Last Updated
：2025/12/04
monoidal category
Noun
(category
theory)
A
category
𝒞
with
a
bifunctor
⊗:𝒞⨯𝒞→𝒞
which
may
be
called
tensor
product,
an
associativity
isomorphism
𝛼_(A,B,C):(A⊗B)⊗C≃A⊗(B⊗C),
an
object
I
which
may
be
called
tensor
unit,
a
left
unit
natural
isomorphism
𝜆_A:I⊗A≃A,
a
right
unit
natural
isomorphism
𝜌_A:A⊗I≃A,
and
some
"coherence
conditions"
(pentagon
and
triangle
commutative
diagrams
for
those
isomorphisms).
Japanese Meaning
圏論におけるモノイダル圏とは、圏𝒞に二項補関手（テンソル積）⊗:𝒞×𝒞→𝒞が定義され、対象同士のテンソル積を考えることができる。また、結合性同型 𝛼_(A,B,C):(A⊗B)⊗C ≃ A⊗(B⊗C) が存在し、対象 I を単位元として、左単位自然同型 𝜆_A:I⊗A ≃ A と右単位自然同型 𝜌_A:A⊗I ≃ A が定義される。さらにこれらの同型が五角形や三角形の整合性条件を満たす。
Sense（1）
(category
theory)
A
category
𝒞
with
a
bifunctor
⊗:𝒞⨯𝒞→𝒞
which
may
be
called
tensor
product,
an
associativity
isomorphism
𝛼_(A,B,C):(A⊗B)⊗C≃A⊗(B⊗C),
an
object
I
which
may
be
called
tensor
unit,
a
left
unit
natural
isomorphism
𝜆_A:I⊗A≃A,
a
right
unit
natural
isomorphism
𝜌_A:A⊗I≃A,
and
some
"coherence
conditions"
(pentagon
and
triangle
commutative
diagrams
for
those
isomorphisms).
（ plural ）
Quizzes for review
(category theory) A category 𝒞 with a bifunctor ⊗:𝒞⨯𝒞→𝒞 which may be called tensor product, an associativity isomorphism 𝛼_(A,B,C):(A⊗B)⊗C≃A⊗(B⊗C), an object I which may be called tensor unit, a left unit natural isomorphism 𝜆_A:I⊗A≃A, a right unit natural isomorphism 𝜌_A:A⊗I≃A, and some coherence conditions
(pentagon and triangle commutative diagrams for those isomorphisms).
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See correct answer
monoidal category
To model parallel composition of processes, we work within a monoidal category where objects represent systems and the tensor product encodes their combination.
See correct answer
To model parallel composition of processes, we work within a monoidal category where objects represent systems and the tensor product encodes their combination.
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