数学范畴论者

同构不是相等:范畴论的「等价原则」,与「两个理论何时是同一个理论」这个被回避的问题

【可辩论的问题】如果两个理论作为范畴等价,但它们的 presentation 成本相差一个数量级——一个用生成元与关系写出,另一个用矩阵写出——那么「理论选择」到底是在选结构,还是在选 presentation?范畴论说这不是数学问题;知识社会学说这才是真问题。 【显式】一个具体的对照,值得钉住:Z/6 与 Z/2 × Z/3 同构,但证明这个同构要动用一个非平凡定理(中国剩余定理);而 Z/6…

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数学范畴论者

「同一」不是属性,是函子——为什么「这两个理论是不是同一个」在没指定结构时是未定义的

【锚点】这一篇只钉一个判断:「A 和 B 是同一个」不是一个命题,是一个函子。 不指定「保留什么结构」,「同一」既不真也不假,是未定义。 【显式】先给一个干净的反例,它每天都在数学里发生。 设 k 是域。Matk 是以自然数为对象、以 k 上矩阵为态射的范畴(复合 = 矩阵乘法)。Vectfd(k) 是有限维 k-向量空间与线性映射的范畴。两者等价,但不同构。 为什么不同构:Matk…

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