begin
theorem Th1:
theorem
theorem Th3:
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th7:
theorem Th8:
theorem Th9:
:: deftheorem defines Functor ISOCAT_1:def 1 :
theorem Th10:
theorem Th11:
theorem Th12:
:: deftheorem Def2 defines " ISOCAT_1:def 2 :
:: deftheorem Def3 defines isomorphic ISOCAT_1:def 3 :
theorem Th13:
theorem
theorem
theorem Th16:
theorem Th17:
:: deftheorem defines are_isomorphic ISOCAT_1:def 4 :
theorem
canceled;
theorem
canceled;
theorem
theorem
theorem
theorem
theorem
:: deftheorem Def5 defines * ISOCAT_1:def 5 :
:: deftheorem Def6 defines * ISOCAT_1:def 6 :
theorem Th25:
theorem Th26:
theorem Th27:
:: deftheorem Def7 defines * ISOCAT_1:def 7 :
theorem Th28:
:: deftheorem Def8 defines * ISOCAT_1:def 8 :
theorem Th29:
theorem Th30:
theorem Th31:
theorem Th32:
theorem Th33:
theorem Th34:
theorem Th35:
theorem Th36:
theorem Th37:
theorem Th38:
theorem Th39:
theorem Th40:
definition
let A,
B,
C be
Category;
let F1,
F2 be
Functor of
A,
B;
let G1,
G2 be
Functor of
B,
C;
let s be
natural_transformation of
F1,
F2;
let t be
natural_transformation of
G1,
G2;
func t (#) s -> natural_transformation of
G1 * F1,
G2 * F2 equals
(t * F2) `*` (G1 * s);
correctness
coherence
(t * F2) `*` (G1 * s) is natural_transformation of G1 * F1,G2 * F2;
;
end;
:: deftheorem defines (#) ISOCAT_1:def 9 :
theorem Th41:
theorem
theorem
theorem
for
A,
B,
C,
D being
Category for
F1,
F2 being
Functor of
A,
B for
G1,
G2 being
Functor of
B,
C for
H1,
H2 being
Functor of
C,
D for
s being
natural_transformation of
F1,
F2 for
t being
natural_transformation of
G1,
G2 for
u being
natural_transformation of
H1,
H2 st
F1 is_naturally_transformable_to F2 &
G1 is_naturally_transformable_to G2 &
H1 is_naturally_transformable_to H2 holds
u (#) (t (#) s) = (u (#) t) (#) s
theorem
theorem
theorem
for
A,
B,
C being
Category for
F1,
F2,
F3 being
Functor of
A,
B for
G1,
G2,
G3 being
Functor of
B,
C for
s being
natural_transformation of
F1,
F2 for
s' being
natural_transformation of
F2,
F3 for
t being
natural_transformation of
G1,
G2 for
t' being
natural_transformation of
G2,
G3 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
(t' `*` t) (#) (s' `*` s) = (t' (#) s') `*` (t (#) s)
theorem Th48:
theorem Th49:
definition
let A,
B be
Category;
pred A is_equivalent_with B means :
Def10:
ex
F being
Functor of
A,
B ex
G being
Functor of
B,
A st
(
G * F ~= id A &
F * G ~= id B );
reflexivity
for A being Category ex F, G being Functor of A,A st
( G * F ~= id A & F * G ~= id A )
symmetry
for A, B being Category st ex F being Functor of A,B ex G being Functor of B,A st
( G * F ~= id A & F * G ~= id B ) holds
ex F being Functor of B,A ex G being Functor of A,B st
( G * F ~= id B & F * G ~= id A )
;
end;
:: deftheorem Def10 defines is_equivalent_with ISOCAT_1:def 10 :
theorem
theorem
canceled;
theorem
canceled;
theorem Th53:
:: deftheorem Def11 defines Equivalence ISOCAT_1:def 11 :
theorem
theorem
theorem Th56:
theorem Th57:
theorem