:: The Composition of Functors and Transformations in Alternative
:: Categories
:: by Artur Korni{\l}owicz
::
:: Received January 21, 1998
:: Copyright (c) 1998-2018 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 RELAT_2, BINOP_1, ALTCAT_1, STRUCT_0, XBOOLE_0, FUNCTOR0,
MSUALG_6, FUNCOP_1, CAT_1, RELAT_1, FUNCT_1, ZFMISC_1, TARSKI, MEMBER_1,
PBOOLE, NATTRA_1, PZFMISC1, REALSET1, FUNCTOR2, VALUED_1, ALTCAT_3,
CAT_3;
notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2,
STRUCT_0, BINOP_1, FUNCOP_1, PBOOLE, MSUALG_3, ALTCAT_1, ALTCAT_2,
ALTCAT_3, FUNCTOR0, FUNCTOR2;
constructors MSUALG_3, ALTCAT_3, FUNCTOR2, RELSET_1;
registrations XBOOLE_0, SUBSET_1, FUNCT_1, RELSET_1, FUNCT_2, FUNCOP_1,
RELAT_1, PBOOLE, STRUCT_0, ALTCAT_1, FUNCTOR0, FUNCTOR2, ALTCAT_4,
ALTCAT_2;
requirements SUBSET, BOOLE;
definitions FUNCTOR0, FUNCTOR2, PBOOLE;
equalities FUNCTOR0, BINOP_1;
expansions FUNCTOR2, PBOOLE;
theorems ALTCAT_1, ALTCAT_3, ALTCAT_4, FUNCTOR0, FUNCTOR2, ZFMISC_1, MSUALG_3,
PBOOLE, FUNCT_1, FUNCT_2, RELAT_1, XBOOLE_0, PARTFUN1;
schemes PBOOLE;
begin :: Preliminaries
registration
cluster transitive associative with_units strict for non empty AltCatStr;
existence
proof
set A = the transitive associative with_units strict non empty AltCatStr;
take A;
thus thesis;
end;
end;
registration
let A be non empty transitive AltCatStr, B be with_units non empty
AltCatStr;
cluster strict comp-preserving comp-reversing Covariant Contravariant
feasible for FunctorStr over A, B;
existence
proof
set o = the Object of B;
take A --> idm o;
thus thesis;
end;
end;
registration
let A be with_units transitive non empty AltCatStr, B be with_units non
empty AltCatStr;
cluster strict comp-preserving comp-reversing Covariant Contravariant
feasible id-preserving for FunctorStr over A, B;
existence
proof
set o = the Object of B;
take A --> idm o;
thus thesis;
end;
end;
registration
let A be with_units transitive non empty AltCatStr, B be with_units non
empty AltCatStr;
cluster strict feasible covariant contravariant for Functor of A, B;
existence
proof
set o = the Object of B;
set I = A --> idm o;
reconsider I as Functor of A, B by FUNCTOR0:def 25;
take I;
thus I is strict feasible;
thus I is covariant;
thus thesis;
end;
end;
theorem
for C being category, o1, o2, o3, o4 being Object of C for a being
Morphism of o1, o2, b being Morphism of o2, o3 for c being Morphism of o1, o4,
d being Morphism of o4, o3 st b*a = d*c & a*(a") = idm o2 & d"*d = idm o4 & <^
o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} & <^o3,o4^> <> {} & <^o4,o3^>
<> {} holds c*(a") = d"*b
proof
let C be category, o1, o2, o3, o4 be Object of C, a be Morphism of o1, o2, b
be Morphism of o2, o3, c be Morphism of o1, o4, d be Morphism of o4, o3 such
that
A1: b*a = d*c and
A2: a*(a") = idm o2 and
A3: d"*d = idm o4 and
A4: <^o1,o2^> <> {} and
A5: <^o2,o1^> <> {} and
A6: <^o2,o3^> <> {} and
A7: <^o3,o4^> <> {} and
A8: <^o4,o3^> <> {};
A9: <^o2,o4^> <> {} by A6,A7,ALTCAT_1:def 2;
<^o1,o3^> <> {} by A4,A6,ALTCAT_1:def 2;
then
A10: <^o1,o4^> <> {} by A7,ALTCAT_1:def 2;
b = b*idm o2 by A6,ALTCAT_1:def 17
.= b*a*(a") by A2,A4,A5,A6,ALTCAT_1:21;
hence d"*b = d"*(d*(c*(a"))) by A1,A5,A8,A10,ALTCAT_1:21
.= (d"*d)*(c*(a")) by A7,A8,A9,ALTCAT_1:21
.= c*(a") by A3,A9,ALTCAT_1:20;
end;
theorem
for A being non empty transitive AltCatStr for B, C being with_units
non empty AltCatStr for F being feasible Covariant FunctorStr over A, B for G
being FunctorStr over B, C, o, o1 being Object of A holds Morph-Map(G*F,o,o1) =
Morph-Map(G,F.o,F.o1)*Morph-Map(F,o,o1)
proof
let A be non empty transitive AltCatStr, B, C be with_units non empty
AltCatStr, F be feasible Covariant FunctorStr over A, B, G be FunctorStr over
B, C, o, o1 be Object of A;
dom(the MorphMap of G) = [:the carrier of B,the carrier of B:] & rng(the
ObjectMap of F) c= [:the carrier of B,the carrier of B:] by PARTFUN1:def 2;
then
dom((the MorphMap of G)*the ObjectMap of F) = dom(the ObjectMap of F) by
RELAT_1:27
.= [:the carrier of A,the carrier of A:] by FUNCT_2:def 1;
then
A1: [o,o1] in dom((the MorphMap of G)*the ObjectMap of F) by ZFMISC_1:87;
then
A2: ((the MorphMap of G)*the ObjectMap of F).[o,o1] = (the MorphMap of G).((
the ObjectMap of F).(o,o1)) by FUNCT_1:12
.= Morph-Map(G,F.o,F.o1) by FUNCTOR0:22;
dom(the MorphMap of F) = [:the carrier of A,the carrier of A:] by
PARTFUN1:def 2;
then [o,o1] in dom(the MorphMap of F) by ZFMISC_1:87;
then [o,o1] in dom((the MorphMap of G)*the ObjectMap of F) /\ dom(the
MorphMap of F) by A1,XBOOLE_0:def 4;
then
A3: [o,o1] in dom(((the MorphMap of G)*the ObjectMap of F)**the MorphMap of
F) by PBOOLE:def 19;
thus Morph-Map(G*F,o,o1) = (((the MorphMap of G)*the ObjectMap of F)**the
MorphMap of F).(o,o1) by FUNCTOR0:def 36
.= Morph-Map(G,F.o,F.o1)*Morph-Map(F,o,o1) by A3,A2,PBOOLE:def 19;
end;
theorem
for A being non empty transitive AltCatStr for B, C being with_units
non empty AltCatStr for F being feasible Contravariant FunctorStr over A, B
for G being FunctorStr over B, C, o, o1 being Object of A holds Morph-Map(G*F,o
,o1) = Morph-Map(G,F.o1,F.o)*Morph-Map(F,o,o1)
proof
let A be non empty transitive AltCatStr, B, C be with_units non empty
AltCatStr, F be feasible Contravariant FunctorStr over A, B, G be FunctorStr
over B, C, o, o1 be Object of A;
dom(the MorphMap of G) = [:the carrier of B,the carrier of B:] & rng(the
ObjectMap of F) c= [:the carrier of B,the carrier of B:] by PARTFUN1:def 2;
then
dom((the MorphMap of G)*the ObjectMap of F) = dom(the ObjectMap of F) by
RELAT_1:27
.= [:the carrier of A,the carrier of A:] by FUNCT_2:def 1;
then
A1: [o,o1] in dom((the MorphMap of G)*the ObjectMap of F) by ZFMISC_1:87;
then
A2: ((the MorphMap of G)*the ObjectMap of F).[o,o1] = (the MorphMap of G).((
the ObjectMap of F).(o,o1)) by FUNCT_1:12
.= Morph-Map(G,F.o1,F.o) by FUNCTOR0:23;
dom(the MorphMap of F) = [:the carrier of A,the carrier of A:] by
PARTFUN1:def 2;
then [o,o1] in dom(the MorphMap of F) by ZFMISC_1:87;
then [o,o1] in dom((the MorphMap of G)*the ObjectMap of F) /\ dom(the
MorphMap of F) by A1,XBOOLE_0:def 4;
then
A3: [o,o1] in dom(((the MorphMap of G)*the ObjectMap of F)**the MorphMap of
F) by PBOOLE:def 19;
thus Morph-Map(G*F,o,o1) = (((the MorphMap of G)*the ObjectMap of F)**the
MorphMap of F).(o,o1) by FUNCTOR0:def 36
.= Morph-Map(G,F.o1,F.o)*Morph-Map(F,o,o1) by A3,A2,PBOOLE:def 19;
end;
Lm1: for I1 being set, I2 being non empty set, f being Function of I1,I2 for A
being ManySortedSet of I1, B being ManySortedSet of I2 for M being
ManySortedFunction of A,B*f holds ((id B)*f)**M = M
proof
let I1 be set, I2 be non empty set, f be Function of I1,I2;
