六至八年级Category Theory计划中
Diagrams where paths agree
Square diagrams of functions in which the two routes from one corner to the opposite corner give the same result, and how 'preserving an operation' can be expressed by such a square, as when converting units respects adding lengths.
math.cat.commutative-diagrams-intro
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前置知识
10 项必需的前置知识（一直追溯到底）
你将学会
- Check whether the two routes around a square of functions agree for every input.
- Express statements such as 'doubling then adding 2 equals adding 1 then doubling' as agreeing paths in a diagram.
- Recognise maps that preserve an operation, f(a + b) = f(a) + f(b), such as converting metres to centimetres, as diagrams whose paths agree.
定义
- A diagram of functions shows sets at its corners and functions as arrows between them.
- A square diagram commutes when the two routes from the starting corner to the opposite corner give the same output for every input.
定理
- Converting metres to centimetres, x ↦ 100x, respects addition: 100(a + b) = 100a + 100b.
- x ↦ x + 1 does not respect addition: (a + b) + 1 ≠ (a + 1) + (b + 1).
- If two commuting squares are placed side by side, the outer rectangle they form also commutes.
常见误解
- Checking that the routes agree for one input and concluding that the diagram commutes.
- Thinking a diagram commutes just because all its arrows can be drawn.
可选发现
- Categories In category theory, commutative diagrams are the main way of stating equations between composites of morphisms.
- Structure-preserving maps: homomorphisms and isomorphisms Homomorphisms are exactly the maps for which the square comparing 'operate, then map' with 'map, then operate' commutes.