Элементы теории категорий. Ершов А.В. - 49 стр.

UptoLike

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СГУ им. Чернышевского | Саратов

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m: X × X → X
u, v ∈ h
X
(Y ),
t: Y → X × X,
[u, v] u, v, w ∈ h
X
(Y ) [u, v, w]
Y → X × X × X π
1
◦ [u, v, w] = u, π
2
◦ [u, v, w] =
v, π
3
◦ [u, v, w] = w, π
i
: X × X × X → X i i =
1, 2, 3.
(X ×X)×X
∼
=
X ×X ×X
∼
=
X ×(X ×X),
[[u, v], w] ↔ [u, v, w] ↔ [u, [v, w]]
uv = µ(Y )(u, v) = m ◦ [u, v] ∀u, v ∈ h
X
(Y ).
h
X
(Y )
m ◦ [m ◦ [u, v], w] = m ◦ [u, m ◦ [v, w]].
X × X
m
//
X
Y
[u, v]
88
r
r
r
r
r
r
r
r
r
r
r
r
α
//
w
&&
L
L
L
L
L
L
L
L
L
L
L
L
L
(X × X) × X
π
1
OO
π
2
m×id
X
//
X × X
π
1
OO
π
2
X
id
X
//
X,
α [[u, v], w]
m × id
X
[m ◦ π
1
, id
X
◦π
2
] = [m ◦ π
1
, π
2
].
(m × id
X
) ◦ [[u, v], w] = [m ◦ [u, v], w].
(id
X
×m) ◦ [u, [v, w]] = [u, m ◦ [v, w]].
m ◦ (m × id
X
) ◦ [[u, v], w] = m ◦ (id
X
×m) ◦ [u, [v, w]].
u, v, w ∈ h
X
(Y ),

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