,f1],y] = [[n,n1],n] & n1 = n + 1 & n <> 0;
[f,f1] = [n,n1] by A120,XTUPLE_0:1;
hence f |> f2 by A93,XTUPLE_0:1;
end;
end;
hence f |> f1(*)f2 by A91,A88;
end;
end;
suppose f2 in X2;
then f2 is right_identity by A48;
hence thesis by A88;
end;
end;
then
A121: C is right_composable;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f |> f1 & f is left_identity
proof
let f1 be morphism of C;
assume A122: f1 in the carrier of C;
per cases by A122,XBOOLE_0:def 3;
suppose
A123: f1 in X1;
then
A124: f1 in NAT & not f1 in {0} by XBOOLE_0:def 5;
reconsider n2 = f1 as Element of NAT by A123;
reconsider n22 = n2 as natural number;
n22 <> 0 by A124,TARSKI:def 1;
then consider n11 be natural number such that
A125: n22 = n11 + 1 by NAT_1:6;
reconsider n1 = n11 as Element of NAT by ORDINAL1:def 12;
A126: [n1,n2] in X2 by A125;
then reconsider f = [n1,n2] as morphism of C by XBOOLE_0:def 3;
take f;
[[[n1,n2],n2],n2] in c1 by A125;
then [[f,f1],f1] in c0 \/ c1 by XBOOLE_0:def 3;
then [[f,f1],f1] in c0 \/ c1 \/ c2 by XBOOLE_0:def 3;
then [[f,f1],f1] in the composition of C by XBOOLE_0:def 3;
hence f |> f1 by XTUPLE_0:def 12;
thus f is left_identity by A126,A48;
end;
suppose
A127: f1 in X2;
set f = f1;
take f;
thus thesis by A127,A48;
end;
end;
then
A128: C is with_left_identities;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f1 |> f & f is right_identity
proof
let f1 be morphism of C;
assume A129: f1 in the carrier of C;
per cases by A129,XBOOLE_0:def 3;
suppose
A130: f1 in X1;
then
A131: f1 in NAT & not f1 in {0} by XBOOLE_0:def 5;
reconsider n1 = f1 as Element of NAT by A130;
A132: n1 <> 0 by A131,TARSKI:def 1;
reconsider n2 = n1 + 1 as Element of NAT by ORDINAL1:def 12;
A133: [n1,n2] in X2;
then reconsider f = [n1,n2] as morphism of C by XBOOLE_0:def 3;
take f;
[[n1,[n1,n2]],n1] in c2 by A132;
then [[f1,f],f1] in c0 \/ c1 \/ c2 by XBOOLE_0:def 3;
then [[f1,f],f1] in the composition of C by XBOOLE_0:def 3;
hence f1 |> f by XTUPLE_0:def 12;
thus f is right_identity by A133,A48;
end;
suppose
A134: f1 in X2;
set f = f1;
take f;
thus thesis by A134,A48;
end;
end;
then
A135: C is with_identities by A128,Def7;
for f1,f2,f3 being morphism of C st f1 |> f2 & f2 |> f3 &
f1(*)f2 |> f3 & f1 |> f2(*)f3 holds f1(*)(f2(*)f3) = (f1(*)f2)(*)f3
proof
let f1,f2,f3 be morphism of C;
assume A136: f1 |> f2;
assume f2 |> f3;
assume A137: f1(*)f2 |> f3;
assume A138: f1 |> f2(*)f3;
A139: C is non empty by A136;
per cases by A139,XBOOLE_0:def 3;
suppose
A140: f1 in X1;
then
A141: f1(*)(f2(*)f3) = f1 & f1(*)f2 = f1 by A136,A138,A68;
thus f1(*)(f2(*)f3) = (f1(*)f2)(*)f3 by A140,A68,A141,A137;
end;
suppose f1 in X2;
then f1 is left_identity by A48;
then f1(*)(f2(*)f3) = f2(*)f3 & f1(*)f2 = f2 by A136,A138;
hence f1(*)(f2(*)f3) = (f1(*)f2)(*)f3;
end;
end;
hence thesis by A87,A121,A135,Def10;
end;
end;
registration
cluster left_composable non right_composable
with_identities associative for CategoryStr;
correctness
proof
set C1 = the non left_composable right_composable with_identities
associative CategoryStr;
set C = C1 opp;
take C;
thus C is left_composable by Th5;
thus C is non right_composable by Th4;
thus C is with_identities by Th6,Th7,Def12;
thus thesis by Th8;
end;
end;
registration
cluster non associative composable with_identities for CategoryStr;
correctness
proof
set X = NAT;
set comp = {[[n1,n2],n3] where n1,n2,n3 is Element of NAT :
(n1 <> n2 implies n3 = n1+n2) & (n1 = n2 implies n3 = 0)};
A1: for x,y1,y2 being element st [x,y1] in comp & [x,y2] in comp holds y1 = y2
proof
let x,y1,y2 be element;
assume [x,y1] in comp;
then consider n11,n12,n13 be Element of NAT such that
A2: [x,y1] = [[n11,n12],n13] & (n11 <> n12 implies n13 = n11+n12) &
(n11 = n12 implies n13 = 0);
assume [x,y2] in comp;
then consider n21,n22,n23 be Element of NAT such that
A3: [x,y2] = [[n21,n22],n23] & (n21 <> n22 implies n23 = n21+n22) &
(n21 = n22 implies n23 = 0);
A4: x = [n11,n12] & y1 = n13 by A2,XTUPLE_0:1;
x = [n21,n22] & y2 = n23 by A3,XTUPLE_0:1;
then
A5: n11 = n21 & n12 = n22 by A4,XTUPLE_0:1;
per cases;
suppose n11 <> n12;
thus y1 = y2 by A2,A3,A5,XTUPLE_0:1;
end;
suppose n11 = n12;
thus y1 = y2 by A2,A3,A5,XTUPLE_0:1;
end;
end;
for x being element st x in comp holds x in [:[:X,X:],X:]
proof
let x be element;
assume x in comp;
then consider n11,n12,n13 be Element of NAT such that
A6: x = [[n11,n12],n13] & (n11 <> n12 implies n13 = n11+n12) &
(n11 = n12 implies n13 = 0);
[n11,n12] in [:X,X:] by ZFMISC_1:def 2;
hence thesis by A6,ZFMISC_1:def 2;
end;
then reconsider comp as PartFunc of [:X,X:],X
by A1,TARSKI:def 3,FUNCT_1:def 1;
set C = CategoryStr(# X, comp #);
A7: for f1,f2 being morphism of C holds f1 |> f2
proof
let f1,f2 be morphism of C;
reconsider n1=f1,n2=f2 as Element of NAT;
per cases;
suppose
A8: n1 <> n2;
reconsider n3 = n1 + n2 as Element of NAT by ORDINAL1:def 12;
[[n1,n2],n3] in the composition of C by A8;
hence thesis by FUNCT_1:1;
end;
suppose
A9: n1 = n2;
set n3 = 0;
[[n1,n2],n3] in the composition of C by A9;
hence thesis by FUNCT_1:1;
end;
end;
A10: for f1,f2 being morphism of C st f1 = 0 holds f1(*)f2 = f2 & f2(*)f1 = f2
proof
let f1,f2 be morphism of C;
assume
A11: f1 = 0;
reconsider n2=f2 as Element of NAT;
[f1,f2] in dom the composition of C by A7,Def2;
then consider y be element such that
A12: [[f1,f2],y] in the composition of C by XTUPLE_0:def 12;
consider n1,n2,n3 be Element of NAT such that
A13: [[f1,f2],y] = [[n1,n2],n3] & (n1 <> n2 implies n3 = n1+n2) &
(n1 = n2 implies n3 = 0) by A12;
[f1,f2] = [n1,n2] & y = n3 by A13,XTUPLE_0:1;
then
A14: f1 = n1 & f2 = n2 by XTUPLE_0:1;
thus f1(*)f2 = (the composition of C).(f1,f2) by Def3,A7
.= (the composition of C).[f1,f2] by BINOP_1:def 1
.= f2 by FUNCT_1:1,A12,A13,A14,A11;
[f2,f1] in dom the composition of C by A7,Def2;
then consider y be element such that
A15: [[f2,f1],y] in the composition of C by XTUPLE_0:def 12;
consider n2,n1,n3 be Element of NAT such that
A16: [[f2,f1],y] = [[n2,n1],n3] & (n2 <> n1 implies n3 = n2+n1) &
(n2 = n1 implies n3 = 0) by A15;
[f2,f1] = [n2,n1] & y = n3 by A16,XTUPLE_0:1;
then
A17: f2 = n2 & f1 = n1 by XTUPLE_0:1;
thus f2(*)f1 = (the composition of C).(f2,f1) by Def3,A7
.= (the composition of C).[f2,f1] by BINOP_1:def 1
.= f2 by FUNCT_1:1,A15,A16,A17,A11;
end;
A18: C is left_composable by A7;
for f,f1,f2 being morphism of C st f1|>f2 holds f|>f1(*)f2 iff f|>f1
by A7;
then
A19: C is composable by A18,Def9;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f|>f1 & f is left_identity
proof
let f1 be morphism of C;
assume f1 in the carrier of C;
reconsider f = 0 as morphism of C;
take f;
thus f |> f1 by A7;
thus f is left_identity by A10;
end;
then
A20: C is with_left_identities;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f1|>f & f is right_identity
proof
let f1 be morphism of C;
assume f1 in the carrier of C;
reconsider f = 0 as morphism of C;
take f;
thus f1 |> f by A7;
thus f is right_identity by A10;
end;
then
A21: C is with_identities by A20,Def7;
A22: for f1,f2 being morphism of C, n1,n2 being Element of NAT st
f1 = n1 & f2 = n2 & n1 = n2 holds f1 (*) f2 = 0
proof
let f1,f2 be morphism of C;
let n1,n2 be Element of NAT;
assume
A23: f1 = n1 & f2 = n2;
assume
A24: n1 = n2;
[f1,f2] in dom the composition of C by A7,Def2;
then consider y be element such that
A25: [[f1,f2],y] in the composition of C by XTUPLE_0:def 12;
consider n11,n22,n33 be Element of NAT such that
A26: [[f1,f2],y] = [[n11,n22],n33] & (n11 <> n22 implies n33 = n11+n22) &
(n11 = n22 implies n33 = 0) by A25;
[f1,f2] = [n11,n22] & y = n33 by A26,XTUPLE_0:1;
then
A27: f1 = n11 & f2 = n22 by XTUPLE_0:1;
thus f1(*)f2 = (the composition of C).(f1,f2) by Def3,A7
.= (the composition of C).[f1,f2] by BINOP_1:def 1
.= 0 by A27,FUNCT_1:1,A25,A26,A24,A23;
end;
A28: for f1,f2 being morphism of C, n1,n2 being Element of NAT st
f1 = n1 & f2 = n2 & n1 <> n2 holds f1 (*) f2 = n1+n2
proof
let f1,f2 be morphism of C;
let n1,n2 be Element of NAT;
assume
A29: f1 = n1 & f2 = n2;
assume
A30: n1 <> n2;
[f1,f2] in dom the composition of C by A7,Def2;
then consider y be element such that
A31: [[f1,f2],y] in the composition of C by XTUPLE_0:def 12;
consider n11,n22,n33 be Element of NAT such that
A32: [[f1,f2],y] = [[n11,n22],n33] & (n11 <> n22 implies n33 = n11+n22) &
(n11 = n22 implies n33 = 0) by A31;
[f1,f2] = [n11,n22] & y = n33 by A32,XTUPLE_0:1;
then
A33: f1 = n11 & f2 = n22 by XTUPLE_0:1;
thus f1(*)f2 = (the composition of C).(f1,f2) by Def3,A7
.= (the composition of C).[f1,f2] by BINOP_1:def 1
.= n1+n2 by A33,FUNCT_1:1,A31,A32,A30,A29;
end;
ex f1,f2,f3 being morphism of C st f1 |> f2 & f2 |> f3 & f1 (*) f2 |> f3
& f1 |> f2 (*) f3 & not f1 (*) (f2 (*) f3) = (f1 (*) f2) (*) f3
proof
reconsider f1=1,f2=2,f3=3 as morphism of C;
take f1,f2,f3;
thus f1 |> f2 & f2 |> f3 & f1 (*) f2 |> f3 & f1 |> f2 (*) f3 by A7;
A34: f1 (*) f2 = 1+2 & f2 (*) f3 = 2+3 by A28;
reconsider f12 = 3, f23 = 5 as morphism of C;
f1 (*) f23 = 1+5 & f12 (*) f3 = 0 by A22,A28;
hence thesis by A34;
end;
hence thesis by A19,A21,Def10;
end;
end;
registration
cluster empty for CategoryStr;
correctness
proof
reconsider X = the empty set as set;
reconsider F = the PartFunc of [:the empty set,the empty set:],
the empty set as PartFunc of [:X,X:],X;
set C = CategoryStr (# X, F #);
take C;
thus thesis;
end;
end;
registration
cluster empty -> left_composable right_composable with_left_identities
with_right_identities associative for CategoryStr;
correctness
proof
let C be CategoryStr;
assume
A1: C is empty;
for f,f1,f2 being morphism of C holds
f1 |> f2 implies (f1(*)f2 |> f iff f2 |> f) by A1;
hence C is left_composable;
for f,f1,f2 being morphism of C holds
f1 |> f2 implies (f |> f1(*)f2 iff f |> f1) by A1;
hence C is right_composable;
thus C is with_left_identities by A1;
thus C is with_right_identities by A1;
for f1,f2,f3 being morphism of C holds
f1 |> f2 & f2 |> f3 & (f1(*)f2) |> f3 & f1 |> (f2(*)f3) implies
f1(*)(f2(*)f3) = (f1(*)f2)(*)f3 by A1;
hence thesis;
end;
end;
registration
cluster strict with_left_identities with_right_identities
left_composable right_composable associative for CategoryStr;
correctness
proof
set C = (the empty CategoryStr) opp;
take C;
thus thesis by Th4,Th5,Th8;
end;
end;
registration
cluster strict with_identities composable associative for CategoryStr;
correctness
proof
set C = the strict with_left_identities with_right_identities
left_composable right_composable associative CategoryStr;
take C;
thus thesis;
end;
end;
definition
mode category is with_identities composable associative CategoryStr;
end;
definition
let C,f;
attr f is identity means
f is left_identity right_identity;
end;
theorem Th9:
C is with_identities implies (f is left_identity iff f is right_identity)
proof
assume
A1: C is with_identities;
hereby
assume
A2: f is left_identity;
for f1 being morphism of C st f1 |> f holds f1 (*) f = f1
proof
let f1 be morphism of C;
assume
A3: f1 |> f;
then C is non empty;
then consider g be morphism of C such that
A4: f |> g & g is right_identity by A1,Def7;
f = f(*)g by A4 .= g by A2,A4;
hence f1 (*) f = f1 by A4,A3;
end;
hence f is right_identity;
end;
assume
A5: f is right_identity;
for f1 being morphism of C st f |> f1 holds f (*) f1 = f1
proof
let f1 be morphism of C;
assume
A6: f |> f1;
then C is non empty;
then consider g be morphism of C such that
A7: g |> f & g is left_identity by A1,Def6;
f = g(*)f by A7 .= g by A5,A7;
hence f (*) f1 = f1 by A7,A6;
end;
hence f is left_identity;
end;
theorem Th10:
C is empty implies f is identity
proof
assume C is empty;
then (for f1 being morphism of C st f |> f1 holds f (*) f1 = f1) &
(for f1 being morphism of C st f1 |> f holds f1 (*) f = f1);
then f is left_identity & f is right_identity;
hence f is identity;
end;
theorem Th11:
for g1,g2 being morphism of the CategoryStr of C
st f1 = g1 & f2 = g2 & f1 |> f2 holds f1 (*) f2 = g1 (*) g2
proof
let g1,g2 be morphism of the CategoryStr of C;
assume
A1: f1 = g1 & f2 = g2;
assume
A2: f1 |> f2;
then
A3: g1 |> g2 by A1;
thus f1 (*) f2 = (the composition of C).(f1,f2) by A2,Def3
.= g1 (*) g2 by A1,A3,Def3;
end;
theorem Th12:
C is left_composable iff the CategoryStr of C is left_composable
proof
hereby
assume
A1: C is left_composable;
for g,g1,g2 being morphism of the CategoryStr of C st g1 |> g2 holds
g1 (*) g2 |> g iff g2 |> g
proof
let g,g1,g2 be morphism of the CategoryStr of C;
reconsider f=g,f1=g1,f2=g2 as morphism of C;
assume g1 |> g2;
then
A2: f1 |> f2;
hereby
assume g1 (*) g2 |> g;
then f1 (*) f2 |> f by A2,Th11;
then f2 |> f by A2,A1;
hence g2 |> g;
end;
assume g2 |> g;
