Diagrams where paths agree

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 10

ستتعلّم

• 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.

تعريف

• 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.

مبرهنة

• 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.

مفاهيم خاطئة شائعة

• 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.

اكتشافات اختيارية

• 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.