Bicategories and 2-Categories
Bicategories and 2-Categories
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双类别和 2 类别
DOI：
10.1093/oso/9780198871378.003.0002
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Donald Yau
中科院分区：
文献类型：
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作者：
Niles Johnson;Donald Yau
⇓θa b f g，连同垂直和水平组合运算a b f g h⇓θ⇓φ⎮⇓φθa b f h⇓ψc r S⇓θa b f g⎮⇓φθa c r of S o g.ψoθ这些组合要求是联合的和统一的；此外，水平组合必须保持垂直单位，并实施以下交换定律。(χψ)o(φθ)=(χoφ)(ψoθ)a b f g h⇓θ⇓φc r S t⇓ψ⇓χ2-范畴的基本例子是猫：对象是(小)范畴，箭头是函子，2-箭头是自然变换。事实上，组成自然变化的基本“五条规则”载于[GT]附录。在实践中出现了一个较弱的2-范畴概念。弱2-范畴或双范畴[bn1]由2-范畴的数据和条件组成，除了用于水平合成的结合性和单位等式被称为结合性和单位约束的2-箭头的可逆自然族的额外数据αλρf f m f f f a m r f f f，，：()，：，：，o⇒⇒⇒11所代替，使得施加结合性五边形(或3-余循环条件)αp mrf()()o=11和单位三角形(或归一化条件)()，11 rf f r m r f oλαρ=。在最近的一些文献中，双范畴被称为2-范畴，2-范畴被称为严格2-范畴。一个单体范畴V可以被认为是单对象双范畴ΣV，其箭头是V的对象，其2-箭头是V的箭头，其水平合成是V的张量积，其垂直合成是合成
⇓ θ a b f g , together with vertical and horizontal composition operations a b f g h ⇓ θ ⇓ φ ⎮ ⇓ φθ a b f h ⇓ ψ c r s ⇓ θ a b f g ⎮ ⇓ φθ a c r o f s o g. ψ o θ These compositions are required to be associative and unital; moreover, horizontal composition must preserve vertical units and the following interchange law is imposed. (χψ) o(φθ) = (χ oφ) (ψ oθ) a b f g h ⇓ θ ⇓ φ c r s t ⇓ ψ ⇓ χ The basic example of a 2-category is Cat: objects are (small) categories, arrows are functors, and 2-arrows are natural transformations. Indeed, the basic "five rules" for composition of natural transformations appeared in the Appendix of [Gt]. There is a weaker notion of 2-category which occurs in practice. A weak 2-category or bicategory [Bn1] consists of the data and conditions of a 2-category except that the associativity and unital equalities for horizontal composition are replaced by the extra data of invertible natural families of 2-arrows α λ ρ f r m f b f a m r f m r f f f f f , , : () () , : , : , o o o o o o ⇒ ⇒ ⇒ 1 1 called associativity and unital constraints, such that the associativity pentagon (or 3-cocycle condition) α α α α α p m r f p m r f p m r f p m r f () () o o o o o = 1 1 and unit triangle (or normalisation condition) () , , 1 1 r f f r m r f o o λ α ρ = are imposed. In some of the recent literature, bicategories are called 2-categories and 2-categories are called strict 2-categories. A monoidal category V can be identified with the one-object bicategory ΣV whose arrows are objects of V, whose 2-arrows are the arrows of V, whose horizontal composition is the tensor product of V, and whose vertical composition is the composition
环空间的几何
DOI：
--
发表时间：
2009
期刊：
影响因子：
--
作者：
H. Irie;T. Otofuji;K.Fukaya;伊藤秀史;S.Koike;T. Funaki;金銅誠之;Yoshiaki Maeda
通讯作者：
Yoshiaki Maeda