then f2 |> f;
then f1 (*) f2 |> f by A2,A1;
hence g1 (*) g2 |> g by A2,Th11;
end;
hence the CategoryStr of C is left_composable;
end;
assume
A3: the CategoryStr of C is left_composable;
for f,f1,f2 being morphism of C st f1 |> f2 holds
f1 (*) f2 |> f iff f2 |> f
proof
let f,f1,f2 be morphism of C;
reconsider g=f,g1=f1,g2=f2 as morphism of the CategoryStr of C;
assume
A4: f1 |> f2;
then
A5: g1 |> g2;
hereby
assume f1 (*) f2 |> f;
then g1 (*) g2 |> g by A4,Th11;
then g2 |> g by A3,A5;
hence f2 |> f;
end;
assume f2 |> f;
then g2 |> g;
then g1 (*) g2 |> g by A3,A5;
hence f1 (*) f2 |> f by A4,Th11;
end;
hence C is left_composable;
end;
theorem Th13:
C is right_composable iff the CategoryStr of C is right_composable
proof
hereby
assume
A1: C is right_composable;
for g,g1,g2 being morphism of the CategoryStr of C st g1 |> g2 holds
g |> g1 (*) g2 iff g |> g1
proof
let g,g1,g2 be morphism of the CategoryStr of C;
reconsider f=g,f1=g1,f2=g2 as morphism of C;
assume g1 |> g2;
then
A2: f1 |> f2;
hereby
assume g |> g1 (*) g2;
then f |> f1 (*) f2 by A2,Th11;
then f |> f1 by A2,A1;
hence g |> g1;
end;
assume g |> g1;
then f |> f1;
then f |> f1 (*) f2 by A2,A1;
hence g |> g1 (*) g2 by A2,Th11;
end;
hence the CategoryStr of C is right_composable;
end;
assume
A3: the CategoryStr of C is right_composable;
for f,f1,f2 being morphism of C st f1 |> f2 holds
f |> f1 (*) f2 iff f |> f1
proof
let f,f1,f2 be morphism of C;
reconsider g=f,g1=f1,g2=f2 as morphism of the CategoryStr of C;
assume
A4: f1 |> f2;
then
A5: g1 |> g2;
hereby
assume f |> f1 (*) f2;
then g |> g1 (*) g2 by A4,Th11;
then g |> g1 by A3,A5;
hence f |> f1;
end;
assume f |> f1;
then g |> g1;
then g |> g1 (*) g2 by A3,A5;
hence f |> f1 (*) f2 by A4,Th11;
end;
hence C is right_composable;
end;
theorem
C is composable iff the CategoryStr of C is composable by Th12,Th13;
theorem
C is associative iff the CategoryStr of C is associative
proof
hereby
assume A1: C is associative;
for g1,g2,g3 being morphism of the CategoryStr of C
st g1 |> g2 & g2 |> g3 & g1 (*) g2 |> g3 & g1 |> g2 (*) g3
holds g1(*)(g2(*)g3) = (g1(*)g2)(*)g3
proof
let g1,g2,g3 be morphism of the CategoryStr of C;
reconsider f1=g1,f2=g2,f3=g3 as morphism of C;
assume g1 |> g2 & g2 |> g3;
then
A2: f1 |> f2 & f2 |> f3;
A3: f1 (*) f2 = g1 (*) g2 & f2 (*) f3 = g2 (*) g3 by A2,Th11;
assume g1 (*) g2 |> g3 & g1 |> g2 (*) g3;
then
A4: f1 (*) f2 |> f3 & f1 |> f2 (*) f3 by A2,Th11;
then f1(*)(f2(*)f3) = (f1(*)f2)(*)f3 by A2,A1;
hence g1(*)(g2(*)g3) = (f1(*)f2)(*)f3 by A4,A3,Th11
.= (g1(*)g2)(*)g3 by A4,A3,Th11;
end;
hence the CategoryStr of C is associative;
end;
assume
A5: the CategoryStr of C is associative;
for f1,f2,f3 being morphism of C st f1 |> f2 & f2 |> f3 &
f1 (*) f2 |> f3 & f1 |> f2 (*) f3 holds f1(*)(f2(*)f3) = (f1(*)f2)(*)f3
proof
let f1,f2,f3 be morphism of C;
reconsider g1=f1,g2=f2,g3=f3 as morphism of the CategoryStr of C;
assume
A6: f1 |> f2 & f2 |> f3;
then
A7: g1 |> g2 & g2 |> g3;
A8: f1 (*) f2 = g1 (*) g2 & f2 (*) f3 = g2 (*) g3 by A6,Th11;
assume
A9: f1 (*) f2 |> f3 & f1 |> f2 (*) f3;
then g1 (*) g2 |> g3 & g1 |> g2 (*) g3 by A6,Th11;
then g1(*)(g2(*)g3) = (g1(*)g2)(*)g3 by A7,A5;
hence f1(*)(f2(*)f3) = (g1(*)g2)(*)g3 by A9,A8,Th11
.= (f1(*)f2)(*)f3 by A9,A8,Th11;
end;
hence C is associative;
end;
theorem Th16:
for g being morphism of the CategoryStr of C st f = g
holds f is left_identity iff g is left_identity
proof
let g be morphism of the CategoryStr of C;
assume
A1: f = g;
hereby
assume
A2: f is left_identity;
for g1 being morphism of the CategoryStr of C st g |> g1
holds g (*) g1 = g1
proof
let g1 be morphism of the CategoryStr of C;
reconsider f1 = g1 as morphism of C;
assume g |> g1;
then
A3: f |> f1 by A1;
then f (*) f1 = f1 by A2;
hence g (*) g1 = g1 by A1,A3,Th11;
end;
hence g is left_identity;
end;
assume
A4: g is left_identity;
for f2 being morphism of C st f |> f2 holds f (*) f2 = f2
proof
let f2 be morphism of C;
reconsider g2 = f2 as morphism of the CategoryStr of C;
assume
A5: f |> f2;
then g |> g2 by A1;
then g (*) g2 = g2 by A4;
hence f (*) f2 = f2 by A1,A5,Th11;
end;
hence f is left_identity;
end;
theorem Th17:
C is with_left_identities iff the CategoryStr of C is with_left_identities
proof
hereby
assume
A1: C is with_left_identities;
for g1 being morphism of the CategoryStr of C st
g1 in the carrier of the CategoryStr of C holds
ex g being morphism of the CategoryStr of C st
g |> g1 & g is left_identity
proof
let g1 be morphism of the CategoryStr of C;
reconsider f1 = g1 as morphism of C;
assume g1 in the carrier of the CategoryStr of C;
then consider f be morphism of C such that
A2: f |> f1 & f is left_identity by A1;
reconsider g = f as morphism of the CategoryStr of C;
take g;
thus g |> g1 by A2;
thus g is left_identity by A2,Th16;
end;
hence the CategoryStr of C is with_left_identities;
end;
assume
A3: the CategoryStr of C is with_left_identities;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f |> f1 & f is left_identity
proof
let f1 be morphism of C;
reconsider g1 = f1 as morphism of the CategoryStr of C;
assume f1 in the carrier of C;
then consider g be morphism of the CategoryStr of C such that
A4: g |> g1 & g is left_identity by A3;
reconsider f = g as morphism of C;
take f;
thus f |> f1 by A4;
thus f is left_identity by A4,Th16;
end;
hence C is with_left_identities;
end;
theorem Th18:
for g being morphism of the CategoryStr of C st f = g
holds f is right_identity iff g is right_identity
proof
let g be morphism of the CategoryStr of C;
assume
A1: f = g;
hereby
assume
A2: f is right_identity;
for g1 being morphism of the CategoryStr of C st g1 |> g
holds g1 (*) g = g1
proof
let g1 be morphism of the CategoryStr of C;
reconsider f1 = g1 as morphism of C;
assume g1 |> g;
then
A3: f1 |> f by A1;
then f1 (*) f = f1 by A2;
hence g1 (*) g = g1 by A1,A3,Th11;
end;
hence g is right_identity;
end;
assume
A4: g is right_identity;
for f1 being morphism of C st f1 |> f holds f1 (*) f = f1
proof
let f1 be morphism of C;
reconsider g1 = f1 as morphism of the CategoryStr of C;
assume
A5: f1 |> f;
then g1 |> g by A1;
then g1 (*) g = g1 by A4;
hence f1 (*) f = f1 by A1,A5,Th11;
end;
hence f is right_identity;
end;
theorem Th19:
C is with_right_identities iff the CategoryStr of C is with_right_identities
proof
hereby
assume
A1: C is with_right_identities;
for g1 being morphism of the CategoryStr of C st
g1 in the carrier of the CategoryStr of C holds
ex g being morphism of the CategoryStr of C st
g1 |> g & g is right_identity
proof
let g1 be morphism of the CategoryStr of C;
reconsider f1 = g1 as morphism of C;
assume g1 in the carrier of the CategoryStr of C;
then consider f be morphism of C such that
A2: f1 |> f & f is right_identity by A1;
reconsider g = f as morphism of the CategoryStr of C;
take g;
thus g1 |> g by A2;
thus g is right_identity by A2,Th18;
end;
hence the CategoryStr of C is with_right_identities;
end;
assume
A3: the CategoryStr of C is with_right_identities;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f1 |> f & f is right_identity
proof
let f1 be morphism of C;
reconsider g1 = f1 as morphism of the CategoryStr of C;
assume f1 in the carrier of C;
then consider g be morphism of the CategoryStr of C such that
A4: g1 |> g & g is right_identity by A3;
reconsider f = g as morphism of C;
take f;
thus f1 |> f by A4;
thus f is right_identity by A4,Th18;
end;
hence C is with_right_identities;
end;
theorem
C is with_identities iff the CategoryStr of C is with_identities
by Th17,Th19;
definition
let C;
attr C is discrete means :Def15:
for f being morphism of C holds f is identity;
end;
Lm1:
(the empty CategoryStr) opp is discrete
proof
set C = (the empty CategoryStr) opp;
for f being morphism of C holds f is identity
proof
let f be morphism of C;
for f1 being morphism of C st f |> f1 holds f (*) f1 = f1;
then
A1: f is left_identity;
for f1 being morphism of C st f1 |> f holds f1 (*) f = f1;
hence f is identity by A1,Def5;
end;
hence thesis;
end;
registration
cluster strict empty discrete with_identities composable associative
for CategoryStr;
correctness
proof
set C = (the empty CategoryStr) opp;
take C;
C is left_composable right_composable with_left_identities
with_right_identities associative by Th4,Th5,Th8;
hence thesis by Lm1;
end;
end;
theorem Th21:
for C being discrete CategoryStr, f1,f2 being morphism of C
st f1 |> f2 holds f1 = f2 & f1 (*) f2 = f2
proof
let C be discrete CategoryStr;
let f1,f2 be morphism of C;
assume
A1: f1 |> f2;
f2 is identity by Def15;
then
A2: f1 (*) f2 = f1 by Def5,A1;
f1 is identity by Def15;
hence thesis by A2,A1,Def4;
end;
registration
cluster discrete -> composable associative for CategoryStr;
correctness
proof
let C be CategoryStr;
assume
A1: C is discrete;
A2: for f,g,h being morphism of C holds
(h(*)g |> f & h |> g implies g |> f) &
(h |> g(*)f & g |> f implies h |> g) &
(g |> f & h |> g implies h(*)g |> f & h |> g(*)f)
proof
let f,g,h be morphism of C;
thus h(*)g |> f & h |> g implies g |> f by A1,Th21;
thus h |> g(*)f & g |> f implies h |> g
proof
assume
A3: h |> g(*)f & g |> f;
then g(*)f = f & f = g by A1,Th21;
hence h |> g by A3;
end;
assume A4: g |> f;
assume A5: h |> g;
thus h(*)g |> f by A4,A5,A1,Th21;
g |> g(*)f & h = g by A4,A5,A1,Th21;
hence h |> g(*)f;
end;
A6: for f,g,h being morphism of C st h |> g & g |> f
holds h(*)(g(*)f) = (h(*)g)(*)f
proof
let f,g,h be morphism of C;
assume A7: h |> g;
assume g |> f;
then
A8: f = g & g (*) f = f by A1,Th21;
thus h(*)(g(*)f) = g by A1,Th21,A7,A8
.= (h(*)g)(*)f by A1,Th21,A7,A8;
end;
A9: C is left_composable by A2;
for f,f1,f2 being morphism of C holds
f1 |> f2 implies (f |> f1(*)f2 iff f |> f1) by A2;
hence thesis by A6,A9,Def9;
end;
end;
definition
let X be set;
func DiscreteCat(X) -> strict discrete category means :Def16:
the carrier of it = X;
existence
proof
per cases;
suppose
A1: X is empty;
set C = the strict empty discrete with_identities composable associative
CategoryStr;
take C;
thus thesis by A1;
end;
suppose X is not empty;
set F = {[[x,x],x] where x is Element of X : x in X};
A2: for x,y1,y2 being element st [x,y1] in F & [x,y2] in F
holds y1 = y2
proof
let x,y1,y2 be element;
assume [x,y1] in F;
then consider x1 be Element of X such that
A3: [x,y1] = [[x1,x1],x1] & x1 in X;
assume [x,y2] in F;
then consider x2 be Element of X such that
A4: [x,y2] = [[x2,x2],x2] & x2 in X;
A5: x = [x1,x1] & y1 = x1 by A3,XTUPLE_0:1;
x = [x2,x2] & y2 = x2 by A4,XTUPLE_0:1;
hence y1 = y2 by A5,XTUPLE_0:1;
end;
for x being element st x in F holds x in [:[:X,X:],X:]
proof
let x be element;
assume x in F;
then consider x1 be Element of X such that
A6: x = [[x1,x1],x1] & x1 in X;
[x1,x1] in [:X,X:] by A6,ZFMISC_1:87;
hence x in [:[:X,X:],X:] by A6,ZFMISC_1:87;
end;
then reconsider F as PartFunc of [:X,X:],X
by A2,TARSKI:def 3,FUNCT_1:def 1;
set C = CategoryStr(# X, F #);
for u being morphism of C holds u is identity
proof
let u be morphism of C;
A7: for f being morphism of C st u |> f holds u (*) f = f
proof
let f be morphism of C;
assume
A8: u |> f ;
u (*) f = F.(u,f) by A8,Def3
.= F.[u,f] by BINOP_1:def 1;
then [[u,f], u (*) f] in F by A8,FUNCT_1:def 2;
then consider x be Element of X such that
A9: [[u,f], u (*) f] = [[x,x],x] & x in X;
[u,f] = [x,x] & u (*) f = x by A9,XTUPLE_0:1;
hence u (*) f = f by XTUPLE_0:1;
end;
for g being morphism of C st g |> u holds g (*) u = g
proof
let g be morphism of C;
assume
A10: g |> u;
g (*) u = F.(g,u) by A10,Def3
.= F.[g,u] by BINOP_1:def 1;
then [[g,u], g (*) u] in F by A10,FUNCT_1:def 2;
then consider x be Element of X such that
A11: [[g,u], g (*) u] = [[x,x],x] & x in X;
[g,u] = [x,x] & g (*) u = x by A11,XTUPLE_0:1;
hence g (*) u = g by XTUPLE_0:1;
end;
then u is left_identity & u is right_identity by A7;
hence u is identity;
end;
then reconsider C as strict discrete CategoryStr by Def15;
A12: now
let f be morphism of C;
assume
A13: f in the carrier of C;
then
A14: f is identity & f in X by Def15;
reconsider x = f as Element of X;
[[f,f],f] in {[[x,x],x] where x is Element of X : x in X} by A13;
then [f,f] in dom the composition of C by XTUPLE_0:def 12;
hence
(ex c being morphism of C st c |> f & c is left_identity) &
(ex d being morphism of C st f |> d & d is right_identity)
by Def2,A14;
end;
C is with_left_identities & C is with_right_identities by A12;
then reconsider C as strict discrete composable associative
with_identities CategoryStr by Def12;
take C;
thus thesis;
end;
end;
uniqueness
proof
let C1,C2 be strict discrete category;
assume
A15: the carrier of C1 = X & the carrier of C2 = X;