let A be ManySortedSet of I1, B be ManySortedSet of I2;
let M be ManySortedFunction of A,B*f;
A1: now
let i be object;
assume
A2: i in I1;
hence
A3: (B*f).i = B.(f.i) by FUNCT_2:15;
((id B)*f).i = (id B).(f.i) & f.i in I2 by A2,FUNCT_2:5,15;
hence ((id B)*f).i = id ((B*f).i) by A3,MSUALG_3:def 1;
end;
now
A4: (id B)*f is ManySortedFunction of B*f, B*f
proof
let i be object;
assume i in I1;
then ((id B)*f).i = id ((B*f).i) by A1;
hence thesis;
end;
let i be object;
assume
A5: i in I1;
then
A6: M.i is Function of A.i, (B*f).i by PBOOLE:def 15;
((id B)*f).i = (id B).(f.i) & f.i in I2 by A5,FUNCT_2:5,15;
then
A7: ((id B)*f).i = id (B.(f.i)) by MSUALG_3:def 1;
(B*f).i = B.(f.i) by A1,A5;
hence (((id B)*f)**M).i = (id ((B*f).i))*(M.i) by A5,A4,A7,MSUALG_3:2
.= M.i by A6,FUNCT_2:17;
end;
hence thesis;
end;
theorem
for A being non empty transitive AltCatStr for B being with_units non
empty AltCatStr for F being feasible FunctorStr over A, B holds (id B) * F =
the FunctorStr of F
proof
let A be non empty transitive AltCatStr, B be with_units non empty
AltCatStr, F be feasible FunctorStr over A, B;
A1: the ObjectMap of ((id B) * F) = (the ObjectMap of (id B))*the ObjectMap
of F by FUNCTOR0:def 36
.= (id [:the carrier of B, the carrier of B:])*the ObjectMap of F by
FUNCTOR0:def 29
.= the ObjectMap of F by FUNCT_2:17;
A2: the MorphMap of F is ManySortedFunction of the Arrows of A, (the Arrows
of B)*the ObjectMap of F by FUNCTOR0:def 4;
the MorphMap of ((id B) * F) = ((the MorphMap of id B)*the ObjectMap of
F)**the MorphMap of F by FUNCTOR0:def 36
.= ((id the Arrows of B)*the ObjectMap of F)**the MorphMap of F by
FUNCTOR0:def 29
.= the MorphMap of F by A2,Lm1;
hence thesis by A1;
end;
theorem
for A being with_units transitive non empty AltCatStr for B being
with_units non empty AltCatStr for F being feasible FunctorStr over A, B
holds F * (id A) = the FunctorStr of F
proof
let A be with_units transitive non empty AltCatStr, B be with_units non
empty AltCatStr, F be feasible FunctorStr over A, B;
A1: the ObjectMap of (F*(id A)) = (the ObjectMap of F)*the ObjectMap of id A
by FUNCTOR0:def 36
.= (the ObjectMap of F)*id [:the carrier of A, the carrier of A:] by
FUNCTOR0:def 29
.= the ObjectMap of F by FUNCT_2:17;
A2: the MorphMap of F is ManySortedFunction of the Arrows of A, (the Arrows
of B)*the ObjectMap of F by FUNCTOR0:def 4;
the MorphMap of (F*id A) = ((the MorphMap of F)*the ObjectMap of id A)**
the MorphMap of id A by FUNCTOR0:def 36
.= ((the MorphMap of F)*id [:the carrier of A, the carrier of A:]) **the
MorphMap of id A by FUNCTOR0:def 29
.= (the MorphMap of F)**the MorphMap of id A by FUNCTOR0:2
.= (the MorphMap of F)**id the Arrows of A by FUNCTOR0:def 29
.= the MorphMap of F by A2,MSUALG_3:3;
hence thesis by A1;
end;
reserve A for non empty AltCatStr,
B, C for non empty reflexive AltCatStr,
F for feasible Covariant FunctorStr over A, B,
G for feasible Covariant FunctorStr over B, C,
M for feasible Contravariant FunctorStr over A, B,
N for feasible Contravariant FunctorStr over B, C,
o1, o2 for Object of A,
m for Morphism of o1, o2;
theorem Th6:
<^o1,o2^> <> {} implies (G*F).m = G.(F.m)
proof
set I = the carrier of A;
reconsider s = (the MorphMap of F).(o1,o2) as Function;
reconsider r = (((the MorphMap of G)*the ObjectMap of F)** the MorphMap of F
).(o1,o2) as Function;
reconsider t = ((the MorphMap of G)*the ObjectMap of F).(o1,o2) as Function;
A1: dom (((the MorphMap of G)*the ObjectMap of F)**the MorphMap of F) = (dom
((the MorphMap of G)*the ObjectMap of F)) /\ (dom(the MorphMap of F)) by
PBOOLE:def 19
.= [:I,I:] /\ (dom(the MorphMap of F)) by PARTFUN1:def 2
.= [:I,I:] /\ [:I,I:] by PARTFUN1:def 2
.= [:I,I:];
A2: dom the ObjectMap of F = [:I,I:] by FUNCT_2:def 1;
A3: [o1,o2] in [:I,I:] by ZFMISC_1:def 2;
assume
A4: <^o1,o2^> <> {};
then
A5: <^F.o1,F.o2^> <> {} by FUNCTOR0:def 18;
then
A6: dom Morph-Map(F,o1,o2) = <^o1,o2^> by FUNCT_2:def 1;
A7: <^G.(F.o1),G.(F.o2)^> <> {} by A5,FUNCTOR0:def 18;
(G*F).o1 = G.(F.o1) & (G*F).o2 = G.(F.o2) by FUNCTOR0:33;
hence (G*F).m = Morph-Map(G*F,o1,o2).m by A4,A7,FUNCTOR0:def 15
.= r.m by FUNCTOR0:def 36
.= (t * s).m by A1,A3,PBOOLE:def 19
.= t.(Morph-Map(F,o1,o2).m) by A4,A6,FUNCT_1:13
.= t.(F.m) by A4,A5,FUNCTOR0:def 15
.= ((the MorphMap of G).((the ObjectMap of F).(o1,o2))).(F.m) by A2,A3,
FUNCT_1:13
.= Morph-Map(G,F.o1,F.o2).(F.m) by FUNCTOR0:22
.= G.(F.m) by A5,A7,FUNCTOR0:def 15;
end;
theorem Th7:
<^o1,o2^> <> {} implies (N*M).m = N.(M.m)
proof
set I = the carrier of A;
reconsider s = (the MorphMap of M).(o1,o2) as Function;
reconsider r = (((the MorphMap of N)*the ObjectMap of M)** the MorphMap of M
).(o1,o2) as Function;
reconsider t = ((the MorphMap of N)*the ObjectMap of M).(o1,o2) as Function;
A1: dom (((the MorphMap of N)*the ObjectMap of M)**the MorphMap of M) = (dom
((the MorphMap of N)*the ObjectMap of M)) /\ (dom(the MorphMap of M)) by
PBOOLE:def 19
.= [:I,I:] /\ (dom(the MorphMap of M)) by PARTFUN1:def 2
.= [:I,I:] /\ [:I,I:] by PARTFUN1:def 2
.= [:I,I:];
A2: dom the ObjectMap of M = [:I,I:] by FUNCT_2:def 1;
A3: [o1,o2] in [:I,I:] by ZFMISC_1:def 2;
assume
A4: <^o1,o2^> <> {};
then
A5: <^M.o2,M.o1^> <> {} by FUNCTOR0:def 19;
then
A6: dom Morph-Map(M,o1,o2) = <^o1,o2^> by FUNCT_2:def 1;
A7: <^N.(M.o1),N.(M.o2)^> <> {} by A5,FUNCTOR0:def 19;
(N*M).o1 = N.(M.o1) & (N*M).o2 = N.(M.o2) by FUNCTOR0:33;
hence (N*M).m = Morph-Map(N*M,o1,o2).m by A4,A7,FUNCTOR0:def 15
.= r.m by FUNCTOR0:def 36
.= (t * s).m by A1,A3,PBOOLE:def 19
.= t.(Morph-Map(M,o1,o2).m) by A4,A6,FUNCT_1:13
.= t.(M.m) by A4,A5,FUNCTOR0:def 16
.= ((the MorphMap of N).((the ObjectMap of M).(o1,o2))).(M.m) by A2,A3,
FUNCT_1:13
.= Morph-Map(N,M.o2,M.o1).(M.m) by FUNCTOR0:23
.= N.(M.m) by A5,A7,FUNCTOR0:def 16;
end;
theorem Th8:
<^o1,o2^> <> {} implies (N*F).m = N.(F.m)
proof
set I = the carrier of A;
reconsider s = (the MorphMap of F).(o1,o2) as Function;
reconsider r = (((the MorphMap of N)*the ObjectMap of F)** the MorphMap of F
).(o1,o2) as Function;
reconsider t = ((the MorphMap of N)*the ObjectMap of F).(o1,o2) as Function;
A1: dom (((the MorphMap of N)*the ObjectMap of F)**the MorphMap of F) = (dom
((the MorphMap of N)*the ObjectMap of F)) /\ (dom(the MorphMap of F)) by
PBOOLE:def 19
.= [:I,I:] /\ (dom(the MorphMap of F)) by PARTFUN1:def 2
.= [:I,I:] /\ [:I,I:] by PARTFUN1:def 2
.= [:I,I:];
A2: dom the ObjectMap of F = [:I,I:] by FUNCT_2:def 1;
A3: [o1,o2] in [:I,I:] by ZFMISC_1:def 2;
assume
A4: <^o1,o2^> <> {};
then
A5: <^F.o1,F.o2^> <> {} by FUNCTOR0:def 18;
then
A6: dom Morph-Map(F,o1,o2) = <^o1,o2^> by FUNCT_2:def 1;
A7: <^N.(F.o2),N.(F.o1)^> <> {} by A5,FUNCTOR0:def 19;
(N*F).o1 = N.(F.o1) & (N*F).o2 = N.(F.o2) by FUNCTOR0:33;
