Grades 6–8Category TheoryPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 10
You will learn to
- 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.
Definition
- 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.
Theorem
- 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.
Common misconceptions
- 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.
Optional discoveries
- 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.