A16: for C being discrete with_identities CategoryStr, x being element
st C is non empty holds
x in the composition of C iff ex f being morphism of C st x = [[f,f],f]
proof
let C be discrete with_identities CategoryStr;
let x be element;
assume
A17: C is non empty;
hereby
assume
A18: x in the composition of C;
then consider y,f be element such that
A19: y in [:the carrier of C,the carrier of C:] & f in the carrier of C &
x = [y,f] by ZFMISC_1:def 2;
reconsider f as morphism of C by A19;
take f;
consider f1,f2 be element such that
A20: f1 in the carrier of C & f2 in the carrier of C &
y = [f1,f2] by A19,ZFMISC_1:def 2;
reconsider f1,f2 as morphism of C by A20;
A21: [f1,f2] in dom the composition of C by A20,A19,A18,XTUPLE_0:def 12;
A22: f1 |> f2 by A20,A19,A18,XTUPLE_0:def 12;
then
f1 = f2 & f1 (*) f2 = f2 by Th21;
then (the composition of C).(f1,f2) = f2 by A21,Def3,Def2;
then (the composition of C).y = f2 by A20,BINOP_1:def 1;
then f = f2 by A18,A19,FUNCT_1:1;
hence x = [[f,f],f] by A22,A20,A19,Th21;
end;
given f be morphism of C such that
A23: x = [[f,f],f];
consider f1 be morphism of C such that
A24: f1 |> f & f1 is left_identity by A17,Def6,Def12;
A25: f is identity by Def15;
A26: f1 = f1 (*) f by A25,A24,Def5 .= f by A24;
f = f (*) f by A24,A26;
then (the composition of C).(f,f) = f by A24,A26,Def3;
then (the composition of C).[f,f] = f by BINOP_1:def 1;
hence x in the composition of C by A23,A24,A26,FUNCT_1:1;
end;
per cases;
suppose the carrier of C1 is empty; hence thesis by A15; end;
suppose
the carrier of C1 is non empty;
then
A27: C1 is non empty & C2 is non empty by A15;
for x being element holds
x in the composition of C1 iff x in the composition of C2
proof
let x be element;
hereby
assume x in the composition of C1;
then consider f be morphism of C1 such that
A28: x = [[f,f],f] by A27,A16;
reconsider f1 = f as morphism of C2 by A15;
x = [[f1,f1],f1] by A28;
hence x in the composition of C2 by A27,A16;
end;
assume x in the composition of C2;
then consider f be morphism of C2 such that
A29: x = [[f,f],f] by A27,A16;
reconsider f1 = f as morphism of C1 by A15;
x = [[f1,f1],f1] by A29;
hence x in the composition of C1 by A27,A16;
end;
hence thesis by A15,TARSKI:2;
end;
end;
end;
registration
cluster strict for category;
correctness
proof
set C = DiscreteCat(the empty set);
take C;
thus thesis;
end;
cluster strict empty for category;
correctness
proof
set C = DiscreteCat(the empty set);
take C;
thus thesis by Def16;
end;
cluster strict non empty for category;
correctness
proof
set C = DiscreteCat(the non empty set);
take C;
thus thesis by Def16;
end;
end;
definition
let C;
func Ob(C) -> Subset of Mor(C) equals
{f where f is morphism of C: f is identity & f in Mor(C)};
correctness
proof
set O = {f where f is morphism of C: f is identity & f in Mor(C)};
for x being element st x in O holds x in Mor(C)
proof
let x be element;
assume x in O;
then consider f be morphism of C such that
A1: x = f & f is identity & f in Mor(C);
thus thesis by A1;
end;
hence thesis by TARSKI:def 3;
end;
end;
definition
let C;
mode Object of C is Element of Ob(C);
end;
registration
let C be non empty with_identities CategoryStr;
cluster Ob(C) -> non empty;
correctness
proof
consider f be element such that
A1: f in Mor(C) by XBOOLE_0:def 1;
reconsider f as morphism of C by A1;
consider c be morphism of C such that
A2: c |> f & c is left_identity by Def12,Def6;
c is identity by A2,Th9;
then
c in {u where u is morphism of C: u is identity & u in Mor(C)};
hence thesis;
end;
end;
theorem Th22:
for C being non empty with_identities CategoryStr, f being morphism of C
holds f is identity iff f is Object of C
proof
let C be non empty with_identities CategoryStr;
let f be morphism of C;
hereby
assume f is identity;
then f in {u where u is morphism of C: u is identity & u in Mor(C)};
hence f is Object of C;
end;
assume f is Object of C;
then f in {u where u is morphism of C: u is identity & u in Mor(C)};
then consider u be morphism of C such that
A1: f = u & u is identity & u in Mor(C);
thus f is identity by A1;
end;
theorem Th23:
for C being non empty with_identities CategoryStr, f,f1 being morphism of C,
o being Object of C st f = o holds
(f |> f1 implies f (*) f1 = f1) &
(f1 |> f implies f1 (*) f = f1) & f |> f
proof
let C be non empty with_identities CategoryStr;
let f,f1 be morphism of C;
let o be Object of C;
assume A1: f = o;
A2: f is identity by A1,Th22;
consider c be morphism of C such that
A3: c |> f & c is left_identity by Def12,Def6;
A4: c = c (*) f by A2,A3,Def5 .= f by A3;
thus f |> f1 implies f (*) f1 = f1 by A2,Def4;
thus f1 |> f implies f1 (*) f = f1 by A2,Def5;
thus f |> f by A3,A4;
end;
theorem Th24:
for C being non empty with_identities CategoryStr, f being morphism of C
st f is identity holds f |> f
proof
let C be non empty with_identities CategoryStr;
let f be morphism of C;
assume f is identity;
then f is Object of C by Th22;
hence thesis by Th23;
end;
theorem Th25:
for C1,C2 being with_identities CategoryStr
st the CategoryStr of C1 = the CategoryStr of C2
holds for f1 being morphism of C1, f2 being morphism of C2 st f1 = f2 holds
f1 is identity iff f2 is identity
proof
let C1,C2 be with_identities CategoryStr;
assume
A1: the CategoryStr of C1 = the CategoryStr of C2;
let f1 be morphism of C1;
let f2 be morphism of C2;
assume
A2: f1 = f2;
hereby
assume
A3: f1 is identity;
A4: for f being morphism of C2 st f2 |> f holds f2 (*) f = f
proof
let f be morphism of C2;
assume
A5: f2 |> f;
reconsider f3=f as morphism of C1 by A1;
thus f2 (*) f = (the composition of C2).(f2,f) by A5,Def3
.= f1 (*) f3 by A5,A2,A1,Def3,Def2
.= f by A5,A3,Def4,Def2,A1,A2;
end;
for f being morphism of C2 st f,f2 are_composable holds f (*) f2 = f
proof
let f be morphism of C2;
assume
A6: f |> f2;
reconsider f3=f as morphism of C1 by A1;
thus f (*) f2 = (the composition of C2).(f,f2) by A6,Def3
.= f3 (*) f1 by A6,A2,A1,Def3,Def2
.= f by A6,A3,Def5,Def2,A1,A2;
end;
then f2 is right_identity;
hence f2 is identity by A4,Def4;
end;
assume
A7: f2 is identity;
A8: for f being morphism of C1 st f1 |> f holds f1 (*) f = f
proof
let f be morphism of C1;
assume
A9: f1 |> f;
reconsider f3=f as morphism of C2 by A1;
thus f1 (*) f = (the composition of C1).(f1,f) by A9,Def3
.= f2 (*) f3 by A9,A2,A1,Def3,Def2
.= f by A9,A7,Def4,Def2,A1,A2;
end;
for f being morphism of C1 st f |> f1 holds f (*) f1 = f
proof
let f be morphism of C1;
assume
A10: f |> f1;
reconsider f3=f as morphism of C2 by A1;
thus f (*) f1 = (the composition of C1).(f,f1) by A10,Def3
.= f3 (*) f2 by A10,A2,A1,Def3,Def2
.= f by A10,A7,Def5,Def2,A1,A2;
end;
then f1 is right_identity;
hence f1 is identity by A8,Def4;
end;
Lm2:
for C being CategoryStr holds C is with_identities iff
for f being morphism of C st f in Mor(C) holds
(ex c being morphism of C st c |> f & c is identity) &
(ex d being morphism of C st f |> d & d is identity)
proof
let C be CategoryStr;
hereby
assume
A1: C is with_identities;
let f be morphism of C;
assume A2: f in Mor(C);
thus (ex c being morphism of C st c |> f & c is identity)
proof
consider c be morphism of C such that
A3: c |> f & c is left_identity by A2,A1,Def6;
take c;
thus c |> f by A3;
thus c is identity by A3,Th9,A1;
end;
consider d be morphism of C such that
A4: f |> d & d is right_identity by A2,A1,Def7;
take d;
thus f |> d by A4;
thus d is identity by A4,A1,Th9;
end;
assume
A5: for f being morphism of C st f in Mor(C) holds
(ex c being morphism of C st c |> f & c is identity) &
(ex d being morphism of C st f |> d & d is identity);
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f |> f1 & f is left_identity
proof
let f1 be morphism of C;
assume f1 in the carrier of C;
then consider f be morphism of C such that
A6: f |> f1 & f is identity by A5;
take f;
thus f |> f1 by A6;
thus f is left_identity by A6;
end;
then
A7: C is with_left_identities;
for f1 being morphism of C st f1 in the carrier of C holds
ex f being morphism of C st f1 |> f & f is right_identity
proof
let f1 be morphism of C;
assume f1 in the carrier of C;
then consider f be morphism of C such that
A8: f1 |> f & f is identity by A5;
take f;
thus f1 |> f by A8;
thus f is right_identity by A8;
end;
hence C is with_identities by A7,Def7;
end;
Lm3:
for C being CategoryStr holds C is composable iff
for f,g,h being morphism of C holds
(h(*)g |> f & h |> g implies g |> f) &
(h |> g(*)f & g |> f implies h |> g) &
(g |> f & h |> g implies
h(*)g |> f & h |> g(*)f)
proof
let C be CategoryStr;
hereby
assume C is composable;
then
A1: C is left_composable & C is right_composable;
let f,g,h be morphism of C;
thus h(*)g |> f & h |> g implies g |> f by A1;
thus h |> g(*)f & g |> f implies h |> g by A1;
assume A2: g |> f;
assume A3: h |> g;
thus h(*)g |> f by A2,A3,A1;
thus h |> g(*)f by A2,A3,A1;
end;
assume A4: for f,g,h being morphism of C holds
(h(*)g |> f & h |> g implies g |> f)
& (h |> g(*)f & g |> f
implies h |> g) & (g |> f & h |> g
implies h(*)g |> f & h |> g(*)f);
then
A5: C is left_composable;
for f, f1, f2 being morphism of C st f1 |> f2 holds
f |> f1 (*) f2 iff f |> f1 by A4;
hence C is composable by A5,Def9;
end;
definition
let C be composable with_identities CategoryStr;
let f be morphism of C;
func dom f -> Object of C means :Def18:
ex f1 being morphism of C st it = f1 & f |> f1 & f1 is identity
if C is non empty otherwise it = the Object of C;
existence
proof
thus not C is empty implies ex o being Object of C,
f1 being morphism of C st o = f1 & f |> f1 & f1 is identity
proof
assume
A1: not C is empty;
then consider f1 be morphism of C such that
A2: f |> f1 & f1 is identity by Lm2;
reconsider o = f1 as Object of C by A2,A1,Th22;
take o,f1;
thus thesis by A2;
end;
assume C is empty;
take the Object of C;
thus thesis;
end;
uniqueness
proof
now
let o1,o2 be Object of C;
assume not C is empty;
assume
ex f1 being morphism of C st o1=f1 & f |> f1 & f1 is identity;
then consider f11 be morphism of C such that
A3: o1=f11 & f |> f11 & f11 is identity;
assume
ex f1 being morphism of C st o2=f1 & f |> f1 & f1 is identity;
then consider f12 be morphism of C such that
A4: o2=f12 & f |> f12 & f12 is identity;
f(*)f11 = f & f(*)f12 = f by A3,A4,Def5;
then
A5: f11 |> f12 by A3,A4,Lm3;
f11 = f11 (*) f12 by A5,A4,Def5 .= f12 by A5,A3,Def4;
hence o1 = o2 by A3,A4;
end;
hence thesis;
end;
correctness;
func cod f -> Object of C means :Def19:
ex f1 being morphism of C st it = f1 & f1 |> f & f1 is identity
if C is non empty otherwise it = the Object of C;
existence
proof
thus not C is empty implies ex o being Object of C,
f1 being morphism of C st
o = f1 & f1 |> f & f1 is identity
proof
assume
A6: not C is empty;
then consider f1 be morphism of C such that
A7: f1 |> f & f1 is identity by Lm2;
reconsider o = f1 as Object of C by A7,A6,Th22;
take o,f1;
thus thesis by A7;
end;
assume C is empty;
take the Object of C;
thus thesis;
end;
uniqueness
proof
now
let o1,o2 be Object of C;
assume not C is empty;
assume
ex f1 being morphism of C st o1=f1 & f1 |> f & f1 is identity;
then consider f11 be morphism of C such that
A8: o1=f11 & f11 |> f & f11 is identity;
assume
ex f1 being morphism of C st o2=f1 & f1 |> f & f1 is identity;
then consider f12 be morphism of C such that
A9: o2=f12 & f12 |> f & f12 is identity;
f11(*)f = f & f12(*)f = f by A8,A9,Def4;
then
A10: f11 |> f12 by A8,A9,Lm3;
f11 = f11 (*) f12 by A10,A9,Def5 .= f12 by A10,A8,Def4;
hence o1 = o2 by A8,A9;
end;
hence thesis;
end;
correctness;
end;
theorem Th26:
for C being composable with_identities CategoryStr, f,f1 being morphism of C
st f |> f1 & f1 is identity holds dom f = f1
proof
let C be composable with_identities CategoryStr;
let f,f1 be morphism of C;
assume
A1: f |> f1 & f1 is identity;
then
A2: C is non empty;
then reconsider o = f1 as Object of C by A1,Th22;
ex f11 being morphism of C st o = f11 & f |> f11 & f11 is identity by A1;
hence dom f = f1 by A2,Def18;
end;
theorem Th27:
for C being composable with_identities CategoryStr, f,f1 being morphism of C
st f1 |> f & f1 is identity holds cod f = f1
proof
let C be composable with_identities CategoryStr;
let f,f1 be morphism of C;
assume
A1: f1 |> f & f1 is identity;
then
A2: C is non empty;
then reconsider o = f1 as Object of C by A1,Th22;
ex f11 being morphism of C st o = f11 & f11 |> f & f11 is identity by A1;
hence cod f = f1 by A2,Def19;
end;
definition
let C be with_identities CategoryStr;
let o be Object of C;
func id- o -> morphism of C equals
o;
correctness
proof
per cases;
suppose C is empty; hence thesis; end;
suppose not C is empty;
then not Ob(C) is empty;
then o in Ob(C);
hence thesis;
end;