hence (N*F).m = Morph-Map(N*F,o1,o2).m by A4,A7,FUNCTOR0:def 16
.= r.m by FUNCTOR0:def 36
.= (t * s).m by A1,A3,PBOOLE:def 19
.= t.(Morph-Map(F,o1,o2).m) by A4,A6,FUNCT_1:13
.= t.(F.m) by A4,A5,FUNCTOR0:def 15
.= ((the MorphMap of N).((the ObjectMap of F).(o1,o2))).(F.m) by A2,A3,
FUNCT_1:13
.= Morph-Map(N,F.o1,F.o2).(F.m) by FUNCTOR0:22
.= N.(F.m) by A5,A7,FUNCTOR0:def 16;
end;
theorem Th9:
<^o1,o2^> <> {} implies (G*M).m = G.(M.m)
proof
set I = the carrier of A;
reconsider s = (the MorphMap of M).(o1,o2) as Function;
reconsider r = (((the MorphMap of G)*the ObjectMap of M)** the MorphMap of M
).(o1,o2) as Function;
reconsider t = ((the MorphMap of G)*the ObjectMap of M).(o1,o2) as Function;
A1: dom (((the MorphMap of G)*the ObjectMap of M)**the MorphMap of M) = (dom
((the MorphMap of G)*the ObjectMap of M)) /\ (dom(the MorphMap of M)) by
PBOOLE:def 19
.= [:I,I:] /\ (dom(the MorphMap of M)) by PARTFUN1:def 2
.= [:I,I:] /\ [:I,I:] by PARTFUN1:def 2
.= [:I,I:];
A2: dom the ObjectMap of M = [:I,I:] by FUNCT_2:def 1;
A3: [o1,o2] in [:I,I:] by ZFMISC_1:def 2;
assume
A4: <^o1,o2^> <> {};
then
A5: <^M.o2,M.o1^> <> {} by FUNCTOR0:def 19;
then
A6: dom Morph-Map(M,o1,o2) = <^o1,o2^> by FUNCT_2:def 1;
A7: <^G.(M.o2),G.(M.o1)^> <> {} by A5,FUNCTOR0:def 18;
(G*M).o1 = G.(M.o1) & (G*M).o2 = G.(M.o2) by FUNCTOR0:33;
hence (G*M).m = Morph-Map(G*M,o1,o2).m by A4,A7,FUNCTOR0:def 16
.= r.m by FUNCTOR0:def 36
.= (t * s).m by A1,A3,PBOOLE:def 19
.= t.(Morph-Map(M,o1,o2).m) by A4,A6,FUNCT_1:13
.= t.(M.m) by A4,A5,FUNCTOR0:def 16
.= ((the MorphMap of G).((the ObjectMap of M).(o1,o2))).(M.m) by A2,A3,
FUNCT_1:13
.= Morph-Map(G,M.o2,M.o1).(M.m) by FUNCTOR0:23
.= G.(M.m) by A5,A7,FUNCTOR0:def 15;
end;
registration
let A be non empty transitive AltCatStr, B be transitive with_units non
empty AltCatStr, C be with_units non empty AltCatStr, F be feasible
Covariant comp-preserving FunctorStr over A, B, G be feasible Covariant
comp-preserving FunctorStr over B, C;
cluster G*F -> comp-preserving;
coherence
proof
let o1, o2, o3 be Object of A such that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {};
A3: <^F.o1,F.o2^> <> {} & <^F.o2,F.o3^> <> {} by A1,A2,FUNCTOR0:def 18;
let f be Morphism of o1, o2, g be Morphism of o2, o3;
A4: (G*F).o1 = G.(F.o1) & (G*F).o3 = G.(F.o3) by FUNCTOR0:33;
A5: (G*F).o2 = G.(F.o2) by FUNCTOR0:33;
then reconsider GFg = (G*F).g as Morphism of G.(F.o2), G.(F.o3) by
FUNCTOR0:33;
<^o1,o3^> <> {} by A1,A2,ALTCAT_1:def 2;
hence (G*F).(g*f) = G.(F.(g*f)) by Th6
.= G.((F.g)*(F.f)) by A1,A2,FUNCTOR0:def 23
.= (G.(F.g))*(G.(F.f)) by A3,FUNCTOR0:def 23
.= GFg*(G.(F.f)) by A2,Th6
.= ((G*F).g)*((G*F).f) by A1,A5,A4,Th6;
end;
end;
registration
let A be non empty transitive AltCatStr, B be transitive with_units non
empty AltCatStr, C be with_units non empty AltCatStr, F be feasible
Contravariant comp-reversing FunctorStr over A, B, G be feasible Contravariant
comp-reversing FunctorStr over B, C;
cluster G*F -> comp-preserving;
coherence
proof
let o1, o2, o3 be Object of A such that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {};
A3: <^F.o2,F.o1^> <> {} & <^F.o3,F.o2^> <> {} by A1,A2,FUNCTOR0:def 19;
let f be Morphism of o1, o2, g be Morphism of o2, o3;
A4: (G*F).o1 = G.(F.o1) & (G*F).o3 = G.(F.o3) by FUNCTOR0:33;
A5: (G*F).o2 = G.(F.o2) by FUNCTOR0:33;
then reconsider GFg = (G*F).g as Morphism of G.(F.o2), G.(F.o3) by
FUNCTOR0:33;
<^o1,o3^> <> {} by A1,A2,ALTCAT_1:def 2;
hence (G*F).(g*f) = G.(F.(g*f)) by Th7
.= G.((F.f)*(F.g)) by A1,A2,FUNCTOR0:def 24
.= (G.(F.g))*(G.(F.f)) by A3,FUNCTOR0:def 24
.= GFg*(G.(F.f)) by A2,Th7
.= ((G*F).g)*((G*F).f) by A1,A5,A4,Th7;
end;
end;
registration
let A be non empty transitive AltCatStr, B be transitive with_units non
empty AltCatStr, C be with_units non empty AltCatStr, F be feasible
Covariant comp-preserving FunctorStr over A, B, G be feasible Contravariant
comp-reversing FunctorStr over B, C;
cluster G*F -> comp-reversing;
coherence
proof
let o1, o2, o3 be Object of A such that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {};
A3: <^F.o1,F.o2^> <> {} & <^F.o2,F.o3^> <> {} by A1,A2,FUNCTOR0:def 18;
let f be Morphism of o1, o2, g be Morphism of o2, o3;
A4: (G*F).o2 = G.(F.o2) & (G*F).o3 = G.(F.o3) by FUNCTOR0:33;
A5: (G*F).o1 = G.(F.o1) by FUNCTOR0:33;
then reconsider GFf = (G*F).f as Morphism of G.(F.o2), G.(F.o1) by
FUNCTOR0:33;
<^o1,o3^> <> {} by A1,A2,ALTCAT_1:def 2;
hence (G*F).(g*f) = G.(F.(g*f)) by Th8
.= G.((F.g)*(F.f)) by A1,A2,FUNCTOR0:def 23
.= (G.(F.f))*(G.(F.g)) by A3,FUNCTOR0:def 24
.= GFf*(G.(F.g)) by A1,Th8
.= ((G*F).f)*((G*F).g) by A2,A5,A4,Th8;
end;
end;
registration
let A be non empty transitive AltCatStr, B be transitive with_units non
empty AltCatStr, C be with_units non empty AltCatStr, F be feasible
Contravariant comp-reversing FunctorStr over A, B, G be feasible Covariant
comp-preserving FunctorStr over B, C;
cluster G*F -> comp-reversing;
coherence
proof
let o1, o2, o3 be Object of A such that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {};
A3: <^F.o2,F.o1^> <> {} & <^F.o3,F.o2^> <> {} by A1,A2,FUNCTOR0:def 19;
let f be Morphism of o1, o2, g be Morphism of o2, o3;
A4: (G*F).o2 = G.(F.o2) & (G*F).o3 = G.(F.o3) by FUNCTOR0:33;
A5: (G*F).o1 = G.(F.o1) by FUNCTOR0:33;
then reconsider GFf = (G*F).f as Morphism of G.(F.o2), G.(F.o1) by
FUNCTOR0:33;
<^o1,o3^> <> {} by A1,A2,ALTCAT_1:def 2;
hence (G*F).(g*f) = G.(F.(g*f)) by Th9
.= G.((F.f)*(F.g)) by A1,A2,FUNCTOR0:def 24
.= (G.(F.f))*(G.(F.g)) by A3,FUNCTOR0:def 23
.= GFf*(G.(F.g)) by A1,Th9
.= ((G*F).f)*((G*F).g) by A2,A5,A4,Th9;
end;
end;
definition
let A, B be transitive with_units non empty AltCatStr, C be with_units
non empty AltCatStr, F be covariant Functor of A, B, G be covariant Functor of
B, C;
redefine func G*F -> strict covariant Functor of A, C;
coherence by FUNCTOR0:def 25;
end;
definition
let A, B be transitive with_units non empty AltCatStr, C be with_units
non empty AltCatStr, F be contravariant Functor of A, B, G be contravariant
Functor of B, C;
redefine func G*F -> strict covariant Functor of A, C;
coherence by FUNCTOR0:def 25;
end;
definition
let A, B be transitive with_units non empty AltCatStr, C be with_units
non empty AltCatStr, F be covariant Functor of A, B, G be contravariant
Functor of B, C;
redefine func G*F -> strict contravariant Functor of A, C;
coherence by FUNCTOR0:def 25;
end;
definition
let A, B be transitive with_units non empty AltCatStr, C be with_units
non empty AltCatStr, F be contravariant Functor of A, B, G be covariant
Functor of B, C;
redefine func G*F -> strict contravariant Functor of A, C;
coherence by FUNCTOR0:def 25;
