Элементы теории категорий. Ершов А.В. - 31 стр.

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СГУ им. Чернышевского | Саратов

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Рубрика: 

F G(ψ
2
◦ ψ
1
) = g(Y
00
)
−1
◦ ψ
2
◦ ψ
1
◦ g(Y ) =
g(Y
00
)
−1
◦ ψ
2
◦ g(Y
0
) ◦ g(Y
0
)
−1
◦ ψ
1
◦ g(Y ) = F (ϕ
2
) ◦ F (ϕ
1
) =
F (ϕ
2
◦ ϕ
1
) = F (G(ψ
2
) ◦ G(ψ
1
))
G
g(Y
0
) ◦ g(Y
0
)
−1
= id
Y
0
ψ
2
◦ id
Y
0
= ψ
2
, id
Y
0
◦ψ
1
= ψ
1
,
F ϕ
i
= G(ψ
i
), i = 1, 2
F G(ψ
2
◦ ψ
1
) = F (G(ψ
2
) ◦ G(ψ
1
)). F
G(ψ
2
◦ ψ
1
) = G(ψ
2
) ◦ G(ψ
1
).
g = {g(Y )}: F G → id
D
Y
ψ
//
Y
0
F G(Y )
g(Y )
OO
F G(ψ)
//
F G(Y
0
),
g(Y
0
)
OO
g : F G → id
D
g(Y ): F G(Y ) → Y D
Y ∈ Ob(D). g
X ∈ Ob(C) g(F (X)): FGF (X) →
F (X) D
F Hom
D
(F GF (X), F (X)) =
Hom
C
(GF (X), X) g(F (X)) = F (f(X))
f(X) ∈ Hom
C
(GF (X), X).
f = {f(X)}: GF → id
C
g X, X
0
∈ Ob(C)
ϕ ∈ Hom
C
(X, X
0
)
F (X)
F (ϕ)
//
F (X
0
)
F GF (X)
g(F (X))
OO
F GF (ϕ)
//
F GF (X
0
)
g(F (X
0
))
OO
F
X
ϕ
//
X
0
GF (X)
f(X)
OO
GF (ϕ)
//
GF (X
0
).
f(X
0
)
OO

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