end;
end;
definition
let C,D be CategoryStr;
mode Functor of C,D is Function of C,D;
end;
reserve C,D,E for with_identities CategoryStr;
reserve F for Functor of C,D;
reserve G for Functor of D,E;
reserve f for morphism of C;
definition
let C,D,F,f;
func F.f -> morphism of D equals :Def21:
F.f if C is non empty otherwise the Object of D;
correctness
proof
thus not C is empty implies F.f is morphism of D
proof
assume
A1: C is non empty;
per cases;
suppose Mor(D) <> {};
hence thesis by A1,FUNCT_2:5;
end;
suppose Mor(D) = {};
hence thesis by SUBSET_1:def 1;
end;
end;
the Object of D is morphism of D
proof
per cases;
suppose Mor(D) <> {};
then D is non empty;
hence thesis by TARSKI:def 3;
end;
suppose Mor(D) = {};
hence thesis;
end;
end;
hence thesis;
end;
end;
definition
let C,D,F;
attr F is identity-preserving means :Def22:
for f being morphism of C st f is identity holds F.f is identity;
attr F is multiplicative means :Def23:
for f1,f2 being morphism of C st f1 |> f2 holds F.f1 |> F.f2 &
F.(f1(*)f2) = (F.f1) (*) (F.f2);
attr F is antimultiplicative means
for f1,f2 being morphism of C st f1 |> f2 holds F.f2 |> F.f1 &
F.(f1(*)f2) = (F.f2) (*) (F.f1);
end;
registration
let C,D;
cluster identity-preserving for Functor of C,D;
correctness
proof
per cases;
suppose
Mor(D) is non empty;
then reconsider D1 = D as non empty with_identities CategoryStr;
A1: Ob(D1) is non empty;
set F = Mor(C) --> the Object of D;
the Object of D in Ob(D) by A1;
then reconsider F as Function of C,D by FUNCOP_1:45;
take F;
for f being morphism of C st f is identity holds F.f is identity
proof
let f be morphism of C;
assume f is identity;
reconsider x = f as element;
per cases;
suppose
A2: C is non empty;
F.f = F.x by A2,Def21 .= the Object of D by A2,FUNCOP_1:7;
then F.f is Object of D1;
hence F.f is identity by Th22;
end;
suppose C is empty;
then F.f is Object of D1 by Def21;
hence thesis by Th22;
end;
end;
hence thesis;
end;
suppose
A3: Mor(D) is empty;
set F = the Function of C,D;
take F;
for f being morphism of C st f is identity holds F.f is identity
proof
let f be morphism of C;
assume f is identity;
D is empty by A3;
hence F.f is identity by Th10;
end;
hence thesis;
end;
end;
end;
registration
let C be empty with_identities CategoryStr;
let D be with_identities CategoryStr;
cluster identity-preserving multiplicative antimultiplicative
for Functor of C,D;
correctness
proof
per cases;
suppose
Mor(D) is non empty;
then reconsider D1 = D as non empty with_identities CategoryStr;
set F = Mor(C) --> the Object of D;
reconsider F as Function of C,D;
take F;
A1: for f being morphism of C st f is identity holds F.f is identity
proof
let f be morphism of C;
assume f is identity;
F.f is Object of D1 by Def21;
hence thesis by Th22;
end;
(for f1,f2 being morphism of C st f1 |> f2
holds F.f1 |> F.f2 & F.(f1(*)f2) = (F.f1) (*) (F.f2)) &
(for f1,f2 being morphism of C st f1 |> f2
holds F.f2 |> F.f1 & F.(f1(*)f2) = (F.f2) (*) (F.f1));
hence thesis by A1;
end;
suppose
A2: Mor(D) is empty;
set F = the Function of C,D;
take F;
A3: for f being morphism of C st f is identity holds F.f is identity
proof
let f be morphism of C;
assume f is identity;
D is empty by A2;
hence F.f is identity by Th10;
end;
(for f1,f2 being morphism of C st f1 |> f2
holds F.f1 |> F.f2 & F.(f1(*)f2) = (F.f1) (*) (F.f2)) &
(for f1,f2 being morphism of C st f1 |> f2
holds F.f2 |> F.f1 & F.(f1(*)f2) = (F.f2) (*) (F.f1));
hence thesis by A3;
end;
end;
end;
registration
let C be with_identities CategoryStr;
let D be non empty with_identities CategoryStr;
cluster identity-preserving multiplicative antimultiplicative
for Functor of C,D;
correctness
proof
per cases;
suppose
A1: C is non empty;
set F = Mor(C) --> the Object of D;
the Object of D in Ob(D);
then reconsider F as Function of C,D by FUNCOP_1:45;
take F;
A2: for f1,f2 being morphism of C st f1 |> f2
holds F.f1 |> F.f2 & F.(f1(*)f2) = (F.f1) (*) (F.f2)
proof
let f1,f2 be morphism of C;
assume f1 |> f2;
reconsider x1 = f1, x2 = f2, x3 = f1(*)f2 as element;
A3: F.f1 = F.x1 by A1,Def21 .= the Object of D by A1,FUNCOP_1:7;
A4: F.f2 = F.x2 by A1,Def21 .= the Object of D by A1,FUNCOP_1:7;
A5: F.(f1(*)f2) = F.x3 by A1,Def21 .= the Object of D by A1,FUNCOP_1:7;
thus
A6: F.f1 |> F.f2 by A3,A4,Th23;
thus F.(f1(*)f2) = (F.f1) (*) (F.f2) by A4,A5,A3,A6,Th23;
end;
A7: for f1,f2 being morphism of C st f1 |> f2
holds F.f2 |> F.f1 & F.(f1(*)f2) = (F.f2) (*) (F.f1)
proof
let f1,f2 be morphism of C;
assume f1 |> f2;
reconsider x1 = f1, x2 = f2, x3 = f1(*)f2 as element;
A8: F.f1 = F.x1 by A1,Def21 .= the Object of D by A1,FUNCOP_1:7;
A9: F.f2 = F.x2 by A1,Def21 .= the Object of D by A1,FUNCOP_1:7;
A10: F.(f1(*)f2) = F.x3 by A1,Def21 .= the Object of D by A1,FUNCOP_1:7;
thus
A11: F.f2 |> F.f1 by A8,A9,Th23;
thus F.(f1(*)f2) = (F.f2) (*) (F.f1) by A9,A10,A8,A11,Th23;
end;
for f being morphism of C st f is identity holds F.f is identity
proof
let f be morphism of C;
assume f is identity;
reconsider x = f as element;
F.f = F.x by A1,Def21 .= the Object of D by A1,FUNCOP_1:7;
hence F.f is identity by Th22;
end;
hence thesis by A2,A7;
end;
suppose
A12: C is empty;
set F = the Function of C,D;
take F;
A13: for f being morphism of C st f is identity holds F.f is identity
proof
let f be morphism of C;
assume f is identity;
F.f = the Object of D by A12,Def21;
hence F.f is identity by Th22;
end;
(for f1,f2 being morphism of C st f1 |> f2
holds F.f1 |> F.f2 & F.(f1(*)f2) = (F.f1) (*) (F.f2)) &
(for f1,f2 being morphism of C st f1 |> f2
holds F.f2 |> F.f1 & F.(f1(*)f2) = (F.f2) (*) (F.f1)) by A12;
hence thesis by A13;
end;
end;
end;
theorem
ex C,D being category, F being Functor of C,D st
F is multiplicative & F is non identity-preserving
proof
set C = the non empty category;
reconsider X = {0,1} as set;
set comp = {[[0,0],0],[[1,1],1]} \/ {[[0,1],1],[[1,0],1]};
A1: for x being element holds x in comp iff
(x = [[0,0],0] or x = [[1,1],1] or x = [[0,1],1] or x = [[1,0],1])
proof
let x be element;
thus x in comp implies
(x = [[0,0],0] or x = [[1,1],1] or x = [[0,1],1] or x = [[1,0],1])
proof
assume x in comp;
then x in {[[0,0],0],[[1,1],1]} or
x in {[[0,1],1],[[1,0],1]} by XBOOLE_0:def 3;
hence thesis by TARSKI:def 2;
end;
assume x = [[0,0],0] or x = [[1,1],1] or x = [[0,1],1] or x = [[1,0],1];
then x in {[[0,0],0],[[1,1],1]} or
x in {[[0,1],1],[[1,0],1]} by TARSKI:def 2;
hence x in comp by XBOOLE_0:def 3;
end;
A2: for x,y1,y2 being element st [x,y1] in comp & [x,y2] in comp holds y1 = y2
proof
let x,y1,y2 be element;
assume
A3: [x,y1] in comp;
assume
A4: [x,y2] in comp;
per cases by A3,A1;
suppose
A5: [x,y1] = [[0,0],0];
then
A6: x = [0,0] & y1 = 0 by XTUPLE_0:1;
per cases by A4,A1;
suppose [x,y2] = [[0,0],0];
hence thesis by A5,XTUPLE_0:1;
end;
suppose [x,y2] = [[1,1],1];
then x = [1,1] & y2 = 1 by XTUPLE_0:1;
hence thesis by A6,XTUPLE_0:1;
end;
suppose [x,y2] = [[0,1],1];
then x = [0,1] & y2 = 1 by XTUPLE_0:1;
hence thesis by A6,XTUPLE_0:1;
end;
suppose [x,y2] = [[1,0],1];
then x = [1,0] & y2 = 1 by XTUPLE_0:1;
hence thesis by A6,XTUPLE_0:1;
end;
end;
suppose [x,y1] = [[1,1],1];
then
A7: x = [1,1] & y1 = 1 by XTUPLE_0:1;
per cases by A4,A1;
suppose [x,y2] = [[0,0],0];
then x = [0,0] & y2 = 0 by XTUPLE_0:1;
hence thesis by A7,XTUPLE_0:1;
end;
suppose [x,y2] = [[1,1],1] or [x,y2] = [[0,1],1] or [x,y2] = [[1,0],1];
hence thesis by A7,XTUPLE_0:1;
end;
end;
suppose [x,y1] = [[0,1],1];
then
A8: x = [0,1] & y1 = 1 by XTUPLE_0:1;
per cases by A4,A1;
suppose [x,y2] = [[0,0],0];
then x = [0,0] & y2 = 0 by XTUPLE_0:1;
hence thesis by A8,XTUPLE_0:1;
end;
suppose [x,y2] = [[1,1],1] or [x,y2] = [[0,1],1] or [x,y2] = [[1,0],1];
hence thesis by A8,XTUPLE_0:1;
end;
end;
suppose [x,y1] = [[1,0],1];
then
A9: x = [1,0] & y1 = 1 by XTUPLE_0:1;
per cases by A4,A1;
suppose [x,y2] = [[0,0],0];
then x = [0,0] & y2 = 0 by XTUPLE_0:1;
hence thesis by A9,XTUPLE_0:1;
end;
suppose [x,y2] = [[1,1],1] or [x,y2] = [[0,1],1] or [x,y2] = [[1,0],1];
hence thesis by A9,XTUPLE_0:1;
end;
end;
end;
for x being element st x in comp holds x in [:[:X,X:],X:]
proof
let x be element;
assume
A10: x in comp;
per cases by A10,A1;
suppose
A11: x = [[0,0],0];
A12: 0 in X by TARSKI:def 2;
then [0,0] in [:X,X:] by ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A11,A12,ZFMISC_1:def 2;
end;
suppose
A13: x = [[1,1],1];
A14: 1 in X by TARSKI:def 2;
then [1,1] in [:X,X:] by ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A13,A14,ZFMISC_1:def 2;
end;
suppose
A15: x = [[0,1],1];
A16: 0 in X & 1 in X by TARSKI:def 2;
then [0,1] in [:X,X:] by ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A15,A16,ZFMISC_1:def 2;
end;
suppose
A17: x = [[1,0],1];
A18: 0 in X & 1 in X by TARSKI:def 2;
then [1,0] in [:X,X:] by ZFMISC_1:def 2;
hence x in [:[:X,X:],X:] by A17,A18,ZFMISC_1:def 2;
end;
end;
then reconsider comp as PartFunc of [:X,X:],X
by A2,TARSKI:def 3,FUNCT_1:def 1;
set D = CategoryStr(# X, comp #);
A19: for f1,f2 being morphism of D st f1 |> f2 holds
(f1 = 0 & f2 = 0 & f1(*)f2 = 0) or
(f1 = 1 & f2 = 1 & f1(*)f2 = 1) or
(f1 = 0 & f2 = 1 & f1(*)f2 = 1) or
(f1 = 1 & f2 = 0 & f1(*)f2 = 1)
proof
let f1,f2 be morphism of D;
assume
A20: f1 |> f2;
assume
A21: not (f1 = 0 & f2 = 0 & f1(*)f2 = 0) &
not (f1 = 1 & f2 = 1 & f1(*)f2 = 1) &
not (f1 = 0 & f2 = 1 & f1(*)f2 = 1);
consider y be element such that
A22: [[f1,f2],y] in the composition of D by A20,XTUPLE_0:def 12;
A23: f1(*)f2 = (the composition of D).(f1,f2) by Def3,A20
.= (the composition of D).[f1,f2] by BINOP_1:def 1
.= y by A22,FUNCT_1:1;
per cases by A22,A1;
suppose [[f1,f2],y] = [[0,0],0];
then [f1,f2] = [0,0] & y = 0 by XTUPLE_0:1;
hence thesis by A23,A21,XTUPLE_0:1;
end;
suppose [[f1,f2],y] = [[1,1],1];
then [f1,f2] = [1,1] & y = 1 by XTUPLE_0:1;
hence thesis by A21,A23,XTUPLE_0:1;
end;
suppose [[f1,f2],y] = [[0,1],1];
then [f1,f2] = [0,1] & y = 1 by XTUPLE_0:1;
hence thesis by A21,A23,XTUPLE_0:1;
end;
suppose [[f1,f2],y] = [[1,0],1];
then [f1,f2] = [1,0] & y = 1 by XTUPLE_0:1;
hence thesis by A23,XTUPLE_0:1;
end;
end;
A24: for f1,f2 being morphism of D holds f1 |> f2
proof
let f1,f2 be morphism of D;
per cases by TARSKI:def 2;
suppose (f1 = 0 & f2 = 0) or (f1 = 1 & f2 = 1) or (f1 = 0 & f2 = 1);
then [[f1,f2],f2] in the composition of D by A1;
hence thesis by FUNCT_1:1;
end;
suppose f1 = 1 & f2 = 0;
then [[f1,f2],f1] in the composition of D by A1;
hence thesis by FUNCT_1:1;
end;
end;
A25: D is left_composable by A24;
A26: for f,f1,f2 being morphism of D st f1 |> f2 holds
f |> f1(*)f2 iff f |> f1 by A24;
A27: for f1 being morphism of D st f1 in the carrier of D holds
ex f being morphism of D st f |> f1 & f is left_identity
proof
let f1 be morphism of D;
assume f1 in the carrier of D;
reconsider f = 0 as morphism of D by TARSKI:def 2;
take f;
thus f |> f1 by A24;
for f2 being morphism of D st f |> f2 holds f (*) f2 = f2
proof
let f2 be morphism of D;
assume
A28: f |> f2;
f2 = 0 or f2 = 1 by TARSKI:def 2;
hence f (*) f2 = f2 by A28,A19;
end;
hence f is left_identity;
end;
for f1 being morphism of D st f1 in the carrier of D holds
ex f being morphism of D st f1 |> f & f is right_identity
proof
let f1 be morphism of D;
assume f1 in the carrier of D;
reconsider f = 0 as morphism of D by TARSKI:def 2;
take f;
thus f1 |> f by A24;
for f2 being morphism of D st f2 |> f holds f2 (*) f = f2
proof
let f2 be morphism of D;
assume
A29: f2 |> f;
f2 = 0 or f2 = 1 by TARSKI:def 2;
hence f2 (*) f = f2 by A29,A19;
end;
hence f is right_identity;
end;
then
A30: D is with_right_identities;
for f1,f2,f3 being morphism of D st f1 |> f2 & f2 |> f3 &
f1 (*) f2 |> f3 & f1 |> f2 (*) f3 holds
f1 (*) (f2 (*) f3) = (f1 (*) f2) (*) f3
proof
let f1,f2,f3 be morphism of D;
assume
A31: f1 |> f2 & f2 |> f3;
assume
A32: f1 (*) f2 |> f3;
assume
A33: f1 |> f2 (*) f3;
per cases by TARSKI:def 2;
suppose
A34: f3 = 0;
per cases by TARSKI:def 2;
suppose
A35: f2 = 0;
then
A36: f2 (*) f3 = 0 by A34,A31,A19;
per cases by TARSKI:def 2;
suppose f1 = 0;
hence f1(*)(f2(*)f3) = (f1(*)f2)(*)f3 by A36,A34,A35;
end;
suppose
A37: f1 = 1;
then
A38: f1 (*) f2 = 1 by A31,A19;
thus f1 (*) (f2 (*) f3) = 1 by A37,A33,A19
.= (f1 (*) f2) (*) f3 by A32,A38,A19;
end;
end;
suppose
A39: f2 = 1;
then f2 (*) f3 = 1 & f1 (*) f2 = 1 by A31,A19;
hence f1 (*) (f2 (*) f3) = (f1 (*) f2) (*) f3 by A39;
end;
end;
suppose
A40: f3 = 1;
then f2 (*) f3 = 1 by A31,A19;
hence f1 (*) (f2 (*) f3) = 1 by A33,A19
.= (f1 (*) f2) (*) f3 by A32,A40,A19;
end;
end;
then reconsider D as category by A26,A27,A30,Def6,Def10,Def12,A25,Def9,
Def11;
take C,D;
reconsider d1 = 1 as morphism of D by TARSKI:def 2;
deffunc F1(element) = d1;