end;
reserve A, B, C, D for transitive with_units non empty AltCatStr,
F1, F2, F3 for covariant Functor of A, B,
G1, G2, G3 for covariant Functor of B, C,
H1, H2 for covariant Functor of C, D,
p for transformation of F1, F2,
p1 for transformation of F2, F3,
q for transformation of G1, G2,
q1 for transformation of G2, G3,
r for transformation of H1, H2;
theorem Th10:
F1 is_transformable_to F2 & G1 is_transformable_to G2 implies G1
*F1 is_transformable_to G2*F2
proof
assume
A1: for a being Object of A holds <^F1.a,F2.a^> <> {};
assume
A2: for a being Object of B holds <^G1.a,G2.a^> <> {};
let a be Object of A;
<^F1.a,F2.a^> <> {} by A1;
then
A3: <^G1.(F1.a),G1.(F2.a)^> <> {} by FUNCTOR0:def 18;
A4: (G1*F1).a = G1.(F1.a) & (G2*F2).a = G2.(F2.a) by FUNCTOR0:33;
<^G1.(F2.a),G2.(F2.a)^> <> {} by A2;
hence thesis by A4,A3,ALTCAT_1:def 2;
end;
begin :: The composition of functors with transformations
definition
let A, B, C be transitive with_units non empty AltCatStr, F1, F2 be
covariant Functor of A, B, t be transformation of F1, F2, G be covariant
Functor of B, C such that
A1: F1 is_transformable_to F2;
func G*t -> transformation of G*F1,G*F2 means
:Def1:
for o being Object of A holds it.o = G.(t!o);
existence
proof
defpred P[object,object] means
ex o being Object of A st $1 = o & $2 = G.(t!o);
set I = the carrier of A;
A2: for i being object st i in I ex j being object st P[i,j]
proof
let i be object;
assume i in I;
then reconsider o = i as Object of A;
take G.(t!o);
thus thesis;
end;
consider IT being ManySortedSet of I such that
A3: for o being object st o in I holds P[o,IT.o] from PBOOLE:sch 3(A2);
IT is transformation of G*F1,G*F2
proof
thus G*F1 is_transformable_to G*F2 by A1,Th10;
let o be Object of A;
( P[o,IT.o])& G.(F1.o) = (G*F1).o by A3,FUNCTOR0:33;
hence thesis by FUNCTOR0:33;
end;
then reconsider IT as transformation of G*F1,G*F2;
take IT;
let o be Object of A;
P[o,IT.o] by A3;
hence thesis;
end;
uniqueness
proof
let X, Y be transformation of G*F1,G*F2 such that
A4: for o being Object of A holds X.o = G.(t!o) and
A5: for o being Object of A holds Y.o = G.(t!o);
A6: G*F1 is_transformable_to G*F2 by A1,Th10;
now
let o be Object of A;
thus X!o = X.o by A6,FUNCTOR2:def 4
.= G.(t!o) by A4
.= Y.o by A5
.= Y!o by A6,FUNCTOR2:def 4;
end;
hence thesis by A1,Th10,FUNCTOR2:3;
end;
end;
theorem Th11:
for o being Object of A st F1 is_transformable_to F2 holds (G1*p
)!o = G1.(p!o)
proof
let o be Object of A;
assume
A1: F1 is_transformable_to F2;
then G1*F1 is_transformable_to G1*F2 by Th10;
hence (G1*p)!o = (G1*p).o by FUNCTOR2:def 4
.= G1.(p!o) by A1,Def1;
end;
definition
let A, B, C be transitive with_units non empty AltCatStr, G1, G2 be
covariant Functor of B, C, F be covariant Functor of A, B, s be transformation
of G1, G2 such that
A1: G1 is_transformable_to G2;
func s*F -> transformation of G1*F,G2*F means
:Def2:
for o being Object of A holds it.o = s!(F.o);
existence
proof
defpred P[object,object] means
ex o being Object of A st $1 = o & $2 = s!(F.o);
set I = the carrier of A;
A2: for i being object st i in I ex j being object st P[i,j]
proof
let i be object;
assume i in I;
then reconsider o = i as Object of A;
take s!(F.o);
thus thesis;
end;
consider IT being ManySortedSet of I such that
A3: for o being object st o in I holds P[o,IT.o] from PBOOLE:sch 3(A2);
IT is transformation of G1*F,G2*F
proof
thus G1*F is_transformable_to G2*F by A1,Th10;
let o be Object of A;
( P[o,IT.o])& G1.(F.o) = (G1*F).o by A3,FUNCTOR0:33;
hence thesis by FUNCTOR0:33;
end;
then reconsider IT as transformation of G1*F,G2*F;
take IT;
let o be Object of A;
P[o,IT.o] by A3;
hence thesis;
end;
uniqueness
proof
let X, Y be transformation of G1*F,G2*F such that
A4: for o being Object of A holds X.o = s!(F.o) and
A5: for o being Object of A holds Y.o = s!(F.o);
A6: G1*F is_transformable_to G2*F by A1,Th10;
now
let o be Object of A;
thus X!o = X.o by A6,FUNCTOR2:def 4
.= s!(F.o) by A4
.= Y.o by A5
.= Y!o by A6,FUNCTOR2:def 4;
end;
hence thesis by A1,Th10,FUNCTOR2:3;
end;
end;
theorem Th12:
for o being Object of A st G1 is_transformable_to G2 holds (q*F1
)!o = q!(F1.o)
proof
let o be Object of A;
assume
A1: G1 is_transformable_to G2;
then G1*F1 is_transformable_to G2*F1 by Th10;
hence (q*F1)!o = (q*F1).o by FUNCTOR2:def 4
.= q!(F1.o) by A1,Def2;
end;
theorem Th13:
F1 is_transformable_to F2 & F2 is_transformable_to F3 implies G1
*(p1`*`p) = (G1*p1)`*`(G1*p)
proof
assume that
A1: F1 is_transformable_to F2 and
A2: F2 is_transformable_to F3;
A3: G1*F1 is_transformable_to G1*F2 & G1*F2 is_transformable_to G1*F3 by A1,A2
,Th10;
A4: now
let a be Object of A;
A5: G1.(F2.a) = (G1*F2).a & G1.(F3.a) = (G1*F3).a by FUNCTOR0:33;
A6: G1.(F1.a) = (G1*F1).a by FUNCTOR0:33;
then reconsider G1ta = (G1*p)!a as Morphism of G1.(F1.a), G1.(F2.a) by
FUNCTOR0:33;
A7: <^F1.a,F2.a^> <> {} & <^F2.a,F3.a^> <> {} by A1,A2;
thus (G1*(p1`*`p))!a = G1.((p1`*`p)!a) by A1,A2,Th11,FUNCTOR2:2
.= G1.((p1!a)*(p!a)) by A1,A2,FUNCTOR2:def 5
.= G1.(p1!a)*G1.(p!a) by A7,FUNCTOR0:def 23
.= G1.(p1!a)*G1ta by A1,Th11
.= ((G1*p1)!a)*((G1*p)!a) by A2,A6,A5,Th11
.= ((G1*p1)`*`(G1*p))!a by A3,FUNCTOR2:def 5;
end;
F1 is_transformable_to F3 by A1,A2,FUNCTOR2:2;
hence thesis by A4,Th10,FUNCTOR2:3;
end;
theorem Th14:
G1 is_transformable_to G2 & G2 is_transformable_to G3 implies (
q1`*`q)*F1 = (q1*F1)`*`(q*F1)
proof
assume that
A1: G1 is_transformable_to G2 and
A2: G2 is_transformable_to G3;
A3: G1*F1 is_transformable_to G2*F1 & G2*F1 is_transformable_to G3*F1 by A1,A2
,Th10;
A4: now
let a be Object of A;
A5: G1.(F1.a) = (G1*F1).a & G3.(F1.a) = (G3*F1).a by FUNCTOR0:33;
A6: G2.(F1.a) = (G2*F1).a by FUNCTOR0:33;
then reconsider s1F1a = (q1*F1)!a as Morphism of G2.(F1.a), G3.(F1.a) by
FUNCTOR0:33;
thus ((q1`*`q)*F1)!a = (q1`*`q)!(F1.a) by A1,A2,Th12,FUNCTOR2:2
.= (q1!(F1.a))*(q!(F1.a)) by A1,A2,FUNCTOR2:def 5
.= s1F1a*(q!(F1.a)) by A2,Th12
.= ((q1*F1)!a)*((q*F1)!a) by A1,A6,A5,Th12
.= ((q1*F1)`*`(q*F1))!a by A3,FUNCTOR2:def 5;
end;
G1 is_transformable_to G3 by A1,A2,FUNCTOR2:2;
hence thesis by A4,Th10,FUNCTOR2:3;
end;
theorem Th15:
H1 is_transformable_to H2 implies r*G1*F1 = r*(G1*F1)
proof
A1: H2*G1*F1 = H2*(G1*F1) by FUNCTOR0:32;
then reconsider m = r*(G1*F1) as transformation of H1*G1*F1, H2*G1*F1 by
FUNCTOR0:32;
assume
A2: H1 is_transformable_to H2;
A3: now
let a be Object of A;
thus (r*G1*F1)!a = (r*G1)!(F1.a) by A2,Th10,Th12
.= r!(G1.(F1.a)) by A2,Th12
.= r!((G1*F1).a) by FUNCTOR0:33
.= (r*(G1*F1))!a by A2,Th12
.= m!a by A1,FUNCTOR0:32;
end;
H1*G1 is_transformable_to H2*G1 by A2,Th10;
hence thesis by A3,Th10,FUNCTOR2:3;
end;
theorem Th16:
G1 is_transformable_to G2 implies H1*q*F1 = H1*(q*F1)
