Функторные морфизмы. Ершов А.В. - 55 стр.

UptoLike

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СГУ им. Чернышевского | Саратов

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S ∈ Ob(Set) G Hom
Set
(S, G)
f, g ∈ Hom
Set
(S, G)
fg : S → G (fg)(x) = f(x)g(x) ∀x ∈ S fg
x ∈ S f, g
f(x), g(x) ∈ G
Hom
Set
(S, G) × Hom
Set
(S, G) → Hom
Set
(S, G), (f, g) 7→ fg
Hom
Set
(S, G).
Hom
Set
(S, G)
f : S → G f(s) = e ∀s ∈ S e ∈ G
G g : S → G
h: S → G h(s) = g(s)
−1
∀s ∈ S. G
Hom
Set
(∗, G), ∗ ∈ Ob(Set)
φ: S
′
→ S
h
G
(φ): Hom
Set
(S, G) → Hom
Set
(S
′
, G)
h
G
(φ)(f) = f ◦ φ, h
G
(φ)(g) = g ◦ φ, h
G
(φ)(fg) = (fg) ◦φ
(fg)(φ(s
′
)) = f(φ(s
′
))g(φ(s
′
)) ∀s
′
∈ S
′
,
h
G
(φ)(fg) = h
G
(φ)(f)h
G
(φ)(g).
α: G → H
h
α
(S): Hom
Set
(S, G) → Hom
Set
(S, H)
h
α
(S)(f) = α ◦f, h
α
(S)(g) = α ◦ g, h
α
(S)(fg) = α ◦ (fg)

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