A41: for x being element st x in the carrier of C
holds F1(x) in the carrier of D;
consider F be Function of the carrier of C, the carrier of D such that
A42: for x being element st x in the carrier of C holds F.x = F1(x)
from FUNCT_2:sch 2(A41);
reconsider F as Functor of C, D;
take F;
for f1,f2 being morphism of C st f1 |> f2 holds
F.f1 |> F.f2 & F.(f1 (*) f2) = (F .f1) (*) (F .f2)
proof
let f1,f2 be morphism of C;
assume f1 |> f2;
thus F.f1 |> F.f2 by A24;
reconsider x1 = f1, x2 = f2, x12 = f1 (*) f2 as element;
A43: F.f1 = F.x1 by Def21 .= d1 by A42;
A44: F.f2 = F.x2 by Def21 .= d1 by A42;
A45: d1 |> d1 by A24;
thus F.(f1(*)f2) = F.x12 by Def21
.= d1 by A42
.= (F.f1)(*)(F.f2) by A43,A44,A45,A19;
end;
hence F is multiplicative;
ex f being morphism of C st f is identity & not F.f is identity
proof
reconsider f = the Object of C as morphism of C by TARSKI:def 3;
take f;
thus f is identity by Th22;
reconsider x = f as element;
A46: F.f = F.x by Def21 .= d1 by A42;
ex d0 being morphism of D st d1 |> d0 & not d1(*)d0 = d0
proof
reconsider d0 = 0 as morphism of D by TARSKI:def 2;
take d0;
thus d1 |> d0 by A24;
hence not d1(*)d0 = d0 by A19;
end;
hence not F.f is identity by A46,Def4;
end;
hence F is non identity-preserving;
end;
theorem Th29:
C is non empty & D is empty implies not ex F being Functor of C,D st
F is multiplicative or F is antimultiplicative
proof
assume
A1: C is non empty;
then reconsider f = the Object of C as morphism of C by TARSKI:def 3;
A2: f |> f by A1,Th23;
assume
A3: D is empty;
assume ex F being Functor of C,D st F is multiplicative
or F is antimultiplicative;
then consider F be Functor of C,D such that
A4: F is multiplicative or F is antimultiplicative;
A5: not F.f |> F.f by A3;
per cases by A4;
suppose F is multiplicative;
hence thesis by A5,A2;
end;
suppose F is antimultiplicative;
hence thesis by A5,A2;
end;
end;
theorem
ex C,D being category, F being Functor of C,D st
F is non multiplicative & F is identity-preserving
proof
set C = the non empty category;
set D = the empty category;
set F = the identity-preserving Functor of C,D;
take C,D,F;
thus thesis by Th29;
end;
definition
let C,D,F;
attr F is covariant means :Def25:
F is identity-preserving & F is multiplicative;
attr F is contravariant means
F is identity-preserving & F is antimultiplicative;
end;
registration
let C be empty with_identities CategoryStr;
let D be with_identities CategoryStr;
cluster covariant contravariant for Functor of C,D;
correctness
proof
set F = the identity-preserving multiplicative antimultiplicative
Functor of C,D;
take F;
thus thesis;
end;
end;
registration
let C be with_identities CategoryStr;
let D be non empty with_identities CategoryStr;
cluster covariant contravariant for Functor of C,D;
correctness
proof
set F = the identity-preserving multiplicative antimultiplicative
Functor of C,D;
take F;
thus thesis;
end;
end;
theorem
C is non empty & D is empty implies not ex F being Functor of C,D
st F is covariant or F is contravariant by Th29;
definition
let C,D be non empty with_identities CategoryStr;
let F be covariant Functor of C,D;
let f be Object of C;
redefine func F.f -> Object of D;
coherence
proof
f in Ob(C);
then consider f1 be morphism of C such that
A1: f = f1 & f1 is identity & f1 in Mor(C);
F.f1 is identity & F.f1 in Mor(D) by A1,Def22,Def25;
then F.f1 in Ob(D);
hence thesis by A1,Def21;
end;
end;
theorem Th32:
for C,D being non empty composable with_identities CategoryStr,
F being covariant Functor of C,D,
f being morphism of C
holds F.(dom f) = dom(F.f) & F.(cod f) = cod(F.f)
proof
let C,D be non empty composable with_identities CategoryStr;
let F be covariant Functor of C,D;
let f be morphism of C;
A1: F is multiplicative by Def25;
consider d be morphism of C such that
A2: dom f = d & f |> d & d is identity by Def18;
F.(dom f) in Ob(D);
then reconsider d1 = F.(dom f) as morphism of D;
A3: F.f |> F.d by A2,A1;
A4: F.d = F.(dom f) by A2,Def21;
d1 is identity by Th22;
hence F.(dom f) = dom(F.f) by A3,A4,Def18;
consider c be morphism of C such that
A5: cod f = c & c |> f & c is identity by Def19;
F.(cod f) in Ob(D);
then reconsider c1 = F.(cod f) as morphism of D;
A6: F.c |> F.f by A5,A1;
A7: F.c = F.(cod f) by A5,Def21;
c1 is identity by Th22;
hence F.(cod f) = cod(F.f) by A6,A7,Def19;
end;
theorem
for C,D being non empty composable with_identities CategoryStr,
F being covariant Functor of C,D,
o being Object of C holds F.(id- o) = id-(F.o) by Def21;
definition
let C,D,E;
let F,G;
assume
A1: (F is covariant or F is contravariant) &
(G is covariant or G is contravariant);
func G (*) F -> Functor of C,E equals :Def27:
F * G;
coherence
proof
set FG = F * G;
reconsider FG as PartFunc of the carrier of C, the carrier of E;
not (C is non empty & D is empty)
& not (D is non empty & E is empty) by A1,Th29;
hence thesis;
end;
end;
theorem Th34:
F is covariant & G is covariant & C is non empty implies (G(*)F).f = G.(F.f)
proof
assume
A1: F is covariant;
assume
A2: G is covariant;
assume
A3: C is non empty;
then
A4: D is non empty by A1,Th29;
then
A5: E is non empty by A2,Th29;
reconsider x = f as element;
A6: dom(G (*) F) = the carrier of C by A5,FUNCT_2:def 1;
A7: F.x = F.f by A3,Def21;
then
A8: G.(F.x) = G.(F.f) by A4,Def21;
A9: dom F = the carrier of C by A4,FUNCT_2:def 1;
dom G = the carrier of D by A5,FUNCT_2:def 1;
then [x,F.x] in F & [F.x,G.(F.x)] in G by A3,A4,A9,A7,FUNCT_1:1;
then [f,G.(F.f)] in F * G by A8,RELAT_1:def 8;
then
A10: [f,G.(F.f)] in G (*) F by A1,A2,Def27;
thus (G(*)F).f = (G(*)F).x by A3,Def21
.= G.(F.f) by A10,A6,FUNCT_1:def 2;
end;
theorem
F is covariant & G is covariant implies G (*) F is covariant
proof
assume
A1: F is covariant;
assume
A2: G is covariant;
set GF = G (*) F;
for f being morphism of C st f is identity holds GF.f is identity
proof
let f be morphism of C;
assume
A3: f is identity;
per cases;
suppose
A4: C is non empty;
A7: F.f is identity by A1,A3,Def22;
(G(*)F).f = G.(F.f) by A1,A2,Th34,A4;
hence GF.f is identity by A2,A7,Def22;
end;
suppose
A8: C is empty;
per cases;
suppose
A9: E is non empty;
GF.f = the Object of E by A8,Def21;
hence GF.f is identity by A9,Th22;
end;
suppose E is empty; hence GF.f is identity by Th10; end;
end;
end;
then
A10: GF is identity-preserving;
for f1,f2 being morphism of C st f1 |> f2 holds
GF.f1 |> GF.f2 & GF.(f1 (*) f2) = (GF.f1) (*) (GF.f2)
proof
let f1,f2 be morphism of C;
assume
A11: f1 |> f2;
then
A12: C is non empty;
A15: GF.f1 = G.(F.f1) by A12,A1,A2,Th34;
A16: G is multiplicative by A2;
F is multiplicative by A1;
then
A17: F.f1 |> F.f2 & F.(f1 (*) f2) = (F.f1) (*) (F.f2) by A11;
then G.(F.f1) |> G.(F.f2) &
G.((F.f1) (*) (F.f2)) = (G.(F.f1)) (*) (G.(F.f2)) by A16;
hence GF.f1 |> GF.f2 by A15,A12,A1,A2,Th34;
thus GF.(f1 (*) f2) = G.(F.(f1 (*) f2)) by A12,A1,A2,Th34
.= (G.(F.f1)) (*) (G.(F.f2)) by A17,A16
.= (GF.f1) (*) (GF.f2) by A15,A12,A1,A2,Th34;
end;
hence G (*) F is covariant by A10,Def23;
end;
definition
let C;
redefine func id C -> Functor of C,C;
coherence;
end;
registration
let C;
cluster id C -> covariant;
correctness
proof
set F = id C;
A1: F = id the carrier of C by STRUCT_0:def 4;
for f1,f2 being morphism of C st f1 |> f2
holds F.f1 |> F.f2 & F.(f1(*)f2) = (F.f1)(*)(F.f2)
proof
let f1,f2 be morphism of C;
assume A2: f1 |> f2;
per cases;
suppose
A3: C is non empty;
reconsider x1=f1, x2=f2, x3=f1(*)f2 as element;
A4: F.f1 = F.x1 by A3,Def21 .= f1 by A1;
A5: F.f2 = F.x2 by A3,Def21 .= f2 by A1;
A6: F.(f1(*)f2) = F.x3 by A3,Def21 .= f1(*)f2 by A1;
thus F.f1 |> F.f2 by A4,A5,A2;
thus F.(f1(*)f2) = (F.f1)(*)(F.f2) by A4,A6,A5;
end;
suppose C is empty;
hence thesis by A2;
end;
end;
then
A7: F is multiplicative;
for f being morphism of C st f is identity holds F.f is identity
proof
let f be morphism of C;
assume A8: f is identity;
per cases;
suppose
A9: C is non empty;
reconsider x=f as element;
F.f = F.x by A9,Def21 .= f by A1;
hence F.f is identity by A8;
end;
suppose C is empty;
hence F.f is identity by Th10;
end;
end;
hence thesis by A7,Def22;
end;
end;
definition
let C,D;
pred C,D are_isomorphic means
ex F being Functor of C,D, G being Functor of D,C
st F is covariant & G is covariant & G (*) F = id C & F (*) G = id D;
reflexivity
proof
let C be with_identities CategoryStr;
set F = id C, G = id C;
A1: F = id the carrier of C & G = id the carrier of C by STRUCT_0:def 4;
take F,G;
G (*) F = (id the carrier of C) * (id the carrier of C) by A1,Def27
.= id ((the carrier of C) /\ (the carrier of C)) by FUNCT_1:22
.= id C by STRUCT_0:def 4;
hence thesis;
end;
symmetry;
end;
notation
let C,D;
synonym C ~= D for C,D are_isomorphic;
end;
begin :: Transform a category in the other
Lm4:
for C being CategoryStr, f,g being morphism of C st g |> f
holds g (*) f = (the composition of C).[g,f]
proof
let C be CategoryStr;
let f,g be morphism of C;
assume g |> f;
hence g (*) f = (the composition of C).(g,f) by Def3
.= (the composition of C).[g,f] by BINOP_1:def 1;
end;
Lm5:
for C being composable CategoryStr holds C is associative iff
for f,g,h being morphism of C st h |> g & g |> f
holds h(*)(g(*)f) = (h(*)g)(*)f
proof
let C be composable CategoryStr;
hereby
assume
A1: C is associative;
A2: C is left_composable & C is right_composable by Def11;
let f,g,h be morphism of C;
assume
A3: h |> g & g |> f;
then h (*) g |> f & h |> g (*) f by A2;
hence h(*)(g(*)f) = (h(*)g)(*)f by A3,A1;
end;
assume for f,g,h being morphism of C st
h |> g & g |> f holds h(*)(g(*)f) = (h(*)g)(*)f;
hence C is associative;
end;
definition
let C be CategoryStr;
func CompMap(C) ->
PartFunc of [:Mor(C), Mor(C):], Mor(C) equals
the composition of C;
correctness;
end;
definition
let C be composable with_identities CategoryStr;
func SourceMap(C) -> Function of Mor(C), Ob(C) means :Def30:
for f being Element of Mor(C) holds it.f = dom f if C is non empty
otherwise it = {};
correctness
proof
A1: not C is empty implies ex F being Function of Mor(C), Ob(C) st
for f being Element of Mor(C) holds F.f = dom f
proof
assume not C is empty;
then reconsider A = Mor(C), O = Ob(C) as non empty set;
defpred P[set,set] means
ex a being Element of Mor(C), o being Element of Ob(C)
st a = $1 & o = $2 & o = dom a;
A2: for x being Element of A ex y being Element of O st P[x,y]
proof
let x be Element of A;
reconsider f = x as Element of Mor(C);
reconsider y = dom f as Element of O;
take y;
thus P[x,y];
end;
consider F be Function of A, O such that
A3: for x being Element of A holds P[x,F.x] from FUNCT_2:sch 3(A2);
reconsider F as Function of Mor(C), Ob(C);
take F;
let f be Element of Mor(C);
P[f,F.f] by A3;
hence thesis;
end;
A4: C is empty implies ex F being Function of Mor(C), Ob(C)
st F = {}
proof
assume C is empty;
then Mor(C) = {} & Ob(C) = {};
then reconsider F = {} as Function of Mor(C), Ob(C);
take F;
thus thesis;
end;
A5: for F1, F2 being Function of Mor(C), Ob(C) holds
(for f being Element of Mor(C) holds F1.f = dom f) &
(for f being Element of Mor(C) holds F2.f = dom f)
implies F1 = F2
proof
let F1,F2 be Function of Mor(C), Ob(C);
assume
A6: for f being Element of Mor(C) holds F1.f = dom f;
assume
A7: for f being Element of Mor(C) holds F2.f = dom f;
for x being element st x in Mor(C) holds F1.x = F2.x
proof
let x be element;
assume x in Mor(C);
then reconsider f = x as Element of Mor(C);
thus F1.x = dom f by A6 .= F2.x by A7;
end;
hence thesis by FUNCT_2:12;
end;
thus thesis by A1,A4,A5;
end;
func TargetMap(C) -> Function of Mor(C), Ob(C) means :Def31:
for f being Element of Mor(C) holds it.f = cod f if C is non empty
otherwise it = {};
correctness
proof
A8: not C is empty implies ex F being Function of Mor(C), Ob(C) st
for f being Element of Mor(C) holds F.f = cod f
proof
assume not C is empty;
then reconsider A = Mor(C), O = Ob(C) as non empty set;
defpred P[set,set] means
ex a being Element of Mor(C), o being Element of Ob(C)
st a = $1 & o = $2 & o = cod a;
A9: for x being Element of A ex y being Element of O st P[x,y]
proof
let x be Element of A;
reconsider f = x as Element of Mor(C);
reconsider y = cod f as Element of O;
take y;
thus P[x,y];
end;
consider F be Function of A, O such that
A10: for x being Element of A holds P[x,F.x] from FUNCT_2:sch 3(A9);
reconsider F as Function of Mor(C), Ob(C);
take F;
let f be Element of Mor(C);
P[f,F.f] by A10;
hence thesis;
end;
A11: C is empty implies ex F being Function of Mor(C), Ob(C)
st F = {}
proof
assume C is empty;
then Mor(C) = {} & Ob(C) = {};
then reconsider F = {} as Function of Mor(C), Ob(C);
take F;
thus thesis;
end;
A12: for F1, F2 being Function of Mor(C), Ob(C) holds