proof
A1: H1*G2*F1 = H1*(G2*F1) by FUNCTOR0:32;
then reconsider m = H1*(q*F1) as transformation of H1*G1*F1, H1*G2*F1 by
FUNCTOR0:32;
assume
A2: G1 is_transformable_to G2;
A3: now
let a be Object of A;
A4: (G1*F1).a = G1.(F1.a) & (G2*F1).a = G2.(F1.a) by FUNCTOR0:33;
thus (H1*q*F1)!a = (H1*q)!(F1.a) by A2,Th10,Th12
.= H1.(q!(F1.a)) by A2,Th11
.= H1.((q*F1)!a) by A2,A4,Th12
.= (H1*(q*F1))!a by A2,Th10,Th11
.= m!a by A1,FUNCTOR0:32;
end;
H1*G1 is_transformable_to H1*G2 by A2,Th10;
hence thesis by A3,Th10,FUNCTOR2:3;
end;
theorem Th17:
F1 is_transformable_to F2 implies H1*G1*p = H1*(G1*p)
proof
A1: H1*G1*F2 = H1*(G1*F2) by FUNCTOR0:32;
then reconsider m = H1*(G1*p) as transformation of H1*G1*F1, H1*G1*F2 by
FUNCTOR0:32;
assume
A2: F1 is_transformable_to F2;
now
let a be Object of A;
A3: (G1*F1).a = G1.(F1.a) & (G1*F2).a = G1.(F2.a) by FUNCTOR0:33;
A4: <^F1.a,F2.a^> <> {} by A2;
thus (H1*G1*p)!a = (H1*G1).(p!a) by A2,Th11
.= H1.(G1.(p!a)) by A4,Th6
.= H1.((G1*p)!a) by A2,A3,Th11
.= (H1*(G1*p))!a by A2,Th10,Th11
.= m!a by A1,FUNCTOR0:32;
end;
hence thesis by A2,Th10,FUNCTOR2:3;
end;
theorem Th18:
(idt G1)*F1 = idt (G1*F1)
proof
now
let a be Object of A;
thus ((idt G1)*F1)!a = (idt G1)!(F1.a) by Th12
.= idm(G1.(F1.a)) by FUNCTOR2:4
.= idm((G1*F1).a) by FUNCTOR0:33
.= (idt (G1*F1))!a by FUNCTOR2:4;
end;
hence thesis by FUNCTOR2:3;
end;
theorem Th19:
G1 * idt F1 = idt (G1*F1)
proof
now
let a be Object of A;
thus (G1*(idt F1))!a = G1.((idt F1)!a) by Th11
.= G1.(idm (F1.a)) by FUNCTOR2:4
.= idm (G1.(F1.a)) by FUNCTOR2:1
.= idm ((G1*F1).a) by FUNCTOR0:33
.= (idt (G1*F1))!a by FUNCTOR2:4;
end;
hence thesis by FUNCTOR2:3;
end;
theorem Th20:
F1 is_transformable_to F2 implies (id B) * p = p
proof
assume
A1: F1 is_transformable_to F2;
now
let i be object;
assume i in the carrier of A;
then reconsider a = i as Object of A;
A2: <^F1.a,F2.a^> <> {} by A1;
thus ((id B) * p).i = (id B).(p!a) by A1,Def1
.= p!a by A2,FUNCTOR0:31
.= p.i by A1,FUNCTOR2:def 4;
end;
hence thesis;
end;
theorem Th21:
G1 is_transformable_to G2 implies q * id B = q
proof
assume
A1: G1 is_transformable_to G2;
now
let i be object;
assume i in the carrier of B;
then reconsider a = i as Object of B;
thus (q * id B).i = q!((id B).a) by A1,Def2
.= q!a by FUNCTOR0:29
.= q.i by A1,FUNCTOR2:def 4;
end;
hence thesis;
end;
begin :: The composition of transformations
definition
let A, B, C be transitive with_units non empty AltCatStr, F1, F2 be
covariant Functor of A, B, G1, G2 be covariant Functor of B, C, t be
transformation of F1, F2, s be transformation of G1, G2;
func s (#) t -> transformation of G1*F1, G2*F2 equals
(s*F2) `*` (G1*t);
coherence;
end;
theorem Th22:
for q being natural_transformation of G1, G2 st F1
is_transformable_to F2 & G1 is_naturally_transformable_to G2 holds q (#) p = (
G2*p) `*` (q*F1)
proof
let q be natural_transformation of G1, G2;
assume that
A1: F1 is_transformable_to F2 and
A2: G1 is_naturally_transformable_to G2;
A3: G1*F1 is_transformable_to G1*F2 by A1,Th10;
A4: G2*F1 is_transformable_to G2*F2 by A1,Th10;
A5: G1 is_transformable_to G2 by A2;
then
A6: G1*F1 is_transformable_to G2*F1 by Th10;
A7: G1*F2 is_transformable_to G2*F2 by A5,Th10;
now
let a be Object of A;
A8: G1.(F1.a) = (G1*F1).a by FUNCTOR0:33;
A9: G2.(F2.a) = (G2*F2).a by FUNCTOR0:33;
then reconsider sF2a = q!F2.a as Morphism of (G1*F2).a, (G2*F2).a by
FUNCTOR0:33;
reconsider G2ta = G2*p!a as Morphism of G2.(F1.a), G2.(F2.a) by A9,
FUNCTOR0:33;
A10: G1.(F2.a) = (G1*F2).a by FUNCTOR0:33;
A11: <^F1.a,F2.a^> <> {} by A1;
A12: G2.(F1.a) = (G2*F1).a by FUNCTOR0:33;
thus ((q*F2) `*` (G1*p))!a = ((q*F2)!a) * ((G1*p)!a) by A7,A3,
FUNCTOR2:def 5
.= sF2a * ((G1*p)!a) by A5,Th12
.= (q!F2.a) * G1.(p!a) by A1,A8,A10,A9,Th11
.= G2.(p!a) * (q!F1.a) by A2,A11,FUNCTOR2:def 7
.= G2ta * (q!F1.a) by A1,Th11
.= G2*p!a * (q*F1!a) by A5,A8,A12,A9,Th12
.= ((G2*p) `*` (q*F1))!a by A6,A4,FUNCTOR2:def 5;
end;
hence thesis by A1,A5,Th10,FUNCTOR2:3;
end;
theorem
F1 is_transformable_to F2 implies (idt id B)(#)p = p
proof
assume
A1: F1 is_transformable_to F2;
then
A2: (id B)*F1 is_transformable_to (id B)*F2 by Th10;
thus (idt id B)(#)p = (idt (id B*F2)) `*` (id B*p) by Th18
.= id B*p by A2,FUNCTOR2:5
.= p by A1,Th20;
end;
theorem
G1 is_transformable_to G2 implies q(#)(idt id B) = q
proof
assume
A1: G1 is_transformable_to G2;
then
A2: G1*(id B) is_transformable_to G2*(id B) by Th10;
thus q(#)(idt id B) = (q*(id B))`*`(idt (G1*id B)) by Th19
.= q*id B by A2,FUNCTOR2:5
.= q by A1,Th21;
end;
theorem
F1 is_transformable_to F2 implies G1*p = (idt G1) (#) p
proof
assume F1 is_transformable_to F2;
then G1*F1 is_transformable_to G1*F2 by Th10;
hence G1*p = (idt (G1*F2))`*`(G1*p) by FUNCTOR2:5
.= (idt G1) (#) p by Th18;
end;
theorem
G1 is_transformable_to G2 implies q*F1 = q (#) idt F1
proof
assume G1 is_transformable_to G2;
then G1*F1 is_transformable_to G2*F1 by Th10;
hence q*F1 = (q*F1)`*`(idt(G1*F1)) by FUNCTOR2:5
.= q (#) idt F1 by Th19;
end;
reserve A, B, C, D for category,
F1, F2, F3 for covariant Functor of A, B,
G1, G2, G3 for covariant Functor of B, C;
theorem
for H1, H2 being covariant Functor of C, D for t being transformation
of F1, F2, s being transformation of G1, G2 for u being transformation of H1,
H2 st F1 is_transformable_to F2 & G1 is_transformable_to G2 & H1
is_transformable_to H2 holds u(#)s(#)t = u(#)(s(#)t)
proof
let H1, H2 be covariant Functor of C, D, t be transformation of F1, F2, s be
transformation of G1, G2, u be transformation of H1, H2;
assume that
A1: F1 is_transformable_to F2 and
A2: G1 is_transformable_to G2 and
A3: H1 is_transformable_to H2;
A4: G1*F2 is_transformable_to G2*F2 & G1*F1 is_transformable_to G1*F2 by A1,A2
,Th10;
A5: H1*s*F2 = H1*(s*F2) & H1*G1*t = H1*(G1*t) by A1,A2,Th16,Th17;
A6: H1*G2*F2 = H1*(G2*F2) & H2*G2*F2 = H2*(G2*F2) by FUNCTOR0:32;
A7: H1*G1*F1 is_transformable_to H1*G1*F2 & u*G2*F2 = u*(G2*F2) by A1,A3,Th10
,Th15;
A8: H1*G1*F1 = H1*(G1*F1) & H1*G1*F2 = H1*(G1*F2) by FUNCTOR0:32;
A9: H1*G1 is_transformable_to H1*G2 by A2,Th10;
then
A10: H1*G1*F2 is_transformable_to H1*G2*F2 by Th10;
A11: H1*G2 is_transformable_to H2*G2 by A3,Th10;
then
A12: H1*G2*F2 is_transformable_to H2*G2*F2 by Th10;
thus u(#)s(#)t = ((u*G2)*F2) `*` ((H1*s)*F2) `*` ((H1*G1)*t) by A11,A9,Th14
.= (u*(G2*F2)) `*` ((H1*(s*F2)) `*` (H1*(G1*t))) by A12,A10,A7,A5,A8,A6,
FUNCTOR2:6
.= u(#)(s(#)t) by A4,Th13;
end;
reserve t for natural_transformation of F1, F2,
s for natural_transformation of G1, G2,
s1 for natural_transformation of G2, G3;
Lm2: now
let A, B, C, F1, F2, G1, G2, s, t;
set k = s(#)t;
assume
A1: F1 is_naturally_transformable_to F2;
then
A2: F1 is_transformable_to F2;
