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In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a cocartesian monoidal category, and any finite coproduct category can be thought of as a cocartesian monoidal category.

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  • In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a cocartesian monoidal category, and any finite coproduct category can be thought of as a cocartesian monoidal category. Cartesian categories with an internal Hom functor that is an adjoint functor to the product are called Cartesian closed categories. (en)
  • 数学の特に圏論と呼ばれる分野において、デカルトモノイド圏(デカルトモノイドけん、英: cartesian monoidal category)あるいは短くデカルト圏は、モノイド積(テンソル積)が圏論的(直)積で与えられるモノイド圏を言う。有限積を持つ任意の圏(有限積圏)はデカルトモノイド圏と見なすことができる。任意のデカルトモノイド圏において、終対象がモノイド単位を与える。双対的に、有限余積を持つ圏において余積が始対象を単位として成すモノイド構造を考えて余デカルト(モノイド)圏が得られ、やはり任意の有限余積圏が余デカルトモノイド圏と見なせる。 デカルト圏の直積を与える関手が随伴となるHom関手を持つとき、デカルト閉圏という。 (ja)
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  • 数学の特に圏論と呼ばれる分野において、デカルトモノイド圏(デカルトモノイドけん、英: cartesian monoidal category)あるいは短くデカルト圏は、モノイド積(テンソル積)が圏論的(直)積で与えられるモノイド圏を言う。有限積を持つ任意の圏(有限積圏)はデカルトモノイド圏と見なすことができる。任意のデカルトモノイド圏において、終対象がモノイド単位を与える。双対的に、有限余積を持つ圏において余積が始対象を単位として成すモノイド構造を考えて余デカルト(モノイド)圏が得られ、やはり任意の有限余積圏が余デカルトモノイド圏と見なせる。 デカルト圏の直積を与える関手が随伴となるHom関手を持つとき、デカルト閉圏という。 (ja)
  • In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a cocartesian monoidal category, and any finite coproduct category can be thought of as a cocartesian monoidal category. (en)
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  • Cartesian monoidal category (en)
  • デカルトモノイド圏 (ja)
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