(for f being Element of Mor(C) holds F1.f = cod f) &
(for f being Element of Mor(C) holds F2.f = cod f)
implies F1 = F2
proof
let F1,F2 be Function of Mor(C), Ob(C);
assume
A13: for f being Element of Mor(C) holds F1.f = cod f;
assume
A14: for f being Element of Mor(C) holds F2.f = cod f;
for x being element st x in Mor(C) holds F1.x = F2.x
proof
let x be element;
assume x in Mor(C);
then reconsider f = x as Element of Mor(C);
thus F1.x = cod f by A13 .= F2.x by A14;
end;
hence thesis by FUNCT_2:12;
end;
thus thesis by A8,A11,A12;
end;
end;
definition
let C be with_identities CategoryStr;
func IdMap(C) -> Function of Ob(C), Mor(C) means :Def32:
for o being Element of Ob(C) holds it.o = id-o if C is non empty
otherwise it = {};
correctness
proof
A1: not C is empty implies ex F being Function of Ob(C), Mor(C) st
for o being Element of Ob(C) holds F.o = id-o
proof
assume not C is empty;
then reconsider A = Mor(C), O = Ob(C) as non empty set;
defpred P[set,set] means
ex o being Element of Ob(C), a being Element of Mor(C)
st o = $1 & a = $2 & a = id-o;
A2: for x being Element of O ex y being Element of A st P[x,y]
proof
let x be Element of O;
reconsider o = x as Element of Ob(C);
reconsider y = id-o as Element of A;
take y;
thus P[x,y];
end;
consider F be Function of O, A such that
A3: for x being Element of O holds P[x,F.x] from FUNCT_2:sch 3(A2);
reconsider F as Function of Ob(C), Mor(C);
take F;
let o be Element of Ob(C);
P[o,F.o] by A3;
hence thesis;
end;
A4: C is empty implies ex F being Function of Ob(C), Mor(C)
st F = {}
proof
assume C is empty;
then Mor(C) = {} & Ob(C) = {};
then reconsider F = {} as Function of Ob(C), Mor(C);
take F;
thus thesis;
end;
A5: for F1, F2 being Function of Ob(C), Mor(C) holds
(for o being Element of Ob(C) holds F1.o = id-o) &
(for o being Element of Ob(C) holds F2.o = id-o)
implies F1 = F2
proof
let F1,F2 be Function of Ob(C), Mor(C);
assume
A6: for o being Element of Ob(C) holds F1.o = id-o;
assume
A7: for o being Element of Ob(C) holds F2.o = id-o;
for x being element st x in Ob(C) holds F1.x = F2.x
proof
let x be element;
assume x in Ob(C);
then reconsider o = x as Element of Ob(C);
thus F1.x = id-o by A6 .= F2.x by A7;
end;
hence thesis by FUNCT_2:12;
end;
thus thesis by A1,A4,A5;
end;
end;
theorem Th37:
for C being non empty composable with_identities CategoryStr
for f,g being Element of Mor(C) holds
[g,f] in dom(CompMap(C)) iff (SourceMap(C)).g = (TargetMap(C)).f
proof
let C be non empty composable with_identities CategoryStr;
let f,g be Element of Mor(C);
hereby
assume [g,f] in dom(CompMap(C));
then
A1: g |> f;
consider u be morphism of C such that
A2: dom g = u & g |> u & u is identity by Def18;
g (*) u = g by A2,Def5;
then
A3: u |> f by A2,A1,Lm3;
thus (SourceMap(C)).g = dom g by Def30
.= cod f by A3,A2,Def19
.= (TargetMap(C)).f by Def31;
end;
assume
A4: (SourceMap(C)).g = (TargetMap(C)).f;
A5: dom g = (SourceMap(C)).g by Def30
.= cod f by A4,Def31;
consider d be morphism of C such that
A6: dom g = d & g |> d & d is identity by Def18;
consider c be morphism of C such that
A7: cod f = c & c |> f & c is identity by Def19;
g(*)d |> f by A6,A7,A5,Lm3;
hence [g,f] in dom(CompMap(C)) by A6,Def5;
end;
theorem Th38:
for C being composable with_identities CategoryStr
for f,g being Element of Mor(C)
st (SourceMap(C)).g = (TargetMap(C)).f holds
(SourceMap(C)).((CompMap(C)).(g,f)) = (SourceMap(C)).f &
(TargetMap(C)).((CompMap(C)).(g,f)) = (TargetMap(C)).g
proof
let C be composable with_identities CategoryStr;
let f,g be Element of Mor(C);
assume
A1: (SourceMap(C)).g = (TargetMap(C)).f;
per cases;
suppose
A2: C is empty;
thus (SourceMap(C)).((CompMap(C)).(g,f)) = {} by A2
.= (SourceMap(C)).f by A2;
thus (TargetMap(C)).((CompMap(C)).(g,f)) = {} by A2
.= (TargetMap(C)).g by A2;
end;
suppose
A3: not C is empty;
then
A4: [g,f] in dom (CompMap(C)) by A1,Th37;
then
A5: (CompMap(C)).[g,f] in rng (CompMap(C)) by FUNCT_1:3;
reconsider a = (CompMap(C)).(g,f) as Element of Mor(C)
by A5,BINOP_1:def 1;
consider d be morphism of C such that
A6: dom a = d & a |> d & d is identity by A3,Def18;
A7: g |> f by A4;
A8: a = (the composition of C).[g,f] by BINOP_1:def 1
.= g (*) f by A4,Def2,Lm4;
then f |> d by A6,A7,Lm3;
then
A9: dom a = dom f by A3,A6,Def18;
thus (SourceMap(C)).((CompMap(C)).(g,f)) = dom a by A3,Def30
.= (SourceMap(C)).f by A9,A3,Def30;
consider c be morphism of C such that
A10: cod a = c & c |> a & c is identity by A3,Def19;
c |> g by A10,A7,A8,Lm3;
then
A11: cod a = cod g by A3,A10,Def19;
thus (TargetMap(C)).((CompMap(C)).(g,f)) = cod a by A3,Def31
.= (TargetMap(C)).g by A11,A3,Def31;
end;
end;
theorem Th39:
for C being composable associative with_identities CategoryStr
for f,g,h being Element of Mor(C)
st (SourceMap(C)).h = (TargetMap(C)).g &
(SourceMap(C)).g = (TargetMap(C)).f
holds (CompMap(C)).(h,(CompMap(C)).(g,f)) =
(CompMap(C)).((CompMap(C)).(h,g),f)
proof
let C be composable associative with_identities CategoryStr;
let f,g,h be Element of Mor(C);
assume
A1: (SourceMap(C)).h = (TargetMap(C)).g &
(SourceMap(C)).g = (TargetMap(C)).f;
per cases;
suppose
A2: C is empty;
thus (CompMap(C)).(h,(CompMap(C)).(g,f))
= (CompMap(C)).[h,(CompMap(C)).(g,f)] by BINOP_1:def 1
.= {} by A2
.= (CompMap(C)).[(CompMap(C)).(h,g),f] by A2
.= (CompMap(C)).((CompMap(C)).(h,g),f) by BINOP_1:def 1;
end;
suppose
A3: not C is empty;
A4: [h,g] in dom(CompMap(C)) by Th37,A3,A1;
then
A5: h |> g;
A6: [g,f] in dom(CompMap(C)) by Th37,A3,A1;
then
A7: g |> f;
A8: h |> g(*)f by A7,A5,Lm3;
A9: h(*)g |> f by A7,A5,Lm3;
A10: (CompMap(C)).(g,f) = (the composition of C).[g,f] by BINOP_1:def 1
.= g(*)f by A6,Def2,Lm4;
A11: (CompMap(C)).(h,g) = (the composition of C).[h,g] by BINOP_1:def 1
.= h(*)g by A4,Def2,Lm4;
thus (CompMap(C)).(h,(CompMap(C)).(g,f))
= (CompMap(C)).[h,g(*)f] by A10,BINOP_1:def 1
.= h(*)(g(*)f) by A8,Lm4
.= (h(*)g)(*)f by A5,A7,Lm5
.= (CompMap(C)).[h(*)g,f] by A9,Lm4
.= (CompMap(C)).((CompMap(C)).(h,g),f) by A11,BINOP_1:def 1;
end;
end;
theorem Th40:
for C being composable with_identities CategoryStr
for b being Element of Ob(C) holds
(SourceMap(C)).((IdMap(C)).b) = b & (TargetMap(C)).((IdMap(C)).b) = b &
(for f being Element of Mor(C) st (TargetMap(C)).f = b
holds (CompMap(C)).((IdMap(C)).b,f) = f ) &
for g being Element of Mor(C) st (SourceMap(C)).g = b
holds (CompMap(C)).(g,(IdMap(C)).b) = g
proof
let C be composable with_identities CategoryStr;
let b be Element of Ob(C);
per cases;
suppose
A1: C is empty;
then
A2: b = {} by SUBSET_1:def 1;
thus (SourceMap(C)).((IdMap(C)).b) = b by A1,A2;
thus (TargetMap(C)).((IdMap(C)).b) = b by A1,A2;
thus for f being Element of Mor(C) st (TargetMap(C)).f = b
holds (CompMap(C)).((IdMap(C)).b,f) = f
proof
let f be Element of Mor(C);
assume (TargetMap(C)).f = b;
A3: f = {} by A1,SUBSET_1:def 1;
thus (CompMap(C)).((IdMap(C)).b,f)
= (the composition of C).[(IdMap(C)).b,f] by BINOP_1:def 1
.= f by A1,A3;
end;
thus for g being Element of Mor(C) st (SourceMap(C)).g = b
holds (CompMap(C)).(g,(IdMap(C)).b) = g
proof
let g be Element of Mor(C);
assume (SourceMap(C)).g = b;
A4: g = {} by A1,SUBSET_1:def 1;
thus (CompMap(C)).(g,(IdMap(C)).b)
= (the composition of C).[g,(IdMap(C)).b] by BINOP_1:def 1
.= g by A1,A4;
end;
end;
suppose
A5: not C is empty;
then
A6: (IdMap(C)).b = id-b by Def32 .= b;
not Ob(C) is empty by A5;
then b in Ob(C);
then reconsider u = b as Element of Mor(C);
A7: u is identity by A5,Th22;
consider d be morphism of C such that
A8: dom u = d & u |> d & d is identity by A5,Def18;
A9: d = u (*) d by A7,A8,Def4 .= u by A8,Def5;
thus
A10: (SourceMap(C)).((IdMap(C)).b) = b by A8,A9,A6,A5,Def30;
consider c be morphism of C such that
A11: cod u = c & c |> u & c is identity by A5,Def19;
A12: c = c (*) u by A7,A11,Def5 .= u by A11,Def4;
thus
A13: (TargetMap(C)).((IdMap(C)).b) = b by A11,A12,A6,A5,Def31;
thus for f being Element of Mor(C) st (TargetMap(C)).f = b
holds (CompMap(C)).((IdMap(C)).b,f) = f
proof
let f be Element of Mor(C);
assume (TargetMap(C)).f = b;
then
A14: [u,f] in dom(CompMap(C)) by A6,A10,A5,Th37;
then
A15: u |> f;
thus (CompMap(C)).((IdMap(C)).b,f) = (CompMap(C)).[u,f]
by A6,BINOP_1:def 1
.= u (*) f by A14,Def2,Lm4
.= f by A7,A15,Def4;
end;
thus for g being Element of Mor(C) st (SourceMap(C)).g = b
holds (CompMap(C)).(g,(IdMap(C)).b) = g
proof
let g be Element of Mor(C);
assume (SourceMap(C)).g = b;
then
A16: [g,u] in dom(CompMap(C)) by A6,A13,A5,Th37;
then
A17: g |> u;
thus (CompMap(C)).(g,(IdMap(C)).b)
= (CompMap(C)).[g,u] by A6,BINOP_1:def 1
.= g (*) u by A16,Def2,Lm4
.= g by A7,A17,Def5;
end;
end;
end;
Lm6:
for C being non empty category holds
CatStr(# Ob(C), Mor(C), SourceMap(C), TargetMap(C), CompMap(C) #)
is Category
proof
let C be non empty composable associative with_identities CategoryStr;
set CC = CatStr(# Ob(C), Mor(C), SourceMap(C),
TargetMap(C), CompMap(C) #);
reconsider CC as non empty non void CatStr;
A1: for f,g being Morphism of CC
holds [g,f] in dom(the Comp of CC) iff dom g = cod f
proof
let f,g be Morphism of CC;
(the Source of CC).g = dom g &
(the Target of CC).f = cod f by GRAPH_1:def 4,def 3;
hence thesis by Th37;
end;
A2: for f,g being Morphism of CC st dom g = cod f
holds dom(g (*) f) = dom f & cod(g (*) f) = cod g
proof
let f,g be Morphism of CC;
assume A3: dom g = cod f;
then
A4: (the Source of CC).g = cod f by GRAPH_1:def 3
.= (the Target of CC).f by GRAPH_1:def 4;
A5: [g,f] in dom the Comp of CC by A3,A1;
thus dom(g (*) f) = (the Source of CC).(g (*) f) by GRAPH_1:def 3
.= (the Source of CC).((the Comp of CC).(g,f)) by A5,CAT_1:def 1
.= (the Source of CC).f by A4,Th38
.= dom f by GRAPH_1:def 3;
thus cod(g (*) f) = (the Target of CC).(g (*) f) by GRAPH_1:def 4
.= (the Target of CC).((the Comp of CC).(g,f)) by A5,CAT_1:def 1
.= (the Target of CC).g by A4,Th38
.= cod g by GRAPH_1:def 4;
end;
A6: for f,g,h being Morphism of CC st dom h = cod g & dom g = cod f
holds h(*)(g(*)f) = (h(*)g)(*)f
proof
let f,g,h be Morphism of CC;
assume A7: dom h = cod g;
then
A8: (the Source of CC).h = cod g by GRAPH_1:def 3
.= (the Target of CC).g by GRAPH_1:def 4;
A9: [h,g] in dom the Comp of CC by A7,A1;
assume A10: dom g = cod f;
then
A11: (the Source of CC).g = cod f by GRAPH_1:def 3
.= (the Target of CC).f by GRAPH_1:def 4;
A12: [g,f] in dom the Comp of CC by A10,A1;
dom h = cod(g(*)f) by A2,A7,A10;
then
A13: [h,g(*)f] in dom the Comp of CC by A1;
dom (h(*)g) = cod f by A2,A7,A10;
then
A14: [h(*)g,f] in dom the Comp of CC by A1;
thus h(*)(g(*)f) = (the Comp of CC).(h,g(*)f) by A13,CAT_1:def 1
.= (the Comp of CC).(h,(the Comp of CC).(g,f)) by A12,CAT_1:def 1
.= (the Comp of CC).((the Comp of CC).(h,g),f) by Th39,A8,A11
.= (the Comp of CC).(h(*)g,f) by A9,CAT_1:def 1
.= (h(*)g)(*)f by A14,CAT_1:def 1;
end;
A15: for b being Element of CC holds (IdMap(C)).b in Hom(b,b)
proof
let b be Element of CC;
reconsider o = b as Element of Ob(C);
reconsider f = o as Morphism of CC by TARSKI:def 3;
A16: (IdMap(C)).b = id- o by Def32 .= f;
(SourceMap(C)).((IdMap(C)).b) = b & (TargetMap(C)).((IdMap(C)).b) = b
by Th40;
then (SourceMap(C)).(id- o) = b & (TargetMap(C)).(id- o) = b by Def32;
then dom f = b & cod f = b by GRAPH_1:def 3,def 4;
hence (IdMap(C)).b in Hom(b,b) by A16,CAT_1:1;
end;
then
A17: for b being Element of CC holds Hom(b,b) <> {};
for a being Element of CC
ex i being Morphism of a,a st
for b being Element of CC holds
(Hom(a,b)<>{} implies for g being Morphism of a,b holds g(*)i = g) &
(Hom(b,a)<>{} implies for f being Morphism of b,a holds i(*)f = f)
proof
let a be Element of CC;
set i = (IdMap(C)).a;
A18: i in Hom (a,a) by A15;
then reconsider i as Morphism of a,a by CAT_1:def 5;
take i;
let b be Element of CC;
thus (Hom(a,b)<>{} implies for g being Morphism of a,b holds g(*)i = g)
proof
assume A19: Hom(a,b)<>{};
let g be Morphism of a,b;
g in Hom(a,b) by A19,CAT_1:def 5;
then
A20: dom g = a by CAT_1:1;
then dom g = cod i by A18,CAT_1:1;
then
A21: [g,i] in dom the Comp of CC by A1;
(the Source of CC).g = a by A20,GRAPH_1:def 3;
then (CompMap(C)).(g,(IdMap(C)).a) = g by Th40;
hence g(*)i = g by A21,CAT_1:def 1;
end;
assume A22: Hom(b,a)<>{};
let f be Morphism of b,a;
f in Hom(b,a) by A22,CAT_1:def 5;
then
A23: cod f = a by CAT_1:1;
then cod f = dom i by A18,CAT_1:1;
then
A24: [i,f] in dom the Comp of CC by A1;
(the Target of CC).f = a by A23,GRAPH_1:def 4;
then (CompMap(C)).((IdMap(C)).a,f) = f by Th40;
hence i(*)f = f by A24,CAT_1:def 1;
end;
hence thesis by A1,A2,A6,A17,CAT_1:def 6,def 7,def 8,def 9,def 10;
end;
::$INSERT: A category defined in \cite{CAT_1.ABS}, to avoid confusion, is called an object-category.