assume
A3: G1 is_naturally_transformable_to G2;
then
A4: G1 is_transformable_to G2;
A5: now
let a, b be Object of A such that
A6: <^a,b^> <> {};
A7: <^(G1*F1).a, (G1*F1).b^> <> {} by A6,FUNCTOR0:def 18;
A8: (G2*F2).a = G2.(F2.a) by FUNCTOR0:33;
then reconsider sF2a = s!(F2.a) as Morphism of (G1*F2).a, (G2*F2).a by
FUNCTOR0:33;
A9: (G2*F2).b = G2.(F2.b) by FUNCTOR0:33;
then reconsider sF2b = s!(F2.b) as Morphism of (G1*F2).b, (G2*F2).b by
FUNCTOR0:33;
<^G1.(F2.b),G2.(F2.b)^> <> {} by A4;
then
A10: <^(G1*F2).b, (G2*F2).b^> <> {} by A9,FUNCTOR0:33;
let f be Morphism of a,b;
A11: (G1*F1).a = G1.(F1.a) by FUNCTOR0:33;
then reconsider
G1tbF1f = G1.(t!b*F1.f) as Morphism of (G1*F1).a, (G1*F2).b by FUNCTOR0:33;
reconsider G1ta = G1.(t!a) as Morphism of (G1*F1).a, (G1*F2).a by A11,
FUNCTOR0:33;
A12: <^G1.(F1.a),G2.(F1.a)^> <> {} by A4;
A13: (G1*F1).b = G1.(F1.b) by FUNCTOR0:33;
then reconsider G1tb = G1.(t!b) as Morphism of (G1*F1).b, (G1*F2).b by
FUNCTOR0:33;
A14: <^F1.b,F2.b^> <> {} by A2;
then <^G1.(F1.b),G1.(F2.b)^> <> {} by FUNCTOR0:def 18;
then
A15: <^(G1*F1).b, (G1*F2).b^> <> {} by A13,FUNCTOR0:33;
A16: <^F1.a,F1.b^> <> {} by A6,FUNCTOR0:def 18;
then
A17: <^F1.a,F2.b^> <> {} by A14,ALTCAT_1:def 2;
reconsider G1F1f = G1.(F1.f) as Morphism of (G1*F1).a, (G1*F1).b by A13,
FUNCTOR0:33;
A18: s!(F2.a) = (s*F2).a by A4,Def2;
A19: G1.(t!b*F1.f) = G1.(t!b)*G1.(F1.f) by A14,A16,FUNCTOR0:def 23
.= G1tb*G1F1f by A11,A13,FUNCTOR0:33;
reconsider G2F2f = G2.(F2.f) as Morphism of (G2*F2).a, (G2*F2).b by A8,
FUNCTOR0:33;
A20: s!(F2.b) = (s*F2).b by A4,Def2;
A21: G1*F2 is_transformable_to G2*F2 by A4,Th10;
A22: <^F2.a,F2.b^> <> {} by A6,FUNCTOR0:def 18;
then
A23: <^G2.(F2.a),G2.(F2.b)^> <> {} by FUNCTOR0:def 18;
A24: <^F1.a,F2.a^> <> {} by A2;
then
A25: <^G2.(F1.a),G2.(F2.a)^> <> {} by FUNCTOR0:def 18;
A26: G1*F1 is_transformable_to G1*F2 by A2,Th10;
hence (k!b)*((G1*F1).f) = ((s*F2)!b)*((G1*t)!b)*((G1*F1).f) by A21,
FUNCTOR2:def 5
.= sF2b*((G1*t)!b)*((G1*F1).f) by A21,A20,FUNCTOR2:def 4
.= sF2b*G1tb*((G1*F1).f) by A2,Th11
.= sF2b*G1tb*G1F1f by A6,Th6
.= sF2b*G1tbF1f by A7,A15,A10,A19,ALTCAT_1:21
.= s!(F2.b)*G1.(t!b*F1.f) by A11,A9,FUNCTOR0:33
.= G2.(t!b*F1.f)*(s!(F1.a)) by A3,A17,FUNCTOR2:def 7
.= G2.(F2.f*(t!a))*(s!(F1.a)) by A1,A6,FUNCTOR2:def 7
.= G2.(F2.f)*G2.(t!a)*(s!(F1.a)) by A22,A24,FUNCTOR0:def 23
.= G2.(F2.f)*(G2.(t!a)*(s!(F1.a))) by A12,A25,A23,ALTCAT_1:21
.= G2.(F2.f)*(s!(F2.a)*G1.(t!a)) by A3,A24,FUNCTOR2:def 7
.= G2F2f*(sF2a*G1ta) by A11,A8,A9,FUNCTOR0:33
.= ((G2*F2).f)*(sF2a*G1ta) by A6,Th6
.= ((G2*F2).f)*(((s*F2)!a)*G1ta) by A21,A18,FUNCTOR2:def 4
.= ((G2*F2).f)*(((s*F2)!a)*((G1*t)!a)) by A2,Th11
.= ((G2*F2).f)*(k!a) by A21,A26,FUNCTOR2:def 5;
end;
thus G1*F1 is_naturally_transformable_to G2*F2
by A2,A4,Th10,A5;
hence s(#)t is natural_transformation of G1*F1, G2*F2 by A5,FUNCTOR2:def 7;
end;
theorem Th28:
F1 is_naturally_transformable_to F2 implies G1*t is
natural_transformation of G1*F1, G1*F2
proof
assume
A1: F1 is_naturally_transformable_to F2;
then
A2: F1 is_transformable_to F2;
thus G1*F1 is_naturally_transformable_to G1*F2 by A1,Lm2;
let a, b be Object of A such that
A3: <^a,b^> <> {};
A4: (G1*F1).b = G1.(F1.b) & <^F1.a,F1.b^> <> {} by A3,FUNCTOR0:33,def 18;
A5: <^F1.b,F2.b^> <> {} by A2;
A6: <^F1.a,F2.a^> <> {} by A2;
reconsider G1ta = G1.(t!a) as Morphism of G1.(F1.a), (G1*F2).a by FUNCTOR0:33
;
reconsider G1tb = G1.(t!b) as Morphism of (G1*F1).b, G1.(F2.b) by FUNCTOR0:33
;
let f be Morphism of a, b;
A7: (G1*F2).a = G1.(F2.a) by FUNCTOR0:33;
A8: <^F2.a,F2.b^> <> {} by A3,FUNCTOR0:def 18;
A9: (G1*F1).a = G1.(F1.a) by FUNCTOR0:33;
then reconsider G1F1f = G1.(F1.f) as Morphism of (G1*F1).a, (G1*F1).b by
FUNCTOR0:33;
A10: (G1*F2).b = G1.(F2.b) by FUNCTOR0:33;
hence (G1*t)!b*(G1*F1).f = G1tb*((G1*F1).f) by A2,Th11
.= G1tb*G1F1f by A3,Th6
.= G1.(t!b*F1.f) by A9,A4,A5,FUNCTOR0:def 23
.= G1.(F2.f*(t!a)) by A1,A3,FUNCTOR2:def 7
.= G1.(F2.f)*G1.(t!a) by A6,A8,FUNCTOR0:def 23
.= (G1*F2).f*G1ta by A3,A7,A10,Th6
.= (G1*F2).f*((G1*t)!a) by A2,A9,Th11;
end;
theorem Th29:
G1 is_naturally_transformable_to G2 implies s*F1 is
natural_transformation of G1*F1, G2*F1
proof
assume
A1: G1 is_naturally_transformable_to G2;
thus G1*F1 is_naturally_transformable_to G2*F1 by A1,Lm2;
let a, b be Object of A such that
A2: <^a,b^> <> {};
A3: <^F1.a,F1.b^> <> {} by A2,FUNCTOR0:def 18;
reconsider sF1a = s!F1.a as Morphism of G1.(F1.a), (G2*F1).a by FUNCTOR0:33;
let f be Morphism of a, b;
A4: (G2*F1).a = G2.(F1.a) by FUNCTOR0:33;
A5: (G2*F1).b = G2.(F1.b) by FUNCTOR0:33;
then reconsider sF1b = s!(F1.b) as Morphism of (G1*F1).b, (G2*F1).b by
FUNCTOR0:33;
A6: (G1*F1).b = G1.(F1.b) & (G2*F1).b = G2.(F1.b) by FUNCTOR0:33;
A7: (G1*F1).a = G1.(F1.a) by FUNCTOR0:33;
then reconsider G1F1f = G1.(F1.f) as Morphism of (G1*F1).a, (G1*F1).b by
FUNCTOR0:33;
A8: G1 is_transformable_to G2 by A1;
hence (s*F1)!b*(G1*F1).f = sF1b*((G1*F1).f) by Th12
.= sF1b*G1F1f by A2,Th6
.= G2.(F1.f)*(s!F1.a) by A1,A7,A6,A3,FUNCTOR2:def 7
.= (G2*F1).f*sF1a by A2,A4,A5,Th6
.= (G2*F1).f*((s*F1)!a) by A8,A7,Th12;
end;
theorem
F1 is_naturally_transformable_to F2 & G1 is_naturally_transformable_to
G2 implies G1*F1 is_naturally_transformable_to G2*F2 & s(#)t is
natural_transformation of G1*F1, G2*F2 by Lm2;
theorem
for t being transformation of F1, F2, t1 being transformation of F2,
F3 st F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to F3
& G1 is_naturally_transformable_to G2 & G2 is_naturally_transformable_to G3
holds (s1`*`s)(#)(t1`*`t) = (s1(#)t1)`*`(s(#)t)
proof
let t be transformation of F1, F2, t1 be transformation of F2, F3 such that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_naturally_transformable_to F3 and
A3: G1 is_naturally_transformable_to G2 and
A4: G2 is_naturally_transformable_to G3;
A5: F1 is_transformable_to F2 by A1;
then
A6: s(#)t = (G2*t)`*`(s*F1) by A3,Th22;
A7: G2*F1 is_transformable_to G2*F2 by A5,Th10;
A8: G3*F1 is_transformable_to G3*F2 by A5,Th10;
G1*F1 is_naturally_transformable_to G2*F2 by A1,A3,Lm2;
then
A9: G1*F1 is_transformable_to G2*F2;
A10: G1 is_transformable_to G2 by A3;
then
A11: G1*F1 is_transformable_to G2*F1 by Th10;
A12: F2 is_transformable_to F3 by A2;
then
A13: s1(#)t1 = (G3*t1)`*`(s1*F2) by A4,Th22;
A14: G3*F2 is_transformable_to G3*F3 by A12,Th10;
A15: G2 is_transformable_to G3 by A4;
then
A16: G2*F1 is_transformable_to G3*F1 by Th10;
G1 is_transformable_to G3 by A10,A15,FUNCTOR2:2;
then
A17: G1*F1 is_transformable_to G3*F1 by Th10;
A18: G2*F2 is_transformable_to G3*F2 by A15,Th10;
F1 is_transformable_to F3 by A5,A12,FUNCTOR2:2;
hence (s1`*`s)(#)(t1`*`t) = (G3*(t1`*`t))`*`((s1`*`s)*F1) by A3,A4,Th22,