definition
let C be non empty category;
func Alter(C) -> strict Category equals
CatStr(#Ob(C), Mor(C), SourceMap(C), TargetMap(C), CompMap(C)#);
coherence by Lm6;
end;
Lm7:
for C being Category holds
CategoryStr(# the carrier' of C, the Comp of C #) is category
proof
let C be Category;
consider o be element such that
A1: o in the carrier of C by XBOOLE_0:def 1;
reconsider o as Object of C by A1;
reconsider cc = CategoryStr (# the carrier' of C, the Comp of C #)
as non empty CategoryStr;
A2: for f,g being morphism of cc, f1,g1 being Element of the carrier' of C
st g = g1 & f = f1 & g |> f holds g(*)f = g1(*)f1
proof
let f,g be morphism of cc;
let f1,g1 be Element of the carrier' of C;
assume
A3: g = g1 & f = f1;
assume
A4: g |> f;
thus g(*)f = (CompMap(cc)).[g1,f1] by A3,A4,Lm4
.= (the Comp of C).(g1,f1) by BINOP_1:def 1
.= g1(*)f1 by A3,A4,CAT_1:def 1;
end;
A5: for f,g,h being morphism of cc holds
(h(*)g |> f & h |> g implies g |> f) &
(h |> g(*)f & g |> f implies h |> g) &
(g |> f & h |> g implies
h(*)g |> f & h |> g(*)f)
proof
let f,g,h be morphism of cc;
reconsider f1=f,g1=g,h1=h as Morphism of C;
thus (h(*)g |> f & h |> g
implies g |> f)
proof
assume
A6: h(*)g |> f;
assume
A7: h |> g;
then
A8: dom h1 = cod g1 by CAT_1:def 6;
h(*)g = h1(*)g1 by A7,A2;
then dom(h1(*)g1) = cod f1 by A6,CAT_1:def 6;
then dom g1 = cod f1 by A8,CAT_1:def 7;
hence g |> f by CAT_1:def 6;
end;
thus (h |> g(*)f & g |> f
implies h |> g)
proof
assume
A9: h |> g(*)f;
assume
A10: g |> f;
then
A11: dom g1 = cod f1 by CAT_1:def 6;
g(*)f = g1(*)f1 by A10,A2;
then
dom h1 = cod(g1(*)f1) by A9,CAT_1:def 6;
then dom h1 = cod g1 by A11,CAT_1:def 7;
hence h |> g by CAT_1:def 6;
end;
assume
A12: g |> f;
then
A13: dom g1 = cod f1 by CAT_1:def 6;
assume
A14: h |> g;
A15: dom h1 = cod g1 by A14,CAT_1:def 6;
dom(h1(*)g1) = cod f1 by A13,A15,CAT_1:def 7;
then [h1(*)g1, f1] in dom(CompMap(cc)) by CAT_1:def 6;
hence h(*)g |> f by A14,A2;
dom h1 = cod g1 by A14,CAT_1:def 6;
then dom h1 = cod(g1(*)f1) by A13,CAT_1:def 7;
then [h1, g1(*)f1] in dom(CompMap(cc)) by CAT_1:def 6;
hence h |> g(*)f by A12,A2;
end;
A16: for f,g,h being morphism of cc
st h |> g & g |> f
holds h(*)(g(*)f) = (h(*)g)(*)f
proof
let f,g,h be morphism of cc;
reconsider f1=f,g1=g,h1=h,gf1=g(*)f,hg1=h(*)g as Morphism of C;
assume
A17: h |> g;
then
A18: dom h1 = cod g1 by CAT_1:def 6;
assume
A19: g |> f;
then
A20: dom g1 = cod f1 by CAT_1:def 6;
A21: h |> g(*)f by A17,A19,A5;
A22: h(*)g |> f by A17,A19,A5;
thus h(*)(g(*)f) = h1(*)gf1 by A21,A2 .= h1(*)(g1(*)f1) by A19,A2
.= (h1(*)g1)(*)f1 by A20,A18,CAT_1:def 8
.= (hg1)(*)f1 by A17,A2
.= (h(*)g)(*)f by A22,A2;
end;
A23: for f being morphism of cc st f in Mor(cc) holds
(ex c being morphism of cc st c |> f & c is identity) &
(ex d being morphism of cc st f |> d & d is identity)
proof
let f be morphism of cc;
assume f in Mor(cc);
reconsider f1=f as Morphism of C;
thus (ex c being morphism of cc st c |> f & c is identity)
proof
set c1 = id cod f1;
A24: dom c1 = cod f1;
reconsider c = c1 as morphism of cc;
take c;
thus c |> f by A24,CAT_1:def 6;
A25: for ff being morphism of cc st c |> ff holds c (*) ff = ff
proof
let ff be morphism of cc;
reconsider ff1=ff as Morphism of C;
assume A26: c |> ff;
then
A27: dom c1 = cod ff1 by CAT_1:def 6;
thus c (*) ff = c1(*)ff1 by A26,A2 .= ff by A27,CAT_1:21;
end;
for gg being morphism of cc st gg |> c holds gg (*) c = gg
proof
let gg be morphism of cc;
reconsider gg1=gg as Morphism of C;
assume A28: gg |> c;
then
A29: cod c1 = dom gg1 by CAT_1:def 6;
thus gg (*) c = gg1(*)c1 by A28,A2 .= gg by A29,CAT_1:22;
end;
hence c is identity by A25,Def4,Def5;
end;
set d1 = id dom f1;
A30: cod d1 = dom f1;
reconsider d=d1 as morphism of cc;
take d;
thus f |> d by A30,CAT_1:def 6;
A31: for ff being morphism of cc st d |> ff holds d (*) ff = ff
proof
let ff be morphism of cc;
reconsider ff1=ff as Morphism of C;
assume A32: d |> ff;
then
A33: dom d1 = cod ff1 by CAT_1:def 6;
thus d (*) ff = d1(*)ff1 by A32,A2 .= ff by A33,CAT_1:21;
end;
for gg being morphism of cc st gg |> d holds gg (*) d = gg
proof
let gg be morphism of cc;
reconsider gg1=gg as Morphism of C;
assume A34: gg |> d;
then
A35: cod d1 = dom gg1 by CAT_1:def 6;
thus gg (*) d = gg1(*)d1 by A34,A2 .= gg by A35,CAT_1:22;
end;
hence d is identity by A31,Def4,Def5;
end;
set CS = CategoryStr (# the carrier' of C, the Comp of C #);
CS is composable by A5,Lm3;
hence thesis by A23,A16,Lm5,Lm2;
end;
definition
let A be Category;
func alter(A) -> strict category equals
CategoryStr(# the carrier' of A, the Comp of A #);
coherence by Lm7;
end;
registration
let A be Category;
cluster alter(A) -> non empty;
correctness;
end;
theorem Th41:
for A being Category, a1,a2 being (Morphism of A),
f1,f2 being morphism of alter(A) st a1 = f1 & a2 = f2 &
[a1,a2] in dom the Comp of A holds a1(*)a2 = f1(*)f2
proof
let A be Category;
let a1,a2 be Morphism of A;
let f1,f2 be morphism of alter(A);
assume
A1: a1 = f1 & a2 = f2;
assume
A2: [a1,a2] in dom the Comp of A;
thus a1(*)a2 = (the composition of alter A).(f1,f2) by A1,A2,CAT_1:def 1
.= f1(*)f2 by A2,A1,Def2,Def3;
end;
theorem Th42:
for A being Category, f being morphism of alter(A)
holds f is identity iff ex o being Object of A st f = id o
proof
let A be Category;
let f be morphism of alter(A);
hereby
assume
A1: f is identity;
A3: f is Object of alter(A) by A1,Th22;
then
A4: for f1 being morphism of alter(A) holds
( f |> f1 implies f (*) f1 = f1 ) & ( f1 |> f implies f1 (*) f = f1 ) &
f |> f by Th23;
reconsider a1 = f as Morphism of A;
[a1,a1] in dom the Comp of A by A4,Def2;
then
A5: dom a1 = cod a1 by CAT_1:15;
set o = dom a1;
reconsider a1 as Morphism of o,o by A5,CAT_1:4;
take o;
for b being Object of A holds
(Hom(o,b)<>{} implies for a being Morphism of o,b holds a(*)a1 = a) &
(Hom(b,o)<>{} implies for a being Morphism of b,o holds a1(*)a = a)
proof
let b be Object of A;
thus (Hom(o,b)<>{} implies
for f1 being Morphism of o,b holds f1(*)a1 = f1)
proof
assume
A6: Hom(o,b)<>{};
let f1 be Morphism of o,b;
f1 in Hom(o,b) by A6,CAT_1:def 5;
then dom f1 = o & cod f1 = b by CAT_1:1;
then
A7: [f1,a1] in dom the Comp of A by A5,CAT_1:def 6;
reconsider f2 = f1 as morphism of alter(A);
f2(*)f = f2 by A4,A7,Def2;
hence f1(*)a1 = f1 by Th41,A7;
end;
assume
A8: Hom(b,o)<>{};
let f1 be Morphism of b,o;
f1 in Hom(b,o) by A8,CAT_1:def 5;
then
A9: dom f1 = b & cod f1 = o by CAT_1:1;
then
A10: [a1,f1] in dom the Comp of A by CAT_1:def 6;
reconsider f2 = f1 as morphism of alter(A);
f |> f2 by A9,CAT_1:def 6;
then f(*)f2 = f2 by A3,Th23;
hence a1(*)f1 = f1 by Th41,A10;
end;
hence f = id o by CAT_1:def 12;
end;
given o be Object of A such that
A11: f = id o;
A12: for f1 being morphism of alter(A) st f |> f1 holds f (*) f1 = f1
proof
let f1 be morphism of alter(A);
assume
A13: f |> f1;
reconsider a2 = f1, a1 = f as Morphism of A;
A14: cod a2 = dom id o by A11,CAT_1:15,A13 .= o;
thus f (*) f1 = a1(*)a2 by A13,Th41
.= f1 by A11,A14,CAT_1:21;
end;
for f1 being morphism of alter(A) st f1 |> f holds f1 (*) f = f1
proof
let f1 be morphism of alter(A);
assume
A15: f1 |> f;
reconsider a2 = f1, a1 = f as Morphism of A;
A16: dom a2 = cod id o by A15,A11,CAT_1:15 .= o;
thus f1 (*) f = a2(*)a1 by A15,Th41
.= f1 by A11,A16,CAT_1:22;
end;
then f is right_identity;
hence f is identity by A12,Def4;
end;
theorem
for A,B being Category, F being Functor of A,B
holds F is covariant Functor of alter(A), alter(B)
proof
let A,B be Category;
let F be Functor of A,B;
reconsider F1 = F as Function of alter(A), alter(B);
for f1,f2 being morphism of alter(A) st f1 |> f2
holds F1.f1 |> F1.f2 & F1.(f1(*)f2) = (F1.f1)(*)(F1.f2)
proof
let f1,f2 be morphism of alter(A);
assume
A1: f1 |> f2;
reconsider a1 = f1, a2 = f2 as Morphism of A;
A2: dom a1 = cod a2 by A1,CAT_1:15;
dom(F.a1) = F.(dom a1) & cod(F.a2) = F.(cod a2) by CAT_1:72;
then dom(F.a1) = cod(F.a2) by A1,CAT_1:15;
then
A3: [F.a1,F.a2] in dom the Comp of B by CAT_1:15;
A4: F1.f1 = F.a1 by Def21;
A5: F1.f2 = F.a2 by Def21;
thus F1.f1,F1.f2 are_composable by A3,A4,Def21;
f1(*)f2 = a1(*)a2 by A1,Th41;
hence F1.(f1(*)f2) = F.(a1(*)a2) by Def21
.= (F.a1)(*)(F.a2) by A2,CAT_1:64
.= (F1.f1)(*)(F1.f2) by A3,A4,A5,Th41;
end;
then
A6: F1 is multiplicative;
for f being morphism of alter(A) st f is identity holds F1.f is identity
proof
let f be morphism of alter(A);
assume f is identity;
then consider o be Object of A such that
A7: f = id o by Th42;
consider o1 be Object of B such that
A8: F.(id o) = id o1 by CAT_1:62;
thus F1.f is identity by Def21,A7,A8,Th42;
end;
hence thesis by A6,Def25,Def22;
end;
theorem Th44:
for C being non empty category, a1,a2 being (Morphism of Alter(C)),
f1,f2 being morphism of C st a1 = f1 & a2 = f2 & f1 |> f2
holds a1(*)a2 = f1(*)f2
proof
let C be non empty category;
let a1,a2 be Morphism of Alter(C);
let f1,f2 be morphism of C;
assume
A1: a1 = f1 & a2 = f2;
assume
A2: f1 |> f2;
thus a1(*)a2 = (the Comp of Alter(C)).(a1,a2) by A1,A2,CAT_1:def 1
.= f1(*)f2 by A1,A2,Def3;
end;
theorem Th45:
for C being non empty category, f1 being morphism of C,
a1 being Morphism of Alter(C) st a1 = f1
holds dom f1 = dom a1 & cod f1 = cod a1
proof
let C be non empty category;
let f1 be morphism of C;
let a1 be Morphism of Alter(C);
assume
A1: a1 = f1;
thus dom f1 = (the Source of Alter(C)).a1 by A1,Def30
.= dom a1 by GRAPH_1:def 3;
thus cod f1 = (the Target of Alter(C)).a1 by A1,Def31
.= cod a1 by GRAPH_1:def 4;
end;
theorem Th46:
for C being non empty category, o1 being Object of C,
o2 being Object of Alter(C) st o1 = o2 holds id- o1 = id o2
proof
let C be non empty category;