FUNCTOR2:8
.= (G3*t1)`*`(G3*t)`*`((s1`*`s)*F1) by A5,A12,Th13
.= (G3*t1)`*`(G3*t)`*`(((s1 qua transformation of G2,G3)`*`s)*F1) by A3,A4,
FUNCTOR2:def 8
.= (G3*t1)`*`(G3*t)`*`((s1*F1)`*`(s*F1)) by A10,A15,Th14
.= (G3*t1)`*`((G3*t)`*`((s1*F1)`*`(s*F1))) by A14,A8,A17,FUNCTOR2:6
.= (G3*t1)`*`((G3*t)`*`(s1*F1)`*`(s*F1)) by A8,A11,A16,FUNCTOR2:6
.= (G3*t1)`*`((s1(#)t)`*`(s*F1)) by A4,A5,Th22
.= (G3*t1)`*`((s1*F2)`*`((G2*t)`*`(s*F1))) by A11,A18,A7,FUNCTOR2:6
.= (s1(#)t1)`*`(s(#)t) by A14,A18,A9,A13,A6,FUNCTOR2:6;
end;
begin :: Natural equivalences
theorem Th32:
F1 is_naturally_transformable_to F2 & F2 is_transformable_to F1
& (for a being Object of A holds t!a is iso) implies F2
is_naturally_transformable_to F1 & ex f being natural_transformation of F2, F1
st for a being Object of A holds f.a = (t!a)" & f!a is iso
proof
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_transformable_to F1 and
A3: for a being Object of A holds t!a is iso;
defpred P[object,object] means
ex a being Object of A st a = $1 & $2 = (t!a)";
set I = the carrier of A;
A4: for i being object st i in I ex j being object st P[i,j]
proof
let i be object;
assume i in I;
then reconsider o = i as Object of A;
take (t!o)";
thus thesis;
end;
consider f being ManySortedSet of I such that
A5: for i being object st i in I holds P[i,f.i] from PBOOLE:sch 3(A4);
f is transformation of F2, F1
proof
thus F2 is_transformable_to F1 by A2;
let a be Object of A;
ex b being Object of A st b = a & f.a = (t!b)" by A5;
hence thesis;
end;
then reconsider f as transformation of F2, F1;
A6: F1 is_transformable_to F2 by A1;
A7: now
let a, b be Object of A such that
A8: <^a,b^> <> {};
A9: <^F1.a,F1.b^> <> {} by A8,FUNCTOR0:def 18;
let g be Morphism of a, b;
A10: ex bb being Object of A st bb = b & f.b = (t!bb)" by A5;
A11: t!b is iso by A3;
A12: <^F2.a,F1.a^> <> {} by A2;
A13: <^F1.a,F2.a^> <> {} by A6;
A14: ex aa being Object of A st aa = a & f.a = (t!aa)" by A5;
then reconsider fa = f.a as Morphism of F2.a, F1.a;
A15: t!a is iso by A3;
A16: <^F1.b,F2.b^> <> {} by A6;
A17: <^F2.b,F1.b^> <> {} by A2;
A18: <^F2.a,F2.b^> <> {} by A8,FUNCTOR0:def 18;
then
A19: <^F2.a,F1.b^> <> {} by A17,ALTCAT_1:def 2;
hence f!b*F2.g = f!b*(F2.g)*(idm (F2.a)) by ALTCAT_1:def 17
.= f!b*(F2.g)*((t!a)*fa) by A14,A15,ALTCAT_3:def 5
.= f!b*(F2.g)*((t!a)*(f!a)) by A2,FUNCTOR2:def 4
.= f!b*(F2.g)*(t!a)*(f!a) by A13,A12,A19,ALTCAT_1:21
.= f!b*((F2.g)*(t!a))*(f!a) by A13,A17,A18,ALTCAT_1:21
.= f!b*((t!b)*(F1.g))*(f!a) by A1,A8,FUNCTOR2:def 7
.= f!b*(t!b)*(F1.g)*(f!a) by A17,A16,A9,ALTCAT_1:21
.= (t!b)"*(t!b)*(F1.g)*(f!a) by A2,A10,FUNCTOR2:def 4
.= (idm (F1.b))*(F1.g)*(f!a) by A11,ALTCAT_3:def 5
.= F1.g*(f!a) by A9,ALTCAT_1:20;
end;
hence F2 is_naturally_transformable_to F1 by A2;
F2 is_naturally_transformable_to F1 by A2,A7;
then reconsider f as natural_transformation of F2, F1 by A7,FUNCTOR2:def 7;
take f;
let a be Object of A;
consider b being Object of A such that
A20: b = a and
A21: f.a = (t!b)" by A5;
thus f.a = (t!a)" by A20,A21;
A22: <^F1.a,F2.a^> <> {} by A6;
A23: <^F2.a,F1.a^> <> {} by A2;
f!a = (t!b)" by A2,A21,FUNCTOR2:def 4;
hence thesis by A3,A20,A22,A23,ALTCAT_4:3;
end;
definition
let A, B be category, F1, F2 be covariant Functor of A, B;
pred F1, F2 are_naturally_equivalent means
:Def4:
F1 is_naturally_transformable_to F2 & F2 is_transformable_to F1 & ex t being
natural_transformation of F1, F2 st for a being Object of A holds t!a is iso;
reflexivity
proof
let F be covariant Functor of A, B;
thus F is_naturally_transformable_to F & F is_transformable_to F;
take idt F;
let a be Object of A;
(idt F)!a = idm (F.a) by FUNCTOR2:4;
hence thesis;
end;
symmetry
proof
let F1, F2 be covariant Functor of A, B such that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_transformable_to F1;
given t being natural_transformation of F1, F2 such that
A3: for a being Object of A holds t!a is iso;
consider f being natural_transformation of F2, F1 such that
A4: for a being Object of A holds f.a = (t!a)" & f!a is iso by A1,A2,A3,Th32;
thus F2 is_naturally_transformable_to F1 by A1,A2,A3,Th32;
thus F1 is_transformable_to F2 by A1;
take f;
thus thesis by A4;
end;
end;
definition
let A, B be category, F1, F2 be covariant Functor of A, B such that
A1: F1, F2 are_naturally_equivalent;
mode natural_equivalence of F1, F2 -> natural_transformation of F1, F2 means
:Def5:
for a being Object of A holds it!a is iso;
existence by A1;
end;
reserve e for natural_equivalence of F1, F2,
e1 for natural_equivalence of F2, F3,
f for natural_equivalence of G1, G2;
theorem Th33:
F1, F2 are_naturally_equivalent & F2, F3
are_naturally_equivalent implies F1, F3 are_naturally_equivalent
proof
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_transformable_to F1;
given t being natural_transformation of F1, F2 such that
A3: for a being Object of A holds t!a is iso;
assume that
A4: F2 is_naturally_transformable_to F3 and
A5: F3 is_transformable_to F2;
given t1 being natural_transformation of F2, F3 such that
A6: for a being Object of A holds t1!a is iso;
thus F1 is_naturally_transformable_to F3 & F3 is_transformable_to F1 by A1,A2
,A4,A5,FUNCTOR2:2,8;
take t1 `*` t;
let a be Object of A;
A7: t1!a is iso by A6;
F3 is_transformable_to F1 by A2,A5,FUNCTOR2:2;
then
A8: <^F3.a,F1.a^> <> {};
A9: t!a is iso by A3;
A10: F2 is_transformable_to F3 by A4;
then
A11: <^F2.a,F3.a^> <> {};
A12: F1 is_transformable_to F2 by A1;
then
A13: <^F1.a,F2.a^> <> {};
(t1 `*` t)!a = ((t1 qua transformation of F2, F3) `*` t)!a by A1,A4,
FUNCTOR2:def 8
.= (t1!a)*(t!a) by A12,A10,FUNCTOR2:def 5;
hence thesis by A13,A11,A8,A7,A9,ALTCAT_3:7;
end;
theorem Th34:
F1, F2 are_naturally_equivalent & F2, F3
are_naturally_equivalent implies e1 `*` e is natural_equivalence of F1, F3
proof
assume that
A1: F1, F2 are_naturally_equivalent and
A2: F2, F3 are_naturally_equivalent;
thus
A3: F1, F3 are_naturally_equivalent by A1,A2,Th33;
let a be Object of A;
A4: F1 is_transformable_to F2 by A1,Def4;
then
A5: <^F1.a,F2.a^> <> {};
F3 is_transformable_to F1 by A3;
then
A6: <^F3.a,F1.a^> <> {};
A7: F2 is_transformable_to F3 by A2,Def4;
then
A8: <^F2.a,F3.a^> <> {};
F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
F3 by A1,A2;
then
A9: (e1 `*` e)!a = ((e1 qua transformation of F2, F3) `*` e)!a by
FUNCTOR2:def 8
.= (e1!a)*(e!a) by A4,A7,FUNCTOR2:def 5;
e1!a is iso & e!a is iso by A1,A2,Def5;
hence thesis by A9,A5,A8,A6,ALTCAT_3:7;
end;
theorem Th35:
F1, F2 are_naturally_equivalent implies G1*F1, G1*F2
are_naturally_equivalent & G1*e is natural_equivalence of G1*F1, G1*F2
proof
assume
A1: F1, F2 are_naturally_equivalent;
then
A2: F2 is_transformable_to F1;
A3: F1 is_naturally_transformable_to F2 by A1;
then reconsider k = G1*e as natural_transformation of G1*F1, G1*F2 by Th28;
A4: F1 is_transformable_to F2 by A1,Def4;
A5: now
let a be Object of A;
A6: (G1*F1).a = G1.(F1.a) & (G1*F2).a = G1.(F2.a) by FUNCTOR0:33;
A7: <^F2.a,F1.a^> <> {} by A2;
k!a = G1.(e!a) & <^F1.a,F2.a^> <> {} by A4,Th11;
hence k!a is iso by A1,A6,A7,Def5,ALTCAT_4:20;
end;
G1*F1, G1*F2 are_naturally_equivalent
by A3,Lm2,A2,Th10,A5;
hence thesis by A5,Def5;
end;
theorem Th36:
G1, G2 are_naturally_equivalent implies G1*F1, G2*F1
are_naturally_equivalent & f*F1 is natural_equivalence of G1*F1, G2*F1
proof
assume
A1: G1, G2 are_naturally_equivalent;
then
G1 is_naturally_transformable_to G2;
then reconsider k = f*F1 as natural_transformation of G1*F1, G2*F1 by Th29;
A2: now
let a be Object of A;
G1 is_transformable_to G2 by A1,Def4;
then
A3: k!a = f!(F1.a) by Th12;
(G1*F1).a = G1.(F1.a) & (G2*F1).a = G2.(F1.a) by FUNCTOR0:33;
hence k!a is iso by A1,A3,Def5;
end;
G1*F1, G2*F1 are_naturally_equivalent
by Lm2,A1,Th10,A2;
hence thesis by A2,Def5;
end;
theorem
F1, F2 are_naturally_equivalent & G1, G2 are_naturally_equivalent
implies G1*F1, G2*F2 are_naturally_equivalent & f (#) e is natural_equivalence
of G1*F1, G2*F2
proof
assume that
A1: F1, F2 are_naturally_equivalent and
A2: G1, G2 are_naturally_equivalent;
A3: G1*F1, G1*F2 are_naturally_equivalent by A1,Th35;
G1 is_naturally_transformable_to G2 by A2;
then reconsider sF2 = f*F2 as natural_transformation of G1*F2, G2*F2 by Th29;
F1 is_naturally_transformable_to F2 by A1;
then reconsider G1t = G1*e as natural_transformation of G1*F1, G1*F2 by Th28;
A4: G1*F2, G2*F2 are_naturally_equivalent by A2,Th36;
f*F2 is natural_equivalence of G1*F2, G2*F2 & G1*e is
natural_equivalence of G1*F1, G1*F2 by A1,A2,Th35,Th36;
then sF2`*`G1t is natural_equivalence of G1*F1, G2*F2 by A4,A3,Th34;
hence thesis by A4,A3,Th33,FUNCTOR2:def 8;
end;
definition
let A, B be category, F1, F2 be covariant Functor of A, B, e be
natural_equivalence of F1, F2 such that
A1: F1, F2 are_naturally_equivalent;
func e" -> natural_equivalence of F2, F1 means
:Def6:
for a being Object of A holds it.a = (e!a)";
existence
proof
A2: for a being Object of A holds e!a is iso by A1,Def5;
F1 is_naturally_transformable_to F2 & F2 is_transformable_to F1 by A1;
then consider f being natural_transformation of F2, F1 such that
A3: for a being Object of A holds f.a = (e!a)" & f!a is iso by A2,Th32;
f is natural_equivalence of F2, F1
proof
thus F2, F1 are_naturally_equivalent by A1;
let a be Object of A;
thus thesis by A3;
end;
then reconsider f as natural_equivalence of F2, F1;
take f;
let a be Object of A;
thus thesis by A3;
end;
uniqueness
proof
let P, R be natural_equivalence of F2, F1 such that
A4: for a being Object of A holds P.a = (e!a)" and
A5: for a being Object of A holds R.a = (e!a)";
A6: F2 is_transformable_to F1 by A1;
now
let a be Object of A;
thus P!a = P.a by A6,FUNCTOR2:def 4
.= (e!a)" by A4
.= R.a by A5
.= R!a by A6,FUNCTOR2:def 4;
end;
hence P = R by A6,FUNCTOR2:3;
end;
end;
theorem Th38:
for o being Object of A st F1, F2 are_naturally_equivalent holds
e"!o = (e!o)"
proof
let o be Object of A;
assume
A1: F1, F2 are_naturally_equivalent;
then F2 is_transformable_to F1;
hence e"!o = e".o by FUNCTOR2:def 4
.= (e!o)" by A1,Def6;
end;
theorem Th39:
F1, F2 are_naturally_equivalent implies e `*` e" = idt F2
proof
assume
A1: F1, F2 are_naturally_equivalent;
then
A2: F1 is_transformable_to F2 & F2 is_transformable_to F1 by Def4;
A3: F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
F1 by A1,Def4;
now
let a be Object of A;
A4: e!a is iso by A1,Def5;
thus (e `*` e")!a = ((e qua transformation of F1, F2) `*` e")!a by A3,
FUNCTOR2:def 8
.= (e!a)*(e"!a) by A2,FUNCTOR2:def 5
.= (e!a)*((e!a)") by A1,Th38
.= idm (F2.a) by A4,ALTCAT_3:def 5
.= (idt F2)!a by FUNCTOR2:4;
end;
hence thesis by FUNCTOR2:3;
end;
theorem
F1, F2 are_naturally_equivalent implies e" `*` e = idt F1
proof
assume
A1: F1, F2 are_naturally_equivalent;
then
A2: F1 is_transformable_to F2 & F2 is_transformable_to F1 by Def4;
A3: F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
F1 by A1,Def4;
now
let a be Object of A;
A4: e!a is iso by A1,Def5;
thus (e" `*` e)!a = (e" `*` (e qua transformation of F1, F2))!a by A3,
FUNCTOR2:def 8
.= (e"!a)*(e!a) by A2,FUNCTOR2:def 5
.= (e!a)"*(e!a) by A1,Th38
.= idm (F1.a) by A4,ALTCAT_3:def 5
.= (idt F1)!a by FUNCTOR2:4;
end;
hence thesis by FUNCTOR2:3;
end;
definition
let A, B be category, F be covariant Functor of A, B;
redefine func idt F -> natural_equivalence of F, F;
coherence
proof
set e = the natural_equivalence of F, F;
e `*` e" = idt F by Th39;
hence thesis by Th34;
end;
end;
theorem
F1, F2 are_naturally_equivalent implies (e")" = e
proof
assume
A1: F1, F2 are_naturally_equivalent;
then
A2: F1 is_transformable_to F2 by Def4;
now
let a be Object of A;
A3: <^F1.a,F2.a^> <> {} by A2;
F2 is_transformable_to F1 by A1;
then
A4: <^F2.a,F1.a^> <> {};
e!a is iso by A1,Def5;
then
A5: e!a is retraction coretraction by ALTCAT_3:5;
thus (e")"!a = (e"!a)" by A1,Th38
.= ((e!a)")" by A1,Th38
.= e!a by A3,A4,A5,ALTCAT_3:3;
end;
hence thesis by A2,FUNCTOR2:3;
end;
theorem
for k being natural_equivalence of F1, F3 st k = e1 `*` e & F1, F2
are_naturally_equivalent & F2, F3 are_naturally_equivalent holds k" = e" `*` e1
"
proof
let k be natural_equivalence of F1, F3 such that
A1: k = e1 `*` e and
A2: F1, F2 are_naturally_equivalent and
A3: F2, F3 are_naturally_equivalent;
A4: F3 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
F1 by A2,A3,Def4;
A5: F1 is_transformable_to F2 & F2 is_transformable_to F3 by A2,A3,Def4;
A6: F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
F3 by A2,A3;
A7: F3 is_transformable_to F2 & F2 is_transformable_to F1 by A2,A3;
then
A8: F3 is_transformable_to F1 by FUNCTOR2:2;
now
let a be Object of A;
A9: <^F1.a,F2.a^> <> {} & <^F2.a,F3.a^> <> {} by A5;
A10: <^F3.a,F1.a^> <> {} by A8;
A11: e!a is iso & e1!a is iso by A2,A3,Def5;
thus k"!a = ((e1 `*` e)!a)" by A1,A2,A3,Th33,Th38
.= (((e1 qua transformation of F2, F3)`*` e)!a)" by A6,FUNCTOR2:def 8
.= ((e1!a)*(e!a))" by A5,FUNCTOR2:def 5
.= ((e!a)")*((e1!a)") by A11,A9,A10,ALTCAT_3:7
.= ((e!a)")*(e1"!a) by A3,Th38
.= (e"!a)*(e1"!a) by A2,Th38
.= ((e" qua transformation of F2, F1)`*` e1")!a by A7,FUNCTOR2:def 5
.= (e" `*` e1")!a by A4,FUNCTOR2:def 8;
end;
hence thesis by A7,FUNCTOR2:2,3;
end;
theorem
(idt F1)" = idt F1
proof
now
let a be Object of A;
thus (idt F1)"!a = ((idt F1)!a)" by Th38
.= (idm(F1.a))" by FUNCTOR2:4
.= idm(F1.a) by ALTCAT_3:4
.= (idt F1)!a by FUNCTOR2:4;
end;
hence thesis by FUNCTOR2:3;
end;