let o1 be Object of C;
let o2 be Object of Alter(C);
assume
A1: o1 = o2;
A2: o1 in Ob(C);
then reconsider f1 = o1 as morphism of C;
reconsider a2 = o2 as Morphism of Alter(C) by A1,A2;
A3: f1 is identity & f1 |> f1 by Th22,Th24;
A4: dom a2 = dom f1 by A1,Th45 .= o2 by A1,A3,Def18;
A5: cod a2 = cod f1 by A1,Th45 .= o2 by A1,A3,Def19;
reconsider a3 = a2 as Morphism of o2,o2 by A4,A5,CAT_1:4;
for b being Object of Alter(C) holds
( Hom (o2,b) <> {} implies
for a being Morphism of o2,b holds a (*) a3 = a ) &
( Hom (b,o2) <> {} implies
for a being Morphism of b,o2 holds a3 (*) a = a )
proof
let b be Object of Alter(C);
thus Hom (o2,b) <> {} implies
for a being Morphism of o2,b holds a (*) a3 = a
proof
assume A6: Hom (o2,b) <> {};
let a be Morphism of o2,b;
reconsider f2=a as morphism of C;
dom a = cod a3 by A5,A6,CAT_1:5;
then
A7: [f2,f1] in dom the composition of C by A1,CAT_1:15;
thus a (*) a3 = f2 (*) f1 by A1,A7,Def2,Th44 .= a
by A3,A7,Def2,Def5;
end;
assume A8: Hom (b,o2) <> {};
let a be Morphism of b,o2;
reconsider f2=a as morphism of C;
dom a3 = cod a by A4,A8,CAT_1:5;
then
A9: [f1,f2] in dom the composition of C by A1,CAT_1:15;
thus a3 (*) a = f1 (*) f2 by A1,A9,Def2,Th44 .= a
by A3,A9,Def2,Def4;
end;
hence thesis by A1,CAT_1:def 12;
end;
theorem Th47:
for C being non empty category, f being morphism of C
holds f is identity iff ex o being Object of Alter(C) st f = id o
proof
let C be non empty category;
let f be morphism of C;
set A = Alter(C);
reconsider a=f as morphism of alter(A);
hereby
assume f is identity;
then a is identity by Th25;
then consider o be Object of Alter(C) such that
A1: a = id o by Th42;
take o;
thus f = id o by A1;
end;
given o be Object of Alter(C) such that
A2: f = id o;
a is identity by A2,Th42;
hence f is identity by Th25;
end;
theorem
for C,D being non empty category, F being covariant Functor of C,D
holds F is Functor of Alter(C), Alter(D)
proof
let C,D be non empty category;
let F be covariant Functor of C,D;
reconsider F1 = F as
Function of the carrier' of Alter(C),the carrier' of Alter(D);
A1: F is identity-preserving & F is multiplicative by Def25;
A2: for a being Object of Alter(C) ex b being Object of Alter(D)
st F1.(id a) = id b
proof
let a be Object of Alter(C);
reconsider a1 = id a as morphism of C;
a1 is identity by Th47;
then consider b be Object of Alter(D) such that
A3: F.a1 = id b by Th47,A1;
take b;
thus F1.(id a) = id b by A3,Def21;
end;
A4: for f being Morphism of Alter(C) holds F1.(id dom f) = id dom (F1.f) &
F1.(id cod f) = id cod (F1.f)
proof
let f be Morphism of Alter(C);
reconsider o1 = dom f as Object of Alter(C);
reconsider o2 = o1 as Object of C;
reconsider f1=f as morphism of C;
A5: F.f1 = F1.f by Def21;
A6: F.o2 = F1.(dom f1) by Th45
.= dom(F.f1) by Th32
.= dom(F1.f) by Th45,A5;
thus F1.(id dom f) = F1.(id- o2) by Th46
.= id-(F.o2)
.= id dom (F1.f) by A6,Th46;
reconsider o3 = cod f as Object of Alter(C);
reconsider o4 = o3 as Object of C;
reconsider f1=f as morphism of C;
A7: F.f1 = F1.f by Def21;
A8: F.o3 = F1.(cod f1) by Th45
.= cod(F.f1) by Th32
.= cod(F1.f) by Th45,A7;
thus F1.(id cod f) = F1.(id- o4) by Th46
.= id-(F.o4)
.= id cod (F1.f) by A8,Th46;
end;
for f,g being Morphism of Alter(C) st dom g = cod f
holds F1.(g(*)f) = (F1.g)(*)(F1.f)
proof
let f,g be Morphism of Alter(C);
assume
A9: dom g = cod f;
reconsider f1=f,g1=g as morphism of C;
A10: [g1,f1] in dom the composition of C by A9,CAT_1:15;
A11: g1 |> f1 by A9,CAT_1:15;
A12: (the composition of C).(g1,f1) = g1(*)f1 by A10,Def2,Def3;
A13: F.g1 = F1.g & F.f1 = F1.f by Def21;
A14: [F1.g,F1.f] in dom the Comp of Alter D by A13,Def2,A11,A1;
thus F1.(g(*)f) = F1.((the Comp of Alter C).(g,f)) by A9,CAT_1:16
.= F.(g1(*)f1) by A12,Def21
.= (F.g1)(*)(F.f1) by A11,A1
.= (the Comp of Alter(D)).(F1.g,F1.f) by A13,Def3,A11,A1
.= (F1.g)(*)(F1.f) by A14,CAT_1:def 1;
end;
hence thesis by A2,A4,CAT_1:61;
end;
theorem Th49:
for C,D being Category, F being covariant Functor of alter(C), alter(D)
holds F is Functor of C, D
proof
let C,D be Category;
let F be covariant Functor of alter(C), alter(D);
reconsider F1 = F as
Function of the carrier' of C,the carrier' of D;
A1: F is identity-preserving & F is multiplicative by Def25;
A2: for a being Object of C ex b being Object of D st F1.(id a) = id b
proof
let a be Object of C;
reconsider a1 = id a as morphism of alter(C);
a1 is identity by Th42;
then consider b be Object of D such that
A3: F.a1 = id b by Th42,A1;
take b;
thus F1.(id a) = id b by A3,Def21;
end;
A4: for f being Morphism of C holds F1.(id dom f) = id dom (F1.f) &
F1.(id cod f) = id cod (F1.f)
proof
let f be Morphism of C;
reconsider f1 = f as morphism of alter(C);
reconsider d1 = id dom f as morphism of alter(C);
dom f = cod (id dom f);
then
A5: f1 |> d1 by CAT_1:15;
A6: F.d1 is identity by Th42,A1;
reconsider d2 = id dom (F1.f) as morphism of alter(D);
dom (F1.f) = cod (id dom (F1.f));
then [F1.f,id dom (F1.f)] in dom the Comp of D by CAT_1:15;
then
A7: F.f1 |> d2 by Def21;
F.d1 = dom (F.f1) by A1,A5,A6,Th26 .= d2 by Th42,A7,Th26;
hence F1.(id dom f) = id dom (F1.f) by Def21;
reconsider c1 = id cod f as morphism of alter(C);
cod f = dom (id cod f);
then
A8: c1 |> f1 by CAT_1:15;
A9: F.c1 is identity by Th42,A1;
reconsider c2 = id cod (F1.f) as morphism of alter(D);
cod (F1.f) = dom (id cod (F1.f));
then [id cod (F1.f),F1.f] in dom the Comp of D by CAT_1:15;
then
A10: c2 |> F.f1 by Def21;
F.c1 = cod (F.f1) by A1,A8,A9,Th27 .= c2 by Th42,A10,Th27;
hence F1.(id cod f) = id cod (F1.f) by Def21;
end;
for f,g being Morphism of C st dom g = cod f
holds F1.(g(*)f) = (F1.g)(*)(F1.f)
proof
let f,g be Morphism of C;
assume
A11: dom g = cod f;
reconsider f1=f,g1=g as morphism of alter(C);
A12: [g1,f1] in dom the composition of alter(C) by A11,CAT_1:15;
A13: g1 |> f1 by A11,CAT_1:15;
A14: (the composition of alter(C)).(g1,f1) = g1(*)f1 by A12,Def2,Def3;
A15: F.g1 = F1.g & F.f1 = F1.f by Def21;
A16: [F1.g,F1.f] in dom the Comp of D by A15,Def2,A13,A1;
thus F1.(g(*)f) = F1.((the Comp of C).(g,f)) by A11,CAT_1:16
.= F.(g1(*)f1) by A14,Def21
.= (F.g1)(*)(F.f1) by A13,A1
.= (the Comp of D).(F1.g,F1.f) by A15,Def3,A13,A1
.= (F1.g)(*)(F1.f) by A16,CAT_1:def 1;
end;
hence thesis by A2,A4,CAT_1:61;
end;
theorem Th50:
for C1,C2 being Category st alter(C1) ~= alter(C2) holds C1 ~= C2
proof
let C1,C2 be Category;
assume alter(C1) ~= alter(C2);
then consider F be Functor of alter(C1), alter(C2),
G be Functor of alter(C2), alter(C1) such that
A1: F is covariant & G is covariant & G (*) F = id alter(C1) &
F (*) G = id alter(C2);
reconsider F1 = F as Functor of C1,C2 by A1,Th49;
A2: dom F = the carrier of alter(C1) by FUNCT_2:def 1;
G (*) F = F * G by A1,Def27;
then F * G = id the carrier of alter(C1) by A1,STRUCT_0:def 4;
then
A3: F is one-to-one by A2,FUNCT_1:31;
A4: F (*) G = G * F by A1,Def27;
A5: the carrier' of C2 = rng(id the carrier' of C2)
.= rng(G*F) by A4,A1,STRUCT_0:def 4;
the carrier' of C2 c= rng F by A5,RELAT_1:26;
then rng F1 = the carrier' of C2 by XBOOLE_0:def 10;
hence C1 ~= C2 by A3,ISOCAT_1:def 4,def 3;
end;
theorem Th51:
for C1,C2 being Category st the carrier' of C1 = the carrier' of C2
& the Comp of C1 = the Comp of C2 holds C1 ~= C2
proof
let C1,C2 be Category;
assume the carrier' of C1 = the carrier' of C2 &
the Comp of C1 = the Comp of C2;
then alter(C1) = alter(C2);
hence C1 ~= C2 by Th50;
end;
theorem
for C being Category holds C ~= Alter(alter(C)) by Th51;
theorem
for C being non empty category holds C ~= alter(Alter(C))
proof
let C be non empty category;
set D = alter(Alter(C));
reconsider X = the carrier of C as set;
reconsider I0 = id C as Function of X, X;
A1: I0 = id X by STRUCT_0:def 4;
reconsider F = id C as Functor of C,D;
reconsider G = id C as Functor of D,C;
for f being morphism of C st f is identity holds F.f is identity
proof
let f be morphism of C;
assume
A2: f is identity;
F.f = I0.f by Def21
.= (id X).f by STRUCT_0:def 4
.= f;
hence F.f is identity by A2,Th16,Th18;
end;
then
A3: F is identity-preserving;
for f1, f2 being morphism of C st f1 |> f2 holds
F.f1 |> F.f2 & F.(f1 (*) f2) = (F.f1) (*) (F.f2)
proof
let f1, f2 be morphism of C;
assume
A4: f1 |> f2;
A5: F.f1 = I0.f1 by Def21
.= (id X).f1 by STRUCT_0:def 4
.= f1;
A6: F.f2 = I0.f2 by Def21
.= (id X).f2 by STRUCT_0:def 4
.= f2;
hence
A7: F.f1 |> F.f2 by A4,A5;
thus F.(f1 (*) f2) = I0.(f1(*)f2) by Def21
.= (id X).(f1(*)f2) by STRUCT_0:def 4
.= (the composition of C).(f1,f2) by A4,Def3
.= (F.f1) (*) (F.f2) by A5,A6,A7,Def3;
end;
then F is multiplicative;
then
A8: F is covariant by A3;
for f being morphism of D st f is identity holds G.f is identity
proof
let f be morphism of D;
assume
A9: f is identity;
G.f = I0.f by Def21
.= (id X).f by STRUCT_0:def 4
.= f;
hence G.f is identity by A9,Th16,Th18;
end;
then
A10: G is identity-preserving;
for f1, f2 being morphism of D st f1 |> f2 holds
G.f1 |> G.f2 & G.(f1 (*) f2) = (G.f1) (*) (G.f2)
proof
let f1, f2 be morphism of D;
assume
A11: f1 |> f2;
A12: G.f1 = I0.f1 by Def21
.= (id X).f1 by STRUCT_0:def 4
.= f1;
A13: G.f2 = I0.f2 by Def21
.= (id X).f2 by STRUCT_0:def 4
.= f2;
hence
A14: G.f1 |> G.f2 by A11,A12;
thus G.(f1 (*) f2) = I0.(f1(*)f2) by Def21
.= (id X).(f1(*)f2) by STRUCT_0:def 4
.= (the composition of D).(f1,f2) by A11,Def3
.= (G.f1) (*) (G.f2) by A12,A13,A14,Def3;
end;
then G is multiplicative;
then
A15: G is covariant by A10;
A16: G (*) F = F*G by A8,A15,Def27
.= id(X /\ X) by A1,FUNCT_1:22
.= id C by STRUCT_0:def 4;
F (*) G = G*F by A8,A15,Def27
.= id(X /\ X) by A1,FUNCT_1:22
.= id D by STRUCT_0:def 4;
hence C ~= alter(Alter(C)) by A8,A15,A16;
end;